Emission, propagation and detection of gravitational waves

Chapter 2
GENERAL THEORY OF RELATIVITY
- PHYSICS OF GRAVITY
2.1. Acceleration and gravity from the point of view of special theory of relativity
2.2. Versatility - a basic property and the key to understanding the nature of gravity
2.3. The local principle of equivalence and its consequences
2.4. Physical laws in curved spacetime
2.5. Einstein's equations of the gravitational field
2.6. Deviation and focus of geodesics
2.7. Gravitational waves
2.8. Specific properties of gravitational energy
2.9.Geometrodynamic system of units
2.10. Experimental verification of the theory of relativity and gravity

2.7. Gravitational waves

Wave propagation - a general natural phenomenon
An important natural phenomenon of waves in material environments and physical fields lies in the propagation of certain changes (commotions, vibrations, oscillations) through space. Wave propagation is generally conditioned by two basic aspects :
1. The mechanism of changes - excitations, oscillating motion - in a given environment or field. Without the dynamic emergence of change, "there would be nothing to spread"...
  At the water surface, the commotion may be caused by the impact of a stone, after which the deflected water particles periodically oscillate up and down under the influence of the Earth's gravitational field. In the environment of elastic materials, under the action of the force the mechanical deformations can occur, which then periodically oscillate around the equilibrium position due to elastic forces. In the electromagnetic field, changes in the intensity of the electric and magnetic fields occur during uneven movements of electric charges and are mutually generated due to the Faraday-Ampere law of electromagnetic induction. In the gravitational field, temporal changes in its intensity, or changes in the curvature of spacetime, are caused by the uneven motion of material bodies; periodic oscillations of the gravitational field (curvature of spacetime) arise mainly during the mutual orbit of massive bodies around a common center of gravity under the influence of gravitational attraction (according to the general theory of relativity, it is again the motion of bodies in curved spacetime) .
2. The final rate of propagation of changes (disturbances) in this environment or field. At an infinite rate of propagation of the interaction, the change would not propagate, but would take effect immediately on all bodies, even very distant; the ripple would not occur...
  In material environments, the commotion and oscillating motion spread to the environment due to the elastic interaction with neighboring and with more and more distant atoms and molecules of the environment, which are gradually set in motion. The rate of this propagation depends on the strength of the elastic interaction (expressed as Young's modulus of elasticity) and on the density of the environment. In air, where there is a relatively weak elastic interaction between adjacent molecules, the speed of propagation - the speed of sound - about 330 m/s., In water about 1500 m/s., in hard solids is significantly higher (eg in steel about 5000 m/s.). However, it is always finite and substantially lower than the speed of light c .
  In an electromagnetic field, the commotion propagates into space at the speed of light c (in vacuum) in the form of electromagnetic waves, where the electric and magnetic fields excite each other by their variability (law of electromagnetic induction - Maxwell's equations, §1.5 "Electromagnetic field. Maxwell's equations."). As we will see below, even in the gravitational field, the commotion propagates at the speed of light in the form of gravitational waves - the oscillating curvature of space-time.
Wave function, wave equation
The propagation of a wave is mathematically expressed by means of a special differential equation between the rate of time change (time derivative) of deflection f and the gradient of spatial change (derivative by coordinates) of this quantity f - using the wave equation. In the simplified one-dimensional case of a plane wave propagating in the direction of the X axis at a phase velocity c *), the wave equation has the form :
            d² f / dt²  =  c. d² f / dx²  .
*) The speed of wave propagation is denoted by c here, but it does not have to be the speed of light in general.
  The solution of the wave equation is a special function f(x, t) of spatial coordinates and time - a wave function that has the general form :
            f (x, t)  =  f (x, t- x/c)  .
If we start from some initial point with the coordinate x_o in time t_o , then the same value of the deviation f as in the point with the coordinate x_o at time instant t_o , will be in all places whose coordinates and time satisfy the equation x - x_o = c. (t - t_o). It thus describes the ripple of the deflection f , gradually propagating through space in the direction of the X-axis at the phase velocity c . The most commonly considered is the harmonic (sine or cosine) time dependence:
            f (x, t)  =  f .cos [ w . (t - x/c)]  ,
where w = 2p f is the circular frequency; waves are often caused by periodic oscillating movements of electric charges (eg in antennas supplied with a high-frequency signal of frequency f) or circular orbiting of gravitational bodies. Even in cases where this is not the case, the resulting wave can be Fourier- decomposed into harmonic components of different frequencies and amplitudes. When using complex (imaginary "i") numbers, harmonic wave functions are often written in the form
            f (x, t)  =  Re (f. e ^{-i w (t- x/c)} )  .
  In the three - dimensional analysis in the coordinates x, y, z, the wave equation has the general form :
          (1/c²) .¶²f/¶t²  =  ¶²f/¶x² + ¶²f/¶y² + ¶²f/¶z²  ,
which is often equivalently written using the Laplace operator D : (1/c²) .¶²f/¶t² = D f . In a 4-dimensional relativistic formulation, then using d'Alembert's operator o s - (1/c²) .¶²/¶t² + ¶²/¶x² + ¶²/¶y² + ¶²/¶z² as o f = 0 .
  The wave equations are derived from the equations of motion of the elements of matter in continuum mechanics, and from the field equations - Maxwell's equations of electrodynamics (§1.5 "   Electromagnetic field. Maxwell's equations ", part" Electromagnetic waves ") and from the Einstein's equations of the gravitational field (shown below in the section "Origin and properties of gravitational waves"). If from these fundamental equations of substances or fields results the wave equations, it means that in the given substance environment or a physical field can propagate waves.
  From a somewhat different perspective, the wave function is widely used in quantum physics. In quantum mechanics, the state of the particle (or a collection of particles, and generally any physical system) is described by so-called wave function y(x, y, z). The physical meaning of the wave function is that the square of the modulus of the wave function ú yú ² determines the probability dW that the particle at a given time t is in the element of volume dV = dx.dy.dz around the point (x, y, z): dW = ú yú ² .dx.dy.dz. Schrodinger's equation and another apparatus of quantum mechanics and quantum field theory then operate with the quantum wave function conceived in this way in order to determine quantum states and transition probabilities between different quantum states. However, this concept of the wave function is already outside the scope of our treatise on physical waves (it is discussed in §1.1, part "Corpuscular-wave dualism" and "Quantum nature of the microworld" in monograph "Nuclear physics and ionizing radiation physics") .
  Graphically, the wave propagation is represented by wavefronts. A wavefront is a geometric location of points in space that oscillate with the same phase when rippling. Waves from a point or spherically symmetric source in a homogeneous and isotropic medium are a spherical (round) wavefront, the points of which lie on a spherical surface. The wavefront that a wave reaches in a given time is called the front wavefront. The perpendicular to the wavefront indicates the direction of wave propagation.
Huygens principle
The propagation of waves using wavefronts is analyzed in an illustrative way by means the so-called Huygens-Fresnel principle : At any moment, each point where the front of the propagating wavefront has reached, can be considered a new source of secondary elementary waves, from which secondary waves propagate again in all directions, while they interfering with the original waves, as well as with all other elementary waves. The total wavefront in the next moment of time then arises as the outer envelope of all elementary wavefronts. We can thus construct a wavefront at a certain moment, if the wavefront is known at a previous point in time. It can be deduced from the shape of the resulting wavefronts the laws of reflection, diffraction and refraction of waves.
  A common general feature of wave propagation - radiation - is the fact that the relevant waves gradually detach from the source and carry some a part of his energy, momentum and angular momentum into space, even without the presence of any distant "receiver" of these waves. In fact, the waves themselves (their fields) actually have energy.
Inductive and wave zone
From the point of view of mutual energy connection between the source and the receiver, the space around the oscillating wave source of frequency f can be divided into two areas :
¨ The inductive zone is a close range of distances r from the source, smaller than the radiation wavelength: r < c/f = l. Here, in the first approximation, the action of the source on the test bodies can be explained by the direct action "at a distance" under the influence of Coulomb's law of electricity or Newton's law of gravity. The loss of energy of the source here significantly depends on the presence of other bodies or systems that "receive" energy from the source - in which movements in the source "induce" by their force action certain movements of charges or gravitational bodies, while performing work. And this induction, in turn, manifests itself in the loss of energy in the source.
¨ The wave zone is the more distant region of several wavelengths r >> c/f = l; often they are places hundreds, thousands, millions of l. If we place a "receiving" system here (electric charges in a coil or antenna for electromagnetic waves, or test bodies for gravitational waves), no amount of energy received by this system will affect the energy ratios in the source. We can say that the waves have already irrevocably took away from this energy source to a remote space, without any feedback, what will hapen there with this energy...
The time course of field and the shape of the wavefront
Time course of oscillation in a wave field generally depends on the dynamics of power, it may not have a regular sinusoidal shape. As we will see below (" Sources of gravitational waves in universe"), the waves from the final phases of the binary system are not exactly sinusoidal, they consist of harmonic waveforms of different frequencies and amplitudes. And when the bodies merge with each other, they even have the aperiodic character of a powerful pulse! On the other hand, using Fourier analysis, each waveform can be expressed as a superposition of harmonic functions (sine or cosine) with different amplitudes, phases and frequencies. In general analysis, therefore, waves are usually drawn as sine waves.
  Also, the shape of the wave propagation can be more complicated. In principle, the waves propagate isotropically over a spherical wavefront. However, near the source, in the inductive or near-wave zone, the field in the wave may have a complex irregular course and also the wavefront may be deformed and time-varying, not necessarily regular in spherical shape (heterogeneity in structure and motions in the source system). However, at greater distances from the source, these irregularities are usually gradually smoothed out and the waves converge to a regular spherical waveform with isotropic propagation and a harmonic (sine - cosine) time course of the field in the wave. And at great distances, the spherical wavefront has such a large radius that its curvature is almost zero, we observe a plane wave.
Longitudinal and transverse waves, polarization of waves
According to the direction in which the wave oscillates with respect to the direction of wave propagation, we distinguish two types of waves :
l Longitudinal wave , in which the amplitude of oscillations in the wave occurs in the same direction in which the wave propagates. Longitudinal waves most often arise in elastic media environments, where due to the binding forces between particles (atoms, molecules) of matter, the deflection of a given particle is transmitted to adjacent and then to more and more particles. The wave is formed by alternating areas of dilution and compaction.
l Transverse wave , where the amplitude of the oscillation in the wave is perpendicular to the direction of wave propagation. The simplest example is waves on the water surface... However, physically important transverse waves arise in fundamental physical fields - electromagnetic and gravitational. Electromagnetic waves are formed by oscillating vectors of electric intensity E and magnetic induction B, which are perpendicular to each other and oscillate in a plane perpendicular to the direction of propagation; thus cause the oscillations of electric charges in directions perpendicular to the direction of wave propagation. In a gravitational wave, the components of the metric tensor of the curved space also oscillate in such a way as to cause the test particles to oscillate in directions perpendicular to the direction of propagation of the wave (although in a more complex way - see below "Plane gravitational waves in linearized gravity").
  For transverse waves, the oscillation vector can be oriented - within a plane perpendicular to the direction of propagation - in different directions. If this oscillation direction randomly and chaotically changing, we are talking about non-polarized waves. In many cases, however, the direction of oscillation along the wave is constant or changes regularly - it is a polarized wave. If the oscillation takes place during the propagation of the wave at the same angle at all times in a plane perpendicular to the direction of propagation, it is referred to as linear polarization. The direction of oscillation while doing so may be inclined in any constant direction - the angle of polarization. In some cases, the direction of oscillation in a plane perpendicular to the propagation of the wave can change regularly and continuously, orbiting along a circle - it is a circular polarization (clockwise or counterclockwise). More generally, elliptical polarization may occur.
 Waves emitted by rotating sources - spiral polarization of waves
A very special polarization can be imparted to the waves by rotation. When waves are emitted by a rotating source, this rotation leaves specific traces on the structure of the emitted waves. At the emission of the wave, the maximum and the minimum during the rotation of the source periodically shift in time relative to each other, so the phases of the wave - the alternation of maximum and minimum - will shift as the wave propagates at speed c. The wavefront takes on the helix shape of a spiral. A kind of "rotating, twisting or spiral radiation" with specific polarization properties is created. If it hits interacting particles, it causes, among other things, their rotation around the direction of propagation of the beam, which can be considered as a manifestation of the angular momentum of this radiation. It therefore behaves like a wave with orbital angular momentum (for electromagnetic waves it is discussed in §1.5, passage "Intrinsic and orbital angular momentum of waves").
Note: In laboratory optical experiments, this light with helically rotating polarization is generated by special rotating sources, cylindrical lenses, lasers, holograms, and measured using spatial light modulators. For radio waves, helical polarization can be achieved by a special spatial configuration of antennas powered by phase-shifted alternating signals, for sound waves analogously by speaker systems. Gravitational waves arising in rotating systems of black holes or neutron stars should also have a component of helical rotating polarization (however, their measurement is complicated, so far it has not been possible).
  Bodies - particles that get into the wave field, oscillate in different ways, depending on the intensity and polarization of the waves. They can also spin around their axis (spin) or rotate around the axis of the wave beam (orbital momentum of waves).

We will now examine how these general physical-wave laws apply to a specific region of the gravitational field - gravitational waves :

Time-varying gravitational field
The gravitational field is excited by matter localized or distributed in space, according to the GTR the distribution of matter curves space-time. If the distribution of matter changes with time (the shape or position of material objects changes), the excited gravitational field also reacts to this: we will observe a time-varying gravitational field, according to GTR the changing curvature of space-time. If the source body moves periodically or the distribution of matter changes periodically, it is reflected in the surrounding space by an oscillating state of gravitational action - oscillating deformations of the curvature of spacetime. How will such a time-varying or oscillating gravitational action and the curvature of space-time behave?
  The gravitational field has many features in common with the electromagnetic field (see §1.4), Einstein's equations of the gravitational field are to some extent constructed "according to the pattern" of Maxwell's equations of electrodynamics. While watching the analogy between electrodynamics and gravity, emerges most important questions :
¨ What is the speed at which gravitational interaction propagate - the gravitational response to changes in the distribution of matter ?
¨ Is there a gravitational analogy of electromagnetic waves - gravitational waves ?
¨ How does gravity mediate energy transfer ?
  We will try to answer the first two questions in this chapter, we will discuss the issue of gravitational energy and its transfer in the following §2.8 "Specific properties of gravitational energy".

Origin and properties of gravitational waves
In principle, gravitational waves should arise wherever the position or shape of a material object changes unevenly, with accelerated motion and non-spherical changes in the distribution of matter.
Similarities and differences of electromagnetic and gravitational waves
Gravitational waves are very similar in nature to electromagnetic waves: both types of waves have a transverse character and propagate at the maximum possible speed of interactions - the speed of light c . Einstein's equations of the gravitational field are analogous in structure to Maxwell's equations of the electromagnetic field. However, there are certain structural differences between gravitational and electromagnetic waves :
× In universality of action - an electromagnetic wave vibrate only electrically charged particles (such as electrons), while a gravitational wave, representing changes in the geometry of space-time, can oscillate any mass.
× In polarization properties - electromagnetic waves have mainly dipole character, while gravitational waves have quadrupole character * ), they represent periodic changes of tidal effects.
*) "Monopole moment" represents the total mass-energy of the system, which is maintained and therefore does not cause radiation. A certain argument why even dipole gravitational waves cannot arise is the basic one itself
the principle of equivalence, according to which gravity is a universal interaction and mass always has the same sign. Thus, unlike an electric dipole, it is not possible to create a real gravitational dipole with different signs. The mass dipole corresponds to the center of gravity of the mass of the system, the first derivative of which corresponds to the momentum, which is also a conserving quantity, so that the mass dipole also does not emit any gravitational radiation. Only oscillations of quadrupole and higher mass distribution moments can emit gravitational radiation, just as oscillating electric and magnetic dipoles and higher multipoles in electrodynamics emit.
× In intensity of radiation
Penetrating the gravitational and electromagnetic waves varies in intensity - "force". Electromagnetic waves of relatively high intensity are generated by electromagnetic interaction during normal natural processes and can be efficiently generated in electronic sources (transmitters). We can also easily receive them and transform their energy. The intensity of electromagnetic wave radiation is determined at the basic level by the Larmor formula (1.61) in §1.5 "Electromagnetic field. Maxwell's equations.". However, gravity is by far the weakest interaction in nature - the bond between the gravitational field and matter is very small compared to electromagnetic or nuclear action. As will be shown below in the section "Sources of gravitational waves", the intensity of gravitational wave radiation is given by the so-called quadrupole formula (2.77), in which there is an extremely small coefficient G/c⁵ ; for the amplitude of the waves, according to formula (2.77b), the coefficient G/c⁴ is. The efficiency of gravitational wave generation and detection is therefore extremely low - under normal circumstances, gravitational waves are very weak, almost unmeasurable. Stronger gravitational waves can only occur with extreme mass accumulation, under the action of very strong gravitational fields on some compact objects in universe (will be discussed below in the section "Sources of gravitational waves").
  Basic different structural property of gravitational waves  (according to the general theory of relativity) can be expressed by the following comparison: Wave usually means the ripples of "something" in space. In gravitational waves, space itself is rippling.

General properties of gravitational propagation in GTR
Consider an isolated material system described by the energy-momentum tensor T^ik in asymptotically planar spacetime. We choose the coordinate system such that at large distances from the material source it continuously changes into an asymptotic inertial (Lorentz) system. The components of the metric tensor can be examined in the form

and h_ik = ^def g_ik - h_ik are deviations from this metric; so far we do not have to assume that the h_ik are small everywhere. We can agree that the indices will be "raised" and "lowered" using h_ik (even if it is not a tensor in the given geometry). If we define modified metric quantities

h  =^def  hⁱ_i = h_ik h^ik   ,   y_ik  =^def  h_ik - ¹/₂ h_ik h       

and we choose the coordinates so that y_ik satisfies the four conditions y^k _{i, k} = 0 everywhere , we can express Einstein's equations of the gravitational field using y^ik :

where t^ik are the quantities of the second and higher order in y^ik (t ^ik are the components of the so-called pseudotensor of energy-momentum of the gravitational field, as will be shown in the next §2.8). The solution of these "forcibly linearized" (or "seemingly linearized") Einstein equations can be expressed in the form of retarded integrals similar to electrodynamics

(2.65)

where R = ÖS(x^a -x'^a)² is the distance between the individual points x'^a of the source system and the reference point x^a, in which we determine the field (Fig.2.8). If t ^ik ¹ 0, this relation is actually an integral equation, because t^ik is a function of y^ik. However, for weak fields in the approximation of the linearized theory, the pseudotenzor t^ik is not present , and the relationship (2.65) passes in the relationship (2.55) in §2.5.

Fig.2.8. The resulting gravitational field at point xa is given by the distribution of the mass (~ energy) of the source, always retarded by the time needed by the field to overcome the distances R from the individual points x'a of the source to point xa .

How fast is gravity?
According to the relation (2.55), resp. (2.65), the resulting gravitational field at each location is given not by the instantaneous distribution of matter ~ energy, but by the delayed distribution - retarded, shifted to the past - always by the time the field needs to overcome the distance R from individual locations x'^a of the source system to the investigated point x^a by speed c (Fig.2.8). Thus, changes in the gravitational field propagate at a finite speed equal to the speed of light. In other words (in the terminology of gravitational waves, see below), gravitational waves move at the same speed as electromagnetic waves - at the speed of light c .
  At first glance, it may seem strange that the gravitational field propagates at the same velocity as an electromagnetic field, as light. However, this is not a miraculous coincidence, because the general theory of relativity, as the physics of gravity, is built on the basis of the special theory of relativity, in which the speed of light plays a decisive role in the structure of spacetime *). Rather than specifically the speed of light, this is the maximum speed of propagation of interactions, which has the value c . The answer to the question about the speed of gravitational waves can be formulated in reverse: Light propagates at the speed of gravitational waves ! Gravity determines the structure of spacetime, and it determines how objects can move - including ligh ...
*) Compare relevant discussions in §1.6 "Four-imensional spacetime and special theory of relativity" and §2.2 "Universality - a basic property and the key to understanding the nature of gravity".
  Direct experimental confirmation of the speed of propagation of the gravitational interaction is still lacking *), we cannot produce detectable disturbances in the gravitational field, so far it has been difficult to capture gravitational waves from distant space objects (see below "Sources of gravitational waves" and "Detection of gravitational waves"). However, since all other experiments and astronomical observations so far support the general theory of relativity as the correct theory of gravity, the light velocity of gravitational propagation is highly probable.
*) So far we have only indirect astronomical methods . The most convincing of these is the observation of tight binary pulsars, showing the effect of accelerating their circulation due to the emission of gravitational waves, as described below in the section "Indirect Evidence of Gravitational Waves". The extent of this effect is very sensitive to the value of the velocity of gravitational waves; the measurement of the binary pulsar PSR1913 + 16 gives the speed of gravity equal to the speed of light with an accuracy of about 1%.
    In principle, astronomical methods for comparing the speed of gravity with the speed of light are applicable. It consists in observing the optical eclipse of a distant strong cosmic source of electromagnetic radiation (such as a quasar) by a near massive moving body (such as a planet or the sun during a total eclipse), analyzing the dynamics of gravitational bending of electromagnetic beams and gravitational lensing (see §4.3, part "Gravitational lenses. Optics of black holes"). This depends on the identity or difference in the speed of the observed electromagnets. waves from a distant source and the speed of gravitational interaction from a "lensing" moving body. Either a slight shift in the image position of the remote source can be observed due to the movement of the test ("lens") body, or a slight time delay in the arrival of electromagnetic waves. These position and time shifts depend on the speed at which the gravitational field propagates from the test body, compared to the speed of the measured electromagnetic waves coming from the remote source object. However, our (radio) telescopic technology has not yet sufficiently to observe these subtle effects...

Versatility - the basic physical property of gravitational radiation
The basic physical property that distinguishes gravitational waves from all other types of radiation in nature is its completely universal action - it interacts in exactly the same way with all kinds of particles and antiparticles, with all forms of matter. It causes periodic changes in the geometric properties (curvature) of spacetime, which affect the movements of all particles and the behavior of all fields in the same way.

Planar gravitational waves in linearized gravity
At sufficiently large distances from the source masses, the gravitational field will be very weak, so that in the relation g_ik = h_ik + h_ik  there will be |h_ik| << 1. We will first assume that spacetime is practically planar with the Minkowski metric, only slightly altered by the gravitational field expressed by the quantities h_ik . In this case, all the nonlinear effects of the field feedback on the metric will be negligibly small, and such a gravitational field can then be investigated (as an independent field) against the background of Minkowski spacetime, much like an electromagnetic field. Linearized theory of the gravitational field we have already outlined in §2.5 as the simplest possibility of solving Einstein's equations. Under suitable gauge conditions (2.53), the linearized Einstein equation (2.54) applies to weak fields. For a vacuum, oy_ik = 0, which is the wave equation (same as in electrodynamics - cf. equation (1.46-47) in §1.5), the solution of which are waves propagating at the speed of light, in this case gravitational waves *).
*) The existence of gravitational waves is not a specific consequence of only the general theory of relativity. Gravitational waves must exist within each relativistic theories of gravity (as a consequence of the finite velocity of disturbance propagation in the gravitational field); only some of their properties may be different.
  The simplest solution of linearized gravitational equations in vacuum
 

describes a monochromatic plane wave with amplitude A_lm and wave vector k_r . From equations (2.54) and 2.53) the relations k_r k^r = 0, A_lm ^km = 0 follow, according to which k is an isotropic vector perpendicular to A; gravitational waves are therefore transverse waves (oscillating bodies only in a plane perpendicular to the direction of propagation) with frequency w = k° = Ö(k_x² + k_y² + k_z²) propagating at the speed of light in the direction k. Harmonic solutions (2.66) form a complete system (basis) of y functions and any solution of wave equations can be composed as superpositions of these solutions.
  Lorentz condition (2.53) reduce the number of variables y_ik from 10 to 6 independent components. Lorentz conditions do not change during the transformation y_ik ® y_ik + f_{i, k} + f_{k, i} , where f_i are four arbitrary functions satisfying the condition f_{i, l} ^l = 0 (and small enough not to violate the condition |y_ik| < <1) . The quantities y can be a suitable choice of f_i  also reduced to only two independent components corresponding to the two polarization states.
  As shown below, the component h_xx oscillates the test particles in ellipses with x, y axes, while the component h_yx oscillates them in a transverse plane rotated by 45^° relative to x, y. The polarization of a gravitational wave is called the "+" and " - " polarizations .
  For a monochromatic plane wave (2.66) the gauge function f_i can be selected so that y_io = 0, y_aa = 0. Then h_ik = y_ik and h_io = 0, h_aa = 0. Such a gauge, which is very advantageous, is called TT-gauge (Transversal Traceless calibration) *). In this TT-gauge, the components of the curvature tensor have a very simple connection with the components h_ik :

*) The procedure of gauge transformation in field theory is generally discussed in §B.6, passage "Calibration - gauge - transformation; calibration - gauge - field".
  If a plane wave propagates in the direction of the X axis, it is described by a tensor

Nonzero are therefore only two components h_ik :

h_yy  =  - h_zz  =  Re ( A₊ .e^-iw(t-x))   ,   h_yz  =  - h_zy  =  Re ( A_´ . e^-iw(t-x))   .    

Note the symmetry properties of the plane gravitational wave when rotated around the propagation axis. During the transition to the new coordinate system S', twisted around the axis of propagation of gravitational waves Z by the angle J, ie during the transformation t' = t, x' = x.cos J + y.sin J , y' = y.cos J -x.sin J , z' = z, the unit vectors of the gravitational wave polarization are transformed according to the relation e ₊ = e₊ cos 2J + e_´ .cos 2J , e'_´ = -e₊ sin 2J + e_´ cos 2J.
The definition of classical spin is as follows: A plane wave y has spin s if, when rotated by an angle J around the propagation direction, it transforms according to the law y' = e ^{i s.J}.y - in other words it remains invariant when rotated by an angle of 2p/s around the propagation axis. This symmetry is closely related to the spin of the quanta, of which, from the point of view of quantum field theory, the respective wave consists. For gravitational waves, therefore, this invariance angle is equal to 180°, so that the gravitational waves have spin s = 2 *). This spin s = 2 should therefore have a quantum of gravitational waves, so far hypothetical gravitons (see below) .
*) The polarization vectors of the electromagnetic wave transform when rotated by an angle J around the direction of propagation: e_x = e_xcosJ + e_ysinJ, e_y = e_ycosJ - e_xsinJ - electromagnetic waves have spin s = 1 , are symmetrical about a 360° rotation around the propagation direction.
Gravitons - a quantum of gravitational waves?
According to the concept of quantum physics, each energy should radiate not continuously, but in quantums. The well-known and experimentally proven quantum of electromagnetic waves are photons (see "Particle-wave dualism"). Although a complete quantum theory of gravity has not yet been developed, the analogous application of the quantum model to gravitational waves has led to the idea of the graviton: a hypothetical quantum of gravitational radiation - elementary particles mediating gravitational force in quantum field theory. The graviton is expected to be a particle with zero rest mass (the gravitational interaction has an unlimited range) and will be a boson with spin s = 2 (related to the quadrupole character of gravitational radiation discussed above); the electric charge of the graviton is, of course, zero (or irelevant). How the graviton arises in the quantum theory of gravity is discussed in §B.5 "Quantization of the Gravitational Field".
 From the point of view of the physical analysis of gravitational waves in universe (perhaps with the exception of the cosmology of the very early universe and unitary field theories), gravitons are pointless. They would only occur at very high frequencies  of gravitational waves of the order of gigahertz and higher (a kind of "gravitational gamma radiation"). Such gravitational waves do not arise anywhere in the universe known to us. Gravitons will perhaps remain permanently only hypothetical or model particles, the direct or indirect demonstration and detection of which is unlikely in the foreseeable future (for hypothetical and model particles in elementary particle physics, see the passage "Hypothetical and model particles")... In our treatise on gravitational waves we will therefore not consider them.

" Gravitationally charged " gravitational waves
Within the linearized theory of gravity, gravitational waves are completely analogous to electromagnetic waves in classical electrodynamics. In reality, however, there must be the important difference between electricity and gravity, which was already mentioned at the beginning of §2.5 "Einstein's equation of the gravitational field". If an electromagnetic wave passes through an area of space in which an electric field acts, there is no effect on the wave through that field; similarly, when the two electromagnetic waves meet, they pass "one over the other" without interfering with each other and will continue to move, as if the second wave was not. In other words, electromagnetic waves are electrically neutral (uncharged). Gravitational waves, however, are not gravitationally neutral: they transfer energy (~ mass), and are therefore both influenced by the gravitational field through which they pass, on the other hand, they (co-) acts as a another source of gravity. This is due to the versatility of gravity. It can be said that gravitational waves are "gravitationally charged", they themselves cause gravity! Below it will be quantified by the so-called Isaacson tensor of energy-momentum of gravitational waves. A hypothetical extreme consequence of this is the model of the gravitational geone (§B.3 "Wheeler's geometrodynamics. Gravity and topology."), or even"gravitational-wave" black hole created by the collapse of massive gravitational waves (mentioned in §4.5 "Black hole has no hair", passage "Uniformity of black holes") .
  Locally (in not very large areas) we can consider gravitational waves as a commotion caused by some uneven motion of matter (eg orbiting, binary supernova, non-spherical gravitational collapse, etc. - see the section "Sources of gravitational waves" below), propagating in plane space-time and it is not necessary to take into account the interaction with the total curvature space-time growth and nonlinear interactions of waves with each other. Globally, however, the curvature of spacetime caused by the distribution of other matter (such as stars and galaxies) will affect the propagation of gravitational waves - it will cause a frequency shift and change the direction of propagation. For this global curvature while also contributing the energy carried by waves themselves (see below). Thus, when propagating gravitational waves, characteristic nonlinear effects will arise [58], eg two gravitational waves will scatter each other.

So let's investigate gravitational waves in general curved spacetime. In order to be able to talk about gravitational waves at all, we must be able to distinguish the rippling part of the curvature caused by gravitational waves from the global curvature of the "background" caused by other influences (the distribution of material bodies). This separation of the global curvature of spacetime from the local fluctuations of the waves is possible in cases where the mean wavelength l is much smaller than the characteristic radius of curvature R of the spacetime against which the waves propagate :

Similarly, we can distinguish the global shape of the Earth from the local unevenness of the terrain, or the shape of an orange from the small local unevenness of its surface. The local curvature of the wave can thereby be significantly larger than the global curvature of spacetime (distinguishing the background from the waves is made possible not by the difference in the value of the curvature, but by the differences in the scales at which the curvature changes) *).
*) But as we will see below, the gravitational waves themselves cause, according to Einstein's equations, a global curvature of space-time proportional to A/l. Therefore, in order to satisfy the basic condition of the shortwave approximation (2.69), the amplitude A of the gravitational waves must also be small enough.

Spacetime satisfying condition (2.69) can then be analyzed both in terms of small scales ("local approach") and in terms of global properties of spacetime. This approximation is called the shortwave approximation and the corresponding method of gravitational wave analysis Isaacson's formalism [140]. The metric tensor (field potentials) can then be re-writen in form

where g_ik^glob is a global space-time metric of background against which the waves h_ik propagate. Similarly, the curvature tensor R_ik can be decomposed in series according to the small dimensionless parameter l/R << 1 :

where R^glob is the global curvature of the background (monotonic over a range of multiple wavelengths).

is the undulating part of the curvature linear in l/R and

is the part of the curvature tensor quadratic in l/R. Triggering and raising indices, as well as covariant derivation ";" is performed everywhere according to the metric g_ik^glob.

The general equations of the field in vacuum R_ik = 0 can then be divided into parts and analyzed from two points of view :

a) Local access
At small scales (in areas comparable to the wavelength l), where the global curvature of space is not directly applied, the linear part R⁽¹⁾_ik induced by the waves must be equal to zero

With the help of the quantities y_ik = ^def h_ik - (1/2) h g_ik ^glob, by choosing a suitable gauge in which y^k _{i; k} = 0 and by omitting members of higher orders, this equation can be rewritten in the form

Equation (2.74) is therefore the equation of the propagation of gravitational waves - the generalization of the wave equation (2.54) to curved spacetime.

From equation (2.74) follows the basic laws of propagation of gravitational waves in curved spacetime, analogous to the "geometric optics" of electromagnetic waves [271], [181 ] :

1. Gravitational waves propagate along zero geodesics (k_i k^j=0, k_i;j k^j=0 - gravitational "rays", which are curves perpendicular to the surfaces of the constant phase, are given by the equation of isotropic geodesics).
2. Polarization vector is perpendicular to the beams and is transmited along them by parallel transmission.
3. The amplitude of wave A with wave vector k forms an adiabatic invariant (A² k^a) _;a= 0 expressing the law of conservation of the "number of rays" (quantum, ie the law of conservation of the number of gravitons) in the propagation of gravitational radiation in space - time, whose global curvature changes slowly compared to the frequency of waves.

Thus, optical effects in GTR, such as redshift or curvature of rays in a gravitational field, also apply to gravitational waves.

Fig.2.9. In the Isaacson shortwave approximation, the global curvature of spacetime ("background") can be distinguished from the local fluctuations of gravitational waves if the wavelength is much smaller than the characteristic radius of curvature of spacetime. This separation is performed by averaging over a region of several wavelengths using a suitable standard weighting function W(z) converging to zero with increasing distance.

b) Global approach
In the global approach, we perform an averaging of "< >" all quantities over an area of dimensions of several wavelengths, to separate the global curvature of spacetime from local fluctuations in waves. All the structure of the fluctuating curvature caused by gravitational waves is smoothed during this averaging - <R⁽¹⁾ _ik> = 0 - while the global curvature of spacetime is practically unchanged: <R_ik^glob > @ R_ik^glob. Appropriate standard weight functions converging to zero with increasing distance (with number of wavelengths) can be used for averaging for number of wavelengths, and parallel transmission to the investigated site along the appropriate geodesy in the metric g_ik^glob [140] - see Fig.2.9. The field equations will then sound R_ik^glob + <R_ik⁽²⁾ )> = 0, which can be adjusted to the form of Einstein's equations

Gikglob  º   Rikglob - 1/2 Rglob gikglob  =   T ikwaves   ,

(2.75)

where the source on the right side

is the so-called Isaacson's tensor of "effective spread" energy-momentum of gravitational waves *).
*) How the source of the global gravitational field appears on the right side (2.75) of the global gravitational field even in "empty" space without material sources, is somewhat analogous to how the Maxwell shear current appears even in a vacuum without currents for a non-stationary electromagnetic field, exciting the magnetic field as well as the current of real electric charges (compare with §1.5, equation (1.34)).
  Equation (2.75) describes how gravitational waves curve space-time globally as they propagate. Thus, T_ik^wave we can interpreted as a tensor of energy-momentum of gravitational waves in global surrounding space-time (it is a tensor only in global geometry g_ik^glob, not in complete metric g_ik = g_ik^glob + h_ik !), for which equations (2.75) follow common the laws of conservation of T_waves^ik_{; k} = 0. The Isaacson tensor plays an important role in the correct understanding of the specific nature of gravitational energy, to which we return in the following §2.8 "Specific properties of gravitational energy".
Note: The remaining members of higher orders in equation R_ik = 0 describes the above-mentioned nonlinear "corrections" and effects, such as distortion of the waveform and the interaction of the waves with each other (wave scattering on the wave, etc.).
  The fundamental issues of gravitational wave energy transfer will also be discussed in more detail in the §2.8, in the context of general aspects of gravitational energy. In this §2.7, we focus on the method of origin (generation) of gravitational waves and on the possibilities of their detection.

Gravitational waves propagation
In the vacuum of cosmic space, gravitational waves propagate in a completely analogous way to electromagnetic waves in a vacuum. In plane spacetime with the Minkowski metric, it propagates rectilinearly at the speed of light in a vacuum, according to the universal wave equation. Graphically, wave propagation is represented using a wavefront, which is a geometric location of points in space that oscillate with the same phase during the wave. For waves from a point or spherically symmetric source, this is a spherical wavefront, the points of which lie on the spherical surface. The perpendicular to the wavefront indicates the direction of wave propagation.
  In curved spacetime, gravitational waves propagate along zero (isotropic) geodesics (2.5b), again analogous to electromagnetic waves: gravity bends - curves the direction of propagation of gravitational waves. Various "optical" effects arise here, including the possibility of gravitational lensing (for light described in §4.3, passage "Gravitational lensing. Black hole optics.") and the spectral shift of the frequency of gravitational waves. Due to the weakness of gravitational waves and the difficulty of their detection, however, these interesting effects will probably not be observed in the foreseeable future.
  During the propagation of gravitational waves through a material-substance medium there is a mechanical oscillation of the particles of this environment, which in general can lead to the absorption of gravitational waves. However, due to the very weak gravitational interaction with matter, this absorption of gravitational waves is normally very small. By analogy with light, we could also think about the slowing down of the speed of gravitational waves in a material medium, which could cause the "refraction" of gravitational waves - hypothetically about the "gravitational index of refraction" in the material environment. However, this has nothing to do with the common optical index of refraction and is practically exactly equal to 1 for all common environments (the difference compared to vacuum is not measurable). Only in an environment with enormous density, such as in neutron stars, would there be a more pronounced interaction between gravitational waves and matter, which could slow down the propagation speed of gravitational waves and generate a non-negligible index of refraction..?..
  A common general property of wave propagation - radiation - is the fact that the corresponding waves detach from the source and carry with them part of its energy, momentum and angular momentum into space, even without the presence of any distant "receiver" of these waves. The waves themselves (their fields) have energy. This is the energy of the gravitational field (whose special features are discussed in §2.8 "Specific properties of gravitational energy").

Gravitational waves emission and sources
Under what circumstances do gravitational waves arise? By analogy with electrodynamics, it can be expected that gravitational waves will be emitted during accelerated (uneven) motions of bodies, when the excited gravitational field changes over time - when the position or shape of material objects changes unevenly.
  The most common type of radiation in electrodynamics is the radiation of an electric dipole, the intensity of which is given by the second derivative of the dipole moment d = _{n = 1}S ^N q _n . r _n systems of N electric charges q_n , located in positions r_n , according to time (§1.5, relation (1.61)). In gravity, the role of the electric dipole moment is played by the dipole moment d = Sm_n . r_n mass distribution in a system of N particles m_n . The first time derivative of this dipole moment d ^. = S m _n . r _n^. º p is equal to the total momentum p system, so its second derivative will be equal to zero due to the law of conservation of momentum. It turns out that dipole gravitational radiation cannot exist, gravitational radiation must have at least a quadrupole character *).
*) This is related to the theorem of classical radiation theory [166], according to which the lowest "multipolarity" of radiation that can be emitted is greater than or equal to the classical spin of a given field. This spin is given by the degree of symmetry in the plane wave: spin s = 360°/(angle of rotation around the axis of propagation maintaining symmetry), so that for electromagnetic field with spin s = 1 the radiation is at least dipole, for gravitational field with spin s = 2 is at least quadrupole.
  Thus, we can generally consider as a source of gravitational waves any physical system with a time-varying mass distribution r(t, x^a), in which the quadrupole moment of the spatial distribution of matter also changes over time - when matter-energy moves in an accelerated non-spherical manner. Temporal changes in the distribution of matter cause corresponding temporal changes in the geometry of the surrounding spacetime - they "vibrate" the curvature of spacetime. These waves spacetime curvature - i.e. gravitational waves - detach from the source system and propagate into the surrounding space, wherein carrying away part of the kinetic energy of the moving matter in the source system.
  To determine the strength - intensity - amplitude of gravitational waves radiated by a certain physical system, we use the general solution of linearized gravitational equations in Lorentz gauge in the form of retarded potentials

formula (2.55) is presented here again for clarity

similarly in electrodynamics, where R = Ö[_{a = 1}S³ (x^a - x'^a)²] is the distance from individual places x'^a the source system to the investigated point x^a (according to Fig.2.8). If the mass-energy distribution (components of the energy-impulse tensor T_ik ) is time-varying, it will excite the time-varying potentials y_ik of the gravitational field, which can describe the radiated gravitational waves.
If the speed of movements in the investigated source system will be small in comparison with c and the gravitational contribution to the total mass-energy will be small, we can express the energy-momentum tensor as using the mass density r and the four-velocity uⁱ : T^ik  = r . uⁱ u^k and using the conservation laws (2.90) of energy-momentum in the source introduce a tensor of quadrupole mass distribution in the source K^ab = _Vn T^oo(t,x) x^a x^b dV = c² _Vn r(t,x) x^a x^b dV.
  In the limit of a weak field (and in the so-called TT-gauge-calibration) then the general relation (2.55) can be expressed using the important quadrupole formula :

expressing the "amplitude" of a gravitational wave - fluctuation of the metric h_ab(t, r) at time t and at a distance R from the source with a time-varying quadrupole moment tensor K_ab mass distribution in the source :

The amplitude of the waves thus decreases with the distance from the source as 1/r and is given by the second time derivative ^..K_ab quadrupole moment of mass-energy distribution in the source system, with retardation R/c (according to Fig.2.8).
   The quadrupole formula (2.77b) is derived within linearized field equations with source T _ik , into whose solution (2.55) or (2.65) in the form of retarded potentials the quadrupole moment (2.78) of mass distribution in the source system is implemented in the above TT-gauge (see the passage "Gravitational radiation of the island system" in §2.8 "Specific properties of gravitational energy") . The quadrupole formula was first derived by A.Einstein in 1916-18.
   To calculate the energy radiated by such a system in the form of gravitational waves (ie the intensity of gravitational waves), the methods outlined in the following §2.8 are used. If the motion of matter in a source is slow compared to the speed of light, the source is small compared to the length of the emitted waves and the field in it is weak enough, the instantaneous amount of energy gravitationally emitted by the system per unit time - "gravitational wave power" - is given again by quadrupole formula (derived in the following §2.8 "Specific properties of gravitational energy", passage "Gravitational radiation of the island system") :

where the dots mean derivatives according to time t - this is the 3rd derivative; K_ab² = K_ab K^ab , adds over a, b = 1,2,3. In astronomical terminology, the quadrupole formula (2.77b) expresses a kind of "gravitational-wave luminosity" of the source system. The intensity of radiation in the direction of the (unit) vector n to the element of the spatial angle dW is given by the relation

(2.79)

Equations (2.77) and (2.79), in agreement with the above arguments, show that only the quadrupole moment of the source is essential for the emission of gravitational waves, which must change with time, while the monopole and dipole moment do not contribute to the radiation.
   The coefficient G/(45.c⁵) shows that the radiation of energy by gravitational waves is a 5th-order effect with respect to 1/c, which, together with the small value of the gravitational constant G, leads to a very small intensity of this effect under all common conditions, except for tight orbit and merging black holes or neutron stars (discussed below in "Sources of gravitational waves in space").

Clasificattion of gravitational wave sources
Sources of gravitational wave can be classified from different points of view. According to the dimensions and location, we can distinguish between laboratory (terrestrial) and astrophysical (space) sources . In terms of the time course of the motion of matter in the source (and thus the frequency spectrum of the emitted waves), we can divide the sources of gravitational waves into two types :

• a) Periodic sources , emitting almost periodic gravitational waves for a very long time - they are various rotating bodies (around an axis that is not the axis of their axial symmetry), binary stars orbiting a common center of gravity, etc. The closer the two circulating components are and the more massive they are, the stronger the generated gravitational waves.
• b) Aperiodic sources of impulse nature, in which gravitational waves arise as a result of various "catastrophic" events - eg supernova explosions, non-spherical gravitational collapse, collision or inelastic scattering of two bodies, etc. The strongest gravitational "ripple" should occur when two neutron stars collide or when fusion of black holes.

However, some astrophysical sources that were originally periodic may become aperiodic over time. E.g. a body orbiting in an almost circular distant path around a black hole will be for a long time practically a periodic source of (weak) gravitational waves until it falls to a limit stable orbit ( §4.3 passage "Emission of gravitational waves when moving in a black hole field"). Then it is quickly absorbed by the black hole, emitting an intense flash of gravitational radiation - it becomes an aperiodic source. However, these phenomena most often occur in close binary stars (see below "Sources of gravitational waves in space", Fig.4.13-GW) .
  The simplest laboratory source of gravitational waves is a rod rotating around the perpendicular axis at an angular velocity w (Fig.2.10a). According to Equation (2.77), such a rotating rod will gravitationally radiate energy

where I is the moment of inertia with respect to the respective axis of rotation. How little energy is radiated in this way can be illustrated by the example of a steel rod 1 m in diameter and 20 m long (total weight almost 500 tons!) rotating at a maximum speed of about 4 revolutions per second (limited material strength), when gravitationally radiating energy dE/dt @ 2.2.10^-29 W; such a neglibigle value is well below the current possibilities to register it in any way. It can be seen from this that laboratory gravitational wave generators (at least in terms of mechanical-based sources) are not yet applicable to gravitational wave experiments.

Sources of gravitational waves in space
A more favorable situation can be expected in some space objects, where incomparably heavier masses come into play than in laboratory generators. An isolated star is able to emit gravitational waves either when it pulsates non-radially, or when it rotates without being axially symmetrical. In the case of a rotating star at an angular velocity w , the formula for the gravitational radiation energy is based on the formula ( 2.77)

where I is the moment of inertia and e = (a-b)/Öab expresses the deviation from axial symmetry (a, b are the principal axes in the equatorial plane). According to the relevant model, the gravitational radiation generated by this mechanism could cause the deceleration of the PSR 0532 pulsar in the Crab Nebula (pulsar has a period of about 33 ms, deceleration rate 1.3.10^-5 s./year, radiated gravitational wave power should be about 10³¹ W [ 89]) .
Note: Supernova is only a weak source of gravitational waves.
In §4.2, the section "Supernova explosion. Neutron star. Pulsars." it is shown that the supernova explosion is the most catastrophic phenomenon in the universe, emitting enormous electromagnetic and corpuscular energy. However, in terms of gravitational wave emission, a supernova is usually only a relatively weak source of gravitational wave pulse. The reason is that the collapse of the nucleus and the subsequent explosion of the supernova usually takes place almost symmetrically, without significant gravitational radiation. However, if this process were to proceed asymmetrically (perhaps due to a previous collision of the original stars in the binary system..?..) , possible detection and analysis of generated gravitational waves could yield valuable information (otherwise unattainable) about the processes in the infernal hearth of the "heart" of the supernova ...

Fig.2.10. The simplest typical examples of gravitational wave sources. a) Rotating rod as a (laboratory) source of gravitational waves. b) A binary system orbiting a common center of gravity is the most frequent source of near-periodic gravitational waves in space.

Binary stellar systems - cardinal sources of gravitational waves
The most important sources of gravitational waves are, however, tight binary systems of compact astronomical objects - neutron stars and black holes. A significant portion (more than half) of stars are part of binary or multiple systems. Individual stars in these binary systems will sooner or later deplete their thermonuclear fuel and reach the final stages of their evolution (§4.2 "Final Stages of Stellar Evolution. Gravitational Collapse. Black Hole Formation.") - and according to their remainig mass, they become white dwarfs, or collapse into neutron stars or black holes. These compact objects will then continue to orbit each other, creating gravitational waves.
  If we have two bodies with masses m₁ and m₂, which are gravitationally attracted (according to Newton's law) and orbit in circular orbits of radius r around a common center of gravity at angular velocity w (Fig.2.10b), this system will be according to quadrupole relation (2.77) radiate energy

in the form of almost monochromatic gravitational waves with frequency f = 2p/w (so far we will not consider the acceleration of rotation due to the approach of both bodies, see below).
  When orbiting along an elliptical orbit with a major half-axis a and an eccentricity e , the gravitationally radiated energy is given by a more complex relation [285]
 

where the function f(e) = (1 + (73/24) e² + (37/96) e⁴ ). (1 - e² )^-7/2 captures the growing influence of eccentricity on radiation intensity. In elliptical motion, the emitted gravitational waves contain not only the second harmonic frequency of the orbital motion (as in circular orbital motion), but also higher harmonics. The intensity of the radiation is highest in the "perihelion" where the two bodies are closest and the acceleration is greatest. This effect leads to a gradual decrease in eccentricity - the elliptical motion slowly changes to circular; overall, the orbital period is shortened.
  Taking away of the energy of the orbital motion by gravitational waves leads to mutual approaching orbiting bodies, shortening the orbital period, increasing the speed of circulation and increasing the frequency and intensity of gravitational waves. This is captured in Fig.4.13-GW (it is a modification of Fig.4.13 from §4.3 passage "Emission of gravitational waves when moving in the field of a black hole") :

Fig.4.13-GW. Time course of amplitude, frequency and intensity of gravitational radiation of a binary system of two compact bodies m1 and m2 orbiting a common center of gravity. Bodies that begin their orbit at time t = t0 on some large radius r0 descend very slowly in a spiral and continuously emit gravitational waves, initially weak (stage I). Even with tight binary systems, it is a process that lasts hundreds of thousands and millions of years. As they approach, the intensity and frequency of the radiation continues to increase. After reaching the circulation distance of several tens of gravitational radii, there is an avalanche-like increase in the intensity and frequency of gravitational waves (stage II). After reaching the limit of innermost stable orbit, the bodies fuse rapidly, sending a short intense flash of gravitational waves ( stage III ). In the upper part of the figure, enlarged sections from the last few cycles are symbolically drawn, during which both horizons are deformed and finally they are connected to the deformed horizon of the resulting black hole. The resulting black hole m1 + m2 is rotating and rapidly relaxes to the stationary axially symmetrical configuration of the Kerr black hole (stage IV) by radiating damped gravitational waves.

The reduction of the radius of circulation r of a binary system of bodies m₁ and m₂ with time t due to gravitational radiation is (in a linearized approximation) given by the relation

The time t _r®0 until the fusion of the two bodies of the binary system *), currently circulating at a distance R , is then comes out :

Using the current orbital period T of the binary system, this can be expressed by the relation :

For conventional binary systems, this fusion time is very long (on the order of billions of years or more), but for tight binary systems of compact objects, it can be relatively short from an astronomical point of view (discussed below).
*) Note: This would be the expected fusion time of idealized material points m₁ and m₂ when the distance r = 0 is reached; for real bodies of finite dimensions, this fusion time is somewhat shorter.
  For the time increase of the frequency df/dt of emitted gravitational waves in the mutual circulation of two bodies with masses m₁ and m₂ in circular orbits (around the common center of gravity) in the post-Newtonian approximation (to the order O[(G.m/r.c²)]) of the quadrupole formula [...] the relation can be derived :

The advantage of relations (2.82d, e) is that they do not explicitly contain parameters of the orbit (radii r), which are astronomically mostly unknown. By analyzing the relationship between the frequency f of the received gravitational waves and its time increase df/dt, it is possible to determine the parameter of the proportion of masses M = (m₁ .m₂ )^3/5 / (m₁ + m₂)^1/5 *) of radiating bodies. From it, in principle, the total weight m₁ + m₂ of the system can be determined and further detailed computer analysis (modeling according to "nonlinearized" general theory of relativity and fitting with the measured course of the signal from the gravitational wave) it is possible to determine in principle the masses of individual components, the radiated power of gravitational waves, or even the rotational angular momentum. .....
*) This mass parameter M = (m₁ . m₂)^3/5 / (m₁ + m₂)^1/5 in gravity-wave slang, it is sometimes called the chirp mass - "chirping matter" because the rapid increase in frequency just before the fusion of the two compact bodies resembles a bird chirp. The value of this mass parameter M is approximately equal to the geometric mean of the masses of the orbiting bodies m₁ and m₂.
  In general, the most important permanent (periodic or quasi-periodic) sources of gravitational waves in the universe are massive bodies that orbit each other (orbit around a common center of gravity). The orbing planets, such as the Earth around the Sun, emit only faint gravitational waves (in the order of fractions or units of Watts). It is different when compact gravitationally collapsed objects orbit each other - neutron stars and especially black holes. Each such body creates a deep gravitational potential pit around itself - a large curvature of spacetime. As these objects revolve around each other, the periodic motion of potential pits causes strong periodic changes in curvature - a kind of "furrow" in space-time, which, as the gravitational waves, detaches from the source binary system and propagates into the surrounding space. Gravitational waves carry the kinetic energy of rotation - as they fly into outer space, according to the law of action and reaction, they "push" back (in the opposite direction) on orbiting bodies, braking them and forcing them to move closer together, with a higher orbital speed. They are slowly approaching each other in a spiral (phases I and II in Fig.4.13-GW). Half of the released gravitational energy is carried away by the waves, the other half increases the orbital velocity (according to the Virial Act ).
Massive sources and flashes of gravitational waves 
As long as the bodies orbit the common center of gravity at great distances (due to their gravitational radius), and thus with a long period, the gravitational radiation according to Equation (2.82) is very weak. E.g. in the solar system during rotation of Jupiter generate gravitational wave carrying scant about 5.10^-2 W, during its orbiting the Earth gravity emits only about 20 W. For distant (visual) binary stars, the flux of gravitational radiation is also relatively low (~10³ -10⁷ W); in the case of tight (eclipsing) binary stars, however, the gravitationally radiated power is already ~10²⁰ -10²⁵ W (even that is quite small for astronomical conditions...).
  However, truly massive sources of gravitational waves may be a binary system of compact gravitationally collapsed objects such as neutron stars or black holes, orbiting close to each other, only a few dozen gravitational radii away *), - the possibility of their occurrence see §4.8 passage "Binary gravitationally bound black hole systems - collisions and fusion of black holes" - phase II in Fig.4.13-GW. A hypothetical binary system of two neutron stars or black holes with masses of the Sun orbiting at a distance of 10⁴ km would gravitationally emit about 3.10³⁶ W, with an orbital radius of 100 km the radiated power would even be about 3.10⁴⁶ W! Such objects would have been only quasiperiodic with a lifetime (the time of falling along the spiral of one body to another - by relation 2.82c) from several years to fractions of a second. During the actual extinction of the binary system (phase III in Fig.4.13-GW), a gigantic flash of gravitational waves with a frequency of tens to hundreds of Hz releases energy reaching a power of up to 10⁴⁷ W; for a few milliseconds, the two collapsing components "shine by gravity" as intensely as the entire observed universe in the electromagnetic field! Gravitational waves will carry about 5% of the total weight of both merging compact objects !
*) Mutual convergence of compact bodies circulation
The problem, however, is how do these circulating compact bodies get so close to each other ? In conventional binary systems, the orbital distance is at least 10⁶ km (close "spectrometric" binary stars), which is more than 100,000 gravitational radii. Should such a massive star collapsed into neutron stars or black holes at their circulation would radiate a relatively weak gravitational waves about 10²⁵ W. They would reach the stage of close circulation by gravitational radiation in a few million years (see formula above (2.82c)). However, most binary stars are much more distant (...-...) - due to gravitational radiation, they would not reach the stage of close orbit and fusion even during the entire existence of the universe! There are two possible mechanisms that compact objects could "converge" in the foreseeable future :
¨ Friction in a large and sufficiently dense cloud of gas surrounding a binary system. If more gas from the envelope of collapsing stars remains inside the binary system, dissipative convergence can occur over several million years. However, the tight binary systems created by the collapse of the oldest stars of the 1st generation, during the more than 10 billion years of the universe, could converge to a phase of intense gravitational radiation and fusion, even when there was almost no gas environment left.
¨ Gravitational interactions with surrounding stars, unless the binary system is isolated, but is located in an environment with a denser concentration of stars, such as globular clusters. To a nearby star approaching, the binary system can transmit kinetic energy through the dynamics of its orbit , bringing the two components closer together. This could also happen in the case of a multiple system.
  In the final stages due to the close approach of both black holes, due to strong mutual gravity, both horizons deform strongly, "they go agains each other" *) and vibrate wildly during rapid orbit, with unusually strong emissions of gravitational waves. Then the two horizons are interconnected into one horizon of the resulting black hole - stage III, during which there is a massive "explosion of gravitational waves". This resulting horizon is rotating and initially strongly deformed. As it rotates, it emits damped, rapidly fading gravitational waves, thereby "relaxing" to the stationary configuration of Kerr's axially symmetric rotating black hole (phase IV in Fig.4.13-GW) - "hair loss", gravitational waves carry away "hair asymmetry" - §4.5 "Theorem" black hole has no hair".
*) In the direction of the junction of the circulating black holes, "bumps" initially appear indistinct and round on their horizons, but when approached more closely, they are already sharp protrusions. These protrusions then connect the two horizons, momentarily in the shape of a rotating "8 - eight". However, due to the massive emission of gravitational waves, this shape merges into the ellipsoidal horizon of the resulting Kerr axially symmetrically rotating black hole in a few tenths of a second (at unit weights or tens of M _¤ ). Gravitational radiation then stops forever...
  To summarize, the dynamics of the orbit of a binary system of compact objects and the emission of gravitational waves can be divided into 4 stages according to Fig.4.13-GW :
I. Distant circulating along almost Kepler orbits, with weak gravitational radiation and very slow spiral approach. The duration of this stage I depends (according to formulas (2.82)) on the initial distance of orbit, it can last several billion years.
II. After approaching a distance of several tens of gravitational radii, the intensity of gravitational radiation increases greatly, which leads to a rapid spiral approach of both bodies, with the emission of increasingly massive gravitational waves with rapidly rising frequencies, from units to several hundred Hz. The final part of this stage II for compact stellar mass objects lasts only on the order of seconds.
In the jargon of hunters gravitational waves, this phase is sometimes called a "chirp", because the rapid increase in frequency just before the fusion of the two compact objects resembles a bird chirp. The dynamics of growth of the amplitude and frequency of gravitational waves is characteristic here for the masses of converging black holes.
III. The merging (fusion, collision) of both compact objects into one resulting rotating black hole, emitting a gigantic flash of gravitational waves, lasting only milliseconds.
IV. Relaxation the resulting black hole, initially strongly deformed, to a stationary axially symmetrical configuration of the Kerr rotating black hole, with a rapid attenuated reverberation of gravitational radiation in fractions of a second. Then already no more gravitational waves are emitted.
The dynamics of this reverberation of gravitational waves (reminiscent of a kind of "bell reverberation") is characteristic for the mass and speed of rotation (angular momentum) of the resulting black hole.
  If the collision and fusion of black holes takes place in a "clean" environment without gases and other bodies, only gravitational waves are emitted. However, when merging in the binary system of neutron stars, in addition to strong gravitational waves, intense emission of electromagnetic waves - gamma, X-rays, visible light, radio waves (it is discussed in §4.8, passage "Collisions and fusion of neutron stars") .
  Other compact objects serving as potential sources of gravitational waves could be binary supermassive black holes in the center of galaxies (see §4.8, section "Quasars"). According to galactic astrophysics, they could form during galaxy collisions in situations where galaxies penetrate each other with a small impact parameter and at a lower mutual speed. Black holes in the center of both galaxies can then form a bound binary system as they "pass". As they circulate, gradually approach, and eventually fuse these giant black holes, massive gravitational waves of low frequencies, milliHertz and lower, would be created... This whole final process would be far slower than the binary black holes of stellar masses. Due to the slowness and long term of these processes, it can be expected that space will be filled with a flood of low-frequency gravitational waves of various intensities and frequencies, from many sources - the stochastic gravitational-wave background.
 The expected frequencies of gravitational waves from tight binary astrophysical sources
are given by the mass of the components :
- Units up to hundreds of M_¤ : frequency 10Hz - 10kHz, possibility of detection by terrestrial interferometers.
- Thousands to millions of M_¤ : frequency 0.0001Hz - 0.1Hz, detectable by space interferometers.
- 10⁸ to 10¹⁰ M_¤ : period of the month up to decades, possibility of detection by monitoring changes in pulsar frequency.
  An medium power source of gravitational waves can also be the gravitational collapse of a star, if it occurs asymmetrically (in the case of a spherical collapse, gravitational waves do not emit - see §4.3 "Schwarzschild static black holes"). An extreme example of such a process is shown in Fig.4.14 in §4.4 "Rotating and electrically charged Kerr-Newman black holes", where during the collapse of a rotating star may occur its fragmentation and re-absorption of individual parts, accompanied (and caused) by very intense emission of gravitational waves. In any case strongly nonspherical collapse stars under the gravity radius is accompanied by a strong flash gravitational waves to carry away a considerable part of the total rest mass [289].
  An already "finished" black hole, when alone, does not emit gravitational waves, but if it forms a binary or multiple system (as mentioned above) or interacts with the surrounding matter, it can become a powerful source of gravitational waves. If a small body of mass m falls directly on a (non-rotating) black hole of mass M , the total amount of energy radiates

in the form of a "flash" of gravitational radiation with a continuous spectrum. When a body orbits a black hole in a circular path, it emits periodic gravitational waves with the total intensity given by (4.19). As a result, it constantly decreases in a spiral, the intensity and frequency of the gravitational waves increasing until the body is finally absorbed. In §4.3 (passage "Emission of gravitational waves while moving in a black hole field") and §4.4 will be shown, that the total amount of energy that can be emitted during this process by gravitational waves, makes for the non-rotating Schwarzschild black hole about 6% od rest mass falling body, while for a rotating black hole can represent up to 40% of its rest weight ! Thus we see that the unfavorable situation of excitation of gravitational waves in laboratory conditions can completely reverse for us in astrophysical scales: not only that there may be the sources of massive gravitational waves with high powers, but also the efficiency of converting rest mass to gravitational radiation can be incomporably higher than the efficiency, with which we can "benefit" from matter here on Earth, eg electricity (the efficiency of thermonuclear power plants will be only about 0.7%).

Primordial gravitational waves
The most powerful source of "all time" gravitational waves was undoubtedly the stormy creation of the universe - the "big bang". Thus, in addition to the gravitational waves of the above-mentioned astrophysical origin, the universe can also be filled with "cosmological" or "primordial" gravitational waves generated by inhomogeneities and turbulences of the super-dense substance in the period around the Big Bang [288]. These gravitational waves emitted in Planck's time, in the inflation phase (§5.5 "Microphysics and Cosmology. Inflation Universe ") , or in the event of inhomogeneities, turbulence and topological defects during symmetry breaking, would probably have the stochastic character of some "gravitational noise".
The primordial gravitational waves have weakened so much during billions of years of universe expansions, that there is probably no hope of their direct detection in the foreseeable future; in contrast to the detection of gravitational waves from relatively close astrophysical sources. An interesting possibility of indirect demonstration of primordial gravitational waves by measuring the polarization of relic microwave radiation will be discussed below in the section "Detection of gravitational waves" (passage "Measurement of polarization of relic microwave radiation").
Also in all high-energy microscopic processes (with elementary particles), in principle, gravitational waves should be emitted, but of very slight intensity, with no hope of measurement...

Detection of gravitational waves
So much therefore for the origin and properties of gravitational waves. This brings us to the last point of this treatise - the issue of gravitational wave detection. If we compare the situation with electrodynamics, then we have at our disposal very strong sources of electromagnetic waves of natural and artificial origin, which we can sensitively detect and receive. In the field of radio waves, these are, in the simplest case, ordinary radio antennas, in which the received signal is generated by electromagnetic induction. Electromagnetic waves from space are very sensitively received by radio telescopic antennas. Visible light is effectively emitted by all heated bodies (eg a filament of a light bulb), discharge lamps, in the universe of a star; and our eyes are a sensitive detector. We also have sensitive radiometers and spectrometers (Detection and spectrometry of ionizing radiation) for shortwave X and gamma radiation. However, for gravitational waves, we only have resources available in a very distant universe, whose waves are naturally very weak here on Earth. The detection of gravitational waves - the construction of a sufficiently sensitive "gravitational receiver" - is therefore an extremely delicate matter !
  To better understand this issue, let us first notice the effect of gravitational waves on the motion of test particles. According to the principle of equivalence (and in context with what was said in §2.6) the local action of gravitational waves on a single isolated particle does not exist. A gravitational wave cannot be detected by a given observer who "vibrates with her". Therefore, we must take two close or more distant test bodies A and B (Fig.2.11a) and observe the periodic changes in the distance between them, caused by the oscillating curvature of the space metric in the gravitational wave. The gravitational wave causes transverse deformations of space.

Fig.2.11. Effect of gravitational waves on test particles.
a) The world lines of two free-falling particles A and B periodically diverge and approach due to gravitational waves.
b) For comparison - the action of a (linearly polarized) plane electromagnetic wave incident perpendicular to the drawing on a set of test charged particles placed on a circle leads to periodic shifts of the whole circle of test particles in the direction dependent on the polarization of the wave.
c) The action of a plane gravitational wave incident perpendicular to a circular arrangement of mass test particles causes periodic deformations of this arrangement into an ellipse alternately in two perpendicular directions given by the polarization of the wave.

With the particles A, which we take as the reference, we connect the reference system, which will be locally inertial along the whole worldline of the particle A. The vector eⁱ in the equation of deviation of geodesics (2.57) will then be equal to the coordinate xⁱ_B of the particle B, so

Because we work in a locally inertial Cartesian system connected to the particle A, the absolute derivatives will change into ordinary derivatives and with the accuracy of the 1st order the coordinate time t coincides with the proper time t. With respect to relation (2.67), the deviation equation takes on a simpler form

If at time t = 0 was h_{a b} = 0 and the particles were at rest with each other, by integrating this equation we can obtain the relation

x^a_B(t)  »  x^b(0) [ d_ab + ¹/₂ h_ab(t, x^g_A=0) ]     ,      

expressing the oscillations of the position of the particle B with respect to A caused by the gravitational wave. The oscillations while doing so show only those components x^a_B(t) which are perpendicular to the propagation vector of the plane wave k^a (gravitational waves are transverse). Fig.2.11c shows the periodic deformation action of a plane gravitational wave on a system of regularly (in a circle) arranged test particles.
  If the monitored test particles A and B are not free, but interact with non-gravitational forces, the deviation equation (2.57) must be replaced by the equation

where Fⁱ is nongravitational 4-force describing the interaction of particles A and B . Such a case is shown in Fig.2.12a. In practice, the force Fⁱ always electromagnetic origin (all the powers of strength and flexibility in the body are caused by electromagnetic forces). As in the previous case, the oscillations of particles A and B can be used to detect gravitational waves. If we also include dissipative processes (viscous friction), we can imagine real bodies as composed of a series of such non-gravitational interacting material parts.

Gravitational wave amplitude
The force of a gravitational wave can be simply and concisely expressed by its amplitude h = DL/L_o , where DL = L_max - L_min is the maximum change in the distance of two test particles, whose original (initial) distance was L_o (Fig.2.11a). It is a dimensionless number expressing how large a relative change in the distance of the two test particles will be caused by the wave passing through it. This number is then also used to characterize the sensitivity of gravitational wave detectors.
Very roughly we can estimate the expected amplitude of the gravitational wave from the above quadrupole formula (2.77b)...
Strength - weakness of gravitational waves from space
To assess the chances, possibilities and methodology of detecting gravitational waves, it is useful to estimate how strong (or unfortunately weak...) gravitational waves can be expected to come from universe? In the passage "Origin and properties of gravitational waves", it was discussed above that gravitational waves are generally very weak. According to the calculations mentioned above in the section "Sources of gravitational waves", as well as in §4.3, passage "Emission of gravitational waves when moving in the field of a black hole" and in §4.4, passage "Movement of particles in the field of a rotating black hole", however, with relativistic mass movements in very strong gravitational fields of compact objects, 5% of the total mass can radiate in the form of gravitational waves in a short time interval (for a rotating black hole it can hypothetically be up to 40%!). In the astronomical vicinity of such objects, we could observe quite strong gravitational waves!
  Unfortunately, nowhere in the around *) do we have any such strong source of gravitational waves... The basic obstacle to successful detection is therefore the extreme weakness of gravitational waves coming to us from space. This is due to the vast distance of probable strong sources of gravitational waves - thousands and millions of light-years.
*) It is perhaps fortunate for us ! If there were such a binary system of closely circulating black holes near a few light-years away, gravitational waves strong enough to come to us, that they could destabilize the solar system !
  The expected amplitudes of the gravitational waves coming to us from the presumed sources in space are therefore very small. The main factor influencing a particular wave strength is the distance of the source r in relation to its gravitational-wave power P_gw , ie to the amount of energy that is transferred to the gravitational waves in a given process. The amplitude of the wave then approximately comes out h » 3.10^{-22 ..? ..} .P _gw/r , where the source distance r is measured in light years. If, for example, a supernova exploded near the center of our Galaxy in such a way as to transmit about 1% of the Sun's energy M_¤.c² to gravitational waves, the amplitude of the gravitational waves measured here on Earth could be estimated at h » 10^-19. For gravitational waves from supernovae in nearby galaxies, their amplitudes are estimated to be 10^-19 -10^-21 ; gravitational waves from a non-spherical supernova explosion would have the character of a pulse. During the collision and fusion of two neutron stars or black holes in distant galaxies, we could capture a flash of gravitational waves with increasing frequency with an amplitude of about 10^-20 -10^-22. The sensitivity of current gravitational wave detectors makes it possible to detect only these "catastrophic" events, accompanied by a powerful "flash" of gravitational waves; previous "quieter" phases with "moderate" gravitational radiation are still well below the detection threshold.
  The intensity of gravitational waves in our environment is therefore estimated at a maximum h » 10^-21, so that a rod with a length of units of meters could vibrate to an amplitude of about one hundred millionths of the diameter of the atomic nucleus. With such small response values, a major obstacle to detection can occur - quantum uncertainty relations (see eg "Quantum Physics") *). As we will see below, the only way to detect weak waves is to increase the distance of the test specimens and use highly sensitive methods of measuring position changes, especially interferometric ones.
*) Below in the section "Interferometric detectors of gravitational waves", Note 2: "Limitation by quantum uncertainty relations? - can they be bypassed!", it is discussed how in interferometric detection of gravitational waves it is possible to "overcome" the usual quantum relations of uncertainty, or more precisely to bypass them...
Disturbing background
As little regular signals as can be expected from gravitational waves from outer space will usually be lost in terrestrial detectors in the ubiquitous chaotic "cacophony" of interfering signals - in the background of the noise. The disturbing background in our terrestrial conditions is formed mainly by seismic waves. It is also noise and vibration from trucks, trains or aircraft flights. Electronic instrument noise and quantum noise caused by statistical fluctuations due to quantum laws of the microworld are also manifests in signal measurement (cf. eg "Quantum Physics" and "Statistical Scattering and Measurement Errors" in the monograph "Nuclear Physics and Ionizing Radiation Physics ). It is the background of interfering signals, which are usually much stronger than the useful signal, that is the basic limiting factor in the detection of gravitational waves...

Gravitational wave detectors - direct detection
Like sources, also gravitational wave detectors can be divided into individual types according to various aspects. As for the basic principle of their operation, we distinguish between mechanical detectors (measuring the movements of bodies caused by gravitational waves) and non- mechanical detectors (analysis of the influence of gravitational waves on electromagnetic fields - not yet practically implemented). Depending on their size and location, they can also be laboratory (terrestrial) and astronomical (space) detectors. Mechanical gravitational wave detectors can be divided into two groups :

• Aperiodic detectors ,
  consisting of two freely moving bodies (Fig.2.11), whose mutual distance is accurately monitored and evaluated using suitable electromagnetic signals (Fig.2.12c).
• Resonance detectors
  - flexible bodies in which time-varying tidal forces (to which each body located in the oscillating field of gravitational forces of the wave is exposed) cause periodic mechanical stresses and deformations that can lead to resonance of the body - if the incident waves contain a frequency equal to the natural mechanical frequency of the body. Using suitable sensors, the vibrations are converted into electrical signals (Fig.2.12b).

Gravitational wave detectors, especially terrestrial waves, should operate at least in pairs at a sufficiently large distance from each other (at least hundreds of kilometers) in coincidence mode This makes it possible to distinguish local disturbances (eg seismic or technical), manifested in only one of the detectors, from the signal of cosmic origin, which is detected simultaneously by both detectors.

Gravitational wave resonant detectors
Let's first notice the resonant detectors. The simplest (model) type of such a gravitational wave detector is drawn in Fig.2.12a, where two mass bodies A and B are connected by a spring. In practice, however, the resonant gravitational wave detector consists of three basic parts (Fig.2.12b) :
1. A flexible body of suitable shape and properties, which responds by mechanical movements - oscillations, vibrations, deformations - to the incoming gravitational waves.
2. A sensor that registers these mechanical oscillations and converts them into electrical signals.
3. Electronic evaluation device which amplifies, processes and records these electrical signals.
  The physics and technique of these detectors is quite complicated (we can refer in detail to the literature [270], [29], [30], [6]) and it is quite similar to the theory of antenns for radiowave reception; for this reason, resonant bodies used in mechanical detectors are also called "gravitational antennas". The basic requirements for these gravitational antennas are sufficient weight and the highest possible parameter of mechanical quality (ie the smallest possible damping of mechanical vibrations by dissipative processes).

Fig.2.12. Detection of gravitational waves.
a) A harmonic oscillator formed by two bodies A and B connected by a spring is the simplest resonant detector of gravitational waves.
b) Resonant detector of gravitational waves formed by a massive (flexible) cylinder in which gravitational waves cause oscillations. Using suitable deformation sensors, these mechanical oscillations are converted into electrical signals and further processed. A detector of this type was designed by J.Weber in 1968.
c) Interferometric gravitational wave detector (for description see "Interferometric detectors" below).

The pioneer in the field of gravitational wave detectors was Joseph Weber [269], [270], who in the 60s-70s designed the first gravitational wave detectors, consisting of aluminum cylinders with a diameter of 66 cm and a length of 153 cm (weight about 1.4 tons , fundamental resonant frequency 1660Hz), suspended in a vacuum and mechanically isolated from the surroundings. The oscillations of the cylinder were registered by piezoelectric deformation sensors. To eliminate local interference during measurement Weber installed two such detectors, one of which was located in the University of Maryland and the second in Aragon Laboratory near Chicago (distance between the two locations about 1000 km). Pulses that occurred simultaneously were considered positive cases of gravitational wave detectionin both detectors. In 1979, Weber actually registered several such coincidences, which he considered to be caused by gravitational waves. However, this optimism was not confirmed in further developments. Subsequent experiments performed with improved detection of higher sensitivity no waves being registered ...
  Furthermore, a sensitivity analysis of Weber cylinders showed that supposedly received gravitational radiation would have to have an intensity of about 1 W/cm²; if the source of this radiation were in the center of the Galaxy (as Weber estimated), then assuming isotropic radiation, the source would gravitationally emit a power of about 10⁴³ W, which corresponds to a mass loss of about 10³ M_¤ per year. Such a large gravitational power could hardly be explained by possible physical processes in the center of the Galaxy. The origin of the pulses detected by Weber is therefore unclear (the vibration of both cylinders may have been caused by disturbances in the Earth's magnetosphere caused by magnetic eruptions on the Sun..?..). However, if we look from above in the passage "Sources of gravitational waves in space", Fig.4.13, hypothetically, a rare event of fusion of two black holes (their fusion or collision) in the binary system in our galaxy could be recorded, during which it radiates in a fraction of a second. colossal energy - a "flash" of gravitational waves..?.. But it would be a big coincidence ...
   Laboratory mechanical gravitational wave detectors were further improved, with the trend being to increase detector quality parameters and noise suppression (instead of aluminum eg sapphire resonators, cooling to fraction of °K, electronic sensing apparatus improvements) rather than increasing detector weight [29], [6]. However, mechanical gravitational wave detectors have two major disadvantages :
¨ Principle limitations of sensitivity resulting from the laws of quantum mechanics: the accuracy of rod vibration measurements is limited by the quantum uncertainty principle *). The vibrations of the cylinder caused by the weak gravitational waves will be very small, of subatomic dimensions, so that quantum phenomena will be significantly applied in their measurement.
*) Below in the section "Interferometric detectors of gravitational waves", Note 2: "Limitation by quantum uncertainty relations? - can they be bypassed!", it is discussed how in interferometric detection of gravitational waves it is possible to "overcome" the usual quantum relations of uncertainty, or more precisely said bypass... According to the original estimates, these vibrations are less than a tenth of the atomic nucleus diameter (later estimates even gave amplitude amplitudes of only »10^-20 m, ie ten millionths of the atomic nucleus diameter! - was discussed above in the section "Amplitude of gravitational waves"). The quantum uncertainty principle shows that the more accurately the sensor measures the position of the ends or circumference of a vibrating cylinder, the stronger and more randomly it influences its vibrations. No sensor can monitor vibrations more accurately than quantum uncertainty relations allow. For Weber-sized cylinders, the smallest detectable vibration amplitude is about 10^-18 cm (100,000 times smaller than the size of an atomic nucleus). This seems fantastic at first glance, but for the detection of gravitational waves from distant space objects (assumed amplitudes 10^-21) probably no technically feasible resonant cylinders, using the best known types of sensors, will not be enough ...
¨ Narrow bandwidth frequency sensitivity - are tuned to a fixed resonant frequency, given the mechanical dimensions and elastic properties of the material used (usually hundreds of Hz or several kHz) and are not capable of efficiently registering signals of other frequencies. This significantly reduces their overall effective sensitivity and potential chance of successful gravitational wave detection.
   To detectively cover the expected variable frequency spectrum of gravitational waves, we would need a kind of "gamelan" composed of many cylinders tuned to different frequencies. However, we would not cover the low frequencies of the Hz unit in this way, and certainly not frequencies less than 1 Hz.

Effect of gravitational waves on electromagnetic waves ?
Since gravitational waves interact not only with material bodies but also with the electromagnetic field, the respective effects can be in principle used to detect gravitational waves in a non-mechanical way. One of the designs of such a detector [30] uses the resonant action of gravitational waves on electromagnetic waves orbiting in a circular (toroidal) waveguide, if the period of circulation of electromagnetic waves through the waveguide is equal to twice the period of incident gravitational waves. Then one part of the electromagnetic wave will still be in the "accelerating" field of the gravitational wave causing the "blue shift", while the other area will be permanently in the "decelerating" gravitational field leading to the "red" frequency shift. This resonant gravitational-electromagnetic interaction will lead to an ever-increasing phase difference and frequency of electromagnetic waves, which with a sufficiently long duration of action could in principle be measured (the waveguide would have to be superconducting). All research proposals are still only in the stage of theoretical projects. The disadvantage of this solution would be the narrow spectral sensitivity, similar to that discussed above for mechanical resonance detectors. .....

Earth and Moon ?
Another type of mechanical detector could be the Earth itself, in which gravitational waves would cause mechanical deformations and oscillations. However, the considerably high seismic background is a problem here. The connection between some earthquakes and intense flashes of gravitational waves is theoretically possible [74], although unproven. As for aperiodic gravitational wave detectors, such a gravitational "antenna" could be used by the Earth-Moon system, the distances of which would be continuously measured, for example, by means of lasers. To do so, however, the accuracy of these measuring methods should be significantly improved ..... ....

Interferometric gravitational wave detectors
In aperiodic gravitational wave detectors, subtle changes in distances between test specimens caused by a gravitational wave are monitored. The most sensitive method we have available for measuring changes in distances between bodies is laser interferometry. The great advantage of these detectors is their broad-spectrum sensitivity - they are able to register gravitational waves of different frequencies, especially low frequencies; such waves should most often come from real space sources.  
     Fig.2.12
The basic arrangement of the interferometric gravitational wave detector is shown in Fig.2.12c (which we have presented here again for clarity). It consists of two free-hanging massive test specimens M1 and M2, on which light-reflecting mirrors are mounted. These test specimens gently "sway" on the gravitational wave - as the gravitational wave passes, the distance between the mirrors increases periodically as the space expands and contracts.
   The geometrically perpendicular arrangement of the test mirrors (measuring arms) is advantageous in terms of better sensitivity, due to the quadrupole nature of gravitational waves, that oscillate space alternately in two perpendicular directions - the incoming gravitational wave slightly stretches one arm and compresses the other in the perpendicular direction. The beam of light emitted by the laser is a semi - transmissive plate S, serving as the separator, divided into two beams which are reflected from the mirrors located on the bodies M1 and M2 and returned to the plate S; here they interfere and the resulting light signal is recorded by a photoelectric detector FD. The passage of the gravitational wave in the direction perpendicular to the plane of the laser beams causes mechanical displacements of the bodies M1 and M2 such that in one half period the distance L1 increases and L2 decreases, while in the other half period L1 decreases and L2 increases. This change in the length of the paths of the interfering rays causes the two light waves to meet at different phases, which is reflected in a change in the intensity of the resulting interference signal measured by a photometer.
   For a measurable effect, a change of only a fraction of the wavelength of the laser light is sufficient. Interference detectors are therefore characterized by high sensitivity and appear to be very promising, especially after the expected improvement of the measuring technique and the achievement of a large length L1 , L2 of the measuring arms. The sensitivity can be further significantly improved by the optical realization of multiple light reflections between pairs of parallel mirrors - Fabry-Perot interferometer.
   The described interferometric detector with two perpendicular arms is basically sensitive to gravitational waves coming from different directions, but with different directional sensitivity. The best sensitivity is for waves coming perpendicularly from above (or below), while waves from the direction of the shoulder plane would be virtually undetected.
   In two notes below, we will try to mention some debatable aspects in the detection of gravitational waves :
Note 1: Elimination of the gravitational frequency shift of light in the interferometer
The passing gravitational wave periodically changes not only the distance of mirrors but also changes the geometry of spacetime, which naturally affects the movement of photons and laser light between the mirrors of the interferometric system. Above all, it causes gravitational frequency shift of light, alternately to lower and higher frequencies, which could be interfering in interferometric measurements. However, the measuring laser beam travels back and forth between the interference plate and the test specimen mirrors in a very fast sequence (even many times using the Fabry-Perot interferometer method), and the transit time and reflection of the beams in the interferometer is incomparably shorter than the detected gravitational wave period. The gravitational red and blue spectral shifts thus cancel each other out immediately and continuously. Thus, in the end, only the avtual change in distance, caused by the oscillation of the gravitational wave, is manifested in the event of interference.
Note 2: Constraints on quantum uncertainty relations? - they can be bypassed !
The need to measure extremely small (subnuclear) shifts the test bodies, caused by weak gravitational waves, naturally raises the question whether it ever allow Heisenberg quantum uncertainty relations..?.. Basic quantum uncertainty relations between the change in position of the Dx particle and its momentum change Dp is given by the product Dx. D p ³ h . If we needed to determine both the change in position and the momentum of the test mirrors at the same time in the interferometric measurement of the gravitational wave, we would have no chance. However, we only need here to measure the longitudinal change in position of mirrors that are free, not their momentum (which becomes unlearned, but we do not need it here). Thus, in this circumstance, it is possible not to violate, but to "bypass" the quantum uncertainty relations ..!..
   Despite all the technical improvements (see below "New experiments for the detection of gravitational waves"), gravitational waves could not be detected directly for many years. The gravitational waves coming from space were apparently weaker than the sensitivity of the existing detectors. In gravitational physics, then, we were in a similar situation as electrodynamics after Maxwell, but before Hertz. In the end, however, it succeed - see the passage bellow "The good news - the direct detection of a gravitational wave by the LIGO device".

Indirect evidence of gravitational waves , indirect detection
However, despite the difficulty of direct detection, we already have some indirect evidence for the existence of gravitational waves. Namely, if a system emits intense gravitationally, considerable energy is carried away from it, which leads to changes in the physical parameters of such a system. E.g. in a tight binary system, the intensity of the gravitational radiation is so great that the two bodies will approach each other in a spiral and the orbital period will be noticeably shortened; this will further increase the radiated power, so that this effect will take place with increasing speed. However, the finding of such a reduction in the period (ie an increase in the orbital frequency) in a binary star does not in inself prove that this is an effect caused by gravitational radiation. Changes in the orbital period can also be caused by the loss of mass of one of the stars, viscous braking in a gas cloud, or the overflow of mass from one component to another due to the close proximity of the two stars. In the case of tight binary systems of ordinary stars, the latter possibilities probably play a dominant role. However, if the components of the binary system are sufficiently compact (eg neutron stars or black holes), then the mutual flow of matter and viscous friction is negligible - the system is "clean" - and the reduction of the orbital period will be caused exclusively by the emission of energy by gravitational waves.
Binary pulsar  
Indeed, in 1974, J.Taylor, H.Russel, J.Weisberg and other collaborators on the large radio telescope of the Arecibo Observatory discovered the binary system PSR 1913+16 containing a pulsar (§4.2, part "Pulsars"), which proved to be very suitable not only for the given purpose, but also for testing of relativistic effects in general - it is therefore often referred to as the "astrophysical relativistic laboratory PSR 1913+16". Careful measurements have shown that the second component is also a compact object and no gas or plasma is present in the orbit of the pulsar, so that subtle relativistic effects are not overlapped by phenomena caused by mass transfer, the viscous braking, tidal forces and the like [243], []. From the point of view of GTR, it is therefore an almost ideal clock (§4.2, passage "Pulsars - precision 'clockworks' in universe") moving in strong gravitational field at a high speed along a considerably eccentric path. In addition to a number of other relativistic effects, at this object was succeeded to measure the rate of change of the period (T = 7.75 hours) of the pulsar circulation, which is about DT @ -6.7.10^-8 s/circulation. This observed rate of change of the pulsar's orbital period agrees very well with the value predicted for a given system by the general theory of relativity due to losses of orbital energy by gravitational radiation. According to the relation (2.82), the kinetic energy of the orbital motion is carried away from the binary system by the emission of gravitational waves, whereby the two components spirally approach each other (in this case by about 3 mm with each cycle, which is about 3 m/year) and according to Kepler's law, the period of their circulation decreases. Other alternative explanations of the observed changes in the pulsar circulation period - e.g. a third body of appropriate mass orbiting at a suitable distance - in the light of the observation data seems considerably unlikely.
   The binary pulsar PSR 1913+16 has the following basic characteristics [243]: the mass of each of the components is about 1.4 M_¤, the elliptical orbit around the common center of gravity has a major half-axis a » 1.9.10⁶ km and an eccentricity e @ 0.62, the circulation time is 7.75 hours, the basic pulse period of the pulsar is 59 milliseconds. The twisting of the periastra here is about 4.2° per year, which is about 10⁷-times faster than Mercury. By analysing of various variables components of periods of the pulses arrival from the pulsar was also able to measure the effect of time dilation (transverse Doppler effect), gravitational red shift and delay of the signal in the gravitational field.
   A number of binary pulsars have been discovered in recent decades, some of which are interesting for studying the dynamic gravitational effects of GTR, including the emission of gravitational waves. We can name eg PSR 1534+12, PSR 033+1715, especially interesting is PSR 0737-3039 (contains 2 pulsars).
   The measured data from the PSR 1913+16 pulsar and others have become very convincing (albeit indirect) evidence for the existence of gravitational waves.

Time modulation of the period of signals from pulsars by gravitational waves
In addition to the above-mentioned dynamic effects in binary systems of compact objects, pulsars can potentially serve also to study "already finished" gravitational waves in another way. Pulsars - fast-rotating, strongly magnetized neutron stars are sources of highly regular pulses of electromagnetic waves in outer space (§4.2, passage "Pulsars - precision 'clockworks' in universe"). As these pulses pass through space containing low-frequency gravitational waves, there is some (albeit very weak) effect on their propagation - long-period modulation of short-period electromagnetic signals from pulsars may occur due to gravitational waves. A gravitational wave of amplitude h would lead to a relative change in the repetition frequency of the pulses n_o by the order of Dn/n_o » h. By sensitive analysis of these radio signals using large radio telescopes or their systems in the future, it will be possible to measure these subtle deviations. A correlation of times would be performed detection of pulses from several pulsars according to their angular distance.
   Microsecond pulsars are particularly suitable for this purpose, in which the effects of "star shaking" and accretion, which may affect the pulse period, are less pronounced. In the future, this phenomenon will hopefully be used in principle to detect long-period gravitational waves in space - systems of a larger number of radio telescopes are being built for this detection using the Pulsar timing array method. This method is sensitive even to very low frequencies of 10^-9-10^-6 Hz gravitational waves, which predominate in space. They occur during the orbit of massive objects (such as large black holes in the centers of colliding galaxies) with an orbital period of several months to years. Low-frequency gravitational waves could also come from space-forming processes (reduced frequency due to cosmological expansion of the universe). Such measurements could be complementary to interferometric methods that are unable to detect low frequency gravitational waves.
   In 2008, the virtual project NANOGrav (North American Nanohertz Observatory for Gravitational Waves) was founded as a consortium of about 190 astronomers who, through regular observations of a set of 68 millisecond pulsars, try to detect low-frequency gravitational waves by means of fine monitoring of changes in the observed frequencies of pulsars. By sophisticated stochastic analysis of large amounts of data, it tries to distinguish the signal from certain localized sources from the stochastic background of random gravity waves from many distant sources. An analysis of the angular correlation of measured pulse deviations between pairs of pulsars in different parts of the sky may be a certain possibility. Other random variances would not show the same correlation.

Measuring the polarization of relict microwave radiation
As mentioned above at the end of the "Sources of Gravitational Waves" section, the most powerful source of gravitational waves was probably the turbulent formation of the universe - the phenomena around the "big bang". Especially with the gigantic and rapid inflationary expansion of the very early universe (§5.5 "Microphysics and cosmology. Inflationary universe.") there should be an intense ripple in the curvature of space-time - massive primordial gravitational waves should emerge, which will then propagate through space. However, with the expansion of the universe, they have become so weak and enormously lengthened their wavelengths at the present time, that there is no hope for the direct detection of these primordial gravitational waves by the methods described above. However, there is an interesting indirect method of finding "traces" *) of primordial gravitational waves, which they may have left in the ubiquitous relict microwave cosmic background at a time when these waves were still relatively strong, at the end of the radiation era. This "trace" could be a partial polarization of the relic microwave radiation that was being created at that time (separated from the substance).
*) We can symbolically compare this to how we observe trilobite prints on some stones. From such imprints, we can relatively realistically reconstruct the sizes and shapes of these ancient animals, even though their organic bodies have decomposed irreversibly and have long since disappeared. Similarly, primordial gravitational waves have now weakened and almost disappeared. However, in earlier times of space, when they were still strong, these "inflationary" gravitational waves left characteristic traces - "imprints" - on microwave relic radiation. And we can now find and measure them in principle, although it is much more difficult and complex than with those trilobite prints..!..
   The origin and properties of the relict microwave cosmic background are discussed in §5.4, passage "Microwave relic radiation - the messenger of early news space" (see also §1.1, part "Methods of nature research"). Most of the relic microwave radiation generated by the chaotic interactions of particles in a hot plasma does not show any regular polarization, their electrical and magnetic vectors E and B oscillate randomly in different planes. At the end of the radiation era, the so-called Thomson scattering occurs during the last scattering of photons on the rest of the free electrons. If photons of different frequencies from different directions collectively interact with electrons in a heterogeneous plasma, the resulting waves with a preferred plane of electric field oscillations can be generated: this is called the E-mode of polarization. However, if there is an interaction of photons that have a changed frequency due to the passage of gravitational waves (with a tensor quadrupole character), the so-called B-mode of polarization arises, which in space shows a certain "arc" or "twist" around the center of fluctuation.
   Strong gravitational waves traveling through space at the time of the formation (separation) of relic radiation should somewhat affect its properties - causing a very weak but characteristic polarization of relic microwave radiation. With sufficient accuracy and sensitivity in measuring this polarization (especially its vortex mode B), hypothetical inflationary primordial gravitational waves could thus be indirectly demonstrated. This will require a significant increase in sensitivity and the resolution of the receiving "antennas" of microwave radiation. A different problem could also be to distinguish the gravitational-wave polarization of relic radiation from the polarization on interstellar dust *). All astronomical observations outside our galaxy necessarily take place through radiation passing through interstellar dust in our galaxy.
*) The polarization of electromagnetic radiation by interstellar dust is caused by the predominant ellipsoidal shape of the dust grains, which are slightly oriented in the galactic magnetic field. The degree of this polarization is about 3%. Another source of polarization may be synchrotron radiation of electrons circling in a magnetic field (due to a very weak magnetic field, it manifests itself only in long-wave regions).

The astrophysical significance of gravitational waves
The prediction of Einstein's general theory of relativity, that at an accelerated moving matter must lose energy by emitting gravitational waves has been confirmed. It was another stimulus for the designers of sophisticated gravitational wave detectors, who could be sure of the ultimate success of their endeavor, because gravitational waves probably exist ! The astrophysical significance of gravitational waves is basically twofold :
¨ Dynamic-evolutionary effects
The emission of gravitational waves affects or is responsible for important astrophysical processes in space, leading to the evolution of many space systems. We will briefly discuss this below in the passage "Gravitational waves and dynamics of space systems ".
¨ Observational - epistemological importance
Gravitational waves are the "messengers" carrying valuable information about their sources. And if these resources are at great distances (cosmological) can carry also information that they "imprinted" on them by the intermediate interacting matter *). The cognitive significance of gravitational waves will be discussed in the following passage.
*) It is similar to the cosmic microwave radiation, in which in passing through the large space structures there is a slight modulation of temperature anisotropy (cf. §5.4, section "Microwave relict radiation - a unique messenger of early space news" passage "The influence of gravitational fluctuations of metrics in the universe to the relic radiation - Sachs-Wolf effect ").
   From the observational point of view, the successful detection of gravitational waves and their practical applications in astronomy is important - to gain a new perspective on the processes in the universe :
Gravity-wave astronomy
The observation and analysis of gravitational wave is of great potential to deepening our understanding of the universe. To reflect on how extremely important a "window" into space is gravitational wave detection and imaging, let's first briefly summarize the important stages of observing the universe with electromagnetic waves (it is also discussed in §1.1, section "Electromagnetic radiation - the basic source of information about space").
   Until the middle of the 20th century all our knowledge of the universe came from the observation of visible light - the narrow spectral range of wavelengths of electromagnetic radiation to which our eye and photographic materials are sensitive. In this optical field, the universe appears to us as a relatively calm system of stars associated in galaxies and planets orbiting smoothly (we observe the planets of our solar system directly; we would probably get a similar picture if we could observe planets around other stars). The properties of stars and planets change significantly in the optical field only over time scales of millions or billions of years (except for rare phenomena such as novae or supernova explosions). The brightest objects observed in the optical field (eyes or optical telescopes) are the Sun, planets and nearby stars, and in the more distant universe nebulae and galaxies. Light with a wavelength of about 0.5 mm is emitted mainly by excited atoms found in the hot atmospheres of stars and planets, or in large gas nebulae. Optical photometry and spectrometry therefore bring us information about temperatures and chemical composition, through Doppler spectrometry as well as the velocities of objects and gas flow.
   Since the 60's we have been observing using radio waves showed another, far more dynamic side of the universe - massive jets of gas from the nuclei of galaxies, quasars with extremely high but fluctuating brightness, pulsars rotating at high speed and emitting narrow cones of radiation. The brightest objects observed by radio telescopes are gigantic intergalactic clouds ("lobes") and jets from galactic nuclei, probably propelled by giant black holes. Radio waves (with a wavelength 10 million times longer than light) are emitted mainly by fast electrons moving at almost the speed of light in spirals in magnetic fields.
   Astronomical observations in the field of X-rays using X-ray telescopes installed on satellites began in the 1970s. Here again, a different picture of the universe appears, showing local turbulent processes around neutron stars and black holes in stellar masses, with the accretion of hot gas in binary systems. X-rays with a wavelength of the order of 1000 times shorter than light are emitted mainly by high-energy electrons in an extremely hot gas, such as those formed in accretion disks around black holes or neutron stars. Due to turbulence and shock waves in the accretion disks, this X-ray radiation has an irregular, rapidly changing intensity. It can be synchrotron radiation emitted by relativistic electrons moving in a strong magnetic field, braking radiation, radiant recombination of atoms in an ionized gas. Flashes of X-rays can occur when thermonuclear ignites hydrogen accumulated by accretion from a red giant to a white dwarf in a tight binary system. During the gravitational collapse and the birth of a black hole in the surrounding shock wave, intense flashes of gamma radiation also occur.
   Thus, astronomical observations in different spectral domains of the wavelengths of electromagnetic radiation provide significantly different images of the universe, which, however, do not contradict each other, but complement each other and compose a "mosaic" of an objective picture of the structure and dynamics of space systems. However, a number of important "stones" are still missing in this mosaic. From some places (such as the interiors of stars or regions of dense gas and dust in the central parts of galaxies), light and no other electromagnetic radiation can penetrate outside. Neutrinos can get out of here, or high energy particles. Therefore, certain hopes are placed in the detection of neutrinos and primary cosmic radiation which, however, is very difficult and is still in its infancy. Some compact objects, such as black holes, do not emit electromagnetic waves at all if they do not have accretion disks; however, if they are part of a tight binary object, they will emit gravitational waves.
   An important missing "mosaic stones" of knowledge of the universe could therefore bring the most difficult "window" into space - gravitational radiation, which - as discussed above in the section "Detection of gravitational waves" - is just beginning to "open"! ... see below "The first direct detection of a gravitational wave by the LIGO device" ...
Analogy with music - "see" and hear the music ?
The basic force, that controls the construction and evolution of the universe, is gravity. So far, we have only observed these gravity-controlled objects in space by analyzing electromagnetic radiation. In this way we learn a lot about the positions and movements of bodies and the behavior of matter in the universe, but we learn only indirectly, vicariously, incompletely about the goverging force of it all - gravity. With considerable exaggeration (but with some concise features) we can compare this situation with an imaginary example of orchestral music :
   Imagine that a concert of classical music for a large orchestra takes place in the exterior, which we observe with a powerful telescope from a hill about 4 km away. By observing the movements of the conductor's baton, violin strings, drumsticks, etc., it would be very difficult even for a good expert to know what musical composition is being played. Only the capture of sound waves by a sensitive directional microphone would help to know whether the P.I.Tchaikovsky !Concerto for Violin and Orchestra in D major" or the L.v. Bethoven's "9th Symphony" are being played. Similarly, capturing the dynamics of gravitational forces in distant cosmic objects by detecting emitted gravitational waves can help to concretize the local dynamic situation...
   However, the very sound of music does not give us complete information about its specific origin in the orchestra, for that we need visual information. Similarly, gravitational waves, due to their long-wavelength, cannot give us a detailed sharp image of astronomical objects. Only future multimodal astronomy, which studies astrophysical objects and events simultaneously using electromagnetic radiation, gravitational waves, and various emitted particles, can provide us with comprehensive knowledge.
   In addition to the astronomy of electromagnetic waves (radio, optical, X-ray and g - astronomy - §1.1, part "Electromagnetic radiation - the basic source of information about the universe"), the detection of neutrinos and cosmic ray particles, the future "gravitational-wave astronomy" is beginning to emerge, which would probably significantly expand our knowledge of the phenomena taking place in space. Detecting gravitational waves, measuring their frequency and intensity, along with showing the direction they come from, will reveal important dynamic processes with compact objects, often invisible in other ways *), including the most tumultuous processes of gravitational collapse and collisions of neutron stars and black holes.
*) They are mainly binary compact objects, which are mostly astromically and optically "silent". During their long-term close orbit, they probably lost their accretion disks (they were "torn down" or black holes had already "consumed" them before), so their fusion is not accompanied by a more powerful electromagnetic flash (radio, optical or gamma). The only way to detect these dramatic astrophysical events is to detect gravitational waves !
Some possibilities of more complex scenarios of fusion of a binary system of compact objects, in which photon radiation could also be emitted, are discussed in §4.8, section "Binary gravitationally coupled black hole systems. Collisions and fusion of black holes".
  Observations in the electromagnetic spectrum and in gravitational waves complement each other : photon radiation, including X and gamma-ray bursts, informs about the material nature of objects and the environment (such as accretion disks), gravitational waves can show dynamics leading to observed turbulent astrophysical phenomena - "make the invisible visible".
New information from gravitational waves
Gravitational waves bring us very different information from that of electromagnetic radiation. This is given by the mechanism of their origin and the properties of their interaction with matter :
l The mechanism of origin
Electromagnetic waves from cosmic sources are emitted (during deexcitation of atoms and electron interactions) individually and independently by a huge number of separate atoms and electrons. These individual electromagnetic waves, each of which oscillates somewhat differently, then fold together in the resulting radiation we observe (this is the case for radio waves and light, for X and g radiation we register individual photons that were emitted by atoms and electrons with a certain probability). Using spectrometric analysis, they carry information about temperature, composition, magnetic fields and density, which act on radiating atoms or electrons (due to the Doppler effect also the speeds of motion) .
   Gravitational waves are excited collectively, by synchronous large-scale motions of a large amount of matter - the collapse of the star's core, the mutual orbit of massive objects (stars, neutron stars, black holes). Therefore, gravitational waves bring us information about the motions of large masses and the dynamics of large curvatures of space.
l Interaction with matter
Although electromagnetic waves pass freely through the almost empty vacuum of interstellar space, but as they pass trough matter, they interact significantly with atoms and electrons, causing them to be absorbed. The areas where the supernova exploded, the gravitational collapse, the collision of black holes, or the big bang at the beginning of the universe, are surrounded by a thick layer of matter which absorbs all electromagnetic waves; neither light nor other electromagnetic waves, potentially carrying information about stormy events at this place, will simply "get out". Astronomically, we can only observe electromagnetic waves coming from weakly gravitational regions (star surfaces, glowing nebulae), which are not overshadowed by clouds of interstellar dust or ionized gas.
   Gravitational waves, which arise most intensively in such places of large mass accumulation, strong gravity and stormy phenomena, on the other hand, easily pass through clouds of gases and dust. Within these areas, they bring information about the dynamics of relativistic processes taking place there.
   Thus, in the direction, amplitude and frequency of gravitational waves (and in temporal changes of amplitude and frequency), information about turbulent processes taking place in the vicinity and inside of massive objects is encoded in some way; this information can be "bring out" only by gravitational waves, as these are areas that do not emit light and are opaque to light and other electromagnetic waves. Monitoring gravitational waves could reveal a lot about dynamic phenomena around compact gravitationally collapsed objects. In the quite characteristic time course of the amplitude and frequency of the strong gravitational waves produced in the final stages of the close orbit and collision (merger) of two neutron stars or black holes, significant information about the course of this dramatic event, involving large masses and extremely strong gravitational fields is encoded - see above the sharp increase in amplitude and frequency in Figure 4.13-GW in the section "Sources of gravitational waves in space".
   Gravitational waves can potentially also provide information about the dynamics of the earliest stages of the universe, when the universe was impermeable to all other forms of radiation *), but the "primordial" gravitational waves derived from that period can in principle be detected. With the help of large-scale space gravitational wave detectors, in the distant future, it would theoretically be possible to even take a "snapshot of the universe" during Planck's time and thus bring "light" into the mechanism of space formation..?.. 
*) We will never see the "Big Bang" in light or other electromagnetic radiation, because it is hidden behind its own powerful flash. However, with the help of gravitational waves, we could be able (at least in principle or theoretically) to "look" into the events of the very beginning of the universe.
   In principle, gravitational waves arise with each accelerated motion of matter, ie also in the orbit of planets around stars or the orbital motions of distant stars around a common center of gravity in binary or multiple stellar systems. However, the gravitational waves generated in this way are extremely weak and also very "slow" (low frequency) - their frequency is given by the period of circulation, it is one cycle in several hours, days or even years. There is no hope for the detection or even astronomical use of these gravitational waves in the foreseeable future (and probably never!)...
What would a gravitational-wave universe look like ?
If, in a hypothetical (or rather sci-fi ) concept, we had "gravitational eyes" sensitive only to gravitational radiation, or were equipped with a gravitational-wave telescope, we would see a completely different image of the universe than looking at the sky (whether night or day) than we know from previous astronomical observations. We would not see the Sun or known bright stars, constellations, nebulae. Instead, we would see numerous objects in places where we do not observe any brighter stars in the optical field. These are tight binary systems of orbiting compact objects - neutron stars and black holes, emitting gravitational waves of high power. These objects would be more numerous in those parts of the galaxy where there is a greater accumulation of older stars (there is a greater probability that many of them have already reached the final stages of their evolution and collapsed into compact objects). When we are patient, from time to time we see dazzlingly bright flashes of gravitational waves. These can be four types of dramatic events :
- The collapse of the star's core, which "detonates" a supernova explosion. If this collapse or explosion is asymmetrical, produces a strong flash of gravitational waves. Here there is a correlation between a short impulse of gravitational waves and a visual astronomical observation of a massive light brightening, which fades for weeks and months.
- Gravitational collapse of a rotating star into a black hole with fragmentation and subsequent fusion of some ejected parts (see Fig.4.14 in §4.4 "Rotating and electrically charged Kerr-Newman black holes").
- The "collision" of two neutron stars or black holes - such a direct collision is probably a very rare phenomenon.
- However, tight rotation and merging of compact objects in the above-mentioned binary system is common, in which gravitational waves have already carried away almost all the kinetic energy of the orbit. These should be the most common and strongest sources of gravitational wave flashes - see above "Sources of gravitational waves in space", Fig.4.13, passage "Massive sources and flashes of gravitational waves".
   If we had a very powerful "sci-fi" gravitational wave telescope, we would see a large number of gravitational flashes from the fusion extinctions of binary systems of compact objects (these events fill the universe with a faint spreading gravitational wave background) in distant space. If this telescope were able to detect even very low frequencies, we could see even more distant orbiting binary sytems and a faint continuous background of relict gravitational waves from the first moments of the origin of universe..?..
Gravitational waves and dynamics of space systems
In addition to observational significance (gravitational-wave astronomy), gravitational waves are also of fundamental astrophysical importance for the dynamics and evolution of many systems in space. Above all, it is the development of massive compact objects and their binary or multiple systems. As an example we can mention the process of fragmentation and reconnection during the collapse of a rotating star in Fig.4.14 in §4.4 "Rotating and electrically charged Kerr-Newman black holes". Without the gravitational waves, there would be no connection of fragments and "completion" of the gravitational collapse, the theorem "Black hole has no hair" would not apply (§4.5 "Theorem "black hole has no hair""). At tight binary systems of orbiting compact objects, it causes gravitational radiation, carrying away orbital kinetic energy, approaching rotating objects and shortening the orbital period until they eventually merge (as discussed above in the "Binary Pulsar" section). However, in distant binary star systems and planets orbiting stars, gravitational radiation is irrelevant: it is so weak that it is completely outweighed by dissipative tidal forces in the orbiting materials and friction when moving in sparse interstellar or interplanetary gas.
   The emission of gravitational waves is probably also important for the evolution of rotating galaxies in the long term.

New experiments for gravitational wave detection - LIGO, VIRGO, GEO, TAMA, LISA -
Despite improving and increasing the sensitivity of Weber-type resonant detectors (eg the detector at Stanford University reaches a sensitivity of 10^-18), interferometric detectors appear to be the most promising gravitational wave detectors. The physical principle and basic arrangement of the interferometric gravitational wave detector was described above in the basic text, section "Gravitational wave detectors" and schematically sketched in Fig.2.12c. The first such detectors, designed in the 70s, with a sensitivity of 10^-15 did not exceed Weber's original detector. Over the years, however, they constantly improved, especially through the development work of physicists and engineers in a group led by R.Weiss, K.Thorne and R.Dever. At the end of the 1980s, a laboratory interferometer MARK2 with an arm length of 40 meters was built at the California Institute of Technology, reaching a peak sensitivity of 10^-18 at that time.

LIGO - large gravitational wave detector
Under the leadership of the above mentioned group was started in 2001 in the USA the construction of the largest and the most sensitive devices for detecting gravitational waves - of LIGO ( Laser Interferometer Gravitational wave Observatory). This major project, built in collaboration with the California Institute of Technology and the Massechussets University of Technology, consists of two remote observatories. One is located in Livingstone (Louisiana), the other of the same type is located in Hanford, near Washington. Sensitivity should be in the order of h @ 10^-21 and after reconstruction even 10^-23 ! Coincidence analysis of signals from remote interferometers makes it possible to eliminate spurious signals originating in local disturbances.  
                     
   A significant increase in sensitivity by several orders of magnitude compared to previous detectors has been achieved through a combination of a number of top technical innovations. On the one hand, they are huge dimensions - the length of the arms of the interferometer is 4 kilometers (which is more than 100 times greater than with previous interferometers). The optical system of both arms is placed in two tubes 4 km long and 120 cm in diameter, in which a high vacuum is maintained. Instead of the usual two test bodies, the LIGO system uses 4 free-hanging bodies with precise mirrors with high reflectivity, two on each arm. The special geometric configuration of the pair of mirrors (and the inlet and outlet openings of the interior mirror) ensures that the laser beam is reflected many times between these parallel mirrors in each arm and only then passes through the opening in the interior mirror to the beam splitter, interfering with its partner from the other arm and hits the photodetector. This multiple reflection on the principle of the so-called Fabry-Perot interferometer allows to effectively extend the optical length of the device by a coefficient equal to the number of reflections. With 100 reflections, the optical length of the arms will be 100 times greater than the physical dimensions, ie as if the arm were 400 km long!
   In the default (idle) state, the interferometer is set so that both output interfering beams meet in antiphase and cancel out - the photodetector window is "dark". Changing the distances of the test specimens changes this phase shift, the photodetector window brightens and the photoelectric sensor sends an electrical signal proportional to the intensity of the interference beam.
   The LIGO system is equipped with a number of other advanced electronic, optical and mechanical conveniences, contributing to the improvement of sensitivity and isolation of disturbing influences - vibrations, tidal forces, thermal noise, pressure changes. The laser beam is "cleaned" into a perfectly coherent shape. Part of the beam is diverted to a frequency modulator, which creates two reference beams with a slightly higher and lower frequency than the main beam; these reference rays pass through part of the optical system, but are not subject to multiple reflections in the arms, but are reflected from the first two mirrors and fall into the photodetector, where they are compared with the interference rays from both arms. Test specimens with mirrors are suspended as pendulums on special suspensions; the hinges are fixed to the frames anchored to the columns in several mechanically insulating layers. The position of the mirrors is finely corrected by magnetic coils.
Improved aLIGO detector
In 2013-2015, a general reconstruction of the instrumentation of both LIGO detectors was carried out in order to significantly increase the sensitivity. Several significant technical innovations have been implemented :
- Increase of laser power from the original 10W to 200W. This significantly reduced quantum photon noise.
- Larger and heavier quartz test optical mirrors, which reduced the effect of thermal noise and radiation pressure of the laser radiation (and thus reduced small random movements of the mirror).
- Magnetic neutral silica fibers were used instead of the original steel wires to hang the mirrors.
- Use of electronic active seismic isolation.
   This advanced detection system, called aLIGO (advanced LIGO), has about 10-times better sensitivity than the original LIGO (sensitivity increase by a factor of 10 leads to an increase in the detectable volume of the universe by a factor of 1000 !). This significantly increased the "radius of action" of detection from many more distant sources, which increased the probability of incidence of gravitational waves; this was actually done by the first successful detection of a gravitational wave shortly after starting aLIGO (see "FirstDetection of Gravity Waves" below).

Is built (or is under construction) several other somewhat smaller terrestrial interferometric gravitational wave detectors are being , eg :
VIRGO (Italian-French project) :
Arm length 3 km, sensitivity 10^-22 at a frequency of 500Hz. The name was chosen from a cluster of about 1,500 galaxies in the constellation Virgo, about 50 million light-years from Earth; there one can expect an increased probability of occurrence of sufficiently strong sources of gravitational waves. The Virgo device has a very well solved active seismic correction. This observatory, the second largest after LIGO, is located in Cascina near the Italian city of Pisa.
GEO 600 (British-German project) :    
Arm length 600 m, declared sensitivity 10^-22 at a frequency of 600Hz. Located near Hanover.
TAMA 300 (Japan) : 
Arm length 300 m, sensitivity 5.10^-21 at a frequency of 700-1000Hz. This device serves as a precursor to a larger system: KAGRA ( KAmioka GRAvitational wave detector) :
(Large-scale Cryogenic Gravitational wave detector)
with an arm length of 3 km (located in close proximity to the famous underground neutrino detector SuperKamiokaNde - see "Neutrino detection", passage "Neutrino detector Kamioka NDE"). It will be part of a worldwide system of gravitational wave detectors.
   The construction of a detector of type LIGO in India is in the project stage.
   Furthermore, the gradual improvement of large detection systems LIGO (-> aLIGO) and VIRGO is planned, where by increasing the laser power, improved active seismic isolation + correction, using more precise mirrors and other state-of-the-art technologies, sensitivity up to h @ 10^-23 should be achieved.

The worldwide network of gravitational wave detectors
Experts have high hopes for the cooperation and electronic interconnection of several gravitational wave detectors located in different parts of the Earth. On the one hand, the simultaneous detection of pulses by independent remote detectors makes it possible to eliminate accidental false vibrations of local origin. Furthermore, as the gravitational wave travels across the earth's surface (at the speed of light), various of these detectors strike at slightly different times (in the order of a few milliseconds). The evaluation of the delayed coincidences of the signals between the individual remote detectors, it will thus allow by triangulation to determine the direction from which the gravitational wave is coming and thus to make an astronomical assignment of a place in the sky.
The interconnection of six large gravitational wave detectors is being prepared: in Hanford (LIGO) and Livigston (LIGO), in Hanover (GEO), in Pisa (VIRGO) and in Japan (TAMA-KAGRA); all are of the interferometric type. It is planned to build another LIGO detector in India.

Cosmic detectors of gravitational waves
One of the main problems limiting the sensitivity of the most advanced terrestrial gravitational wave detectors, especially in the low frequency range, is the "turbulent Earth" - a seismic background of natural origin (geological, atmospheric, tidal) as well as man-made disturbances ( crossings of heavy cars, earthworks and mining work, overflights of aircrafts). The ubiquitous seismic background makes it impossible for terrestrial instruments to detect mainly gravitational waves with frequencies less than 1Hz. For technical and geological reasons, it is also no longer possible to increase the arm length of terrestrial interferometers. Future large gravitational wave detectors will therefore have to be built in space - a network of satellites connected by laser interferometers.
LISA - cosmic gravitational wave observatory
NASA and the European Space Agency is preparing the project of detection of gravitational waves located in the universe, called LISA (Laser Interferometer Space Antenna). Three space probes equipped with lasers are to be launched into orbit around the Sun, creating a triangular interferometric system with an arm spacing of 5 million kilometers (about 10 times the Earth-Moon distance). The system of these three probes is to be launched around 2011 and will orbit the Sun at a distance of 1 astronomer units (such as Earth). As they orbit the Sun, these three probes will maintain a constant distance between them with an accuracy of one micrometer. To avoid non-gravitational effects on the movement of the probes, these probes will be maintained in an ideal geodetic path using active correction so that the position of the free-moving test specimen floating in the cavity inside the probe remains constant. The probes will emit and use special mirrors to reflect laser beams, the interference of which will be detected by detectors and transmitted to Earth.
   The LISA system will achieve much higher sensitivity and will also be able to detect gravitational waves at a much lower frequency (and therefore long wavelengths) than terrestrial detectors - frequencies from 1Hz to 10^-4 Hz. Such (and even longer) gravitational waves should predominate in the gravitational-wave spectrum from space. It will make it possible to record, among other things, the movement of neutron stars or black holes in compact binary systems (even longer before they merge) and massive black holes (weighing millions to billions of M_¤), which probably orbit around the center of galaxies and generate slow gravitational wave frequencies. In this way, it may be possible to capture primordial gravitational waves..?..
The LISA project has not yet been launched, NASA has withdrawn from it.. .... Was designed a reduced eLISA project. ....
DECIGO ( Deci-hertz Interferometer Gravitational wave Observatory)
- Japanese Project Space gravitational wave detector. Length of the arms 1000 km, max. sensitivity range 0.1 - 10 Hz. ......

The first direct detections of gravitational waves
Large and highly sensitive systems for the detection of gravitational waves have been "silent" for many years, except for noise and accidental fluctuations, no signal was recorded that would correspond to the detection of a gravitational wave.
  The turnaround occurred on September 14, 2015, when at 9:50:45 UTC both detectors in the Laser Interferometric Gravity-Wave Observatory LIGO simultaneously recorded a short but significant signal from passing through the gravitational waves whose frequency during 0.45 sec. increased from 35 to 250 Hz; then the signal dropped quickly and virtually disappeared. At the top of the peak, the amplitude reached h @ 1.10 ^-21, the signal-to-noise ratio was 24. It was shortly after the equipment was improved to increase sensitivity (advanced LIGO). The aLIGO staff called this newly detected gravitational-wave source (event, signal) GW150914 (according to the date of discovery). A detailed article on this first successful detection, signed by a team of almost 1,000 researchers and technicians, was published in February 2016 in the leading physics journal Physical Review Letters 116, 061102 (2016) .

Signal processing GW1504914 from the first successful detection of a gravitational wave by the LIGO system. The signal was detected simultaneously by an interferometric detector in Hanford (left) and Livingston (right) in coincidence with a time difference of 7 milliseconds, corresponding to a distance of 3000 km from both detectors.   At the top of the figure is the primary captured signal in both interferometers (only with a 35-350Hz baseband frequency filter) .   In the middle, this signal is fitted by a computer-modeled waveform for binary systems of two black holes. The narrow graph below it shows the differences between the actual and best suited modeled signal.   At the bottom of the figure is a two-dimensional time-frequency spectrogram (diagram) of the signal, color and brightness modulated by its amplitude. On the horizontal axis is time, on the vertical axis is frequency, color and brightness express the amplitude of the signal. It clearly shows the increase in frequency ("chirp") during the detection time. < - Phys.Rev.Lett. 116, 061102 (2016)

Interpretation
The detected signal has a similar shape as the theoretical course of radiated waves in the mutual circulation of two massive compact bodies m₁ and m₂ just before and during their fusion in the above figure 4.13-GW in the basic text, passage "Sources of gravitational waves in space" (only the increase in the amplitude of the gravitational wave before fusion is not as sharp as it appears in Fig.4.13-GW - it is because the measured signal captures only a very narrow spatial and temporal region of only a few (about 8) cycles just before fusion; to capture previous slower cycles, detection sensitivity is not sufficient) . We will present this picture again for clarity :

Fig.4.13-GW. Time course of amplitude, frequency and intensity of gravitational radiation of a binary system of two compact bodies m1 and m2 orbiting a common center of gravity. Bodies that begin their orbit at time t = t0 on some large radius r0 descend very slowly in a spiral and continuously emit gravitational waves, initially weak (stage I). Even with tight binary systems, it is a process that lasts hundreds of thousands and millions of years. As they approach, the intensity and frequency of the radiation continues to increase. After reaching the circulation distance of several tens of gravitational radii, there is an avalanche-like increase in the intensity and frequency of gravitational waves (stage II). After reaching the limit of innermost stable orbit, the bodies fuse rapidly, sending a short intense flash of gravitational waves ( stage III ). In the upper part of the figure, enlarged sections from the last few cycles are symbolically drawn, during which both horizons are deformed and finally they are connected to the deformed horizon of the resulting black hole. The resulting black hole m1 + m2 is rotating and rapidly relaxes to the stationary axially symmetrical configuration of the Kerr black hole (stage IV) by radiating damped gravitational waves.

The character of the captured signal thus corresponds to the gravitational waves emitted in a binary system during a close approach and connection ("collision", fusion) of two orbiting massive compact objects. It can be said that this detected signal carried a true "signature" or "imprint" of its origin, visible even at from a visual viewing: it is a rapid increase in frequency and amplitude (after conversion into an acoustic signal resembling a bird "beep" - chirp ) and after reaching the maximum, then a sudden drop and rapid decay of the amplitude.
  The detected signals were subjected to a very careful highly sophisticated computer analysis. Using the above formula (2.82e) (in the passage "Gravitational wave sources in space") based on the frequency and dynamics of the frequency increase, a basic estimate of the total mass of the source M = m₁ + m₂ > @ 70 M_¤. Of course, the binary system cannot be smaller than corresponds to the sum of the Schwarzschild radii of the two binary components, which here it gives 2GM/c² > @ 210 km. In order to achieve an orbital frequency of 75 Hz (half the measured frequency of the gravitational wave max. 150Hz), objects m₁ and m₂ had to orbit very close to each other (which is only possible when they are very compact), at a distance of about 350 km from each other. In the final stage, the orbital speeds reached up to 2/3 of the speed of light!
  These parameters, derived from signal analysis, place significant limitations on the nature of the binary source. Pairs of neutron stars, which are compact, would not have the required mass. For a pair of black holes and a neutron star with the required total mass, a neutron star (@ 2 M_¤) with a large black hole (@ 60 M_¤) would combine at a significantly lower frequency. Thus, black holes are here the only known compact objects that, when circulating with each other, can reach an orbital frequency of 75 Hz without being in contact before their connection. After a sudden drop in the signal behind the peak corresponding to the connection of the two black holes, smaller waves appear with rapidly decreasing amplitude, corresponding to damped oscillations of the resulting deformed black hole as it transitions to a stationary axially symmetric Kerr configuration (gravitational waves carry away "asymmetry hair" - §4.5 "The theorem "black hole has no hair"").
  This was followed by a complex computer search of the parameters of the source from which the detected gravitational waves came. In the range of weights of individual components 1-99 M_¤ and total weight up to 99 M_¤ binary systems with different circulation parameters were modeled using post-Newtonian approximations, perturbation analysis of black holes and other methods of numerical theory of relativity. An entire "atlas" of many thousands of theoretical binary sources with various parameters was created. The results of this modeling were fitted with the measured signal curves and the deviations were assessed by statistical chi- square methods and Bayesian coherence analysis. This detailed analysis of the detected signals led to the following conclusions :
  The detected signal GW1504914 came from gravitational waves emitted by a binary object of two black holes in the last phase of their close mutual circulation and merging - fusion (collision). Specified parameters of the source system :

The total value of energy carried away by gravitational waves is remarkable - three masses of our Sun have radiated! And absolutely colossal is the instantaneous gravitational-wave power - the gravitational "luminosity" of the source in the final phase at fusion - 200 M_¤ c² per second, which is 10 times more than the radiant power of all stars in all galaxies in the universe!
  From the coincidence analysis of the time difference of the 6.9 ms signal between the detectors in Hanford and Livingston, it was possible to determine by triangulation only a very rough approximate position (direction, angle) of a source in the sky that does not allow accurate astronomical assignment; it is an area of about 600 square angular degrees in the southern sky, approximately in the direction of the Magellanic Clouds (but the source was much further than these smaller neighboring galaxies). More detectors would be needed to more accurately locate a place in the sky (Virgo detector in Italy, preparated KAGRA in Japan and LIGO India) . But even then we would probably not see anything at this place *), because the black holes probably lost their accretion disks during their long-term close orbit (they were discarded, or the black holes had already "consumed" them before), so their fusion is not accompanied by a more powerful electromagnetic flash. However, a significant optical effect can be expected for the fusion of white dwarfs and neutron stars.
*) Although there was a report that at the same time a faint flash of gamma rays from about the same place in the sky was registered; however, due to the uncertainty of the position, it was probably a coincidence.
However, if this side effect of photon emission were confirmed in further observations, it would be interesting to speculate what the X or gamma-ray burst might cause, when the original accretion disks were probably ejected at high angular velocities during long-term orbits (and may have been long before consumed "by black holes). Perhaps a "common accretion disk" could have formed there around a tight binary system ..?.. Or it is a multiple system of two black holes and a white dwarf or neutron star , which in they destruction could supply a binary black hole system with a substance (gas) that interacted with hard photon radiation during a collision...?.. Some possibilities of fusion scenarios of a binary system of compact objects are discussed in §4.8, passage "Binary gravitationally coupled black hole systems. Collisions and Mergers of Black Holes".
A breathtaking story from the distant universe 
So we can tell a fascinating story that took place in ancient times in a very distant place in the Universe; however, the gnoseological denouement had "here and now" on our Earth..!.. Somewhere in the immense depths of space, at a vast distance (about 1 billion light-years), there is an unnamed galaxy in space, which would appear even in the largest astronomical telescopes only like a tiny speck (although it contains hundreds of billions of stars). About 10 billion years ago (when neither the Sun nor our solar system yet existed) two 1st-generation stars with masses of about 30-50 solar masses formed near each other from a dense gas-dust cloud, which orbited each other in a tight binary star system. These stars consumed hydrogen, helium, carbon and heavier elements very quickly in intense thermonuclear reactions (cf. §4.1, section "Thermonuclear reactions inside stars") and in about 1 million years exploded as supernovae and subsequently collapsed into black holes. The resulting black holes around each other (around the common center of gravity) continued to orbit several million kilometers away - as a binary system of compact objects. At first, they had accretion disks from the remaining gases from the cloud around them (there is a section on them in §4.8 "Accretion disks around black holes"), but gradually they "consumed" them (or discarded them in the final stages). These black holes orbited each other for billions of years, emitting relatively weak gravitational waves at first. This caused a gradual decrease in the radius of circulation, initially very slow (only about millimeters per year). Over time - billions of years, with a gradual approach, however, the gravitational radiation intensified and the approach accelerated. As the two black holes orbited several hundred thousand kilometers in their orbits, the intensity of the gravitational waves began to increase avalanche, along with an increase in the frequency of the orbit, leading to an increasingly rapid helical decrease in orbit. This "death spiral" then very quickly resulted in a "collision" - merger, fusion - of both black holes. In this final phase of the last few orbits and mergers, a colossal amount of gravitational wave energy was radiated. It was a lucky coincidence, that these gravitational waves had arrived to our Earth right now, when the immense efforts of many hundreds of physicists, engineers and workers managed to construct such a sensitive detector of gravitational waves (LIGO), that could detect these gravitational waves even over a hughe distance..!.. So that was the signal GW1504914 .

Significance of direct detection of gravitational waves
The gnoseological significance of the first direct detection of gravitational waves GW1504914 can be summarized in 5 points :
¨ 1. Direct proof of the "physical" existence of gravitational waves and the properties of their interaction with bodies.
¨ 2. The measurement shows the existence of binary systems of black holes of stellar mass, confirming the correctness of the ideas of stellar relativistic astrophysics about the evolution of massive stars and their binary systems (more frequent occurrence of such massive double black hole systems, however, was not expected ..!..).
¨ 3. It is the first observation of a "catastrophic process" of close circulation and the fusion of two black holes to emit a colossal flash of gravitational energy.
¨ 4. Further confirmation of the correctness of the general theory of relativity, under very "exotic" conditions of very strong time-dynamic gravity and highly relativistic velocities (all previous tests were based on sensitive analysis of subtle relativistic effects in weak gravitational fields).
¨ 5. This success is likely to stimulate the upgrading of existing detectors and the construction of new ones - building a denser global network of gravitational wave detectors, enabling accurate coincidence-triangulation determination of the position of detected sources in the sky and thus their astronomical assignment. And perhaps even the construction of large space gravitational wave detectors with many times higher sensitivity and spectral range. This will open a new "window into space" - gravitational-wave astronomy (outlined above in the section "Astrophysical Significance of Gravitational Waves").
Pitfalls and doubts in the detection of gravitational waves
When measuring such subtle effects (at the limit of detectability) as gravitational waves provide, there are naturally many pitfalls and technical difficulties. And even after overcoming them, some problems with interpretation and doubts about the validity of the obtained results remain ...
A certain "disadvantage" of the first direct detection of gravitational waves GW1504914 is loneliness - the fact that it could not be verified by other independent measurements or correlated with any particular astronomically observed object. Old binary compact objects are astromically and optically silent. During their long-term close orbit, they have already lost their accretion disks (discarded them or "consumed" them before), so their fusion is not accompanied by a more powerful electromagnetic flash (radio, optical or gamma). The only way to detect these dramatic astrophysical events is to detect gravitational waves. Their direct astronomical assignment is usually not possible ...
  This is a rare astrophysical event *), which has been secretly "prepared" for millions or billions of years (according to the above formula (2.82c) in the section "Space sources of gravitational waves") during the orbiting of compact stars in a binary system under weak gravitational radiation, which was well below the sensitivity of our detectors. It is only possible to capture it completely the last phase of this process - a close approach, several dozen last orbits and the interconnection of the two black holes, emitting a huge "flash" of gravitational waves.
*) A rarity event ?
The " rarity " is only relative from a global perspective . With a huge number of stars in the astromically observed universe, estimated at @ 10²² (our galaxy has about 2.10¹¹ stars, in the field of view of large astronomical telescopes there are about 10¹¹ galaxies), the gravitational collapse of more massive stars, which are mostly part of a binary or multiple system, very often occurs, creating black holes (mass greater than @ 5 M_¤). It can be expected that for billions of years, fusion in the binary systems of orbiting compact objects probably occurs several times a day, emitting massive gravitational waves (space is as if "flooded" with a weak background of gravitational-wave "noise" from these sources) - see "Gravitational-wave universe". Usually, however, it is too far away for us to detect them with today's detectors. In the 1Megaparsek circuit, the frequency of fusion of binary compact objects is estimated at about 2-400/year. Only the strongest ones have a chance to detect; if it were to succeed again repeatedly in the near future, it would make it possible to specify this hitherto very indeterminate range of incidence of mergers of compact objects. This has already succeeded, below are listed other successful detections of gravitational waves.
The fact that the aLIGO system detected a significant gravitational-wave signal GW1504914 so soon after its improvement is certainly a big coincidence on the one hand... The original LIGO detector would not detect it either at all or with such a large disturbing background that a more detailed analysis would not be possible. The original LIGO would perhaps in a few decades significantly capture another, much stronger, fusion signal in a much closer binary system of compact objects..?.. Reconstruction and improvement of the LIGO system significantly increased the "radius of action" - the probability of detecting gravitational waves from much more distant binary resources, or from closer weaker sources. The incidence of gravitational wave capture on aLIGO could thus be significantly increased. .......
  Thus, even if GW1504914 is very well physically and technically substantiated, the possibility of some unknown interfering effect *) cannot yet be ruled out with absolute certainty..!.. Eg. a weak but extensive variable magnetic field from disturbances in the Earth's magnetosphere caused by eruptions on the Sun might also be able to slightly vibrate the metal components in the arms of the interferometric detector..?.. However, such fields are monitored around LIGO.
*) How was it probably in 1979 with Weber's cylinders..?.., see "Gravitational wave detectors".
  However, astronomers hoped, that in the near future a similar stormy astrophysical events will occur in our or some nearby galaxy, whose gravitational waves will be detected independently by several systems - and hopefully can be assigned astronomically..?.. It has already succeeded - see below "Further direct detection of gravitational waves".
Further direct detections of gravitational waves
Following the signal GW1504914, another weaker signal was detected at aLIGO on 12.10.2015, which could also probably come from gravitational waves emitted during close circulation and fusion of two compact objects, LVT151012 (a more pronounced signal was recorded in Livigston). However, the amplitude was more than 10 times weaker than that of GW1504914 and it could not be reliably evaluated against the background of the noise signal.

	Computer evaluation of gravitational-wave signal LVT151012
	Computer evaluation of gravitational-wave signal GW151226 PhysRevLett.116.241103

On December 26, 2015, another gravitational-wave signal GW151226 was detected. Although this signal was more pronounced than LVT151012, but in comparison with the first signal GW1504914 it was also not very good - with the naked eye we would not find a gravitational wave on the graph of the measured signal; only very careful computer analysis was able to "extract" useful data from it.
  The third direct detection of gravitational waves occurred on January 4, 2017, the signal was named GW170104. This was again a typical signal from the fusion of a binary black hole system :

Computer evaluation of gravity-wave signal GW170104 : Weight 1: 31 M ¤ Weight 2: 19 M ¤ Total weight: 50 M ¤ Weight of the resulting black hole: 48 M ¤ Gravitationally radiated energy: 2 M ¤ . c 2 Luminosity distance: 880 Mpc Source: LIGO

The fourth direct detection of gravitational waves from the fusion of black holes was successful on August 14, 2017, the signal was named GW170814 . It is remarkable that for the first time a gravitational wave signal was detected simultaneously by three instruments: first LIGO Livingstone, 8 milliseconds apart LIGO Hanford and finally Virgo in Italy after 14 ms. (from Livingstone). This allows to more accurately locate the place in the sky where the gravitational waves are coming from. For GW170814, this stereotactic analysis showed an area of the sky as small as 60 degrees² in the constellation Eridanus in the southern sky. The localization here is about 10 times better than just two LIGO detectors; however, no astronomical (optical, radio) side effect was observed here, which is not to be expected with the fusion of black holes. The different orientations of the arms of these three detectors further make it possible to analyze the polarization of the gravitational wave (the measured data corresponded to the above-mentioned polarization alternating in two perpendicular directions according to Fig.2.11) .

LIGO Hanford	LIGO Livingstone	Virgo (Italy)
			Computer evaluation of gravity-wave signal GW170814 : Weight 1: 31 M ¤ Weight 2: 24 M ¤ Total weight: 56 M ¤ Weight of the resulting black hole: 53 M ¤ Gravitationally radiated energy: 2.8 M ¤ . c 2 Luminosity distance: 540 Mpc Source: LIGO

GW170817 - Merging neutron stars
Three days later, on August 17, 2017, another gravitational-wave signal GW170817 was detected by LIGO and Virgo detectors , which was interpreted as the last phase of the orbit and merging of two neutron stars. Two observed facts led to this remarkable conclusion :
1. Analysis of the course of the gravitational-wave signal .
The gravitational wave signal was observable for about 100 seconds, with a low amplitude starting at 30Hz, and during about 3000 cycles, the amplitude and frequency increased to about 400Hz; then the signal stopped. This course corresponded to the collision and merging of both compact objects with a smaller weight and a diameter larger than a black hole. A detailed analysis of the course of this gravitational-wave signal determined the masses of orbiting and converging compact bodies in the range of about 1.2-1.6 M_¤ and the total weight of the binary system 2.75 M_¤ . This corresponds to the masses of astronomically observed neutron stars.
The signal first came to the Virgo detector in Italy, then about 22ms. later to the LIGO-Livingstone detector and for another 3ms. to the LIGO-Hanford detector. These three detections made it possible to triangulate the source to an area of 30 degrees² in the southern sky in the region of the constellation Hydra.
2. Emissions of electromagnetic radiation .
For the first time when capturing gravitational waves, the optical-electromagnetic counterpart in the form of a gamma-ray burst GRB170817A (1.7 s. after merging) was astronomically registered here and after about 10 hours also in the optical and infrared field - object SSS17a in galaxy NGC4993, in the region localized by gravitational wave detection. After a few days, the object was observed with X-ray cameras Chandra, then in the area of radio waves at the VLA. The spectral maximum of electromagnetic radiation quickly shifted from the gamma, X-ray and UV regions to the optical and infrared regions. These observations in the electromagnetic region correspond to the situation at the fusion of two neutron stars, when the ejected material, rich in neutrons, explosively "nucleonizes" and transforms radioactively into the nuclei of heavy elements and glows intensely (§4.8, passage "Collisions and fusion of neutron stars"), which observationally manifested itself similarly to the nova explosion *).
*) Such an astronomically observed event is sometimes called a "kilonova" - it can be up to 1000 times stronger than a normal nova, especially if viewed from the direction of the rotational axis of the binary system. However, this was not the case, the optical flash was relatively weak despite the relative proximity of the source. This can be explained by the fact that the axis of rotation was inclined at least 30 ° from the viewing direction.
  However, the fusion of neutron stars is a completely different process that has nothing to do with the explosion of a nova. Therefore, we do not use the misleading name kilonova in our treatises ...

	Computer evaluation of gravity-wave signal GW170817 : Weight 1: 1.3-1.6 M ¤ Weight 2: 1.2-1.4 M ¤ "Chirp" weight 1.2-1.4 M ¤ Total weight: 2.7-2.8 M ¤ Weight of the resulting compact object: 2.7 M ¤ Gravitationally radiated energy: 0.025 M ¤. c2 Luminosity distance: 40 Mpc Source: PhysRevLett.119.161101 (2017)	Upper: Time-frequency diagram of detected GW170817 signals from individual detectors. Unfortunately, a graph of the primary detected gravitational-wave signals has not yet been published (perhaps due to the disturbing short-term electronic fluctuation that occurred at LIGO-Livingstone about 1 second before the maximum ...).
Bottom: Fusion of neutron stars. a) Two neutron stars orbiting in a binary system at a great distance descend very slowly in a spiral and continuously emit gravitational waves, initially faint. b) As you approach, the intensity and frequency of gravitational radiation continue to increase. c) Upon close approach, deformation occurs and eventually a collision and fusion of the two neutron stars occurs. d) During rapid rotation during fusion, a large amount of neutron substance can be ejected, which immediately nucleonizes to form predominantly heavy nuclei, followed by radioactive decay. e) The resulting object, after the instabilities disappear, is either a neutron star or a black hole (depending on the remaining mass). This resulting object will have only a small accretion disk around it (since most of the substance has been ejected away by the enormous energy released during explosive nucleonization). (Source: AstroNuclPhysic §4.8, passage "Collisions and fusion of neutron stars")

Neutron stars emit weaker gravitational waves than black holes as they orbit and merge. However, compared to previous gravitational wave detections, the GW170817 event was much closer - at 130 million instead of billions of light-years, so it could be detected. The final product of the observed fusion neutron stars is probably a black hole, but it could be even larger neutron star..?..
  The case of detection of neutron stars fusion detection is a harbinger of multimodal - multimessenger astronomy - here dual-modality [gravitational-wave + electromagnetic]. It will also be a new way to study the relationships between matter, gravity and electromagnetism. The third modality is also perspective - detection of neutrinos, if the merging of neutron stars occurs at a closer distance (cf. §1.2, part "Detection of neutrinos"; promising here is mainly the Antarctic glacier detection system IceCube, see passage "Detection of neutrinos in glaciers").
  GW170817 has important astrophysical significance. This is the first direct observation of neutron star fusion, in which a large amount of neutron material is ejected followed by explosive nucleonization to form a large number of heavy elements - see the figure in §4.8 above, passage "Collisions and fusion of neutron stars". These heavy elements (including gold, platinum, uranium, ...) have enriched outer space. Other such observations will help refine estimates of how common these events are in space and the extent to which neutron star fusions are involved in the cosmic nucleogenesis of heavier elements; together with stellar synthesis and supernova explosions (cf. §4.1, part 4 "Evolution of stars" and §4.2, part "Astrophysical significance of supernovae").
Note: Overestimated share of nucleogenesis from neutron star fusion 
When the first multimodality detection of gravitational waves from the fusion of neutron stars was achieved, in the first enthusiasm, opinions began to emerge that most (or even all?) of the heavy elements in the universe came from fusions of neutron stars. However, more sober analyzes have shown that these estimates are strongly overestimated. The fusion of neutron stars is not so frequent as to explain the observed number of heavy elements in space. It is an important part, but the main source of heavy elements is probably stellar nucleosynthesis and supernova explosions...
 Gravitational "Standard Siren"
The detection of gravitational waves from the fusion of neutron stars, together with the simultaneous detection of electromagnetic radiation from their optical counterparts, also has astrophysical-cosmological significance for the independent determination of the distance of the respective objects. This measurement makes it possible to combine the determination of the distance to the source derived from the gravitational wave analysis with the receding velocity derived from the redshift measurement using electromagnetic signal spectrometry. This approach not requires the use of a cosmic distance "ladder" (discussed in §4.1, passage "Determining the distance of cosmic objects - a fundamental condition of astrophysics"). The analysis of gravitational + electromagnetic waves can thus be a measurement of the direct determination of the relationship distance <--> luminosity in cosmological scales, without the use of intermediate distances, with sometimes problematic continuity. This independently determined value of the Hubble constant is here 70.0_+12.-8 km/s/Mpc, in good agreement with existing measurements by other methods. The uncertainty of about 10% here is mainly caused by the uncertainty of the orientation of the axis of rotation of the binary system with respect to the Earth. Gravitational wave detection is therefore sometimes metaphorically referred to as a "standard siren" - a gravitational-wave analogue of the electromagnetic "standard candle" (cepheids, Ia supernovae) used to determine large cosmological distances...
 GW190425 - another neutron star merger ?
The next detection of gravitational waves from the merger of neutron stars occurred on April 25, 2019. Unfortunately, they were full-fledged detected only by the LIGO Livingstone detector (the second detector at Handford was currently shut down, the Virgo detector had a low signal-to-noise ratio). Therefore, localization of the source in the sky could not be done. Analysis of the signal showed that it was probably the collision of two atypically massive neutron stars of 1.6-2 M_¤ and 1.4-1.7 M_¤, total mass 3.4 M_¤ (it is also considered that they could be even small black holes..?..), luminosity distance 160 Mpc. No emission of electromagnetic radiation was found to correlate with the detection of gravitational waves. It could therefore be the situation mentioned in §4.8 (passage "Collisions and mergers of neutron stars"), when during the collision of more massive neutron stars, the resulting object can almost immediately collapse into a black hole, which quickly eats up almost all the mass of both neutron stars. No radiation will then escape from the collision site - the kilonova is not observed...
GW190521
Another interesting gravitational wave detection event was recorded on May 21, 2019 from the direction of constellation Coma Berenices. A detailed analysis of the results showed that it was a merger of the two largest black holes so far, weighing 85 and 66 M_¤, which took place 17 billion light-years away :

Computer evaluation of gravity-wave signal GW190521 : Weight 1: 85 M¤ Weight 2: 66 M¤ Total weight: 150 M¤ Weight of the resulting black hole: 142 M¤ Gravitationally radiated energy: 8 M¤.c2 Luminosity distance : 5,3 Gpc Source: LIGO PhysRevLett.125,101102 (2020)

From an astrophysical point of view, this event is interesting due to the relatively large masses of black holes involved and the resulting black hole. Black holes with masses greater than about 60-70 M_¤ should be rare according to current astronomical knowledge, a kind of "gap" is observed in masses between 60 and 100,000 M_¤. Nuclear-astrophysical analyzes show that massive stars with a residual mass greater than about 65 M_¤ in the final stage of evolution during contraction, they probably do not collapse into a black hole, but before reaching the horizon they succumab to the so-called electron-positron pair instability (see §4.1, passage "Electron-positron pair instability"). In this process, the star is smashed - scattered when the supernova explodes, leaving no black hole behind. If this is indeed the case, medium-weight black holes should be rare. Measurements of GW190521 show that the final stages of massive stars may produce black holes of higher masses, and even more massive black holes form when merging between pairs of smaller black holes. This process of multiple merging can then continue hierarchically ...
  Another point of interest here is the capture of a flash of light (at the Palomar Observatory), which could be related to this event. Since no light is emitted during the actual merging of the black holes (discussed above), it has been hypothesized that the resulting black hole could have entered the path through the accretion disk of a nearby supermassive black hole, in the material of which a light effect could occur..?. .
Gravitational-wave astronomy 
The ever-increasing number of other detected signals - despite their rather weak signal-to-noise ratio - show that the first successful detection of the GW151226 gravitational wave was not a coincidence, but that the possibility of "gravitational-wave astronomy" is beginning to develop promisingly..!.. And also multimodal - "multi-messengers" - astronomy.
  So far, existing ground-based gravitational wave detectors are only able to detect powerful one-time gravitational waves from catastrophic events such as collisions - mergers - of black holes or neutron stars. And that only from the last stages of these fusions, lasting around one minute, with relatively high frequencies. Future detection systems (such as the planned LISA), in addition to higher sensitivity, will also be able to detect low-frequency gravitational waves - from many hundreds or even thousands of close binary star systems.

2.6. Deviation and focus of geodesics		2.8. Gravitational energy

Vojtech Ullmann