AstroNuclPhysics ® Nuclear Physics - Astrophysics - Cosmology - Philosophy | Gravity, black holes and physics |
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 (disturbance, oscillations) through space . Wave propagation is generally
conditioned by two basic aspects :
1. The mechanism of changes - disturbances, 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 elastic material environments,
mechanical deformations can occur by force, 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, however
distant; 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
330m / s., In water about 1500m / s., In hard solids is
significantly higher (eg in steel about 5000m / s. with.).
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
- by wave equations . 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 :
d ^{2 }f
/ dt ^{2} = c.d ^{2 }f / dx ^{2} .
The solution of the wave equation is a special function of
spatial coordinates and time - a wave function
that has the general form: f (x, t) = f
(x, tx / c). If we start from some starting point about the
coordinate x _{o} at the time t _{o} , then the same value of the deviation f , as at the point o coordinate x _{o} at time 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 = 2 p f is the circular
frequency ; waves are often caused by periodic
oscillating movements 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 decomposed by
Fourier 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 }^{(tx
/ c)} ).
*) The speed of wave
propagation is denoted by c here , but it does not
have to be the speed of light.
^{ }In the three - dimensional analysis in the coordinates
x, y, z, the wave equation has the general form :
(1 / c ^{2} ) .¶ ^{2 }f / ¶ t ^{2} = ¶ ^{2 }f / ¶ x ^{2} + ¶ ^{2 }f / ¶ y ^{2} + ¶ ^{2 }f / ¶ z ^{2} ,
which is often equivalently written using the Laplace
operator D : (1 / c ^{2} ) .¶ ^{2 }f / ¶ t ^{2} = D f . In a 4-dimensional relativistic
formulation, then using d'Alembert's operator oş - (1 / c ^{2} ). ¶ ^{2} / ¶ t ^{2} + ¶ ^{2} / ¶ x ^{2} + ¶ ^{2} / ¶ y ^{2} + ¶ ^{2} / ¶ z ^{2} as of = 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 Einstein's
equations of the gravitational field (shown below in the section" Origin and properties of gravitational
waves ") .
If these fundamental equations of substances or fields result in
wave equations, it means that in the given substance environment
or a physical box 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 (ie. a
collection of particles, and generally any physical system) so
described. 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 y ú ^{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 y ú ^{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 " 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 rippled
. 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 is clearly analyzed using wavefronts
using 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, They are superimposed
with the original waves, as well as 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^{ }
laws of reflection, diffraction and
refraction of waves .
^{ }A common general
feature of wave propagation - radiation
- is the fact that the waves in question detach from the
source and carry some of its energy, momentum and momentum
into space, even without the presence of any distant "receiver" of
these waves. 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
specimens 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
>>
^{ } = 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 specimens for gravitational waves)
, no amount of energy received by this
system will affect the energy balance in the
source. We can say that the waves have already irrevocably took
away from this energy source to a remote area, without any
feedback, what's with this energy becomes ...
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 , not have a regular
sinusoidal shape. As we will see below ("
VesmirneZdrojeGravitVln "), 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 they 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 ( longitudinal
), 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 wool is formed by alternating areas of dilution and
compaction.
l Transverse
wave ( transverse ), 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 ; they
cause 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 cross wave vector can be vibration
- within the plane perpendicular to the direction of propagation
- oriented in different directions. If this oscillation direction
randomly and chaotically changing, we are talking about non-polarized
waves . In many cases, however, along The direction of
oscillation is constant or changes regularly - it is a polarized
wave . If the oscillation occurs during the propagation
of the wave at the same angle in a plane
perpendicular to the direction of propagation, it is called linear
polarization . polarization angle^{ }. In some cases, the direction of oscillation in a plane
perpendicular to the propagation of the wave can change
regularly and continuously , circling in a circle - it is a circular
polarization (clockwise or
counterclockwise) . More generally, elliptical
polarization may occur .
^{ }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
spreads the gravitational interaction - gravity 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 oscillates 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 do.
×^{ }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 / c ^{5} ;
for the amplitude of the waves, according to formula (2.77b), the
coefficient G / c is ^{4} . 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
space (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) we can express it by the following
comparison: Wave usually means the wave
of "something" in space. In gravitational waves, space
space itself waves.
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
g _{ik} = h _{ik} + h _{ik} , | (2.63) |
where h _{ik} = | / | -1 | 0 | 0 | 0 | \ | is a Minkowski metric |
| | 0 | 1 | 0 | 0 | | | ||
| | 0 | 0 | 1 | 0 | | | ||
\ | 0 | 0 | 0 | 1 | / |
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} h^{i}_{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} :
y ^{ik }_{, lm} h ^{lm} = - 16 p (T ^{ik} + t ^{ik} ),^{ } | (2.64) |
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' ^{and the} source system and the reference point x ^{a} , in which we determine the field (Fig.2.8). If t ^{ik} ą 0, this relation is actually an integral equation, because t ^{ik} is a function of y ^{ik} . However, the weak field in approximation theory is linearized t pseudotenzor ^{also} present, and the relationship (2.65) changes in the relationship (2.55) in §2.5.
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 cover the distance R
from individual locations x ' ^{and
the} source
system to the investigated point x ^{and} 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 speedas 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. It 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 fabric of space * ). 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 light ...
*) Compare relevant discussions in §1.6
"Four- dimensional 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, we have not yet been able to capture
gravitational waves from 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 is not enough 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 fieldw e have already outlined in §2.5
as the simplest possibility of solving Einstein's equations.
Under suitable calibration 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
^{ }
y _{lm} = Re (A _{lm} . e ^{i. k}^{ r }^{x }^{r} ) | (2.66) |
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 _{x }^{2} + k _{y }^{2} + k _{z
}^{2} )
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 calibration 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 calibration, which is very advantageous , is called TT-calibration ( T ransversal T
raceless) . In this
TT-calibration, the components of the curvature tensor have a
very simple connection with the components h _{ik} :
R _{a }_{o }_{b }_{o} = R _{o }_{b }_{o }_{a} = - R _{a }_{oo }_{b} = - R _{o }_{ab }_{o} = - (1/2) h _{ab }_{, oo} = - (1 / 2c ^{2} ). ¶ ^{2} h _{ab} / ¶ t ^{2} . | (2.67) |
If a plane wave propagates in the direction of the X axis, it is described by a tensor
h _{ik} = | | | 0 | 0 | 0 | 0 | | | . |
| | 0 | 0 | 0 | 0 | | | ||
| | 0 | 0 | h _{yy} | h _{zz} | | | ||
| | 0 | 0 | h _{yz} | -h _{yy} | | |
Nonzero are therefore only two components h _{ik} :
h_{yy} = - h_{zz} = Re ( A_{+} .e^{-i}^{w}^{(t-x)}) , h_{yz} = - h_{zy} = Re ( A_{´} . e^{-i}^{w}^{(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 ', rotated
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 pol and the gravitational wave realization
are transformed according to the relation e' _{+}
= e _{+} cos2 J
+ e _{´} .cos2 J
, e ' _{´} = -e _{+} sin2 J + e _{´} cos2 J.
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
^{is }^{J} .y
- in other words it
remains invariant when rotated by an angle of 2 p /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 r is based on 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 _{x} cos J
+ e _{y} sin J
, e _{y} = e _{y} cos J
- e _{x} sin J
- 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 pointless). 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 space (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^{ }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 setkaj s undergo a
"one over the other" without interference and continue
their movement, as if the second wave was not. In other words,
electromagnetic waves are electrically
neutral
(nanabité). Gravitational waves, however, are
not
gravitationally neutral: they transmits energy (~ ´mass),
and secondly because they are influenced by the gravitational
field through which they pass, either (co-) act as a source of gravity . This is due to the versatility of gravity . It can be said that gravitational
waves are " gravitationally charged ", they themselves show 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 :
l << R. | (2.69) |
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 (to distinguish from the background of
waves is possible n ikoliv difference of the curvature, but
the differences in the scales where 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.
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 broken down in form
g _{ik} = g _{ik }^{glob} + h _{ik} , | (2.70) |
where g _{ik }^{glob} is a global space-time metric against which h _{ik} waves propagate. Similarly, the curvature tensor R _{ik} can be decomposed in series according to the small dimensionless parameter l / R << 1:
R _{ik} = R _{ik }^{glob} + R ^{(1) }_{ik} + R ^{(2) }_{ik} ) + F [ l / R) ^{3} ] , | (2.71) |
where R ^{glob} is the global curvature of the background (monotonic over a range of multiple wavelengths).
R_{ik}^{(1)} = ^{1}/_{2} (-h_{;ik} - h_{ik;l}^{l} +h_{lk;i}^{l}+ h_{li;k}^{l}) | (2.72) |
is the undulating part of the curvature linear in l / R a
R_{ik}^{(2)} = (1/2) [^{1}/_{2}_{ }h_{lm;i} h^{lm}_{;k} + h^{lm}(h_{lm;ik} + h_{ik;lm} - h_{li;km} - h_{lk;im}) + h_{k}^{l;m}(h_{li;m} - h_{mi;l}) - (h^{lm}_{;m} - ...... no longer fit on line - will come to add_{ } | (2.73) |
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
in small scales (in areas comparable to the wavelength l ), wherein the global curvature of space
directly does not claim to be the linear portion R ^{(1) }_{ik} induced waves
equals zero
R ^{(1) }_{ik} = 0. | (2.74) |
With the help of the quantities y _{ik} = ^{def} h _{ik} - (1/2) h g _{ik }^{glob} , by choosing a suitable calibration in which y ^{k }_{i; k} = 0 and by omitting members of higher orders, this equation can be rewritten in the form
y _{ik; l }^{l} + 2.R ^{glob }_{likm} y ^{lm} = 0. | (2.74 ') |
Equation (2.74) is therefore the equation of the propagation of gravitational waves - the generalization of the wave equation (2.54) to curved spacetime.
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 ] :
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
G_{ik}^{glob} ş R_{ik}^{glob} - ^{1}/_{2 }R^{glob} g_{ik}^{glob} = T _{ik}^{waves} , | (2.75) |
where the source is on the right
T_{ik}^{waves} = - (c^{4}/8pG) [<R_{ik}^{(2)}> - ^{1}/_{2} g_{ik}^{glob}. <R^{(2)}>] | (2.76) |
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 (compare
with §1.5, equation (1.34)) exciting the
magnetic field as well as the current of real electric charges.
^{ }Equation (2.75) describes how gravitational waves curve space-time globally as they propagate T _{ik }^{wave}^{ }we can therefore interpret it 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 following §2.8, in the context of general aspects
of gravitational energy. Here we focus on the method of origin (generation) of gravitational waves and
on the possibilities of their detection .
Radiation
and sources of gravitational waves
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 = S m _{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 }^{.
}ş 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 science [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 "wave"
the curvature of spacetime. These waves spacetime curvature -
i.e. gravitational waves - the detach
from the source system and spread to the surrounding space,
wherein the take away part of the kinetic energy of the moving
mass 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 calibration in the form of retarded potentials
formula (2.55) is presented here again for clarity |
similarly in
electrodynamics, where R = Ö
[ _{a = 1} S ^{3} (x ^{a} - x ' ^{a} ) ^{2} ] is the distance from individual places x' ^{and} 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 u ^{i}
: T ^{ik} = r . u ^{i} 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}
= _{V }ň
T ^{oo} (t, x) x ^{a} x ^{b} dV = c ^{2 }_{V }ň r (t, x ) x ^{a} x ^{b} dV.
^{ }In the limit of a weak field^{ }(and in the so-called
TT-calibration) then
the general relation (2.55) can be expressed using the important quadrupole formula :
h _{ab} (t, R) = [ (2G / c ^{4} ). ^{..} K _{ab} (t - R / c) ] _{/ }_{R} , | (2.77a) |
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:
K _{ab} (t) = ň r (t, x) · (3 x _{a} x _{b} - d _{ab} x _{g} x ^{g} ) dV. | (2.78) |
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-calibration (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 quadrupole
again formula (derived in the
following §2.8 " Specific properties of
gravitational energy ",
passage " Gravitational radiation of the island system
") :
d E / dt = - (G / 45.c ^{5} ) ^{...} K _{ab }^{2} , | (2.77b) |
where the dots mean derivatives according to time t - this is the 3rd derivative; K _{ab }^{2} = 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 argument, 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.
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 :
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
d E / dt = - (32.G / 5.c ^{5} ) I ^{2} w ^{6} , | (2.80) |
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; so slight value is far below the current possibilities it any registered 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 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)
d E / dt = - (288.G / 45.c ^{5} ) I ^{2} e ^{2} w ^{6} , | (2.81) |
where I is the moment of inertia and e = (ab) / Ö ab 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 10 ^{31} W [ 89] ) .
Note: A supernova is only a weak source of
gravitational waves. In §4.2, the
section " Supernova
explosion. Neutron star. Pulsary. " 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 ..? ..) , or 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. a ) Rotating rod as a (laboratory)
source of gravitational waves. |
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 become their mass depending
on 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
m _{1} and m _{2} , 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^{ }
d E / dt = - (32.G / 5.c ^{5} ) [ m _{1} . m _{2} / (m _{1} + m _{2} ) ] ^{2} r ^{-4} w ^{6} , | (2.82a) |
in the form of almost
monochromatic gravitational waves with frequency f = 2 p / w (apart from
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]
^{ }
d E / dt = - (32.G / 5.c ^{5} ) [m _{1 }^{2} .m _{2 }^{2} / (m _{1} + m _{2} )] a ^{-5} . f (e), |
where the function f (e) = (1 + (73/24) e ^{2} + (37/96) e
^{4} ). (1 -
e ^{2} ) ^{-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
.
^{ }The removal 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 m _{1} and m _{2} orbiting a
common center of gravity. Bodies that begin their orbit at time t = t _{0} on some large radius r _{0} 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 you approach, the intensity and frequency of the radiation continue 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 holem _{1} + m _{2} is rotating and rapidly relaxes to a 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 m _{1} and m _{2} with time t due to gravitational radiation is (in a linearized approximation) given by the relation
dr / dt = - (64.G ^{3} /5.c ^{5} ) [ m _{1} . m _{2} . (m _{1} + m _{2} ) ] / r ^{3} . | (2.82b) |
The time t _{r }_{® }_{0} until the fusion of the two bodies of the binary system *), currently circulating at a distance R , is then based on:
t _{r }_{® }_{0 } @ ( 5.c ^{5} /256.G ^{3} ) . R ^{4} / [ m _{1} . m _{2} . (m _{1} + m _{2} ) ] . | (2.82c) |
Using the current orbital period T of the binary system, this can be expressed by the relation:
t_{r}_{®}_{0} @ (5.c^{5}/256) . (T/2p)^{8/3}/[G^{5/3}.(m_{1}.m_{2})/(m_{1}+m_{2})^{3}] » 10^{7}[year]. T[hour.]^{8/3}.{[(m_{1}.m_{2})^{3/5}/(m_{1}+m_{2})^{1/5}]/M_{¤}}^{-5/3}. | (2.82d) |
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 m _{1} and m _{2} at a distance r = 0; 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 m _{1} and m _{2} in circular orbits (around the
common center of gravity) in the
post-Newtonian approximation (to the order
O [(Gm / rc ^{2} )]) of the quadrupole formula [...] the relation can be derived
:
^{df}/_{dt} @ (m_{1}.m_{2})/(m_{1}+m_{2})^{2/5}.G^{-3/5}.c^{-12/5}.(96/5)p^{8/3}.f^{11/3} , which
can be adjusted to the form: (m_{1}.m_{2})^{3/5}/(m_{1}+m_{2})^{1/5} @ (c^{3}/G). [(5/96)p^{-8/3}. f^{ }^{-}^{11/3}.(df/dt)]^{3/5} . |
(2.82e) |
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 = (m _{1} .m _{2} ) ^{3/5} / (m
_{1} + m _{2} ) ^{1/5} *) of radiating bodies. From it, in
principle, the total weight m _{1} + m _{2 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 momentum. .....
*) This mass parameter M = (m _{1} . m _{2} ) ^{3/5} / (m
_{1} + m _{2} ) ^{1/5} in gravity-wave
slang sometimes called the chirp mass -
" chirping matter " because the rapid growth
rate just before the merger of two compact objects reminiscent
can cvrliknutí. the value of this mass parameter M is
approximately equal to the geometric mean of the masses of the
orbiting bodies m_{1} and m _{2} .
^{ }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 orbits of
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,
like 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 IIin
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 remote (visual) stream of
binary stars is also gravitational radiation P relatively
low extraction (~ 10 ^{3} -10 ^{7}^{ }
W); in the case of
tight (eclipsing) binary stars, however, the gravitationally
radiated power is already ~ 10 ^{20} -10 ^{25} W (even
that is not enough for astronomical conditions ...) . 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
around him
, only a few dozen gravitational radius *), -
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 10 ^{4} km would gravitationally emit
about 3.10 ^{36} W, with an orbital radius of 100
km the radiated power would even be about 3.10 ^{46} W! Such objects would have been only quasiperiodic with a lifetime (the time of the fall of
the spiral body on the second one - 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 10 ^{47}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 !
*) Approximation
of the circulation of compact bodies^{ }
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 10 ^{6} 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 10 ^{25}
^{ }
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 " approximate "
in the foreseeable future:
¨ Friction
in a large and sufficiently dense cloud of gas
surrounding a binary system. In the event that inside a binary
system remains greater quantities of gas from the envelope
collapsing stars can to dissipativne^{ }convergence
will 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 , " meet
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 symmetricrotating black holes (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 closer closer, 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
orbit 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 withrapidly
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"
- " chirp " because the rapid growth rate just
before the merger of two compact objects reminiscent can
cvrliknutí. 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 holes, initially strongly deformed, to a
stationary axially symmetrical configuration of the Kerr black
hole with a rapid attenuated reverberation of
gravitational radiation in fractions of a second. Then no more
gravitational waves are emitted . The dynamics of this reverberation of gravitational
waves (reminiscent of a kind of " bell reverberation
") is characteristic of the mass and speed of rotation
(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, intenseemission
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.
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.
- 10 ^{8} to 10^{10} M _{¤} :
period of the month up to decades, possibility of detection by
monitoring changes in pulsar frequency.
^{ }An intense 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 leads to the fragmentation and re-absorption
of individual parts, accompanied by (and caused by) a very
intense emission of gravitational waves. In any case strongly
nonspherical collapse stars under gravity the 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
D E » 0.0025 m ^{2} c ^{2} / M | (2.83) |
in the form of a "flash" of gravitational radiation with a continuous spectrum. When a body orbits a black hole in orbit, 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 the field of black holes ") and §4.4 will be shown that the total amount of energy during this process can emit g ravitačními waves, makes the non-rotating black hole Schwarzschild about 6% 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 we are in astrophysical scales can completely turn: not only can be the source of massive gravitational waves high performance, but also the efficiency of the conversion of the rest mass to the gravitational radiation can be much higher than the efficiency, with which we can "benefit" from matter here on Earth, for example, 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 ,
are likely to the
stochastic character of some "gravitational noise".
The primordial gravitational waves have weakened
so much during billions of expansions of the universe that there
is probably no hope of their direct detection in the foreseeable
future; even if gravitational waves from relatively close
astrophysical sources were soon detected. 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 about 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 (perhaps 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 light
lines of two free-falling particles A and B periodically recede 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 particles A , which is taken as the reference, connect the reference system which will be locally inertial world lines along the time tuples A . The vector e ^{i} in the equation of deviation of geodesics (2.57) will then be equal to the coordinate x^{i}_{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 it was h _{a b} = 0 and the particles were at rest with each other, we can obtain the relation by integrating this equation
x ^{and }_{B} (t) » x ^{b} (0) [ d _{b} + ^{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 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 F ^{i} 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 F ^{i} 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 include more dissipative processes (viscous friction), we can represent a real body 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 = D L / L _{o} , where D L = 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 then characterizes 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 space? 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 vicinity *) 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 possible for usfortunately
! 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 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 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 _{¤} c ^{2} 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 ...
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
Like sources, 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
implemented) .
Depending on the scope and location, these can also be laboratory (terrestrial) and astronomical (space) detectors . Mechanical
gravitational wave detectors can be divided into two groups :
Gravitational wave detectors, especially terrestrial waves, should operate at least in pairs at a sufficiently large distance from each other (at least hundreds of kilograms) 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) Gravitational wave interferometer (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 intaloval Weber two such detectors, one of which was
located in the University of Maryland and one 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 / cm ^{2} ; 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 10 ^{43} W, which corresponds to a mass
loss of about 10 ^{3} 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, ° K fractional cooling, 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 of 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 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. .....fill
in?.........
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 ..... add ......
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 M 1 and M 2 , 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 M 1 and M 2 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 M 1 and M 2 such that in one half period the
distance L 1 increases and L 2 decreases, while in the other
half period L 1 decreases and L 2 increases. This change in the
length of the paths of the interfering rays causes the two light
waves to meet atdifferent 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 L 1 , L 2 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
. In the end, only the real 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 D x particle and its
momentum change D p is given by the product D x. D p ł 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 to measure
the longitudinal change in position here mirrors
that are free, not their momentum (which becomes unlearned, but
we do not need it here). In this circumstance, it is possible not
to break, but to " bypass " the quantum
uncertainty relation ..! ..
^{ }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
However, despite the difficulty of direct
detection, we
already have some indirect evidence
for the
existence of gravitational waves. 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 the
binary system is tight gravitational radiation intensity is
sufficiently great so that the two bodies will spiral closer to
each other and the orbital period b ude noticeably shorter; this will
increase the radiated power even further, 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 has not yet in itselfand
the effect caused by
gravitational radiation. This is because 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 solely by radiating 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, which proved to be very suitable not only
for the 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 orbiting pulsar absent or gas plasma so subtle
relativistic effects n e are overlapped 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 (pulsar) moving in strong gravitational
field at a
considerable speed along a very eccentric path *). In addition to
a number of other relativistic effects, the rate of change of the
period (T = 7.75 hours) of the pulsar circulation, which is about
D T @ -6.7.10 ^{-8} s / circulation, was measured
for this object . This observed ry c Hlosta changes orbital period
pulsar agree very well with the value for the system predicts the
general theory of relativity as a result of loss
of orbital energy 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 approach each other in a spiral approach
(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 - n etc. the
third body of the 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 » 1.9.10 ^{6} 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 filling of the periastra here
is about 4.2 ° per year, which is about 10 ^{7}-times faster than Mercury. Analysis of
various periods of variable components of arrival of pulses from
pulsar was also able to measure e f ekty time dilation (transverse Doppler)
gravitational red shift and delay of the signal in the
gravitational field.
^{ }The measured data from the PSR
1913 + 16 pulsar became very convincing (albeit indirect) evidence for the existence of gravitational
waves .
Possibility of
using modulation of signals from pulsars
In addition to the above-mentioned dynamic effects in
binary systems of compact objects, pulsars can potentially be
used to study gravitational waves in other ways. Pulsars - fast-rotating, strongly magnetized
neutron stars are sources of highly regular pulses of
electromagnetic waves in outer space. 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 by Doppler effectsn _{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.
This method, sensitive to even very low
frequencies of gravitational
waves (which predominate in space), could be complementary to
interferometric methods.
Measuring
the polarization of relict microwave radiation
As mentioned above at the end of the " Sources
of Gravitational Waves
" section, the most massive 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 nowadays, they have become so weak
and enormously lengthened their wavelengths that for the direct
detection of these There
is no hope of 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 clearly 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 sensitivityand
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 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
fast-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 "pressed"
intermediate interacting mass *). The distinctive importance
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
"arcade"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^{ }
Observations and gravitational wave
analysis is of great potential to deepen our understanding of the
universe. To reflect on how extremely important a
" window "
into space
would be gravitational wave detection and imaging , let us 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 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 m m
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 wavesshowed 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 began in
the 1970s installed on satellites. 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, bremsstrahlung, 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 . Some places (such as the
interiors of stars or regions of dense gas and dust in the
central parts of galaxies) do not get light or any other
electromagnetic radiation. Neutrins can get out of here, or high
energy particles. Therefore, certain hopes are placed in the detection
of neutrinos and primary cosmic radiationwhich,
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"! ... but see below " The first direct detection of a gravitational
wave by the LIGO device "
...
Analogy with music
- "see" and hear 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
, indirectly, incompletely about the controlling force of it all
- gravity . With considerable exaggeration (but
with some concise features) we can compare
this 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 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 PI Tchaikovsky Concerto for Violin and
Orchestra in D major or the LvBethoven'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 and 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, they pass significantly with
atoms and electrons as they pass through matter ,
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 a certain way ; this information can only
be "carried out" 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. Significant
information
is encoded in the completely characteristic time course of the amplitude and
frequency of
massive gravitational waves, which arise in the final stages of close orbit (fusion) of two neutron stars or black
holes.about the course of this dramatic event with the
participation of large masses and extremely strong gravitational
fields - see above the sharp increase in amplitude and frequency
in Fig.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 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 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} .
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
, 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 deviceby 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 initial (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
beamswith a slightly higher and lower frequency than the
main papilla; 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 ) .
Several other smaller terrestrial
interferometric gravitational wave detectors are being built (or
under construction), 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, indicated 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 ( KA
mioka GRA vitational 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 LIGO
detector 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 thus makes it possible to determine in
triangulation 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
gravitational wave detectors
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
(heavy crossings). cars, earthworks and mining work, aircraft
flights). 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 legs 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.
^{ }Turnover occurred on September 14, 2015
, when in 9:50:45 hours. UTC both detectors L
Asher I nterferometrické G
ravitačně wavelength- O bservatoři CME
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. The amplitude
reached the top of the peakh @ 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 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 m _{1} and m _{2} 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 m _{1} and m _{2} orbiting a
common center of gravity. Bodies that begin their orbit at time t = t _{0} on some large radius r _{0} 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 you approach, the intensity and frequency of the radiation continue 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 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 holem _{1} + m _{2} is rotating and rapidly relaxes to a 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
"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 =m_{ 1} + m_{ 2} > @ 70M_{¤}. 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 / c^{ 2 }> @
210 km. In order to achieve an orbital frequency of 75Hz (half
the measured frequency of the gravitational wave max. 150Hz),
objects m_{ 1} and m_{ 2} had to orbit very close to each other (which is only possible when they are very compact) , at a distance of about 350km
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 ( @ 2M _{¤} )
with a large black hole ( @ 60M _{¤} ) would combine at a significantly lower frequency.
Thus, black holes are 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" 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 1-99M _{¤} and total weight up to 99M _{¤}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 holesin the last phase of their
close mutual circulation and merging - fusion (collision).
Specified parameters of the source system :
Weight of black hole m _{1} | _{-4 }36 _{+5} M _{¤} |
Weight of black hole m _{2} | _{-4 }29 _{+4} M _{¤} |
Weight of the resulting black hole M | _{-4 }62 _{+4} M _{¤} |
Rotational momentum (spin) J / M of the resulting black hole | _{-0.07 }0.67 _{+0.05} |
Total energy radiated by gravitational waves | _{-0.5 }3.0 _{+0.5} M _{¤} c ^{2} |
Peak power radiated by gravitational waves during fusion | _{-20 }200 _{+30} M _{¤} c ^{2} / s |
Luminosity distance of a binary source | _{-180 }410 _{+160} Mpc |
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 _{¤} c ^{2} 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 6.9ms 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 could destroy 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 outer space^{ }
So we can tell a fascinating story that took place in ancient
times in a very distant place in the Universe; however, the
gnoseological outcome 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 orbiting
in a close binary star system formed near each
other from a dense gas-dust cloud.. 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,
amalgamation - of both black holes. In this final phase
of the last few orbits and mergers, a colossal amount of
gravitational wave energy was radiated . Happy coincidence
would have it, these gravitational waves 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), he was able to
even the enormous distance of these gravitational waves register..!..
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 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. 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 completelythe
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.
^{ }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 for Weber's
cylinders ..? .., see " Gravitational wave detectors
".
^{ }All that remained was to hope, that in the near future a similar stormy astrophysical event
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".
**) 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 @ 10
^{22} (our galaxy has about
2.10 ^{11} stars, in the field of view of large
astronomical telescopes there are about 10^{11}
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 @
5M _{¤} ). 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 soon time again, it would make it possible to specify
this so far very indeterminate range of incidence of fusions of
compact objects ...
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.
Further direct
detection 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, "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 :
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 you 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 degrees ^{2}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) | ||||||||||||||||||||||
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 1.2-1.6M _{¤} and the
total weight of the binary system 2.75M _{¤} . 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 degrees^{2} 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 localized region. 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 moved rapidly
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 due to 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 ...
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 larger neutron star..? ..
^{ }The case of detection of neutron stars merge
heralds 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, the first
enthusiasm began to reveal 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...
GW190521^{ }
This last interesting event of gravitational detection 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 :
Source: LIGO |
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 are unlikely to collapse into a
black hole, but before the horizon reach the so-called electron-positron
pair instability (see §4.1, passage "Electron-positron
pair instability"). In this process, the star is
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^{ }
These additional captured signals - despite their somewhat weaker
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 developing..!.. And also multimodality
- multimessenger - astronomy.
Gravity, black holes and space-time physics : | ||
Gravity in physics | General theory of relativity | Geometry and topology |
Black holes | Relativistic cosmology | Unitary field theory |
Anthropic principle or cosmic God | ||
Nuclear physics and physics of ionizing radiation | ||
AstroNuclPhysics ® Nuclear Physics - Astrophysics - Cosmology - Philosophy |