Gravity - acceleration - relativity

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

2.1. Acceleration and gravity from the point of view of special theory of relativity

The special theory of relativity, despite its enormous contribution to almost all areas of fundamental physics, has two essential limitations (which, as we shall see below, are interrelated) :

• l. Describes the laws of physics only in terms of inertial frames of reference.
• 2. It does not allow a consistent description of gravity.

In practice, we are often forced to observe physical processes from non-inertial frames of reference. It turns out that only formal natural generalization can be made under the special theory of relativity to study the laws of physics as well as in non-inertial systems, so the first shortage would not be so serious. The second disadvantage is more serious: in the presence of gravity, there are no inertial frames of reference, and the special theory of relativity is globally inapplicable here. But as we will see below, due to the close connection between gravity and inertia, the special theory of relativity in the gravitational field retains local validity; this makes it possible to build a relativistic physics of gravity and spacetime - Einstein's general theory of relativity.

All physical measurements and observations of natural phenomena in general are essentially based on the determination of spatial and temporal relations, ie on the determination of spatio-temporal coincidences between events. Each elementary event (spacetime point) is assigned four numbers - the coordinates of that point - this assignment mutually unambiguous and continuous. The coincidence of two spacetime events will then be expressed by the equality of their coordinates. If two spacetime events thus coincide in one coordinate system, these events must be identical in any other coordinate system. The system of space-time coordinates is a matter of choice and therefore has no relation to the physical content of the theory. We can thus express the natural requirement that the laws of physics valid in one frame of reference also apply to every other frame of reference: "all the laws of physics have the same form in any frame of reference". This is Einstein's general principle of relativity (generalization of the special principle of relativity), which expressed mathematically reads: "All physical laws can be writen in the form invariant (covariant) with respect to transformations of spacetime coordinates".

The special theory of relativity, which operates mostly with inertial frames of reference, is able to describe the properties also of unevenly moving objects (relativistic dynamics). The reference systems associated with non-uniformly (accelerated) moving bodies are not inertial, so the Lorentz transformation is not directly applicable between them. However, the "ideal" *) (sufficiently strong and durable) clock with which the observer in such a non-inertial system, it measures time, does not change its structure and properties due to acceleration, so that this clock will go as fast as an inertial clock moving at a given moment with the observer. Similarly, the "ideal" (sufficiently rigid) measuring rods of a non-inertial observer will show the same lengths as the rods currently moving inertially with it. Thus space and time intervals measured in the same location by different observers depends on their (immediate) relative speeds, not on their acceleration. From the point of view of the observer in the inertial system, the non-inertial (accelerated) frame of reference can be considered as a continuous sequence of individual inertial systems, so that the relationship between space-time coordinates will be given by Lorentz transformations with continuously variable velocities.
*) By ideal clocks and measuring rods we mean that the clock "ticking" speed and the length of the rods are not affected by non-universal influences such as temperature or applied forces, or a correction is made to these non-universal influences.

The fact that the laws of special theory of relativity apply even at enormous accelerations higher than about 10³⁰ cm/s², ie that spacetime relations, mass, energy, momentum, etc., do not depend on instantaneous acceleration, but only on instantaneous velocity, with high accuracy is confirmed experimentally in the scattering of high energy particles [229].

General principle of relativity, curvilinear coordinates, metric tensor
The general principle of relativity therefore states that not only systems of (pseudo)Cartesian spacetime coordinates connected with inertial frames of reference, but also any other systems of spacetime coordinates, are completely equivalent for the formulation of physical laws. At first glance it seems that the principle of relativity contradicts the experience, because in non-inertial reference frames appear "fictitious" forces of inertia, affecting the movement of bodies and course of physical processes. However, the origin of such forces can be easily explained within STR, and the properties of these apparent forces (their effect on physical phenomena) can be expressed by generally invariant equations.

In the inertial frame of reference S^~ in Cartesian coordinates x^{~ i} , the space-time interval element has the form

ds ² = h _ik dx^{~ i} dx^{~ k} .            

If we move to any system of space-time coordinates xⁱ in a generally non-inertial frame of reference S using the transformation xⁱ = xⁱ (x^{~ k} ), in these new coordinates xⁱ the space-time interval will have the form

where

Since the reference systems S^~ and S move with respect to each other with acceleration, the transformation S ® S^~ will not be a fixed Lorentz transformation and the quantities ¶ x^{~ m} / ¶ x ^k will generally be functions of place and time. From a mathematical point of view, non-inertial reference systems are actually systems of curvilinear spacetime coordinates. Instead of the constants h_ik, new quantities g_ik(x^j) appear here, whose functional dependences on the coordinates x^j characterize the relationship of the system S with the new coordinates x^j to the original inertial system S^~ with Cartesian coordinates x^{~ j}. Because these quantities g_ik give a formula for measuring actual distances in space-time using coordinate differences, they are called metric tensor in differential geometry.
The metric tensor is of key importance in the general theory of relativity - it describes gravity as the geometry of curved spacetime. Essentially the same applies to it as was stated in §1.6 (note in the section "Geometry of spacetime. Four-tensors") about the spacetime interval: If we know the spacetime interval, ie the dependence of the element ds² on coordinates, we know "everything" about space-time, and we can use it to study how bodies (particles) will move in it and light (photons) will propagate. In other words, we know the metric tensor g_ik and the equations of geodetic lines - trajectories of free particles in the gravitational field (§2.4 "Physical laws in curved spacetime").

The equation of motion of a free test particle in the inertial system S^~ : d² x^{~ i} / ds² = 0, expressed in the general reference system S (ie in curvilinear space-time coordinates x ⁱ ) has the form

d²xⁱ/dt² + (¶xⁱ/¶x^~m).(¶²x^~m/¶x^k¶x^l).(dx^k/dt).(dx^l/dt)   =   0   .

This equation is invariant with respect to any transformation of the coordinates xⁱ ® x' ⁱ (when using the system of coordinates x'ⁱ we get an equation of the same form, in which x are replaced by x'). The equation of motion of a free mass particle using general (curvilinear) space-time coordinates x ⁱ (in a generally non-inertial frame of reference) therefore has the form

(2.2a)

where the quantities G ⁱ_{k l} can be expressed using the metric tensor :

(2.2b)

These quantities (describing the effect of "apparent" inertial forces on the motion of a particle) contain the components of the metric tensor and its derivatives; they are called Christoffel coefficients of affine connection (their meaning will be shown in §2.4). Equation (2.2a) is called the equation of geodesy (here, of course, it is a line expressed only in curvilinear coordinates).

Thus we see that using the value g_ik (i.e. with the help of metric) can be captured even apparent forces acting on the material body in non-inertial reference systems S. Similarly, all other laws of physics (eg the laws of electrodynamics) can be expressed in a generally invariant form valid in any frame of reference S - see §2.4. In the equations written in this way, the components of the metric tensor g_ik and their derivatives according to the coordinates also explicitly appear. The presence of components g_ik in the laws of physics it expresses the influence of apparent inertial forces, such as their action on the motion of bodies or on electromagnetic phenomena. Within the special theory of relativity, it is possible to build a generally invariant theory of apparent inertial forces and their influence on all general laws of physics (except gravity).

However, the general principle of relativity taken in isolation is physically meaningless, because any physical law can be formally rewritten to conform to the general principle of relativity (it has the same form in all reference frames) without extending or deepening the physical meaning of this law. Only connection with the principle of equivalence, which put fictitious inertial forces in non-inertial systems on the same level as the actually existing gravitation forces, gives the general principle of relativity profound physical meaning: it leads from the generally invariant theory of apparent inertial forces to Einstein's theory of gravitation.

After the emergence of a special theory of relativity, it became clear that every physical law that wants to claim generality and correctness must comply with the principles of STR. Einstein reformulated the fundamental laws of classical Newtonian dynamics right when compiling his special theory of relativity; in addition to such phenomena as time dilation and and lengths contraction, appeared next interesting effects - the dependence of mass on velocity, the relationship between equivalence of mass and energy, and other related consequences. The laws of electrodynamics did not need any reformulation, Maxwell's equations already in his classic form are Lorentz-invariant (based on electrodynamics actually STR was formed...). However, the special theory of relativity has made it possible to better understand the deep connections between the various laws of electrodynamics (see §1.5 and 1.6).

Relativistic generalization of Newton's gravity ?
Another situation is with Newton's law of gravitation, which first is not relativistically invariant, but is also in direct conflict with the second basic principle of STR - the principle of the existence of a finite maximum possible velocity of interactions. Newton's law of gravitation expressed by Poisson's equation Dj = 4p G r does not contain any time derivatives. According to this law, the gravitational field j(x, y, z, t) at some point (x, y, z) at time t is given by the instantaneous distribution of matter r(x, y, z, t) in the whole space at the same time t . A change in the distribution of matter at some point is immediately reflected in the gravitational field throughout space - the commotion in Newton's gravitational field spreads infinitely fast. Newton's law of gravitation is based on the idea of instantaneous action at a distance and is therefore necessarily only an approximate law applicable only at low speeds and not too high masses (so that the gravitational field is weak).

Efforts to relativist reformulation of Newton's theory of gravitation date back to the beginnings of the theory of relativity. Naturally, this was based on Maxwell's electrodynamics, which was (and is still to this day) a model of consistent, perfectly elaborated and comprehensive relativistic invariant field theory, moreover, well verified in practice. In §1.4 we have shown a close analogy between Newton's law of gravitation and Coulomb's law of electrostatics. Since Maxwell's electrodynamics is a generalization of Coulomb's electrostatics, it can be expected that a similar generalization of Newton's law will lead to a correct relativistic theory of gravitation. All attempts to include gravitational phenomena in the special theory of relativity are more or less based on this assumption.

A straightforward way of relativistic rewriting of Newton's law of gravitation is a modification (generalization) of Poisson's equation (1.28 '). The simplest version tries to preserve of the scalar character of Poisson equations: the gravitational field is describeds by the space-time scalar potential j(x), while its source there will also be a scalar quantity characterizing the distribution of the rest mass. The density of the rest mass distribution r_o(x) is equal to r_o = T ⁱ _i / c² in the special theory of relativity, so that the gravitational field will be excited in scalar theory of gravity according to the equation G(j, j_{, k} , j_{, k, l}) = K.T ⁱ _i . The shape of the function G is determined from the Lorentz invariance requirement, the validity of the superposition principle and the validity of Newton's law for the static case. The law of gravitation in this Nordström theory of gravity [191] has the form of d'Alembert's equation

oj   = - ( 4p G / c² ) . T ^j _j   ,       

(where oş h^ik ¶² / ¶ xⁱ ¶ x^k is d'Alembert's differential operator), which is a generalization of Poisson's equation (1.28 '). The solution to this d'Alembert equation is therefore the retarded potential

with time variable not t , but t-r'/c , expressing the final speed of propagation of the gravitational field, equal to the speed of light, similarly to electrodynamics - let's compare with relation (1.47) in §1.5.

The inapplicability of Nordtröm's scalar theory of gravity stems mainly from the fact that the gravitational field is excited in it only by the rest mass r_o = T ⁱ _i , which contradicts the principle of equivalence. We know that, for example, the electromagnetic energy contained in a body contributes to its gravity (Eötvös's experiment - see the following paragraph). Also the known observation of the bending of light rays in a gravitational field shows that photons with zero rest mass (having only electromagnetic energy) have a non-zero gravitational mass and show a gravitational interaction. Furthermore, the equations of motion in scalar theory lead to the twisting of Mercury's perihelion, but this twisting is about six times smaller and mainly in the opposite direction than observed [16].

Another way of generalizing Newton's law of gravitation comes from Lorentz, who developed it even before the emergence of a special theory of relativity. This vector theory of gravitation is an almost exact copy of electrodynamics and is sometimes referred to as the "Maxwell's theory of gravity". The gravitational field is described herein by the gravitation 4-potential yⁱ = (j, y) containing normal gravitational potential j as one of the components, which are similarly tied electrodynamics Lorentz calibration conditions y^k _,k = 0. From this 4-potential is forming antisymmetric tensor intensity of the gravitational field G_ik = ( ¶y _k / ¶x ⁱ ) - ( ¶y _i / ¶x ^k ). The source of this gravitational field is incoherent mass dust, in which the density of rest mass is described by a scalar field r_o(x ^j) and the distribution of motion by a field of four velocities Vⁱ (x _j). Four-vector (density) of the mass flow is then j ⁱ (x) = r₀ .V ⁱ = ( r , r.v), where r(x) = r₀ / Ö(1-v²/c²) is the mass density, v(x) the velocity at a given point x . In vector theory, the gravitational field is excited by a four-stream of mass j ⁱ according to the equations

¶ G _ik / ¶ x ^k = (4p G / c) j ⁱ .               

From these equations, as in electrodynamics, follow d'Alembert's wave equations žyⁱ = (4p G/c) j ⁱ and thus the existence of gravitational waves propagating at velocity c . The equations of motion of the test particle d²x ⁱ / dt² = (G / c) G^ik dx_k / dt show, that all material bodies move in the given gravitational field with the same acceleration.

The outlined vector theory of the gravitational field has two major disadvantages. The first is that only incoherent material dust can be used as a source of gravitational field, or point material resources; for other models of sources (eg fields) it is not possible to define the four-vector mass flow j ⁱ well and unambiguously. Even more serious is the second drawback: formulate if the laws of conservation of energy and momentum of the system composed of the material excited by it, and the dust (according to the equations above) density of energy of the gravitational field is negative; the same goes for the energy transmitted by gravitational waves. Even apart from the logical difficulties and inconsistencies already occurring at the classical level, this result is unacceptable from the point of view of relativistic quantum field theory, which requires a positive energy definite of tensor fields describing particles with integer spins.

The common cause of the failure of the scalar and vector theory of gravitation was mainly that only a part of the energy-momentum tensor was taken as the source. If we take the whole energy-momentum tensor T^ik as the source of the gravitational field and put it on the right side of the generalized Poisson equation (1.28 ''), the excited gravitational field should also be described by the symmetric tensor y^ik. In such a tensor theory of gravity (still within STR) the gravitational field will be excited according to an invariant system of tensor equations

where T^ik _(m) is the energy-momentum tensor of all bodies, substances and fields with the exception of the gravitational field itself. As in electrodynamics, additional conditions y ^ik _{, k} = 0 analogous to the Lorentz condition are introduced. The equations of motion of the test particle are

Even the specially relativistic tensor theory of gravity is not yet perfect and correct: it implies the differential conservation law T^ik _{(m), k} = 0, which does not take into account the gravitational field and could only apply if there were no gravitational action. In reality, however, the law of conservation (T^ik _(m) + T^ik _(g) )_{, k} = 0 must apply for the total energy-momentum tensor, which also includes the energy and momentum of the gravitational field. Thus, the source on the right side of equation (2.3b) should be the total energy-momentum tensor including the gravitational field. Gupta [111] showed that this path can already lead to a satisfactory relativistic theory of gravitation. Since the exact expression for the energy-momentum tensor of the gravitational field could be determined only on the basis of exact field equations, which are not known in the given theory, it is necessary to use methods of successive approximations. Energy-momentum tensor of the gravitational field ^oT^ik _(g), calculated from equation (2.3a) in the zero approximation, is added to the right side of (2.3a), the following equation in first approximation, to calculate a new (advanced) tenzor energy-momentum ¹T^ik _(g), using which in equations (2.3a) gives a second approximation with an even more accurate energy-momentum tensor of the gravitational field ²T^ik _(g) and so on. By this procedure it is possible to obtain (although there are problems with uniqueness) the correct Einstein equations of the gravitational field, which are - in contrast to the initial equations (2.3a) - already nonlinear (see §2.5).

However, this does not mean that gravitational phenomena can be satisfactorily described in the special theory of relativity. The equation of motion of test particles in the gravitational field (2.3b) shows here that the special relativistic condition h^ik (dxⁱ / dt) (dx^k / dt) = - c² is not met on these world lines and the actual time is no longer related here with space-time coordinates by the relation d t ² = - (1 / c) .h _ik dx ⁱ dx ^k . The element of the space-time interval (and thus the actual time) is given by the relation ds² = -c²dt² = g_ik(x)dxⁱdx^k , where g _ik = h_ik - (2G / c² ) y_ik . Thus, the time (running of standard clocks) at a given place depends not only on the speed of their movement, but also on the potentials of the gravitational field at that place; similarly, the gravitational field is affected by length scales. In geometric terminology, this means that the gravitational field affects the geometric properties of spacetime - spacetime becomes a generally curved Riemannian one. We come to the conclusion that a satisfactory interpretation of gravitational phenomena requires a revision of the basic physical concepts of the special theory of relativity, including a new view of the structure of space and time [247].

In the next we will see that this conclusion (which follows from a relatively complicated analysis of especially relativistic theory of gravity) can be reached directly on the basis of simple considerations of the universality of gravity and only then derive the correct equations of the gravitational field - Einstein's equations.

Non-inertial frames of reference - "bridge" between special and general theory of relativity
Let's go back to non-inertial systems. From the point of view of the inertial frame of reference, STR has space (three-dimensional) of Euclidean geometry. However, even in Minkowski's pseudo-Euclidean spacetime, the special theory of relativity becomes non-Euclidean in non-inertial reference systems of the geometry of three-dimensional space! It can easily be shown on a rotating reference frame (Fig.2.1). Let us initially have a non-rotating straight disk, the center O of which forms the beginning of the inertial frame of reference S. An observer who measures the dimensions of this circular disk with (sufficiently short) ideal measuring rods, measures its radius r and circumference l = 2pr in accordance with Euclidean geometry. Then we rotate the disk around its center with an angular velocity w with respect to the inertial frame of reference. If the measuring rods and the disk itself are sufficiently rigid ("ideal"), the centrifugal expansion can be neglected and the observer can measure the same radius r of the disk on the rotating disk using radially applied measuring rods as if there were no rotation. However, if the inertial observer from S^~ observes a measuring rods, which the experimenter attached on a rotating disc to the periphery in order to measure the length of circumferece l , then these rods are moved in the direction of its length with circumferential speed w.r . According to the special theory of relativity, each such rod Ö(1- w² r² / c²) - times shorter than at rest. The observer on the rotating disk therefore finds that there is a relationship between the radius and the circumference of the circular disk

2 p r
l = ----------------------- .
Ö(1 - w ²r ²/c ²)

Fig.2.1. Origin of non-Euclidean geometry of space in a non-inertial frame of reference.
a) The usual relation l = 2p r between the radius and the circumference of a circular non-rotating disk .
b) When measuring on a rotating disk, the circumferential measuring rods (unlike unchanged radial rods) are shortened due to Lorentz contraction of lengths; therefore more of them "fit" on the circumference and the ratio between the length of the circle and its radius will be greater than 2p - the spatial measurement shows a non-Euclidean geometry.

The ratio between the length of the circle and its radius is different from 2p, the spatial geometry of the rotating disk is non-Euclidean. The outer inertial observer explains this kinematically using Lorentz contractions of the special theory of relativity, while the inner observer rotating with the disk (for which all parts of the disk will be at rest) will consider it a consequence of "inertial" forces acting on all bodies. Declares that these inertial forces deviate the geometry of space from Euclidean, while the degre of this non-eucleidicity (curvature) of the space is given by the magnitude of these inertial forces, i.e. the centrifugal force w² r.

Thus, spatial geometry in non-inertial frame of reference is not generally Euclidean, it is not possible to construct a Cartesian system of spatial coordinates. However, the geometry of four-dimensional space-time here remains Euclidean - with a suitable transformation it is always possible to return to an inertial system with (psedo) Cartesian space-time coordinates. Even in the special theory of the relativity of non-inertial systems, spacetime remains a fixed "scene" in which physical processes take place, while these physical processes do not affect the properties of spacetime itself.

Fig.2.2. Dependence of the speed of the passage of time at the different places in non-inertial systems.
a) Clocks located in different places of the non-rotating disk go at the same speed and show the same time.
b) If (identical) clocks are mounted at different distances from the center of the rotating disk, they will move at different circumferential speeds and show different relativistic time dilation (clock A will go as without rotation, clock B will go slower and C even slower) - from the point of view ot the inner observer, time in different places flows at different speeds.

In order for two hours of identical construction to go at different speeds in an inertial system, they need to move relative to each other. However, in a non-inertial frame of reference, clocks *) located in different places can move at different speeds even when they are at rest relative to each other.
*) We mean here the so-called ideal clocks, the operation of which is not affected by any non-universal influences. Thus, a pendulum or hourglass clock would be completely unusable here (whose running speed is directly determined by gravity, it stops in a weightless state); similarly, other mechanical clocks could be affected by mechanical deformations of their components. Atomic clocks are considered to be the most suitable in this respect .
   Again, this can be clearly seen in the example of a rotating reference system (Fig.2.2). Consider a disk rotating speed w (which will be non-inertial reference frame S ) on which are mounted at the center of rotation of clocks A , at the distance r_B clock B and an even greater distance r_C from the center of the next clock C. From the point of view of the inertial system S^~ (starting at the center of rotation), clock A will be at rest, clock B will have velocity w .r _B and clock C will have an even higher velocity w .r _C . According to Lorentz transformation (time dilation) , the clock B will go slower (and the clock C even slower) than clock A : t = t_A . Ö(1-w²r²/c²). From the point of view of the rotating reference system S , all three clocks will be at rest with respect to each other, but each clock will run at such a speed, as the corresponding inertial clocks, moving at a given moment together with the clock being examined.

Therefore, the same will apply to the operation of clocks A, B, C as from the point of view of the inertial system - with respect to clock A, clock B will go slower and clock C will go even slower. The observer in the inertial system S^~ explains this by the dilation of time caused by their different motion, while the observer in the rotating system (where all three hours are at rest) must consider this as a consequence of the field of inertial forces with universal effects, which he observes in his system.

By applying STR to a rotating disk, we obtain non-Euclidean spatial geometry (and different course of time in different places) and at the same time centrifugal forces will arise - the non-Euclidean geometry and the field of centrifugal forces have a common cause in disk rotation. The internal observer, if he did not know about the rotation of the disk, could consider these anomalous phenomena in the geometry of space and the passage of time as a consequence of the field of centrifugal forces; or he could reverse it and consider the field of centrifugal forces as a manifestation of curved geometry of space and inhomogeneity of time. This imaginary experiment thus induces a new way of thinking about the connections between space and time with motion and gravity, which resulted in the general theory of relativity (GTR) as the physics of spacetime and gravity.
   Analysis of imaginary experiments with non-inertial systems - with rotating (Fig. 2.1 and 2.2) and with accelerated rockets (Fig. 2.2 and 2.3 in the following §2.2) - shows that in order for the relevant kinematic effects to occur, gravity must affect time and space in the same way, how the speed of movement in STR affects them. This means that gravity can be described as the geometry of time and space, combined in 4-dimensional spacetime. And that is the filling of GTR...

1. Gravity and its place in physics		2.2. Versatility of gravity

Vojtech Ullmann