Physics of curved spacetime and gravity

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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

2.4. Physical Laws in Curved Spacetime
Gravity can be studied in essentially two ways :
1.
Either consider a "physical" gravitational field in planar spacetime (within STR);
and or :
2.
Introduce curved spacetime without gravity.
In the first mode, we consider the universality of the gravitational interaction to be a coincidence , but its consistent application in the equations of the gravitational field ultimately leads to nonlinear Einstein equations and the need generalization of the special theory of relativity, as mentioned in 2.1. The second approach, which from the very beginning draws appropriate consequences from the universality of gravitational action, identifies the gravitational field with the geometric properties of spacetime. The gravitational force is the result of "deepening" and distortion in the cosmic structure of space and time.
The implementation of this procedure is actually the content of the general theory of relativity. According to Newton's classical physics, planets orbit the Sun in a circular (elliptical) orbit because they are immediately attracted to it by gravitational force , which causes a centripetal acceleration, curving orbits that would otherwise be straight. According to the general theory of relativity, however, between the sun and planets no gravitational force does not - the orbits of the planets are curved because the very space and time in which they move is deformed (curved) by the presence of the massive Sun and automatically forces the planets to move in the appropriate "geodetic" orbit.
In terms of general relativity, the movement of the test particles in a gravitational field inertial (particle free) and any special features are not due to "gravity" acting on the particle , but the metric of space ; it is similar with all physical phenomena in the presence of gravity.
The gravitational field thus "disappeared", instead the generally curved Riemannian spacetime remained.. And the problem of finding the physical laws governing natural phenomena in the gravitational field thus translates into the question of determining the physical laws in the curved Riemann spacetime (without gravity).

Using the principle of equivalence, it is possible to generalize all the physical laws of the special theory of relativity (where the Minkowski spacetime is planar) to curved spacetime, ie to the presence of a gravitational field. The method described in the previous paragraph leads directly to this: to divide spacetime into sufficiently small areas in which the curvature can be neglected, to apply physical laws of planar spacetime in these areas (ie STR formulated for general reference frames) and finally to fold this "mosaic" in the resulting global situation.

We can illustrate this general procedure with simple examples. Consider a test particle (mass m , which are not explicitly not applied) moving in the gravitational field, that is at time t is in svtobod P . If we introduce at point P a locally inertial reference system S ~ with Cartesian space-time coordinates x ~ i related to the eigenvalue t of the test particle by the relation d t 2 = - (1 / c 2 ) h ik dx ~ i dx ~ k, there will be a "weightlessness" state without gravity field in this system at time t in the vicinity of the test particle, and a special theory of relativity will apply locally. The equations of motion of the test particle in this locally inertial system will therefore be (uniform rectilinear motion)

d 2 x ~ i / d t 2 = 0 . (2.5a)

If we go as in 2.1 from the system S ~ to the general non-inertial frame of reference S with space-time coordinates x i related to the time itself by the relation

ds 2 = - c 2 d t 2 = g ik dx i dx k ; g ik (x j ) = h lm . ( x ~ l / x i ). ( x ~ m / x k ) ,

the equation of motion (2.5a) is transformed into the form of the geodetic equation

(2.5b)

wherein G ki l = ( 1/ 2) The g im ( g mk / x L + g mL / x to + g kl / x m ) as shown in 2.1, equation (2.2a , b). We can do this procedure at each point of the test line's worldline and we always get the equation of shape (2.5b). Equation (2.5b) is thus the general equation of motion of a test particle in a gravitational field (in curved spacetime), which is invariant(covariant) with respect to any transformation of spacetime coordinates. Here, this equation of geodesy serves only as an example of the general procedure of finding the laws of physics in the presence of gravity; we will return to its physical significance below. As a second example, we take the differential law of conservation of energy and momentum , which has in the form T ik , k T ik / x k = 0 in STR (see 1.6). It will have the same wording in every locally inertial frame of reference S ~ moving freely in a gravitational field: T ik (x ~ ) / x ~ k
= 0. After transformation into a general (non-inertial) frame of reference S, this law takes on a covariant form

T ik / x k + G m i k T mk + G m k k T im = 0 , (2.6)

which represents the formulation of the law of conservation of energy and momentum in curved spacetime, ie in the gravitational field. The physical aspects of this law will again be discussed below (in 2.8).
From these two cases we can already deduce general laws. According to the principle of equivalence, the physical laws in the gravitational field are locally the same as the physical laws in non-inertial frame of reference without gravity. The non-inertial frame of reference is then mathematically equivalent to a curvilinear system of space-time coordinates. Thus, it can be expected that the generalization of physical laws to the presence of a gravitational field (ie their formulation in curved spacetime) will simply consist in writing these laws in
general curvilinear coordinates. The difference compared to the planar spacetime (situation without gravity) is then only that that flat spacetime can be suitable transformations always return to the laws of special theory relativity in Cartesian global inertial system, while for the curved spacetime this is not possible, the global inertial system here does not exist, there are only "curvilinear" coordinates.

Parallel transfer of vectors, connections, covariant derivation
Physics studies the course of natural processes
in different places in space and in different times - at different points in space-time. It describes the physical processes by the relevant physical quantities in these places, which leads to certain physical fields . In common situations of Euclidean space, or Minkowski spacetime, we do not have to worry about the differences in the geometric properties of space in different places - these are identical Euclidean (or pseudoeuclidean). However, in the general theory of relativity, which implies more complex geometric properties of curved spacetime, non-trivial relationships arise between different places in space and time, which can affect the values ??of physical fields. These relationships between quantities at different points in space (and spacetime) are quantified by the geometric-topological concept of connection (from the Latin connectio = connection, intercourse, binding ) . The connection analyzes what needs to be done with the values ??of the components of the vectors and tensors describing the physical field - what correction to express the objective values ??of the fields, independent of the local geometric conditions and the coordinate system used.
The laws of physics are expressed by differential equations between vector and tensor fields in spacetime. Ordinary partial derivative of vector field A i according to coordinates x k

(2.7)

is usually a measure of how the vector field A i changes with location (from a point with x coordinates to a "neighboring" point x k + D x k ). However, when using curvilinear coordinates for objectively comparing vectors and tensors entered at different points in spacetime, their components calculated with respect to the local base cannot be used immediately , as they may be different at different points. The components of vectors and tensors (vector and tensor fields) can change from point to point using curvilinear coordinates for two reasons :
a
) First, because a given vector field actually (physically) changes with place.
b ) Furthermore, also because there is a different vector base in each place, with respect to which the components of vectors and tensors are determined.
Normal partial derivatives of (2.7) are then objectively not actual changes in vector and tensor fields, as e.g. also constant vector field will have to curvilinear coordinates variable components, and therefore a non-zero partial derivatives of its components. In addition, A i , k is not transformed as a tensor, because the difference between the vectors Ai is at different points where there may be different transformation coefficients.
An appropriate correction to these circumstances must be made - take into account the connection : first transfer the vectors in parallel to one common point and then compare their components. While parallel transfer vector with its components in a Cartesian coordinate system not changes. Using a curvilinear coordinate system, however, when parallel transfer vector Ai from the point with coordinates x k a nearby point x k + D x k components of the vector changes by

d A i = - G k i l A l . D x k , (2.8)

where the quantities G k i l (which depend on the coordinate system) are the Christoffel coefficients of the affine connection , which we have already encountered in 2.1, relation (2.2b). It is clear that the quantities G k i l cannot form a tensor, because by transitioning from the Cartesian system, where they are all equal to zero, to the curvilinear system they become nonzero (and conversely with nonzero G k i l all components can be canceled by system at a given point). It follows from the requirement that d A i in the law of parallel transmission (2.8) be transformed as a vector transformation relation for affine connection coefficients :

(2.9)

From this it can be seen that the connection coefficients behave as tensors only in linear coordinate transformations (such as transformations between Cartesian coordinate systems).
The resulting components of the vector A i ( x k ) transferred in parallel to the point x k + D x k will be A i ( x k ) x k + D x k = A i ( x k ) + d A i. The requirement that the rules of tensor algebra be preserved during parallel transfer follows from (2.8) for the parallel transfer of the general tensor T i r j s . . . . . . . law

d T i r j s . . . . . . . = - G m i n T m r j s . . . . . . . . D x n - G m i n T i r m s . . . . . . . . D x n - ....
+
G r m n T i m j s . . . . . . . . D x n + G s m n T i r j m . . . . . . . . D x n + .....
(2.10)

The "correction" of the partial derivative (2.7) to the change of the vector base caused by the "curvature" of the coordinates then consists in the fact that during the derivation the parallel transfer of the vector A i ( x k + D x k ) from the point x k + D x k back to point x k and then the appropriate limit is made :

(2.11)

It is thus achieved that the difference between the components of the vectors calculated at a single point x k and thus related to the same base is taken . This so - called covariant derivative (it is a partial derivative "," corrected on the connection - denoted by a semicolon ";") already expresses real changes of physical quantities (variability of vector and tensor fields) and has tensor transformation properties [214], [155]. According to (2.7) and (2.11), the covariant derivative of vector A i is equal to

A i ; k = A i / x k + G k i m A m = A i , k + G k i m A m , similarly A i ; k = A i / x k + G i m k A m , (2.12)

If we replace in (2.11) the vector A i with the general tensor T i r j s . . . . . . . , we get based on the law of parallel transfer (2.10) a general rule for covariant derivation of tensors (tensor fields):

T i r j s . . . . . . . ; w = T i r j s . . . . . . . , w + G w i m T m r j s . . . . . . . + G w j m T i r m s . . .. . . . + ....
- G r m w T i m j s . . . . . . . - G s m w T i r j m . . . . . . . + .....
(2.13)

The situation is the same when deriving vector and tensor fields along a given curve (worldlines) C with the parametric equation x k = x k ( l ), ie when deriving vector fields according to the parameter l :

(2.14)

where A i ( l ) A i (x k ( l )) are the components of the vector A i at the point of the curve C given by the value of the parameter l . To this derivative expressed real change vector field along the curve C , we must also correct for connection, to form the absolute derivative of the vector A i along the curve C (x k = x k ( l )):

(2.15)

If the vector field A i is defined not only on the curve C , but also on the surrounding space, the relationship between absolute and covariant derivation is as follows:

DA i / d l = A i , k + G k i l A l dx k / d l = A i ; k dx k / d l , (2.16)

where, for simplicity, it is no longer explicitly indicated that it is calculated at the point of the curve C with the parameter l = l o (can be done at any point). Analogously for absolute derivatives of higher order tensors.
An important equation can be easily derived

g ik ; l = g ik ; l = 0 ; (2.17)

the metric tensor is covariantly constant , so for example, it does not matter whether we raise and lower the tensor indices before or after the covariant derivation.

Symmetry of spacetime, Killing vectors and conservation laws
In analytical mechanics and field theory, it is shown that the Lagrangian symmetries of the physical system lead to the conservation laws of certain quantities (integrals of motion, especially energy and momentum). Even using curvilinear coordinates and curved spacetime, differential geometry can, in certain cases, express
spacetime symmetries leading to conservation laws.
This is a situation where the components of the metric g
ik in a certain coordinate system do not depend on one of the coordinates x K , so the derivative g ik / x K = 0. In this case, we can transform any curve by coordinating all its points by d x K in the direction of the x K coordinate , while the length of the new curve will be identical to the length of the original curve. Thus, no geometric measurement can determine that there has been a shift in the direction of the x K coordinate - metric space (here spacetime) appears in this direction K isometry . To describe this isometry, the so-called Killing vector x K / x K is introduced in the differential geometry. , expressing the components of infinitesimal translation, preserving length. The distribution of this vector at each point of the variety forms the Killing vector field of infinitesimal isometric generators. This field satisfies the covariant Killing equation x i; k + x k; i = 0. If the spacetime has certain properties of symmetry (eg spherical, axial or plane symmetry) expressed by the existence of the respective Killing vectors x k , then the vector P i = T ik x k , for which thanks Killing equations d m the relation P i , i = T ik x k; i = (1/2) T ik ( x k; i + x i; k ) = 0 expressing the law of conservation P i , resp. K-th covariant momentum value, calculated in coordinate base . Depending on whether the Killing vector x k is of temporal or spatial type, this can be interpreted as the law of conservation of energy or momentum.

Curvature of space. Curvature tensor .
In differential geometry plays an important role in the concept of
curvature , which generalizes and formalize our intuitive experience with curved objects - lines (curves) or surfaces. During the development of differential geometry, several expressions of curvature were introduced (external - internal curvature, is briefly discussed in 3.1, section " Connection-Metrics ") .
The components of the connection coefficients G k i l and the metric tensor g ik depend on the coordinate system and at first glance we do not know whether they correspond to planar space (where only curvilinear coordinates are used) or a truly curved space. However, the analysis of the properties of parallel transmission makes it possible to find a general criterion of flatness of space and to determine quantitative quantities expressing the degree of curvature of space (all considerations apply to general space with connection and metrics, ie also especially to spacetime, ordinary three-dimensional space, or even two-dimensional area) .
A space is called Euclidean if Euclidean axioms hold in it and thus there is a Cartesian coordinate system: metric form in general coordinates ds 2 = g ik dx i dx k it can be converted to the form ds 2 = S i (dx i )2 of the "Pythagorean theorem" by a suitable transformation . More generally, under non- curved (planar, flat) space we mean a space in which the metric form can be converted to the form ds 2 = i S k i by appropriate transformation of coordinates . (Dx i ) 2 , where individual constant coefficients k i can take values either +1 or -1.

The criterion of non-curvature of space is therefore the possibility of introducing a global Cartesian or pseudo-Cartesian coordinate system. If we have such a Cartesian coordinate system introduced in flat space, then a vector transferred in parallel from one point to another does not change its components. The curvilinear system, the components of the vector with parallel transfer varies but from the existence of the Cartesian coordinate system follows that, in a planar space parallel transmission not depend on the route along which takes place - changes of the components depend only on the starting and ending points. Thus, if we transfer vector along any closed curve, then after returning to the starting point, the components of the transferred and the initial vector will merge. Affine connections having this property are called integrable. It can be easily shown that (with symmetrical connection) the integrability of the affine connection is a necessary and sufficient condition for the space to be flat (non-curved).


Fig.2.6. Parallel transmission in a curved space.
a ) In a curved space (eg on a spherical surface), the result of the parallel transfer of vector A from a given point A to point B depends on the path along which the transfer takes place.
b ) This non-integrability of the affine connection in the curved space causes the vector transferred in parallel along the closed curve C to differ from the original vector after returning to the starting point.

In the general case, however, G k i l are functions of coordinates and parallel transmission according to equation (2.8) will depend on the path (Fig.2.6) - the connection will no longer be integrable :

where A i (C1) are the components of the vector A i transferred in parallel from point A to point B along the curve C 1 , A i (C2) is the result of the transfer between the same points along the curve C 2 . Perform if in this case said vector parallel translation along a closed curve, we return to the starting point of generally another vector (fig.2.6b). The size of this vector will be the same (the invariance of the size of the vector in parallel transmission is a basic requirement of the relationship between connection and metric in Riemann space), its direction will change. Deviation of this transferred vector from the original (relative to a unit of area surrounded by a closed curve along which the transfer was performed), is then a measure of the nonintegrability of the connection and characterizes the difference in geometric properties from Euclidean - it is a measure of the curvature of space .
A different degree of curvature of space is based on the responsibility of the circle , respectively. sphere . We construct the locus of points having the same distance r (measured along the shortest path) from the fixed point O . In the two-dimensional case it will be a "circle" and any difference in its length from 2 p r is a measure of the curvature of the space (area); if the length of the resulting curve is less than 2 p r, the curvature is positive (eg spherical surface), if this length is greater than 2 p r, the curvature is negative ("saddle" surfaces). Analogously, in three-dimensional space geometric place the points having a distance r from the center is formed by a closed surface, whose contents are compared with the content of Euclidean balls 4 p R 2 ; similarly for higher dimensions. Criterion based on a parallel transfer, however, is more general since it does not requires metrics, connection are enough here.

The change of the vector in the parallel transfer of vector A along the closed curve C is

The line integral generally not transferable to planar integrals using Stokes sentence because the values of the vectors A i the points of the respective surface (inside the curve C) can not be unambiguously determined - depend on the path that the expansion of the vector field with a vector A i at that point we come . However, if the curve C is sufficiently small (infinitesimal), the limit switch, this ambiguity does not apply (corresponding error is up to second order) and Stokes theorem gives (variability of the vector field A i the point here is only through connections, so A i / x l = - G k i l A k)

(2.18)

where D S lm is the tensor of the area bounded by an infinitesimal closed curve C. A more detailed derivation can be found eg in [214], [166]. The tensor R i klm , which quantifies the difference between the geometric properties of a given space from the planar one (nonintegrability of the aine connection), is called the Riemann-Christoffel curvature tensor .

In flat space, the curvature tensor is zero everywhere, because you can choose a Cartesian coordinate system in which all Gk i l are zero everywhere , so that even R i klm = 0 ; thanks to the tensor character Ri klm, this also applies to any other (perhaps curvilinear) coordinate system. Conversely, if R i klm = 0 everywhere , the parallel transfer is unambiguous and path independent, so that a locally Cartesian coordinate system introduced at one point can be transferred in parallel and extended to all other points, ie construct a global Cartesian system space is planar. Equation

R i klm = 0 (2.19)

is therefore a unequivocal criterion of whether the space described (using any coordinate system) by the given fields G k i l , or g ik , is flat or curved .

Here are some properties of the curvature tensor. From the definition of the curvature tensor contained in relation (2.18) it can be seen that the tensor R i klm is antisymmetric in the indices l, m:

R i klm = - R i kml . (2.20)

In addition, the tensor R i klm is cyclically symmetric in its three covariant indices, ie.

R i klm + R i mkl + R i lmk = 0. (2.21)

Other algebraic relations (identities) hold for the covariant curvature tensor R iklm = g ij R j klm , obtained by reducing the index i:

R iklm = - R kilm = - R ikml , R iklm = R lmik ; (2.22)

according to these relations, those components of the curvature tensor which have i = k or l = m are equal to zero.
A 4th order tensor in N-dimensional space generally has a total of N
4 components (in four-dimensional space-time it is 256 components); however, due to the algebraic identities (2.20) - (2.22), the number of algebraically independent components of the curvature tensor is only N 2 (N 2 -1)/12 (ie only 20 independent components in four-dimensional space-time).

By narrowing the tensor R i klm in the indices i and l (which according to identities (2.20) and (2.22) is the only narrowing giving a non-zero result) we get the so-called Ricci curvature tensor R ik

R ik = def R m imk = g ml R milk , (2.23)

which is symmetrical. By further narrowing we get an invariant, which is called the scalar curvature R of the given space:

R = def g ik R ik = g il g mk R milk . (2.24)

In addition to algebraic symmetries, the curvature tensor also satisfies important differential relations, the so-called Bianchi identities, between covariant derivatives of the curvature tensor:

R i klm ; j + R i klj ; m + R and kmj ; l = 0. (2.25a)

Narrowing this equation in the indices i and l and by multiplying g jk we get, given the covariant constant of the metric tensor g jk ; n = 0, relation (R j l - d j l R / 2) ; j = 0, which can be written in the form

G ik ; k = 0, where G ik = def R ik - 1 / 2 g il R. (2.25b)

This narrowed Bianchi identity, according to which the covariant four-divergence of the Einstein curvature tensor G ik is identically equal to zero, plays a key role in the gravitational field equations, as we will see in 2.5.

Curvature tensor figures in all phenomena, in which the applied curved s space (space-time). We will mention two such situations. In flat space, the second partial derivatives of the vectors according to the coordinates are commutative (A i , k , l = A i , l , k ), as well as the covariant derivatives: A i ; k ; l = A i ; l ; k . In the general case, however, according to (2.13) applies

(2.26)

so the covariant derivatives are generally non-commutative and the measure of this non-commutativity is the curvature tensor R iklm .

In planar space, two lines passing parallel through two points remain parallel at all times. In a curved space, however, two geodesics (playing the role of lines here), originally parallel in one place, gradually deviate from each other due to the curvature of space. The equation of this deviation of geodesics (2.57) also shows the tensor of curvature, as we will see in 2.6 " Deviation and focus of geodesics ".

Generalization of physical laws to curved spacetime
Because the actual gravitational field is actually curved spacetime, the extraordinary importance of the space-time curvature tensor in gravity physics is obvious, where this curvature tensor expresses the inhomogeneity of the gravitational field . Already from Newton's theory of gravity we know , the inhomogeneity of the gravitational field is closely related to the excitation source of the gravitational field. In 2.5 we will see that in Einstein's theory of gravity the equations of gravitational field generation relate the curvature of space-time to the distribution of excitation masses, ie they describe how matter curves space-time in its vicinity .

It is precisely such "corrections" on the connection as in (2.11) and in (2.15) that we have actually made in both examples at the beginning of the generalization of physical laws to curved spacetime. The equation of motion of a particle (2.5a) d 2 x i ( l ) / d l 2 du i ( l ) / d l = 0 says that along the world line of a free test particle, the four-velocity u i dx i / d l is constant. In general coordinates, the derivative du i / d l must be replaced by the absolute derivative (2.15), which gives equation (2.5b) :

0 = Du i / d l d u i / d l + G k i l u k dx l / d l = d 2 u i / d l 2 + G k i l (dx k / d l) ( dx l / d l) ,  

And in the equation of the law of conservation of energy and momentum T ik , k = 0, when transcribed into general (curvilinear) coordinates of normal partial fourdivergence must be replaced by covariant fourdivergence, which leads to equation (2.6), which according to the notation in (2.11) can be written in the form

T ik ; k = 0. (2.6 ')

We can therefore state the general rule of the relationship between the laws of nongravity and gravitational physics :

Theorem 2.3
The generalization of the physical laws valid in planar spacetime (ie the laws of special theory of relativity without gravity) to curved spacetime (presence of a gravitational field) consists in the fact that ordinary partial derivatives according to coordinates are replaced by covariant derivatives .

In addition, the Minkowski tensor h ik transitions to the general metric tensor g ik . Theorem (2.3) is sometimes abbreviated as a " swap commas with semicolons " rule .

Let us now return to the physical meaning of the geodetic equation (2.5b). In the limit case of small velocities and weak fields (moreover, the field must be weak so that the particle does not gain high velocity in it), the general equations of motion of the particle in the gravitational field must pass to the corresponding non-relativistic equation (1.29b). To clarify the physical importance of geometric quantities of space compared therefore equation (2.5b) with Newton's equation of motion for a situation where the gravitational field is still sufficiently weak. In this case, the tensor g ik does not depend on the time coordinate x , g o a = 1 ( a = 1,2,3) and there is a reference system in which the metric tensor can be decomposed into

g ik (x) = h ik + h ik (x), | h ik | 1,

where h ik are small deviations from the Minkowski metric. Such a system is approximately inertial with Cartesian coordinates around the test particle. Assuming that the motion of a particle in this frame of reference is not very fast (| v | c, where v v a = dx a / dt is the velocity of the particle), the eigenvalue t will be approximately equal to the coordinate time t = x / c, so in the geodetic equation we can put dx b / d t dx b / dt = v b , dx / d t dx / dt = c. If we limit ourselves to first-order members vh ik , will be the only nonzero components of the affine connection G b oo = G b o = - (1/2) h oo / x b ( b = 1,2,3). In this approximation , the spatial part of equation (2.5b) has the form

d 2 x and / dt 2 - (c 2 /2) h oo / x a = 0 .

If we compare it with Newton's equation of motion in the gravitational field (2.4) rewritten in the form

d 2 x a / dt 2 + j / x a = 0 ,

we see that Newton's equation of motion is a special case of the general equation of motion of geodesy (2.5b), where the relationship between the usual gravitational potential j and the metric tensor is h oo = - 2 j / c 2 , or

g oo = - (1 + 2 j / c 2 ). (2.27)

Again, this shows that the components of the metric tensor have the physical significance of the gravitational field potentials ; Christoffel's affinity connection coefficients then express the gravitational forces acting .

If the "test particle" has zero rest mass and moves at the speed of light (eg photon), its motion in the local and inertial system will be given by the equations d 2 x i / d l 2 = 0, ds 2 = c 2 d t 2 = h ik dx i dx k = 0, where L is afinne parameter replacing the proper time t (which is not applicable because it is equal to zero). In general curved spacetime (in the gravitational field), the light propagation equation has the form

d 2 x i / d l 2 + G k i l (dx i / d l) ( dx k / d l ) = 0, ds 2 = g ik dx i dx k ,

where the second equation can also be written in the form (ds / d l ) 2 = g ik (dx i / d l ) (dx k / d l ) = 0. The worldlines along which photons move freely are called light , isotropic, or zero geodesics (along them, the intrinsic time dt and the four-dimensional distance ds are equal to zero).

Gravitational electrodynamics and optics
Using the rule contained in Theorem 2.3, it is easy to generalize the especially relativistic equations of electrodynamics (derived at the end of 1.6) so that they apply in curved spacetime, ie in the gravitational field. The electromagnetic field intensity tensor F
ik = A k / x i - A i / x k will be defined here as F ik = A k; i - A i; k , but it can be easily shown that A k; i - A i; k = A k, i - Ai, k . The relationship between the four-potential A i and the electromagnetic field tensor F ik therefore does not change. Similarly, the first "pair" of Maxwell's equations retains its (four-dimensional) shape :

F ik, l + F li, k + F kl, i = F ik; l + F li; k + F kl; i = 0. (2.29)

If we replace the charged test particles (mass m and charge q) in the electromagnetic field in the Lorentz equation of motion m.c (du i / d t ) = (q / c) F ik .u k derive du i / d t by an absolute derivative, we get the equation of motion of a charged particle in an electromagnetic and gravitational field shaped

mc. (du i / d s + G k i l u k u l ) mc. D u i / d s = (q / c) F ik u k . (2.30)

The continuity equation j i , i = 0 will have a general form in curved spacetime

j i ; i = 0 (2.31)

and a second portion of the Maxwell equations F ik , k = - (4 p /c).j i in a gravitational field will be generalized to

F ik ; k = - (4 p / c). j i . (2.32)

Thanks antisymatrii tensor F ik again from this equation implies continuity equation (2.31). The four-current current density vector is defined in STR as j i = r .dx i / dt, where r = dQ / dV is the charge distribution density in space. After transformation into curvilinear coordinates, the element of volume dV passes into ( g) dV (where g is the determinant of the spatial metric tensor g ab and dV = dx 1 dx 2 dx 3 ) and the four-current in general equations (2.31) and (2.32) is given by

j i = ( r .c / g oo ). dx i / dx o . (2.33)

To clarify the effect of gravity on electromagnetic phenomena, it is interesting to break down equations (2.29) and (2.32) in three-dimensional form [166]. If we introduce quantities

E a F o a , D a (g oo ) F a , B ab F ab , H ab (g oo ) F ab ,

in them equations (2.29) and (2.32) will have the form after the separation of spatial and temporal components

(2.29 ')


(2.32')

If the gravitational field is static, these equations can be rewritten in common (three-dimensional) vector symbolism:

(2.29 '')

(2.32 '')

where vector H has components H a = - (1/2) ( g ) e abg H bg and vector B has components B a = - (1/2 g ) e abg B bg . If we look at equations (2.29 ") and (2.32") from the point of view of non-gravitational physics, these equations have the form of Maxwell's equations of the electromagnetic field not in vacuum, but in a material environment with dielectric constant and permeability

e = m = 1 / g oo . (2.34)

I see that the gravitational field (curved spacetime) has a similar effect on the electromagnetic field as the electrically and magnetically "soft" matter - optical environment . The electromagnetic wave, which is the wave solution of Maxwell's equations, will therefore propagate unevenly and curvilinearly in the inhomogeneous gravitational field , as can be seen from the equation of zero geodesy (2.28) describing the motion of photons. Due to the versatility of gravitational interaction, there is no dispersion ; however, in contrast to conventional optics of material environments, a frequency shift is manifested in the gravitational field (see " Gravitational spectral shift " below) .
Therefore, we can expect interesting optical phenomena in strong inhomogeneous gravitational fields - a kind of gravitational "fata morgana" - similarly to optically inhomogeneous material environments. The propagation of light in the gravitational field of a black hole and the effect of a "gravitational lens" are mentioned in 4.3, section " Gravitational lenses. Optics of black holes ".

Space and time in a gravitational field
Gravitational time dilation
Remains to be clarified the relationship between the actual time intervals and spatial distances in space events and their coordinates x
i in the general frame of reference S . We start from the expression for an invariant space-time interval

ds 2 = - c 2 d t 2 = g ik dx i dx k

and introduce an inertial frame of reference S ~ such that it is currently at rest with respect to frame of reference S (with respect to its clock and measuring rods) at a given point. Then both the lengths of sufficiently short (infinitesimal) measuring rods and the time intervals will be the same in the system S and S ~ . In this locally inertial system S ~ with the coordinate i x ~ i is

ds 2 = - c 2 d t 2 = h ik dx ~ i dx ~ k = - (dx ~ ) 2 + dx ~ a dx ~ a ,

where the relation between h ik and the metric tensor g ik is given by the transformation relation

g ik = h lm . ( x ~ l / x i ). ( x ~ m / x k ) = ( x ~ a / x i ). ( x ~ a / x k ) - ( x ~ / x i ). ( x ~ / x k ). (2.35)

The inertial system S ~ is locally at rest with respect to the general system S so that x ~ a / x = 0 and the transformation relation dx ~ i = ( x ~ i / x k ) dx k has a separate form dx ~ = ( x ~ / x k ) dx k , dx ~ a = ( x ~ a / x b ) dx b. The relationship between the time coordinate x and the proper time t can be easily determined by taking two events that occurred in short succession from the point of view of the reference system S at the same place. The interval between these events is then ds 2 = - c 2 d t 2 = g ik dx i dx k , and since dx a = 0, ds 2 = - c 2 d t 2 = g oo dx o2 , ie

d t = (1 / c). (-g oo ) dx o . (2.36)

For a weak gravitational field using the relation (2.27) we get

d t = (dx / c). (1 + 2 j / c 2 ) (1 + j / c 2 ) dt. (2.36 ')

Thus, the proper time with respect to the coordinate time (which corresponds to the zero gravitational potential) flows the slower , the higher the value of the gravitational potential j at a given location (the gravitational potential is negative) . The clock located in the gravitational field is delayed compared to the same clock located outside the field, resp. in a place with a weaker field.

Gravitational dilation of time.
The clock located in the gravitational field is delayed compared to the same clock located outside the field (or in a place with a weaker field).

In the vicinity of material bodies (compared to distant places) time flows more slowly *), there is a " slowing down of the flow of time through the gravitational field " - a gravitational dilation of time . The consequences of this phenomenon (such as the gravitational redshift mentioned below) are crucial in the final stages of gravitational collapse and the formation of black holes (see 4.2,4.3).
*) If we stood on the anthropocentric point of view, we could say with a bit of exaggeration that "bodies fall in the gravitational field because they are trying to reach the place where they will age the slowest " ..? ..
This gravitational dilation of time is related to non-inertial accelerated frames of reference, according to the principle of equivalence. The value of the gravitational time dilation at a given location in the gravitational field is the same as the STR time dilation (1.72) caused by a velocity equal to the escape velocity from that location.
Time dilation inside gravitational bodies
Ordinary gravitational bodies - planets, stars - have their mass distributed in space of approximately spherical shape of radius R with density
r (r), where r is the distance from the center r = 0. For a spherically symmetric distribution, the Newtonian gravitational force F (field strength - force acting on a unit mass of the test particle) is given by the law of inverted squares (1.1). Outside the body (r> R) is simply F (r) = GM / r 2 , inside (r <R) is F (r) = G. ( 0 r 4pr (r) r 2 dr ) / r 2 . There is therefore a depth of gravity inside the body smaller - is given only by the gravitational mass contained between the given place r and the center r = 0. And in the middle r = 0 the gravitational force is zero - the gravity of the outer layers, acting in opposite directions, is canceled (it was also discussed in 1.2, passage " Gravitational bodies ") . However, it does not follow that it would "disturb" itself here and the gravitational dilatation of time would also disappear!
The gravitational dilation of time does not depend on the gravitational force, but on the gravitational potential (2.36 ' ), which is j (r) = - r F (r) dr. Although there is no force in the center of a gravitational body, it has the gravitational potential there in reversethe maximum size - and therefore the gravitational dilation of time there will be relatively the largest ! We can imagine that a certain velocity is also needed from the center to reach the surface, which is then added to the escape velocity from the surface .
From the point of view of GTR, the problem of space-time geometry outside and inside gravitational bodies is analyzed by the so-called internal Schwarzschild solution (3.13b).

Spatial metric
The element dl of the spatial distance
cannot generally be determined as the interval between two infinitely close events occurring at the same point in time by putting dx = 0 in the expression for ds 2 , because the relationship between the proper time t and the time coordinate x is in different places different. For get a relationship between the actual length and the spatial coordinates x a ( a = 1,2,3) we must therefore look for elementary length measuring rod in resting locally inertial system S ~ dl ~ 2 = a =1 S 3 (dx ~ a ) 2 = dx ~a dx ~ a transform into a general non-inertial system S . By describing the transformation relationship (2.35) for the metric tensor we get

Since x ~ a / x = 0, x ~ / x a = - g o a / -g oo ; for the proper length of the infinitely short measuring rod then we get the relation *)

(2.37)

and for the interval of the proper time we get d t 2 = - (1 / c 2 ) g oo dx 2 in accordance with (2.36). The expression in parentheses (2.37) therefore indicates the metric of three-dimensional space in the presence of gravity (or in a non-inertial frame of reference), ie the three-dimensional metric g ab "induced" by the space-time metric g ik .
*) Another derivation of the relation (2.37) by light signal propagation analysis ("radar" distance) is given in [162], [135], [166].

By separating the spatial terms in the identity g ik g ik = 0, the relationships between the metric of space and spacetime can be derived:

g ab = - g ab ; g oo g = - g, (2.38)

where g is the determinant composed of g ik and g is the determinant of the components g a b .
In order for a reference system corresponding to the metric tensor g ik to be physically feasible (using real bodies), the three-dimensional metric form (2.37) must be positive definite and according to (2.36) g oo <0. These conditions, expressed using determinants and subdeterminants of the metric tensor, are called Hilbert conditions [162]:

det |
|
g oo
g 10
g 01
g 11
|
|
< 0, | g oo g 01 g 02 | > 0 , g < 0. (2.39)
det | g 10 g 11 g 12 |
| g 20 g 21 g 22 |

Static and stationary gravitational field
If there is a reference system in which the ingredients metric tensor g
ik depend on the time coordinate x , the respective gravitational field is called stationary . In addition, if there is a frame of reference in a stationary field in which all "mixed" components of the metric tensor g o a are equal to zero, these are statica gravitational field in which both directions of time flow are equivalent. It follows from Newton's (as well as General Einstein's) law of gravitation that static gravitational fields are excited by the static distribution of matter; in 3.4 it will be shown that the gravitational field in vacuum of spherically symmetrical body it is a static even when this body radially pulses (expands or collapses). In practice, a stationary gravitational field can only be excited by a compact isolated body, because in a system of several free bodies their gravitational interactions will cause mutual movements and the resulting gravitational field will be variable. An example of a stationary field is a gravitational field around an axially symmetric body evenly rotating around its axis; however, this field is not static, because both directions of time flow are not equivalent here (when the direction of time is reversed, the sign of the angular velocity of rotation of the body changes). Indeed, according to Einstein's equations, the rotation of the source body leaves "traces" in the form of nonzero components g oa metric tensor on the metric of the surrounding spacetime , see 2.5. Some interesting effects taking place in the gravitational field of rotating objects (especially in the vicinity of rotating black holes) will be discussed in 4.4.

Gravitational spectral shift
We will mention another important consequence of the
gravitational dilation of time , the relationship between the interval of proper and coordinate time (2.36) - the gravitational spectral shift , which we have already mentioned above. According to the relation (2.36) in two places with different gravitational potential, different intervals of the proper time will correspond to the same coordinate time interval. Let stationary gravitational field at point P 1 is a light source which transmits two light impulses discrete interval Dt (P 1 ) of its own time; the coordinate time interval between these events will then be D t (P 1 ) = (1 / c) D x (P 1 ) = (1 / c) (-g oo (P 1 )). Dt (P 1 ). These light signals will propagate through space and be captured by the observer at point P 2 . Because in a stationary gravitational field component metric tensor depend on the time coordinate, the interval of coordinate time D t (P 2 ) between the moment of receipt of both pulses the same as in point deployment, i.e. D t (P 2 ) = D t (P 1 ). Because Dt (P 2 ) = (-g oo(P 2 )). D t ( P 2 ), will be

(2.40)

Similarly, if a periodic radiation-emitting process (eg excited light-emitting atoms) takes place in a stationary gravitational field at point P 1 , then the number of oscillations per unit coordinate time will be the same at all points of the propagating radiation trajectory and the ratio between periods T (P 1 ) and T (P 2 ) radiation at P 1 and P 2 will again be given by (2.40). The frequency ratio will therefore be

(2.41)

In a weak gravitational field, g oo (P) - (1 + 2 j (P) / c 2 ), so




(2.41 ')

If light comes from a place with a higher gravitational potential to a place with a lower potential, its frequency decreases - it is a gravitational redshift . Conversely, when radiation propagates from places with a lower gravitational potential to places with a stronger gravitational field, a blue shift occurs - the frequency of light increases.
Thus, the gravitational frequency shift results from the geometric interpretation of the gravitational field in the general theory of relativity. However, the same conclusion, including the relation (2.41 '), can be reached by more elementary procedures. The first of these is
the kinematic interpretation using principle of equivalence : a situation where the light source and the receiver (observer) are located in places with different gravitational potential, we replace the equivalent state in which the light source and the observer are located in different places of a uniformly accelerated reference system (for example, in the rocket cabin as in Fig.2.3b). From the time moment t = 0 the light send from point P 1 to the moment D t = D L / c of its registered receiver located at a distance D L in the direction of the acceleration obtained by the receiver relative velocity v = a. D L / c. Therefore, due to the Doppler effect, the wavelength of the received light will appear different from the wavelength l of the radiation emanating from the source, this deviation expressed by the frequency w will be in the first order

Dw / w v / c = a. D l / c 2 .

Returning now to the initial situation with the gravitational field, then the quantity a. D l means the difference of the gravitational potentials Dj between the source and the receiver, so when overcoming the difference of the gravitational field potential Dj = a. D l the wavelength of light changes by Dj / c 2 in accordance with the formula (2.41 ').
Similarly, the
gravitational dilation of time can be determined according to the principle of equivalence in a uniformly accelerated frame of reference as STR dilation of time when we substitute the above-mentioned relative velocity into the relation (1.70) in 1.6 v = a. D l / c :
D t '= D t / (1 - a 2 D l 2 / c 4 ).
The change in frequency
Dw / w resulting from this time expansion will be in the first order approximation ~ a. D l / c 2 , which, expressed by the potential difference, again gives Dj / c 2 .
The gravitational frequency shift can also be easily derived as a consequence of the
law of conservation of energy during the motion of photons in the classical gravitational field. We consider light of wavelength l and frequency w to be a stream of photons with energy E = h. w = hc / l and mass m = E / c 2. When the potential difference Dj in the gravitational field is overcome , the energy of the photons changes by D E = Dj .m = Dj .E / c 2 , so that the relative change in wavelength is again Dj / c 2 .

Pound-Rebka experiment
Although the gravitational frequency shift is insignificant in terrestrial conditions and does not apply in practical life, the gravitational red shift was
experimentally demonstrated and measured even in the Earth's gravitational field . In 1960, R.V.Pound and G.A.Rebka [208] used the Mssbauer effect *) of resonant nuclear absorption of g- radiation with an energy of 14.4 keV of the excited level of a 57 Fe iron core . Source - b + g radioisotope 57 Co with mechanical displacement (by electro-mechanical movement of the loudspeaker diaphragm) and receiver (absorber 57 Fe with spectrometric radiation detector g) were located in the water tower at Harvard with a height difference of only 22.5 meters. Two measurements were made while exchanging the positions of the transmitter and receiver (ie, both red and blue shifts were measured) to exclude non-gravitational shifts of the spectral line. The measured values of the relative frequency shift of about 2.5.10 -15 agreed with the formula (2.41 ') originally with an accuracy of about 10%, in the improved variant of Pound and Snyder the agreement improved to 1% [209].
*) The Mssbauer effect is described in more detail in the book " Nuclear Physics and Physics of Ionizing Radiation ", 1.6 " Ionizing Radiation ", part " Interaction of gamma radiation ", passage " Resonant nuclear absorption - Mssbauer effect ".

Gravity Probe A
With even greater accuracy, the gravitational frequency shift was measured in 1976 on a strongly eccentric elliptical orbit (so that the probe passed the largest possible difference in gravitational potential) of the disposable satellite Gravity Probe A
(maximum height of the suborbital orbit, ie a turning point of 10,000 km above Earth, flight time 55min, end of runway in the Atlantic) . An accurate clock equipped with a hydrogen MASER was installed on the satellite, the same MASER was placed in a control center on the Earth's surface. By comparing the registered data, the gravitational redshift was measured, which agreed with the GTR with a relative accuracy of 2.10 -4 .

Astronomical measurements of the gravitational redshift
In addition, there are
astronomical verifications of the gravitational redshift. The light emitted by atoms from the surface of the Sun arrives on Earth according to the relation (2.41 ') with a redshift of Dw / w @ 2.10 -6 , which is a few percent of the width of the Fraunhofer lines. This effect is well measurable by spectroscopic methods, but in addition to the Doppler shift correction by due to the relative radial motion of the Earth and the Sun, the flow of hot gases on the surface of the Sun is significantly manifested here. Hot gases rise, cool at the surface and fall back. The value of the redshift corresponding to the relativistic formula (2.41 ') is measured directly only from the areas at the edge of the solar "disk", where the radial streams of glowing hot gases are observed perpendicularly. In contrast, the red shift of radiation from a central region of the solar disc is very small and the spectral lines are Doppler shift due to the rising their hot and descending cooler streams of gas glowing extended and asymmetrical, with the center position is shifted to higher frequencies. Using a detailed analysis of the peak shape of these spectra [216], it was possible to correct the effect of the Doppler effect radial streams of hot gases on the surface of the Sun and show that the gravitational redshift is approximately the same in all places of the solar disk and coincides well with the relation (2.41 ') (accuracy about 5%).
A significantly higher redshift can be expected for
massive compact stars such as white dwarfs (see 4.2), on the surface of which the gravitational potential is one to two orders of magnitude stronger. For measurement and comparison with the relativistic prediction, only bright white dwarfs, which are part of binary stars, are suitable for determining their mass. However, the radius of a white dwarf of known mass cannot be determined directly, but only on the basis of the theory of the structure of white dwarfs; therefore, the measurement of the red displacements of white dwarfs can be considered as a test of the theory of their internal structure rather than as a verification of the relativistic relation for the gravitational red displacement. E.g. for Sirius B, which has a mass approximately the same as the Sun, the theoretical radius is about 10 -2 solar radius and the redshift is thus 2.10 -4 ; however, interferes with accurate measurement of radiography strong light from nearby Sirius A. Somewhat favorable situation is with the binary 40 Eridani a greater distance of the two components, wherein the white dwarf B measurement leads to an agreement about 20% of a theoretical value of red shift [207].
More complex effects of frequency shifts can be expected with relic radiation , which on its long journey through vast cosmic spaces passes through areas with a large accumulation of matter in galaxy clusters and, conversely, large areas of "emptiness". These fluctuations in gravitational potential and space-time metrics may slightly modulate the anisotropy of relic microwave radiation (see 5.4, section " Microwave relic radiation ", section " Influence of gravitational fluctuations in space metrics on relic radiation - Sachs-Wolf effect ") .

2.3. Local principle of equivalence   2.5. Einstein's equations of the gravitational field

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
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Vojtech Ullmann