# Physics of curved spacetime and gravity

 AstroNuclPhysics ® Nuclear Physics - Astrophysics - Cosmology - Philosophy Gravity, black holes and physics

2.4. Physical Laws in Curved Spacetime
Gravity can be examine in essentially two ways :
Either:
1.
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 forces acts   - the orbits of the planets are curved because the actual space and time in which they move is deformed (curved) by the presence of the massive Sun and automatically forces the planets to move along the respective "geodetic" orbit.
In terms of general relativity, the movement of the test particles in a gravitational field is inertial (particle is free) and its possible peculiarities are caused not by the "gravitational force" acting on the particle, but by the space-time metric; it is similar with all physical phenomena in the presence of gravity.
The gravitational field thus "disappeared", instead of him the generally curved (Riemannian) spacetime here 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 to a straightforward approach: to divide spacetime into sufficiently small areas in which the curvature can be neglected, to apply physical laws spacetime in these areas (ie STR formulated for general reference frames) and finally to fold this of planar "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 a given gravitational field, which at time t is in world-point P. If we introduce at point P a locally inertial reference system S~ with Cartesian space-time coordinates x~ i related to the owen time t of the test particle by the relation dt2 = - (1/c2) hik 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 own time by the relation

ds2  =  - c2dt2 = gik dxi dxk ;   gik(x j)  =  hlm.(x~l/xi).(x~m/xk)   ,

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

wherein Gkil = (1/2) gim (gmk/xl + gml/xk + gkl/xm) as shown in §2.1, equation (2.2a,b). We can do this procedure at any point of a test particle world-lines 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 Tik, k ş ¶Tik/xk = 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: Tik(x~)/x~ k = 0. After transformation into a general (non-inertial) frame of reference S, this law takes on a covariant form

 ¶Tik/¶xk + Gmik Tmk + Gmkk Tim   =   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 in the flat spacetime can be by 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 make 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 Ai according to coordinates xk (2.7)

is usually a measure of how the vector field Ai changes with location (from a point with xk coordinates to a "neighboring" point xk + Dxk). 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 - this can lead to different values.
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, Ai , 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. At parallel transfer vector, its components in the Cartesian coordinate system not changes. Using a curvilinear coordinate system, however, when parallel transfer vector Ai from the point with coordinates xk to a nearby point xk + Dxk components of the vector changes by

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

where the quantities Gk 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 vice versa, with nonzero Gk i l , all components can be canceled by going to the Cartesian system at a given point). From the requirement, that dAi in the law of parallel transmission (2.8) be transformed as a vector, for affine connection coefficients follows the transformation relation : (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 Ai(xk) transferred in parallel to the point xk + Dxk will be Ai(xk)®xk+Dxk = Ai(xk) + dAi. 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 irjs....... the law :

 d T irjs.......   =   - Gmin T mrjs........Dxn - Gmin T irms........Dxn - .... + Grmn T imjs........Dxn + Gsmn T irjm........Dxn + ....   . (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 Ai(xk + Dxk) from the point xk + Dxk back to point xk is performed, and only then the appropriate limit be made : (2.11)

It is thus achieved that is taken the difference between the components of the vectors calculated at a single point xk, and thus related to the same base. 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 , . According to (2.7) and (2.11), the covariant derivative of vector Ai is equal to

 Ai;k = ¶Ai/¶xk + Gkim Am = Ai,k + Gkim Am , similarly Ai;k = ¶Ai/¶xk + Gimk Am  . (2.12)

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

 T irjs.......;w   =   T irjs.......,w + Gwim T mrjs....... + Gwjm T irms....... + .... - Grmw T imjs....... - Gsmw T irjm....... + ....   . (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 Ai(l) ş Ai(xk(l)) are the components of the vector Ai 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 Ai along the curve C (xk = xk(l)) : (2.15)

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

 DAi /dl   =   Ai,k + Gkil Al dxk/dl   =   Ai;k dxk/dl   , (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 = lo (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 xK, so the derivative gik/xK = 0. In this case, we can transform any curve using the coordinate shift of all its points by dxK in the direction of the xK 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 xK coordinate - metric space (here spacetime) appears in this direction the K isometry. To describe this isometry, the so-called Killing vector xK ş ¶/xK is introduced in the differential geometry, expressing components of infinitesimal translation, preserving length. The distribution of this vector at each point of the manifold forms the Killing vector field of infinitesimal isometric generators. This field satisfies the covariant Killing equation xi; k + xk; 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 xk , then the vector P i = Tik xk , for which thanks Killing equations the relation applies Pi;i = Tikxk;i = (1/2)Tik(xk;i + xi;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 xk 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-Metric ").
The components of the connection coefficients Gk 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 ds2 = g ik dxi dxk it can be converted to the form ds2 = Si (dxi)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 ds2 = iS ki .(dx i)2 by appropriate transformation of coordinates, where individual constant coefficients ki 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. In a curvilinear system, the components of a vector change during parallel transfer, but from the existence of the Cartesian coordinate system follows that, in a planar space parallel transmission not depend on the path along which it 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, Gk 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 Ai (C1) are the components of the vector Ai transferred in parallel from point A to point B along the curve C1, Ai (C2) is the result of the transfer between the same points along the curve C2. In this case, if we perform a parallel transfer with a given vector along a closed curve, we return to the starting point generally with 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 measure of curvature of space is based on the properties of the circle, respectively sphere. We construct the set 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 2p r is a measure of the curvature of the space (area); if the length of the resulting curve is less than 2p r, the curvature is positive (eg spherical surface), if this length is greater than 2p 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 4p R2; 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 This curve integral generally cannot be converted to a area integral using the Stokes theorem, because the values of the component of vector Ai at the points of the respective area (inside the curve C) cannot be unambiguously determined - they depend on the path that the expansion of the vector field with a vector Ai at that point we come. However, if the curve C is sufficiently small (infinitesimal), this ambiguity does not apply in the transition to limit (the corresponding error is up to second order) and Stokes theorem gives (the variability of the vector field Ai with place is there only due to the connection, so Ai/xl = - Gk i l Ak) (2.18)

where DSlm is the tensor of the area bounded by an infinitesimal closed curve C. A more detailed derivation can be found eg in , . The tensor R i klm , which quantifies the difference between the geometric properties of a given space from the planar one (nonintegrability of the afine 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 Ri klm = 0; thanks to the tensor character Ri klm, this also applies to any other (perhaps curvilinear) coordinate system. Conversely, if Ri 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 Ţ 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 Gk i l , or gik, is flat or curved.

Here we mention 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 Ri klm is antisymmetric in the indices l, m :

 R i klm   =   - R i kml   . (2.20)

In addition, the tensor Ri 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 Riklm = gij Rj klm , obtained by reducing the index i:

 Riklm = - Rkilm = - Rikml   ,   Riklm = Rlmik   ; (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 N4 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 N2 (N2 -1)/12 , ie only 20 independent components in four-dimensional space-time.

By narrowing the tensor Ri 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 :

 Riklm;j + Riklj;m + Rikmj;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 (Rj l - dj l R/2) ; j = 0, which can be written in the form

 Gik;k   =   0 ,   where  Gik   =def Rik - 1/2 gik R   . (2.25b)

This narrowed Bianchi identity, according to which the covariant four-divergence of the Einstein's 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.

The curvature tensor figures in all phenomena, in which the curvature of space (space-time) is applied. We will mention two such situations. In flat space, the second partial derivatives of the vectors according to the coordinates are commutative (Ai, k , l = Ai, l , k), as well as the covariant derivatives: Ai ; k ; l = Ai ; 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 Riklm .
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 gfocus 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 put into context 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) d2xi(l)/dl2 ş dui(l)/dl = 0 says, that along the world line of a free test particle, the four-velocity ui ş dxi/dl is constant. In general coordinates, the derivative dui/dl must be replaced by the absolute derivative (2.15), which gives equation (2.5b) :

 0  =  Dui /dl ş   dui /dl + Gkil uk dxl/dl   =   d2ui /dl2 + Gkil (dxk/dl) (dxl/dl)   .

And in the equation of the law of conservation of energy and momentum Tik, k = 0, when transcribed into general (curvilinear) coordinates of normal partial four-divergence must be replaced by covariant four-divergence, 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 non-gravitational 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 hik passes to the general metric tensor gik. Theorem (2.3) is sometimes abbreviated as the rule "replace commas with semicolons".

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 significance of geometric quantities of space-time, we compare 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 gik does not depend on the time coordinate x°, goa = 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 ş va = dxa/dt is the velocity of the particle), the proper time t will be approximately equal to the coordinate time t = x°/c, so in the geodetic equation we can put dxb/dt » dxb/dt = vb, dx°/dt » dx°/dt = c. If we limit ourselves to first-order members in h ik, will be the only nonzero components of the affine connection Gb oo = G°b o = - (1/2) hoo/xb (b = 1,2,3). In this approximation, the spatial part of equation (2.5b) has the form

d2xa/dt2 - (c2/2) hoo/xa  =  0  .

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

d2xa/dt2 + j /xa  =  0   ,

we see that Newton's equation of motion is a special case of the general equation of motion of geodesy (2.5b), whereas the relationship between the usual gravitational potential j and the metric tensor is hoo = - 2 j/c2 , 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 inertial system will be given by the equations d2xi/dl2 = 0, ds2 = c2 dt2 = hik dxi dxk = 0, where l is afinne parameter replacing the proper time t (which is not applicable here, because it is equal to zero). In general curved spacetime (in the gravitational field), the light propagation equation has the form

d2xi /dl2 + Gkil (dxi/dl) (dxk/dl) = 0  ,   ds2 = gik dxi dxk   ,

where the second equation can also be written in the form (ds/dl)2 = gik(dxi/dl)(dxk/dl) = 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 specially 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 in the Lorentz equation of motion of charged test particles (mass m and charge q) in the electromagnetic field m.c.(dui/dt) = (q/c) Fik .uk the derivative dui/dt by the absolute derivative "Dui /ds", we get the equation of motion of a charged particle in an electromagnetic and gravitational field in the form

 mc . (dui /ds + Gkil uk ul)  ş  mc . Dui /ds  =  (q/c) Fik uk   . (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 Fik , k = - (4p/c).j i in a gravitational field will be generalized to

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

Thanks to the antisymmetry of the tensor Fik, the continuity equation (2.31) follows again from this equation. The four-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 Ö(g) dV (where g is the determinant of the spatial metric tensor gab and dV = dx1 dx2 dx3 ) and the four-current in general equations (2.31) and (2.32) is given by

 j i  =  (r.c/Ögoo) . dxi/dxo   . (2.33)

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

Ea ş Foa ,   Da ş Ö(goo) F°a ,   Bab ş Fab ,   Hab ş Ö(goo) Fab   ,

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 the usual (three-dimensional) vector symbolism : (2.29 '') (2.32 '')

where vector H has components Ha = - (1/2) Ö(g) eabg H bg and vector B has components Ba = - (1/2 Ög) eabg Bbg . If we look at equations (2.29") and (2.32") from the point of view of non-gravitational physics - electrodynamics, 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 / Ö g oo   . (2.34)

Thus we see that the gravitational field (curved spacetime) has a similar effect on the electromagnetic field as the electrically and magnetically "soft" substance - optical environment. The electromagnetic waves, which are the wave solution of Maxwell's equations, will therefore propagate unevenly and curvilinearly in the inhomogeneous gravitational field, as can be seen, by the way, also 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
It still remains to clarifiy 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 coordinatei x~ i is

ds2  =  - c2 dt2  =  hik dx~i dx~k  =  - (dx~°)2 + dx~a dx~a    ,

where the relation between hik and the metric tensor gik is given by the transformation relation

 gik  =  hlm.(¶x~l/¶xi).(¶x~m/¶xk)  =  (¶x~a/¶xi).(¶x~a/¶xk) - (¶x~°/¶xi).(¶x~°/¶xk)  . (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/xk) dxk has a separate form dx~° = (x~°/xk)dxk, dx~a = (x~a/xb)dxb. 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 ds2 = - c2 dt2 = gik dxi dxk, and since dxa = 0, ds2 = - c2 dt2 = goo dxo 2 , ie

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

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

 d t   = (dx ° / c). Ö (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 took the anthropocentric stand, 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 / r2, inside (r <R) is F(r) = G. (0 ň r 4p r(r) r2 dr) /r2. Inside the body, therefore, gravitational force is smaller in depth - 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 the gravitational dilation of time is also "cancel" here and 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 ň Ą F(r) dr. Although there is no force in the center of a gravitational body, it has the gravitational potential there in reverse the maximum size - and therefore the gravitational dilation of time there will be relatively the largest ! We can imagine it 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 ds2, 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 xa (a = 1,2,3) we must therefore look for elementary length measuring rod in resting locally inertial system S~ dl~ 2 = a=1S3(dx~a)2 = dx~adx~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~°/xa = - goa/Ö-goo, 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 dt2 = - (1/c2) goo 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 gab "induced" by the space-time metric gik .
*) Another derivation of the relation (2.37) by light signal propagation analysis ("radar" distance) is given in , , .

By separating the spatial terms in the identity gik gik = 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 gik and g is the determinant of the components ga b .
In order for a reference system corresponding to the metric tensor gik to be physically feasible (using real bodies), the three-dimensional metric form (2.37) must be positive definite and according to (2.36) goo <0. These conditions, expressed using determinants and subdeterminants of the metric tensor, are called Hilbert conditions  :

 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 components of metric tensor g
ik do not 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 goa are equal to zero, it is a static 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 goa of 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, to the same coordinate time interval will correspond the different intervals of the proper (own) time. Let stationary gravitational field at point P1 is a light source which transmits two light impulses discrete interval Dt(P1) of its own time; the coordinate time interval between these events will then be Dt(P1) = (1/c)Dx°(P1) = (1/c)Ö(-goo(P1)).Dt(P1). These light signals will propagate through space and be captured by the observer at point P2 . Because in a stationary gravitational field component metric tensor do not depend on the time coordinate, the interval of coordinate time Dt(P2) between the moment of receipt of both pulses the same as at the sending point, i.e. Dt(P2) = Dt(P ). Because Dt(P2) = Ö(-goo(P2)).Dt(P2), 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 P1, 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(P1) and T(P2) radiation at P1 and P2 will again be given by (2.40). The frequency ratio will therefore be (2.41)

In a weak gravitational field, goo(P) » -(1 + 2j(P)/c2), 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 of sending light from point P1 to the moment Dt = DL/c of its registration by a receiver located at a distance DL in the direction of acceleration, this receiver obtains the relative velocity v = a. DL/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.Dl 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. Dl, the wavelength of light changes by Dj/c2 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 dilatation of time when we substitute the above-mentioned relative velocity into the relation (1.70) in §1.6 v = a. Dl/c :
Dt ' =   Dt / Ö (1 -   a2 Dl 2 / c4 ) .
The change in frequency
Dw/w resulting from this time dilatation will be in the first order approximation ~ a. Dl/c2, which, expressed by the potential difference, again gives Dj c2.
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. The light of wavelength l and frequency w we consider as a stream of photons with energy E = h.w = h l and mass m = E/c2. When the potential difference Dj in the gravitational field is overcome, the energy of the photons changes by DE = Dj .m = Dj .E/c2, so that the relative change in wavelength is again Dj/c2 .

Pound-Rebka experiment
Although the gravitational frequency shift is completely insignificant in terrestrial conditions and does not manifest in practical life, the gravitational red shift was succeeded
experimentally demonstrated and measured even in the Earth's gravitational field. In 1960, R.V.Pound and G.A.Rebka  for this used the Mössbauer effect *) of resonant nuclear absorption of g- radiation with an energy of 14.4 keV of the excited level of a 57Fe iron nuclei. Source - b + g radioisotope 57Co with mechanical shift (periodical shift by electro-mechanical movement of the loudspeaker diaphragm) and receiver (absorber 57Fe 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% .
*) The Mössbauer 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 - Mössbauer 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 Earth) 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