# Equation of excitation of gravitational field by matter and energy

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

2.5. Einstein's equations of the gravitational field

If we are building a field theory, we are basically interested in two areas of questions :

• First, we ask how such a field affects matter , various physical phenomena (for example, we monitor the movement of test particles in this field). While the monitored field we consider to be given - already "finished" - and we are not interested in its creation yet. We have done this for the gravitational field so far (§2.2-2.4).
• In the second set of questions, we are interested in how and in what way the monitored field is excited (generated) and what are the internal properties of this field - for example, what energy does it contain, how fast are changes propagate in it, and so on.

In this section we will try to show how the gravitational field is generated by its sources, ie to "derive" Einstein's equations of the gravitational field *). We will first proceed in an inductive manner similar to the way Einstein arrived at his equations; this way is heuristical the most beneficial. Some other procedures leading to Einstein's equations [111], [166], [181], which arose much later (and probably only thanks to the knowledge of Einstein's equations and his general theory of relativity) will be mentioned in the following.
*) Specifically, a new fundamental law of physics, such as the law of gravitation in GTR, cannot be derived in the mathematical sense of the word, ie as a direct consequence of some other previously known relations and laws (eg theorem 2.3 based on the principle of equivalence cannot be used, because in the special theory of relativity we have no law of gravitation). Gravity law rather looking ("constructs") on the basis of certain physical requirements and postulates.

The universality of gravitational excitation
First,
we extend the universality of the gravitational interaction. So far, we have understood by the universality of gravity the universal effect of the "already finished" gravitational field on all matter~energy. This allowed a geometric description of the gravitational field as a curved spacetime. However, the universality of gravity has a broader framework and also applies to the excitation of the gravitational field :

 Theorem 2.4 (universal excitation of gravity)
 The gravitational field (curvature of spacetime) is excited universally by all matter ~ energy, or active gravitational mass = passive gravitational mass = inertial mass .

The simplest argument in favor of this statement is the law of action and reaction. If we have two bodies A and B placed so far apart that Newtow's law of gravitation applies to its mutual gravitational attraction, it must be

since this must apply to bodies of any structure, the active and passive gravitational masses are the same for each body (or both types of mass are proportional to each other with the universal gravitational constant G).
Thus, the intensity of the gravitational field that excites a body around it does not depend at all on its composition and nature, it is given only by its total inertial mass (ie the resistance it would put to acceleration by non-gravitational forces). It does not matter whether they are solid bodies, the gas, cluster of elementary particles, or if only the electromagnetic field. Its own gravitational field is also generated by the electromagnetic waves (light, radio waves, X-rays), globally even by gravitational waves (§2.7 "Gravitational waves", see also Appendix B, §B.3 "Wheeler geometrodynamics. Gravity and topology.").

If our task is to determine the mass of a large body - such as a planet or a star, we can proceed in two ways. The first method is based on non-gravitational physics, where the mass of the body is given by the volume integral M = ò T°° dV. However, in order to accurately determine the energy-momentum tensor, it is necessary to know in detail the internal structure of a star - what particles it consists of, the nature of interactions between them, the mechanisms of energy conversion and transfer, etc. Figuratively speaking, we would have to either "to count" all the particles the star contains, determine their masses, make corrections to the mass difference caused by their motions and bonds in the respective fields, and also gain the mass of radiation, or calculate the energy-momentum tensor based on certain assumptions arising from the internal construction of a star. In this complex way, we could perhaps theoretically (but never practically!) determine the mass of a given star.
However, in practice we determine this mass precisely and at the same time incomparably more easily by means of gravity: we analyze the properties of the motion of other bodies in the external gravitational field of the observed stars, eg at a distance r we put a small test body in orbit around it and determine the mass of the star from Kepler's law M = w2 .r3/G. This method is completely reliable, there is no danger (as in the first method) that we would "overlook" any contribution to the total weight. To determine the total mass of a star or other physical system, it is not necessary to know what is happening inside, just simple Newtonian measurements in a sufficiently distant region.
Exciting the gravitational field (gravitationally attracting other bodies) is a common universal property of all material formations, of every form of matter. Therefore, the most objective method of determining the mass of some body is to measure how strong the gravitational field around this body excites. In the following, we will understand the mass of some system just as this gravitationally measured mass.

Gravitational field excitation equations
In §1.4, where we followed the analogy between Newton's and Coulomb's law, we said that the principle of superposition applies to both Coulomb's electrostatics and Newton's gravity, so the corresponding equations are linear. But now we see that in this gravity applies only approximately, for weak fields within Newton's gravitational law. Even without knowledge of the exact form of the gravitational field equations, we can make the following statement as a direct consequence of theorem 2.4 (universality of excitation): the equations of the gravitational field must be fundamentally
nonlinear.
To clarify this important aspect of gravity, let's compare the situation with electrodynamics once again. The electromagnetic field is excited using Maxwell's equations

 F ik ; k    =  - (4 p / c) j i    .      á field á  á source á (2.42)

The source of the field here are electric charges and their currents j i , while the electromagnetic field itself does not transmit an electric charge (it is uncharged) and is therefore not the source of another electromagnetic field - Maxwell's equations are linear and the principle of superposition applies.
In contrast, the source of the gravitational field is all matter (~energy) and, because the gravity field itself is also a source of energy (and momentum), evokes some "complementary" gravitational field. This "self -gravity" leads to the principal nonlinearity of gravity, because the excited gravitational field contributes back to its own source (cf. also §2.1).

The equations (of generations) of each physical field have the character *) :

 an object describing an field = an object describing the source
(2.43)

E.g. for the electromagnetic field the object describing the field is the tensor of the intensity of the electromagnetic field Fik, resp. its four-divergences Fik; k . The source of the field is electric charges and the object describing the source is a four-current j i indicating the distribution and movement of electric charges. The field generation is then given by equation (2.42), which is an electromagnetic variant of general equation (2.43). Let's us look for a gravitational variant !
*) We stand here on the classical position, according to which the field is excited by a certain external source (which is of a different character than the excited field) and we ask: "How is the field excited by its source?". The position of unitary field theory (whose equations have no external source) is the opposite - there is only a field with sufficiently rich internal properties and the question is solved: "How is what we considered to be a source, composed of its field?"- see Appendix B "
Unitary field theory and quantum gravity".
We know about the gravitational field, that it is expressed by the geometry of spacetime and that it is excited universally by all matter (~energy). Therefore, for the gravitational field, the general equation (2.43) should read :

 an object describing the geometry of spacetime = an object describing the distribution of matter and energy
(2.44)

whereas for weak gravitational fields it must give the correct Newtonian limit, ie the Poisson equation Dj = 4p.r.G with the potential j related to the metric tensor by relation (2.27).

The object, which exhaustively and independently of the specific structure describes the distribution and motion of matter and energy in the physical system, is the energy-momentum tensor Tik (introduced in §1.6 "Fout-dimensional spacetime and special theory of relativity"). If we denote the left side describing the geometry of spacetime as G ik , the equations of the gravitational field should have the form

 G ik   =   K. T ik   , (2.45)

where K is a constant. In order to be able to specify the quantity Gik more precisely, we will impose several more or less reasonable physical requirements on it :

1. G ik must be a symmetric 2nd order tensor (to be consistent with Tik).
2. G ik is an object describing the gravitational field and thus the geometry of spacetime, so it should be composed of the metric tensor g ik and the curvature tensor Ri klm .
3. Due to the local conservation of energy and momentum of the source (equation (2.6')), the covariant four-divergence G ik ; k = 0.
4. In planar spacetime (where there is no gravity), is G ik = 0.
5. G ik is a linear function of the curvature tensor, so it contains the derivatives of the metric tensor g ik only up to the second order, while these 2nd derivatives are contained linearly.

Requirements 1, 2 and 3 are quite clear and do not raise serious doubts. The least justified seems to be condition No. 5, which is to some extent a condition of simplicity in the sense of Einstein's credo. Its justification is given on the one hand by the fact that 2nd order equations predominate in physics (Cauchy's problem), and on the other hand by the transition to the Newtonian limit. For weak static gravitational field, which according to (2.27) related to the metric tensor potential j relation goo = - (1 + 2 j/c2), the required gravitational equation must pass in to Newton's law of gravitation

 D j   = 4 p G r   , (2.46)

where r is the density of the mass distribution in the source. Since the left-hand side of this equation contains the second derivatives j, it is natural to require that Gik also contain derivatives gik at most to the second order. Due to the weakness of the gravitational field, a coordinate system can be chosen at each point, which is approximately Galilean. In order for equations (2.45) sought in such a system to pass into the Newtonian limit (2.46), Gik must be linear in 2g/xixk with coefficients that are functions only of gik, not their first derivatives.

Condition No.4 looks physically very reasonable. Nevertheless, it was precisely this condition that Einstein temporarily gave up in connection with cosmological problems - see Chapter 5, §5.1 "Basic starting points and principles of cosmology", §5.2 "Einstein's and deSitter's universe. Cosmological constant.". Although recently there are good reasons again for thinking of the cosmological member in global cosmological problems, in our text (except Chapter 5) we will not use the cosmological member yet, it does not affect most of the other issues discussed.
Conditions 1. to 5. allow unambiguous finding of the quantity Gik. In differential geometry it is shown [214], [155] that a quantity satisfying the conditions 1., 2. and 5. must have the form Gik = A.Rik + B.gik R + C.gik , where A, B, C are constants. According to requirement 4., C = 0 (if we omit condition 4, C would be equal to the cosmological constant).
From requirement 3. with respect to Bianchi's identities (2.25), B = -A/2. Thus A. (Rik - (1/2) gik R) = K.Tik. Constant A can be included in the constants K (put down K/A), so we get

 G ik    º    R ik - (1/2) g ik R  =  k. T ik   . (2.47)

The tensor Gik is called the Einstein tensor (it was mentioned in the previous paragraph, equation (2.25b)). The constant k is determined from the condition that for weak fields the general equations change into Newton's law of gravitation. Therefore, in equations (2.47) we make a limit transition to nonrelativistic mechanics: we assume that the gravitational field is weak and the velocities of all bodies are small (both of these requirements are in fact related, because in order for velocities to remain small, the field must be weak). As the source of the gravitational field use incoherent material powder (which is most simplest classical model unstructured mass) with energy and momentum tensor Tik = r.c2.Vi Vk. Assuming low velocities, we can set Vi = (1,0,0,0), so the nonzero component of Tik will be only T°° = r.c2. Einstein's field equations in the form Rik = k.(Tik - (1/2) gik R) (arising from equations (2.47) by a simple algebraic modification) are here then reduced to one equation

 R oo   = (1/2) k. r .c 2   ; (2.48)

the other equations are zero in our approximation. We calculate R°° from relation (2.18), omitting the second order members due to the weakness of the field. We get R°° = ¶Ga oo/xa, which using the relation (2.27) goo » - (1 + 2 j/c2), valid for weak fields, gives R°° » (1/c2).Dj. The field equation (2.48) then reads Dj = (1/2) k.r.c4. This will Poisson equation (2.46) then, when we put

 k  =  8 p G / c 4   ; (2.49)

Then Dj = 4p G.r and its solution will be j = - G. ò (r/R) dV. And this is Newton's law as a special case of Einstein's gravitational equations for a very weak field.
In this way, therefore, Newton's gravitational constant G appeared in the general gravitational equations GTR. We get very important Einstein's equations of the gravitational field in the form

 Einstein's equations of gravitational field :
 .
(2.50a)

By narrowing with the metric tensor gik we get R - 2R = T.8p G/c4, so Einstein's equations can also be written in equivalent form

 R ik    =    (8 p G / c 4 ). (T ik - 1 / 2 g ik T)   , (2.50b)

where T º T i i .

Einstein's equations of the gravitational field generally show that the presence of any matter (with a certain mass and thus energy) curves spacetime with everything present in it. These nonlinear partial second order differential equations for the components of the metric tensor gik determine the spatio-temporal distribution of the metric tensor, i.e. the spatial distribution and temporal evolution of the gravitational field excited by the system of sources described by the energy-momentum tensor Tik. They are also referred to as "geometrodynamic" equations because they describe the dynamics of space-time geometry.
Structure of the energy-momentum tensor Tik, which completely describes the distribution and motion of energy and momentum in a gravitational source, was described in §1.6 "Four-dimensional spacetime and special theory of relativity", part "Energy-momentum tensor". The most important and dominant component of the energy-momentum tensor is T00, representing the density of energy~mass (it is also discussed in §2.6 "Deviation and focusing of geodesics"). For a common substance, this component is given by the mass density r : T00 = r .c2. The other three diagonal components T11, T22, T33 for a common substance (modeled as an ideal liquid) are equal to the pressure p. In such a case, Einstein's equations roughly say that the curvature of spacetime (sum of curvature values in three directions, scalar curvature) is proportional to the mass density r .c2 + three times the pressure p of the substance: R ~ r .c2 + 3.p .
Under normal conditions (on Earth, inside planets, sun and stars) the pressure of matter is negligible with respect to its density (quantity
r .c2) and therefore the pressure almost does not contribute to the curvature of space-time (gravity) - it is given almost exclusively weight of the source. Only inside neutron stars and during gravitational collapse (§4.2 "Final stages of stellar evolution. Gravitational collapse. Black hole formation."), or during cosmological evolution of the universe (§5.1 "Basic principles of cosmology" and the following chapters), pressure is important contributor to the gravitational curvature.

Einstein's equations generally describe how matter curves space-time in its surroundings, which manifests itself as gravity. Significant solutions of Einstein's equations, such as for a spherically symmetric body
(§3.4 "Schwarzschild geometry"), show that the presence of a larger amount of matter~energy - a more massive object - curves space-time in its surroundings more markedly and in a greater range than a light object.

 A material gravitational body curves space-time in its vicinity (example of a spherically symmetric body)

In our terrestrial gravity conditions, it can be clearly illustrated by horizontally tensioned flexible curtain (which represents not curved space without gravity), on which lay the material sphere M. The recess created by the load represents the curvature of the space. When we place a smaller ball m1 (rest) here, will roll into a depression and lands on a larger ball. However, if we give the ball m2 a suitable circumferential speed, it will orbit a heavy sphere, which is the source of curvature, much like a planet orbits a gravitational star or a moon-satellite around a planet. At a higher ball speed m3, its path in the recess only curves and the ball continues to move at a different angle.

The local differential equations of field excitation express the global integral behavior of the field
In field theory, the laws of its excitation by their sources, as well as its internal dynamic properties, are expressed by means of differential equations. These equations have a local character - they relate the local distribution of matter or electric charges with the derivatives of the field intensity or the curvature of spacetime in the same place. However, this does not mean that the sources determine the values of the quantities describing the field only locally or in the immediate vicinity of the field sources. By integrating the local differential equations of the field, we obtain the values of the field intensity or the curvature of spacetime in the whole space surrounding the source system. This happens so often through boundary conditions, as we will see, for example, in §3.4 "
Schwarzschild geometry".

The equations of motion of resources as a consequence of the field equations
Covariant 4-divergence left part G
ik Einstein's equations (2.50) is identically equal to zero. This is due Bianchi identities (2.25) for the curvature tensor, a manifestation of the principle of geometric-topological "boundaries boundary equals zero" - in this case the oriented two-dimensional boundary of the three-dimensional boundary of the four-dimensional region of space [180] [181] (see also §3.1 "Geometrical -topological properties of spacetime"). Therefore, if we take Einstein equations (2.50) as a basis, it follows them automatically Tik;k = 0 - the local law of conservation of energy and momentum of the source *). This law of conservation leads to the equations of motion of the source system described by the corresponding energy-momentum tensor Tik , so in this sense "the equations of motion flow from the Einstein equations of the gravitational field ". Einstein gravitation equation thus determine not only the gravitational field for a given mass distribution, but they also determine the motion of this source.
*) It is an expression of the general regularity between the source and the field it evokes: "the source creates a field around itself so that it preserves itself " - the field of such properties, from which the conservation laws of this source automatically follow. E.g. for the electromagnetic field from the Maxwell equations Fik ; k = 4p j i/c due to the antisymetry of the electromagnetic field tensor Fik, the relation j i ; i = 0 follows identically, which is the equation of current continuity, i.e. the law of conservation of electric charge. Maxwell's equations thus limit the "freedom" of resources only in terms of electrical (not eg mechanical), while Einstein's gravitational equations affect all forms of resource motion.
Unlike all other field theories, the general theory of relativity has the specific property that it is not necessary to postulate separately (enter "from outside") equation the motion of test particles in a given gravitational field - these equations of motion can be obtained as a consequence of the field equations. Indeed, the application of the law of conservation of energy and momentum (2.6) of a test particle, resulting from Einstein's equations (2.50a), leads to its geodetic motion [78] described by equation (2.5). Gravity and mechanics are inextricably linked ("unification of gravity and mechanics") in contrast to Newton's theory, where they are completely independent. Similarly, if we have a free electromagnetic field in a vacuum, then from the Einstein equations (on the right side of which is the energy-momentum tensor of the electromagnetic field) of the gravitational field excited by this electromagnetic field, result the Maxwell's equations Fik ; k = 0 of free electromagnetic field; an interesting interpretation of this fact for unitary field theory is in the geometrodynamics of Wheeler and Misner, see Appendix B, §B.3 "Wheeler's Geodynamics. Gravity and Topology.".

Solving Einstein's equations
Let us now turn to the question of how to determine the structure of spacetime in given physical situations, ie how
to solve Einstein's gravitational equations. The straightforward use of Einstein's equations to determine the evolution of a physical system based on suitable initial conditions - the so-called Cauchy problem - will be outlined in §3.3 "Cauchy problem, causation and horizons". Einstein's gravitational equations (2.50) are tensor nonlinear partial differential equations of second order, so their general solution is very difficult and in most cases can not be found analytically. There are several ways to simplify the solution :

• a ) In the case of weak gravitational fields, it is possible to neglect members of higher orders and obtain the so-called linearized equations of the gravitational field, the solution of which is then an approximate solution of Einstein's equations. We will deal with this first ("Weak gravitational field - linearized theory of gravity").
• b ) If the physical nature of the problem indicates certain symmetries (eg central, axial or planar symmetry, homogeneity and isotropy), the number of independent equations is sharply reduced and it allows in certain cases to determine an accurate analytical solution (this will be discussed in §3.4 -3.6, §5.2 and 5.3). For the centrally symmetric case it is §3.4 "Schwarzschild geometry" and §3.5 "Reissner-Nordström geometry", for the homogeneity and isotropy of the universe in cosmology it is §5.2 "Einstein's and deSitter's universe. Cosmological constant." and especially §5.3 "Friedman 's dynamic models of the universe".
• c ) If we have a certain exact solution of gravitational equations (eg for a suitable symmetric case), we can observe close more general cases using the so-called perturbation analysis. Here we consider a slight change (failure) of the conditions of the original exact solution and we observe how this perturbation manifests and develops. If this perturbation remains small, we can neglect members of higher orders and determine the appropriate more general solution.
• d ) Finally, it is the observation of certain global properties of gravitational equations (geometric-topological properties), without having to know their exact specific solution. An example can be the theorems of singularities (§3.8), where only certain inequalities for the components of the energy-momentum tensor are enough for us to obtain very valuable statements about the evolution of spacetime based on the properties of Einstein's equations (without the need for their specific solution).

Weak gravitational field - linearized theory of gravity
Appropriate approximations of the general theory of relativity - the so-called
linearized theory of gravity - can be used for weak gravitational fields. In the linearized theory of gravity, Einstein's equation and equation of motion are formulated and solved as if spacetime is substantially planar, with only a very small curvature characterizing gravitational phenomena. For sufficiently weak fields and in a suitable (almost inertial) frame of reference, the metric tensor can be expressed in the form

 g ik   =   h ik + h ik ;   | h ik | << 1   , (2.51)

where hik is the Minkowski flat space-time tensor and hik is a small "correction" expressing a weak gravitational field. The components hik are proportional to the Newtonian gravitational potential, as we determined for hoo in §2.4, relation (2.27), and for the other components h we find it below.
We then decompose the nonlinear Einstein's gravitational equations in a series according to the powers hik, and due to their smallness we can neglect their products and powers without much loss of accuracy and leave only the members of the first order; thus we get the sought linearized equations. Linearized connection coefficients for metric (2.51) are Gi kl = (1/2) (hik, l + hil, k - h, ikl) and Ricci's curvature tensor with 1st order precision in hik has the form Rik = (1/2) (hl i, kl + hl k, il - hl ik, l - h, ik), where h º hi i = hik hik; while the indices at hik are "raised" and "triggered" by hik , not the whole gik (contributions from hik they are second order and neglected). Einstein's equations in this approximation are then

hil,kl + hkl,il - hik,ll - h,ik - hik(hlm,lm - h,ll)   =   (16pG/c4) . Tik   .

If we introduce quantities

 y ik   =   h ik   - 1/2 h ik h   , (2.52)

these equations have the form

- yik,ll - hikylm,lm + yil,kl + yik,ll   =   (16pG/c4) . Tik   .

One can show that, without loss of generality it is possible for the variable yik introduce calibration - gauge conditions *) [271]

 y ik , k = 0 (2.53)

analogous to Lorentz gauge conditions Aik, k = 0 in electrodynamics.
*) The procedure of gauge transformation in field theory is generally discussed in §B.6, passage "Calibration - gauge - transformation; calibration - gauge - field".
The linearized Einstein equations
then take on a simple form

 - yik,ll  º  o yik   =   (16p G/c4) . Tik   , (2.54)

where oº ¶2/x2 - (1/c2) 2/t2 is the d'Alembert operator. The general solution of these linearized gravitational equations in Lorentz gauge (2.53) can be expressed in the form of retarded potentials

 (2.55)

similarly to electrodynamics, where R = Ö[a = 1S3 (xa - x'a)2] is the distance from individual places x'a the source system to the investigated point xa (similar to Fig.1.4a). The retarded solution (2.55) shows that changes in the gravitational field propagate at the speed of light. The significance of this solution for gravitational waves will be discussed in §2.7 "Gravitational waves" (where in the passage "How fast is gravity?", general questions of the speed of propagation of changes in the gravitational field will be briefly discussed).

Here we will consider a situation where the gravitational field is excited by a source for which Newtonian physics applies with sufficient accuracy, ie the velocities are small and Too << |Ti a|. Moreover, if found close enough to the source, or if the resource is static (i.e. we located in the "inductive" zone r << c.T, where T is a characteristic time changes in the distribution of mass in the source), retardation can be neglected and solution (2.55 ) has the form

yoo =  - 4j/c4 ,   yoa =  0 ,   yab  = 0   ,

where j(t,xa) = -G.ò(Too(t,x'a)/R)dx'1dx'2dx'3 is the ordinary Newton's potential. In this case, the metric (2.51) is

 ds2   »   - c2(1 + 2j/c2)dt2 + (1- 2j/c2)(dx2+dy2+dz2)   , (2.56a)

i.e.

 gik » / -(1 + 2j/c2) 0 0 0 \ (2.56b) ½ 0 1 - 2j/c2 0 0 ½ ½ 0 0 1 - 2j/c2 0 ½ \ 0 0 0 1 - 2j/c2 /

The expression (2.27) for the time component of the metric tensor for weak fields is thus supplemented by other components. At distances r substantially greater than the dimensions of the source, R»r can be approximately lay down and the metric (2.56a) can be expressed by the total mass M = ò T°° d3x of the source system :

 (2.56c)

This metric is an approximate expression of the Schwarzschild geometry (3.13) derived in §3.4 "Schwarzschild geometry".

Rotating gravity
In the more general case, when the velocities at the source of the gravitational field can be large and the components of the stress tensor T
ab and the momentum density T°a can be comparable to the mass-energy density T°°, a weak gravitational field at a sufficient distance from the source can be approximately determine by distributing the retarded potentials (2.55) into a Taylor series according to the powers of x'/R. In the rest frame of reference with the origin in the center of gravity (ie Pa = òad3x = 0, òxaT°°d3x = 0) then after suitable gauge we get with an accuracy of 1/r :

 (2.56d)

where Ja = òeabgxbTg°d3x is the intrinsic (rotational) angular momentum of the source body.
Gravitational-wave members in metrics we will not be analyzed here, their meaning and properties will be discussed in §2.7 "Gravitational waves". Here we will mention some gravidynamic effects. In polar coordinates with the polar axis oriented in the direction of the angular momentum vector J, the external gravitational field of the rotating body will be described by an approximate metric

 (2.56e)

which is a special case of general Kerr geometry (3.37) for a small angular momentum J (§3.6 "Kerr and Kerr-Newman geometry").

In Newton's theory, the gravitational field is given only by the distribution of matter and does not depend at all on the instantaneous velocity of individual parts of the source or on its possible rotation (unless, however, it leads to changes in the mass distribution). In GTR, however, the rotation of the source leaves characteristic "traces" on the external gravitational field (ie on the space-time metric) in the form of non-diagonal members *) .
*) This metric cannot be diagonalised without the explicit dependence of the components of the metric tensor on the time t .
These off-diagonal members dj.dt lead to the effect, that a certain additional force acts on the bodies (in the geodesic equation the d2j/dt2 ¹ 0 becomes non-zero), which causes entrainment of the local inertial system ( frame dragging) - entrainment of free bodies by a rotating gravitational field in the direction of rotation of the source. It is similar to a sphere rotating in a viscous liquid, which drivess the liquid near its surface into rotation. This phenomenon is called the Lense-Thirring effect according to the authors who first studied it [248]. For common rotating bodies (macroscopic objects, planets, ordinary stars, etc.) the effect of entrainment is very small, but it can be crucial for rotating black holes (in accretion disks and especially in the so-called ergosphere), as will be shown in §4.4 "Rotating and electrically charged Kerr-Newman black holes".

 Hydrodynamic analogy of the influence of the rotation of the source body on the properties of the excited gravitational field. Left: In Newton's theory, the gravitational field of a body is given only by the distribution of matter and does not depend at all on its possible rotation (unless it leads to changes in the distribution of matter). Similarly, a smooth and symmetrical body (such as a sphere) rotating in an ideal liquid without viscosity does not cause the liquid to move around it. Right: In the general theory of relativity, however, the rotation of the source leaves characteristic "traces" on the external gravitational field (on the space-time metric) - local inertial systems are entrained - free bodies are entrained by the rotating gravitational field in the direction of the source rotation.  Similarly, a body rotating in a viscous liquid entrains the liquid near its surface into rotation.

Magnetogravity - gravitoelectromagnetism ?
For these gravidynamic effects, a certain analogy can be traced with magnetism in electrodynamics. In §1.4 "Analogy between gravity and electrostatics" we have shown that the Newtonian gravitational field is, from the formal point of view of the mathematical description, completely analogous to the electric field. More generally, it can be shown that there are formal analogies between the equations of electromagnetism (Maxwell's equations) and special approximations of Einstein's gravitational equations in GTR. This analogy is sometimes referred to as gravitoelectromagnetism - some specific kinematic effects of gravity are analogous to the magnetic effects of moving charges. This is mainly the above-mentioned effect of entraining bodies in the direction of rotation of the source of the gravitational field (Lense-Thirring effect), which is somewhat reminiscent of magnetism. Using special "purpose" transformations, Einstein's gravitational equations can be modified into the form of electromagnetism equations.
From an objective point of view, however, these analogies are only formal, with little physical significance. Phenomena seemingly reminiscent of magnetism are of the second and higher order in comparison with the primary gravitational ("gravistatic") action. Real physical magnetism caused by the interaction of moving "charges" - the sources of field - is not contained in gravity ...
Note 1: For the magnetism of gravity could be considered well-known Coriolis force Fc = -2 m. [v x w], which resemble the magnetic Lorentz force Fm = (1/c) .q. [v x B] applied perpendicularly with electric charge q moving speed v in the magnetic field intensity (induction) B. However, these forces are in fact only a kinematic effect in a rotating frame of reference (angular velocity w), occurring even in classical Newtonian mechanics ...
In the field of electricity, magnetic phenomena are well manifested in normal laboratory conditions because the electric force effects of positive and negative electric charges are annulled on average, so they do not overlap dynamic magnetic effects. A metal wire (conductor) is generally electrically neutral even when a stream of charged electrons passes through it; the resulting magnetic field can thus exert an undisturbed force on the second (also uncharged) conductor with an electric current.
In the field of gravity, however, the attractive "gravistatic" forces add up, so that in this strong static field, the subtle gravidynamic effects are normally completely everlaid by static gravity. Ordinary macroscopic bodies, planets and stars can never move or rotate at high (relativistic) speeds in bound systems, so they can excite strong gravity, but only minimal dynamic effects. Only in compact gravitationally collapsed objects, neutron stars and especially black holes, the rotational motion can be relativistic, as a result of which gravidynamic effects can almost equal gravitational forces and can manifest significant astrophysical effects - §4.4 "
Rotating and electrically charged Kerr-Newman black holes" and §4.8 "Astrophysical significance of black holes".
Note 2:  In this context, we can also mention analogy of electromagnetic and gravitational waves - §2.7 " Gravitational waves ". It is also interesting to note that even in "empty" space without material sources, a source of the global gravitational field appears on the right-hand side of Einstein's equations - Isaacson's tensor of "effective distributed" energy-momentum of gravitational waves (2.76). This is somewhat analogous to how the Maxwell shear current (cf. §1.5, equation (1.34)) excites the magnetic field as well as the current of real electric charges, even in a vacuum without currents for a non-stationary electromagnetic field. However, even this analogy is formal, without a direct connection with magnetism ...
Thus, it cannot be expected, that gravidynamic phenomena could allow, even in principle, some "gravitronics" - the gravitational equivalent of electronics on a cosmic scale!

Possibilities of verification of the effects of rotation
Despite the very small effect of rotation on the excited gravitational field, however, in the 1960s several experts from the University of Stanford
(L.Schiff, G.Pugh, R Conannon, W.Fairbank, F.Everitt, N.Roman) proposed, but for a long time not performed *), highly sensitive experiments that could demonstrate this effect and measure even in the weak gravitational field of the Earth by accurately monitoring changes in the direction of the rotational axis of the gyroscope - precession, orbiting in polar orbit. The rotation axis of such flywheel will change during orbit due to two GTR effects :
a) The gyroscope orbits in the curved spacetime of the Earth's gravitational field - geodetic effect, connections, change of vector direction during parallel transmission - see §2.4. This effect geodetic precession should be dominant and lead to a change in the axis of rotation in the direction of movement of the probe in orbit by about 6''/year.
b) Due to the entrainment by the Earth's rotational angular momentum, the rotational axis of the gyroscope should show a slight "anomalous" precession - twisting in the Earth's direction of rotation (for a polar orbit perpendicular to the plane of orbit) at an angular velocity proportional to the Earth's angular momentum and indirectly proportional to the cube of the orbital radius. The expected value of this anomalous Lense-Thirring precession is only a few hundredths of an angular second per year (approx. 0.04''/year for the proposed orbit approx. 600 km above the Earth).
Both of these deviations a), b) are perpendicular to each other. To objectively demonstrate the effect, it is necessary to compare the rotational axes of at least two gyroscopes rotating in opposite directions.
Another possibility is an accurate analysis of the orbit of special satellites **) and an analysis of the orbital dynamics of tight binary pulsars ***).
*) Gravity Probe B
This experiment was at the project stage for a long time. After overcoming a number of technical difficulties and long-term testing (team led by F.Everitt), the Gravity Probe B satellite was launched on April 20, 2004 , in o polar orbit at a height of 640 km, which carried 4 precision gyroscopes with a diameter of 3.8 cm, rotating at a speed of 10,000 rpm. Two rotate in one direction, two in the opposite. Their surface is coated with a superconducting layer of niobium. During its rotation, this superconducting layer generates a magnetic field, which is monitored by electromagnetic induction in the so-called SQUID (Superconducting QUantum Interference Device) electronic device, which with high sensitivity (10
-4'') detects the deviation of the axis of rotation of the gyroscope. The case with the gyroscopes is connected to a small pointing telescope focused on the star IM Pegasi, which ensures the reference direction of the rotational axes of the gyroscopes. To increase the sensitivity of the measurement (signal-to-noise ratio), the entire measuring system is built inside a Dewar vessel with 2400 liters of liquid helium, which cools the measuring box to 1.8 °K during an operating time of more than 1.5 year. Data transmitted by the satellite were collected until February 2006.

The high sensitivity of the installed equipment gave hope that during the planned approximately 18 months of measurements, the peculiar very fine dynamic-kinematic effects of the general theory of relativity will be verified with high accuracy. During operation the Gravity Probe B , encoutered difficulties. Disorders from the solar protuberances caused disturbing deviations in the positions of the rotational axes of the gyroscopes, and other noises appeared. From the native data it was not possible to accurately prove the analyzed effects (especially the Lense-Thirring effect). The whole two years took a complicated processing of results - data filtering and removal of various types of faults and noise. The resulting measured values after this treatment were finally: geodetic precession (6601.1 ± 18.3) x10-3 ''/year, compared to the GTR predicted value 6606.1x10-3 ''/year; Lense-Thirring precession (37.2 ± 7.2) x10-3 ''/year, compared to the GTR predicted value of 39.2x10-3 ''/year. However, the excessive complexity of data processing somewhat reduced the validity of the results ..?..
**) Satellite orbit analysis
Both of these subtle "gravidynamic" GTR effects can also be detected with comparable accuracy by measuring the orbit of the Laser Geodynamics Satellite (LAGEOS). This satellite consists of a metal sphere with a diameter of 60 cm, equipped with 426 passive laser mirror reflectors (so-called retro-reflectors). It orbits the Earth in low orbit at an altitude of 5,900 km. Measurements are made using reflections of laser pulses from many ground stations - the time intervals between sending the beam and receiving the reflected pulse are evaluated, which allows you to measure instantaneous distances very accurately. These measurements of the exact positions of the satellite relative to different places on the earth's surface make it possible to measure the shape of the earth's geoid and to study the movements of tectonic plates and terrestrial continents.
However, by measuring the differences between the orbits of many successive orbits, the contribution of the Lense-Thirring effect - the relativistic precession caused by the rotational angular momentum of the Earth's - can also be determined here. This gravitomagnetic action will change the point of the closest approach of the satellite to the Earth (perigee) by about 3 meters / year. However, there are a number of disturbing influences such as the pressure of solar radiation, the gravitational action of the Moon, tides, geological inhomogeneities, ....
***) Dynamics of binary pulsars
Another possibility is the observation of binary pulsars, here especially PSR J0737 + 3039 and PSR J1757-1854. These two orbiting neutron stars are rapidly rotating massive compact objects at a short distance from each other in a massive gravitational field, so that all generally relativistic effects are much stronger here than in the faint gravitational field around Earth. Measurement of the periastron shift in relation to the moment of inertia of the double pulsar can assess the contribution of the Lense-Thirring effect (gravito-magnetic spin-orbital precession of the periaston). ....... .......

Three aspects of space-time curvature
In the general theory of relativity, thus there are three basic ways of space-time curvature due to different mass-energy distribution and its temporal dynamics :
× Space curvature
×
Time deformation
×
Rotational motion of space

Do material bodies create space and time ?
The cause (essence) of gravity in the general theory of relativity is the geometric properties - curvature - of spacetime. And this curvature of spacetime is generated by material bodies. The implications [material bodies -> gravity -> spacetime] are sometimes shortened and simplified by the formulation that "bodies create spacetime", "spacetime is an integral part of bodies". From there, it is only a step to the claim "without material bodies there would be no space and time", which is sometimes mentioned in the interpretation of the general theory of relativity. However, this claim is already misleading! Einstein's equations (2.50) have a planar spacetime in empty space without the presence of material bodies as a simple solution of Minkowski flat spacetime, in which the test particles move according to the laws of a special theory of relativity. And at the early stages of the evolution of the universe, when there were no bodies
(and initially no particles or particular fields), existed here a rapidly expanding space from which our later universe came into being (§5.4 "Standard Cosmological Model. The Big Bang. Shaping the Structure of the Universe."). Only at the very origin of the universe ("big bang"), the standard cosmological model does assume, that space and time were also created here, which "did not exist" before... - §5.4, passage "The beginning of time?".
Thus, in interpreting and discussing the general theory of relativity, we would recommend not using the misleading statement that "material bodies create spacetime", but only a physically justified formulation that "material bodies curve spacetime around them". And not only material bodies, but also physical fields...
Note: From a philosophical-gnoseologic point of view, however, it is possible to discuss whether without the existence of particles or bodies the terms space and time have real meaning..?.. We should not measure "by what" neither "nothing real" ...

Variational Derivation of Einstein's Equations of the Gravitational Field
The culmination of the mathematical structure of
every physical theory is the formulation of its laws using Hamilton's variational principle of least action [165], [166]. This approach consists in constructing a Lagrange function (Lagrangian) L for the investigated physical system, such that its integral - action - is extreme for real motion (trajectory, evolution), ie the variation of the action is zero. From this then follows the basic Lagrange equations of motion of a given physical systems. The main benefit of the variational method is that it helps to clarify some structural laws of the theory, such as the relationship between the principles of symmetry and conservation laws, the uniqueness of equations of motion and the like.

For the system [source bodies + excited field], the total quantity of the action can be considered as the sum of three terms: S = Sm + Sf + Smf , where Sm is the action of the source bodies (particles), Sf is the action of the field itself and Smf expresses the mutual interaction of particles with the field. For a physical field in the theory of relativity, the action is given by the integral of the Lagrangian Lf , which is a function of the field potentials and their derivatives over the investigated 4-dimensional spacetime region W: Sf = ò Lf (j, j, i) dW , d W = dtdxdydz. In relativistic physics, it is also advantageous to write the quantities Sm and Smf in the form of integrals (Sm = ò Lm dW over a 4-dimensional space-time region), ie to use the approach of continuum physics. When determining the possible shapes of the Lagrangian, resp. integral of the action, is based on certain general physical requirements for the resulting field equations, such as relativistic invariance (general covariance), resp. linearity (superposition principle), symmetry, the degree and the like. From the group of possible Lagrangians thus defined, it is then often selected according to "aesthetic" criteria of simplicity.
For example, for an electromagnetic field, the described vectors of electric and magnetic field intensities, it follows from the requirement of linearity of field equations (superposition principle) that the Lagrangian must be a quadratic function of the field intensities; the simplest scalar (relativistic invariance) of these properties, created from components of electric and magnetic intensities, is the summation product Fik Fik of the electromagnetic field tensor component (1.114), so the integral of the action for the electromagnetic field has the form Se = k. ò Fik Fik dW. The total Lagrangian of the charged particle system and the electromagnetic field is then [166] L = -rmc2 + (1/16pc) FikFik + (1/c2)Aij i . From the variational principle dS = d òL dW = 0, then we can get both the equations of motion of particles in the electromagnetic field (if the electromagnetic field is considered as a given and varying the particle trajectories), and Maxwell's equations of electromagnetic field (while varying the 4-potential components at the specified distribution and movement of charges).
For the gravitational field in GTR, when the investigated physical system consists of a system of source (material) bodies and an excited gravitational field, the total action will be given by the sum of S = Sm + Sg, where Sm= òLm(qa,qa,i)Ö(-g) dW is the integral of the action of the source part described by the generalized coordinates qa (a = 1,2, ..., N is serial number of the generalized coordinate) and Sg= òLg(gik)Ö(-g) dW is the action of the gravitational field itself described by the components of the metric tensor gik. The factor Ö(-g) comes from curvilinear coordinates - it guarantees that the product Ö(-g) dW behaves as an invariant when integrated over a 4-dimensional volume. The interaction term is not here, because it is implicitly contained in the term Sm (by describing the source by physical laws in the curvilinear coordinates of curved spacetime, its interaction with the gravitational field is also expressed). The Lagrangian Lg must be a scalar function of the metric tensor gik and its derivative, so that the variations of the resulting field equation contain derivatives not higher than the 1st order. The simples such scalar is a scalar curvature R spacetime (2.24); if we also admit the presence of the constant term C = const., the Lagrangian of the gravitational field could have the shape Lg = k.R + C. In order to obtain the Einstein equation directly with the usual form of constant factors, writes this Lagrangian in the form Lg = (c3/8p G) (R - 2 .L ), where G is Newton's gravitational constant and L is the cosmological s constant. The variation principle dS = d (Sg + Sm) = 0 at complete variation then after adjustments gives the relation [166]

is a tensor of energy - momentum of a source. The variation of the metric gik gives Einstein's equations of the gravitational field

Rik - (1/2) gikR - L.gik   =   (8pG/c4) Tik   ,

while the variation of the source variables qa leads to the equations of motion of the source system (non-gravitational fields):

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Vojtech Ullmann