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Chapter 2
GENERAL THEORY OF
RELATIVITY
 PHYSICS OF GRAVITY
2.1. Acceleration and gravity from the point
of view of special theory of relativity
2.2. Versatility
 a basic property and the key to understanding the nature of
gravity
2.3. The
local principle of equivalence and its consequences
2.4. Physical
laws in curved spacetime
2.5. Einstein's equations of the gravitational field
2.6. Deviation
and focus of geodesics
2.7. Gravitational
waves
2.8. Specific
properties of gravitational energy
2.9.Geometrodynamic system of units
2.10. Experimental
verification of the theory of relativity and gravity
2.5. Einstein's equations of the gravitational field
If we are building a field theory, we are basically interested in two areas of questions :
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 method is the most comprehensive heuristic. Some
other procedures leading to Einstein's equations [111], [ 1 66],
[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,
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
nongravitational 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 also generate with electromagnetic waves (light, radio
waves, Xrays), globally even 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
nongravitational physics, where the mass of the body is given by
the volume integral M = ò T °° dV. However, in order to
accurately determine the energymomentum 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 energymomentum 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 = w ^{2} .r ^{3} / 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 most objective method of determining the mass pf some
body is to measure how strong the gravitational field around this
body excites. In the following, we will understand the mass of a
system 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 *) :

(2.43) 
E.g. for the
electromagnetic field the object describing the field is the
tensor of the intensity of the electromagnetic field F ^{ik} , resp. its fourdivergences F ^{ik }_{; k} . The source of the field is electric
charges and the object describing the source is a fourcurrent 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 us seek of 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 :

(2.44) 
whereas for weak
gravitational fields it must give the correct
Newtonian limit, ie the Poisson equation Dj = 4 pr G with the
potential j related to the metric tensor 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
energymomentum tensor T _{ik}
(introduced in §1.6 "Four 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 G _{ik} more precisely , we will impose several more or less reasonable physical requirements on it :
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 g_{oo} =  (1 + 2 j / c ^{2} ), the required gravitational equation must pass in to Newton's law of gravitation
Dj = 4 p G r ,  (2.46) 
where r is the density of the mass distribution in
the source. Since the lefthand side of this equation contains
the second derivatives j , it is natural to require that G
_{ik }also contain derivatives g_{ik }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), G _{ik }must be
linear in ¶ ^{2} g / ¶ x ^{i} ¶ x ^{k} with coefficients that are functions only of g _{ik} , 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 the recently again some grounds for thinking
cosmological element in the global cosmological problems in our
text (except Section 5) currently cosmological
member will
not be used on the most discussed problems are not affected.
^{ }Conditions 1 to 5 allow unambiguous finding of
the quantity G _{ik} . In differential geometry it is
shown [214], [155] that a quantity satisfying the conditions 1.,
2. and 5. must have the form G _{ik} = AR _{ik} + B.g _{ik} R + Cg _{ik} , 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. (R _{ik}  (1/2) g _{ik} R) = KT _{ik} . Constant A can be included in the constants
K (refer to Kº K
/ A), so we get^{ }
G _{ik} º R _{ik}  (1/2) g _{ik} R = k. T _{ik} .  (2.47) 
The tensor G _{ik} 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 T ^{ik} = r .c ^{2} V ^{and} V ^{k} . Assuming low velocities, we can set V ^{i} = (1,0,0,0), so the nonzero component T ^{ik} will be only T °° = r .c ^{2} . Einstein's field equations in the form R ^{ik} = k. ( T ^{ik}  (1/2) g ^{ik} T) (arising from equations (2.47) by a simple algebraic modification) are 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 °° = ¶G ^{a }_{oo} / ¶ x ^{a} , which using the relation (2.27) g _{oo} »  (1 + 2 j / c ^{2} ), valid for weak fields, gives R °° » (1 / c ^{2} ). Dj . The field equation (2.48) then reads Dj = (1/2) k. r .c ^{4} . This will Poisson equation (2.46) then , when we put
k = 8 p G / c ^{4} ;  (2.49) 
Then Dj = 4 p 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.
^{ }Thus, 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 g _{ik} we get R  2R = T.8 p G / c ^{4} , 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 in it. These nonlinear partial second order differential
equations for
the components of the metric tensor g _{ik} determine the spatiotemporal
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 energymomentum tensor T _{ik} . They are also referred to as " geometrodynamic " equations because they describe the dynamics of spacetime geometry.
^{ }Structure ^{ }of
the energymomentum tensor T _{ik} , 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 " Energymomentum tensor ". The most important and dominant component of
the energymomentum tensor is T _{00} , representing the energy density ~ 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 : T _{00} = r .c ^{2} . The other three
diagonal components T _{11} , T_{22} , T _{33} are equal to the pressure p for a
common substance (modeled as an ideal liquid) . 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 .c ^{2} + three times the pressure p of the substance: R ~ r .c ^{2} + 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 .c ^{2} ) and therefore the pressure
almost does not contribute to the curvature of
spacetime (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 and
principles of cosmology "
and the following chapters) , pressure is
important contributor to the gravitational curvature. ^{
}Einstein's equations generally describe how matter
curves spacetime 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
spacetime 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 m _{1} (rest) here, it rolls into a depression and lands on a larger ball. However, if we give the ball m _{2 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 moonsatellite around a planet. At a higher ball speed m _{3} , its path in the recess only curves and the ball continues to move at a different angle.
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 4divergence 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
geometrictopological "boundaries boundary equals zero"
 in this case oriented twodimensional boundarydimensional
fourdimensional region of space boundaries [180] [181] (see also §3.1 " Geometrical topological
properties of spacetime
") . Taking
therefore Einstein equation (2.50) as a basis, it follows them
automatically T ^{also }_{k} = 0  local Act ofn
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 energymomentum tensor T _{ik} , so in this sense "the equations of motion flow from the Einstein
equations of the gravitational field ". Einstein gravitation
equation determine not only the gravitational field of the mass
distribution, but also determines the movement of the 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 "
 an array of such properties from which the laws
of conservation of this
source automatically follow . E.g. for the electromagnetic field
from the Maxwell equations F ^{ik }_{;
k} = 4 p j ^{i} / c due
to the antisymetry of the electromagnetic field tensor F ^{ik, the}
relation j ^{i }_{; i} = 0 follows identically , which is the
equation of current continuity or 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 energymomentum tensor of the
electromagnetic field) of the gravitational field excited by this
electromagnetic field, result the Maxwell's equations F ^{ik }_{; 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 socalled 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 :
Weak gravitational field  linearized theory
of gravity
Appropriate approximations of the general theory of relativity 
the socalled linearized theory of
gravity 
can be used for weak gravitational fields . The linearized gravity theory the Einstein equation and the equation of
motion formulate and solve 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 h _{ik} is the Minkowski flat spacetime
tensor and h _{ik} is a small
"correction" expressing a weak gravitational field. The
components h _{ik} are proportional to the
Newtonian gravitational potential, as we determined for h _{oo} 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 h _{ik,} 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 G ^{i }_{kl} = (1/2)^{ }(h
^{i }_{k, l} + h ^{i }_{l, k}  h ^{,
i }_{kl} ) and Ricci's curvature tensor with 1st
order precision in h _{ik} has the form R _{ik} = (1/2) (h ^{l }_{i,
kl} + h ^{l }_{k, il}  h ^{l }_{ik,
l}  h _{, ik} ), where h º h ^{i }_{i} = h ^{ik} h _{ik} ; while the indices at h _{ik} are "raised" and
"triggered" by h _{ik} , not the whole g _{ik} (contributions from h _{ik}they are second order and neglected).
Einstein's equations in this approximation are then
h _{il, k }^{l} + h _{kl, i }^{l}  h _{ik, l }^{l}  h _{, ik}  h _{ik} ( h _{lm }^{, lm}  h _{, l }^{l} ) = (16 p G / c ^{4} ). T _{ik} .
If we introduce quantities
y _{ik} = h _{ik}  ^{1} / _{2} h _{ik} h,  (2.52) 
these equations have the form
 y _{ik, l }^{l}  h _{ik} y _{lm }^{, lm} + y _{il, k }^{l} + y _{ik, l }^{l} = (16 p G / c ^{4} ). T _{ik} .
One can show that, without loss of generality it is possible for the variable y_{ik} introduce calibration conditions [271]
y ^{ik }_{, k} = 0  (2.53) 
analogous to Lorentz calibration conditions A ^{ik }_{, k} = 0 in electrodynamics. The linearized Einstein equations then take on a simple form
 y _{ik, l }^{l} º o y _{ik} = (16 p G / c ^{4} ). T _{ik} ,  (2.54) 
where oº ¶ ^{2} / ¶ x ^{2}  (1 / c ^{2} ) ¶ ^{2} / ¶ t ^{2} is the d'Alembert operator . The general solution of these linearized gravitational equations in Lorentz calibration (2.53) can be expressed in the form of retarded potentials
(2.55) 
similarly to electrodynamics, where R = Ö [ _{a = 1} S ^{3} (x ^{a}  x ' ^{a} ) ^{2} ] is the distance from individual places x' ^{and} the source system to the investigated point x ^{a} (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 T _{oo} <<  T _{i }_{a} . Moreover, if found close enough to the source, or if the resource is static (i.e. located in the "inductive t Participative zone" r << CT, where T is a characteristic time changes in the distribution of mass in the source) can be neglected and the retardation solution (2.55 ) has the shape
y _{oo} =  4 j / c ^{4} , y _{o }_{a} = 0 , y _{ab} = 0,
where j (t, x ^{a} ) = G. ò (T _{oo} (t, x ' ^{a} ) / R) dx' ^{1} dx ' ^{2} dx' ^{3} is the ordinary Newton's potential. In this case, the metric (2.51) is
ds ^{2} »  c ^{2} (1 + 2 j / c ^{2} ) dt ^{2} + (1 2 j / c ^{2} ) (dx ^{2} + dy ^{2} + dz ^{2} ),  (2.56a) 
i.e.

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, approximately R » r can be laid down (2.56a) can be expressed by the total mass M = ò T °° d ^{3} x 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 ° ^{and} can be comparable to the massenergy
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 P ^{a} = ò T ° ^{and} d ^{3} x = 0, ò x ^{and} T °° d ^{3} x = 0) then after suitable
calibration we get with an accuracy of 1 /
r:
(2.56d) 
where J _{a} = ò e _{abg} x ^{b} T ^{g} ° d ^{3} x is the
intrinsic (rotational)
momentum of the source body.
^{ }We will not consider gravitationalwave
terms in metrics 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 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 momentum J (§3.6
" Kerr and KerrNewman 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, of course, this leads to changes
in the distribution of matter). In GTR, however, the rotation of the source leaves characteristic
"traces" in the form of nondiagonal terms *) on the
external gravitational field (ie on the spacetime metric).
*) This metric cannot be
diagonalised without the explicit dependence of the components of
the metric tensor on the time t .
^{ }These offdiagonal
members d j. dt lead to the fact that the body exerts a
certain additional force acts (in the geodesic equation to d ^{2} j / dt ^{2} ¹ 0 becomes nonzero), which causes entrainment of the local inertial system ( frame dragging )  entrainment loose bodies rotating the
gravitational field in the direction of rotation of the source.
It is similar to a sphere rotating in a viscous liquid entraining
a liquid near its surface. This phenomenon is called the LenseThirring effect according to the authors who
first studied it [ 248 ]. For common rotating bodies
(macroscopic objects, planets, shining stars, etc.) the effect
of entrainment is very small, but it can be crucial for rotating black holes in the socalled ergosphere , as will be shown in §4.4 " Rotating and electrically charged KerrNewman
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 spacetime 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. 
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 called gravitoelectromagnetism  some specific kinematic
effects of gravity are analogous to the magnetic effects of
moving charges. This is mainly the abovementioned effect of entraining bodies in the direction of rotation of
the source of
the gravitational field ( LenseThirring
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  the gravity is not included ...
Note 1: For the magnetism
of gravity could be considered wellknown Coriolis
force F _{c} = 2 m. [ v
´ w ] , which resemble
the magnetic Lorentz force F _{m} = (1 / c) .q. [ v
´ B
] applied perpendicularly with electric charge q moving speed of 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 completelycovered 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 KerrNewman black holes "and §4.8" Astrophysical significance of
black holes ".
Note 2:^{ }
In this context, we can also mentionanalogy 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 righthand side of Einstein's equations  Isaacson's
tensor of " effective distributed " energymomentum
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 nonstationary electromagnetic field .
However, even this analogy is formal, without a direct connection
with magnetism ...
It is not possible^{ }therefore,
to expect 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 slight 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 not yet
implemented *) , 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. Axis
of rotation of the flywheel during such a revolution will change
due to two effects GTR :
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 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 momentum and indirectly proportional to the cube radius.
The expected value of this anomalous LenseThirring
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 longterm
testing (team led by F.Everitt), the Gravity Probe B
satellite was launched on April 20, 2004 , carrying in polar
orbit at a height of 640 km carrying 4 precision gyroscopes with
a diameter of 3.8 cm . 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 socalled 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 (signaltonoise
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 dynamickinematic
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 LenseThirring 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; LenseThirring 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 (socalled
retroreflectors). 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 LenseThirring effect  the relativistic
precession caused by the rotational momentum of the Earth's
momentum  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 J17571854. 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 LenseThirring effect (gravitomagnetic
spinorbital precession of the periaston). ....... .......
Three
aspects of spacetime curvature
In the general theory of relativity, three basic methods of
spacetime curvature are applied due to different
massenergy distribution and its temporal dynamics :
× Space
curvature
× Time deformation
× Rotational motion of space
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
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 = S _{m} + S _{f} + S _{mf} , where S _{m} is the action of the source
bodies (particles), S _{f} is the action of the field
itself and S _{mf} 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 L _{f} , which is a function of the
field potentials and their derivatives over the investigated
4dimensional spacetime region W : S _{f}
= ò L _{f} ( j , j _{, i} ) d W ,, d W
= dtdxdydz. In
relativistic physics, it is also advantageous to write the
quantities S _{m} and S _{mf} in the form of integrals (S _{m}
= ò L _{m} d W over
a 4dimensional spacetime 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 F ^{ik} F _{ik} of the electromagnetic field
tensor component (1.114), so the integral of the action for the
electromagnetic field has the form S _{e} = k. ò F ^{ik} F_{ik} d W .
The total Lagrangian of the charged particle system and the
electromagnetic field is then [166] L =  r _{m} c^{2} + (1/16 p c) F
^{ik} F _{ik} + (1 / c ^{2} ) A _{i} j ^{i} . From the variational principle
dS = d òL dW = 0, then we can get both zhe
equations of motion of particles in the electromagnetic field (if
the electromagnetic field is considered as a given and variances
particle trajectories) partly Maxwell's electromagnetic field
equations (while varying the components 4pote n tial 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 = S _{m}^{ }+ S _{g} , where S _{m} = ò L _{m} (q ^{a} , q ^{a
}_{, i}
) Ö ( g) d W
is the integral of
the action of the source part described by the generalized
coordinates q ^{a} (a = 1,2, ..., N is serial
number of the generalized coordinate) a S _{g} = ò L _{g} (g ^{ik} ) Ö ( g)
d W is the action of the gravitational field
itself described by the components of the metric tensor g ^{ik} . The factor Ö ( g)
comes from curvilinear coordinates  it guarantees that the
product Ö ( g) d W behaves as an invariant when integrated
over a 4dimensional volume. The interaction term is not here,
because it is implicitly contained in the term S _{m}
(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 L _{g}
must be a scalar function of the metric tensor g ^{ik} 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 L _{g} = K + C. In order to obtain the
Einstein equation directly with the usual form of constant
factors, writes this Lagrangian in the form L _{g}
= (c ^{3} /8 p G) (R  2 .L )
where G is Newton's gravitational constant and L is the cosmological s
constant. The
variation principle d S = d (S _{g}
+ S _{m} ) = 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 g ^{ik} gives Einstein's equations of the gravitational field
R _{ik}  (1/2) g _{ik} R  L .g _{ik} = (8 p G / c ^{4} ) T _{ik} ,
while the variation of the source variables q^{a} leads to the equations of motion of the source system (nongravitational fields):
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