Gravitational fourpotential is a fourvector function (4vector), by
which the properties of gravitational field are determined in the Lorentzinvariant theory of gravitation, ^{[1]} as well as in the covariant theory of gravitation. ^{[2]} The gravitational 4potential includes the
scalar and vector potentials of gravitational field. During the gauge
transformation the potentials of the gravitational field can change their form
so that noncoinciding 4potentials with different dependence on the
coordinates and time can correspond to the same gravitational field.
Contents

Gravitational 4potential, like any 4vector, consists of
the scalar and vector parts, which in sum give 4 components:
The timelike component of the 4potential is the scalar
potential , divided by the speed of gravitation . The spacelike component of the 4potential
is represented by the vector potential of gravitational field , which has three components.
Definition of the 4potential in
the covariant representation with a lower index is preferred to the
contravariant representation (with an upper index), as it makes the solution of
equations easier.
In the transition from one reference frame to another the
4potential is transformed in accordance with the axioms of the metric theory of relativity. In case of Minkowski
space of the special theory of relativity transformations of the 4potential
are performed from one inertial frame to another using Lorentz transformations.
In the international system of units SI the gravitational
4potential is
measured in m/s, in the system of physical units CGS – in cm/s.
By the gravitational 4potential the gravitational tensor is determined, for
this purpose a 4rotor is used:
The antisymmetric tensor
contains only 6 components, three
of which are associated with the vector of gravitational
field strength , and the other three components – with the vector of the gravitational torsion field . In Cartesian coordinates these vectors
are obtained as follows:
From the latter relation we see that the torsion field
depends only on the vector potential. At the same time, contribution into the
gravitational field strength is made not only by the gradient of the scalar
potential, but also by the rate of change in time of the vector potential.
The most
convenient is the gauge, when the 4divergence of the 4potential is zero:
In
the special relativity the covariant derivative becomes a
partial derivative . This allows us
to represent the gauge condition explicitly as follows:
In the
Lorentzinvariant theory of gravitation the Heaviside equations for
gravitational field are written in the fourdimensional form:
It can be shown
that the gauge condition of the 4potential (2) follows from the definition of
the gravitational tensor (1) and the field equations (3) and (4). If we apply
the partial derivative to (4),
then its action in the first two terms in (4) on the tensor can be
considered with the help of (3). For the third term in (4) we obtain the
relation , where is the 4d'Alembertian operator:
here
the Laplace operator is applied, which in Cartesian coordinates has
the form
Substituting in
(4) with its expression
according to (1), from (4) as a
simplest option we obtain the wave equation for the 4potential, the source of which is the mass
4current : ^{[3]}
where is
the gravitational constant.
On the other hand, if in (3) we substitute with its
expression according to (1), we obtain again the wave equation (5) for the
4potential, but only given the gauge conditions of the 4potential (2) is met. Thus, due to the symmetry of
fields, this calibration can simplify the field equations.
Note that if we
take the partial derivative of both
sides in (3), then taking into account (1) the left side is equal to zero. Then
from the equality of the right side to zero the continuity
equation follows for the mass 4current:
If we subtract
from the gravitational 4potential the gauge
4vector of the form , depending on a
scalar gauge function , then provided
that the function satisfies
the wave equation
for the new
4potential the gauge condition (2) will remain in force,
and the gravitational tensor according to (1) will not change its form. Thus,
the Lorentzinvariant theory of gravitation and built upon it the covariant
theory of gravitation are gauge theories.
In the covariant theory of
gravitation the Heaviside equations (3) and (4) for gravitational field are
generalized for the curved spacetime and are written in such form: ^{[2]}
The continuity
equation for the mass 4current becomes dependent on Ricci tensor : ^{ [4]}
Wave equation
instead of (5) is as follows:
In the curved
spacetime in the equation we should take into account mixing of the vector
components. In particular, the scalar potential of gravitational field becomes
the function not only of the mass density , but also of the
mass current density , where is the
velocity of the matter.
In the special theory
of relativity, the Christoffel symbols are zero, and then the solution of the wave equation can be simplified
so that it can be represented as follows: ^{[2]}
where the gravitational 4potential at
the time point at the point of the space determined by the radius vector , is found by integrating over the volume,
containing the mass 4current (or 4vector of mass current density) . In this case integration over the volume
is carried out for an earlier point in time
,
where is
the radius vector that specifies the location of the mass 4current at the
earlier time, is the speed of gravitation.
From the given solution
for the timelike components of 4vectors we can see that the scalar potential
depends on the mass density of a certain moving particle at the earlier point
in time and on the distance from this particle to the point where the potential
is measured. In turn, the vector potential also depends on the speed of the
particle at the earlier point in time. The presence of the volume integral
implies that for the potentials of gravitational field the superposition
principle holds, and for calculating the total 4potential we should take into
account all sources of the field.
The 4potential of the proper gravitational field of a single solid point particle can be obtained in another way – by
multiplying the scalar gravitational potential
around this particle, calculated in the
reference frame comoving with this particle, by the 4velocity of the point
particle:
For the observer, relative to which the point particle is
moving, according to the Lorentz transformations the scalar potential changes
due to the motion of the particle: , and also the vector potential appears, which
is equal to . This
gives the ordinary definition of the gravitational 4potential in the form
Indeed, the 4potential of any vector field for a single
particle, inside which the vector potentials of the fields are absent, can be
represented as follows: ^{[5]}^{ [6]}
where
for electromagnetic field and for other fields, and are the mass density and
accordingly charge density in comoving reference frame, is the field energy density of
the particle, is the covariant fourvelocity.
For
gravitational field ,
, and assuming equality of the speed of
light and the speed of gravity , we arrive to formulas for the 4
potential , shown above.
In a system of a set of point particles composing
material bodies, in order to find the total 4potential we should add
4potentials of all point particles, taking into account the differences in
their 4velocities and their different location in space. As a result the total
vector potential of the system of particles only indirectly reflects the total
scalar potential of the given system of particles, in contrast to the direct
relation between the scalar and vector potentials of an individual point
particle. In the case of calculating the total 4potential of a massive solid
body, given the different distances from the parts of the body to the point
where the 4potential is defined, we obtain the gravitational Liénard–Wiechert
potential. ^{[7]}^{ [8]}
The
gravitational 4potential is part of the Lagrangian for the matter in gravitational
and electromagnetic fields that allows us to write the corresponding action
function: ^{[9]} ^{[4]}
where is Lagrangian, is the time differential of the
reference frame used, is a certain coefficient, is the scalar curvature, is the cosmological constant,
which characterizes the energy density of the considered system as a whole and
therefore is the function of the system, is the speed of light as a
measure of the propagation speed of electromagnetic and gravitational
interactions, the electromagnetic 4potential
where is the scalar potential, and is the vector potential, – electric fourcurrent, is the electric constant, – electromagnetic tensor, – acceleration tensor, and are some constants, is fourpotential of pressure field, –
pressure field tensor, is the invariant 4volume expressed through
the differential of the time coordinate , through the product of differentials of the space
coordinates, and through the square root of the determinant of the metric tensor taken with a
negative sign.
In the action integral the gravitational 4potential is
present within the invariant , and as well as part of the gravitational
field tensor and its invariant . In the first case, the 4potential
determines the function of the binding energy of the substance and the field,
and in the second case it determines the energy function of the field as an
independent object. The variation of the action function leads to the determination
of the gravitational stressenergy tensor
of the gravitational field, sets the gravitational field equations (3) and (4),
the equation of the substance motion in the field and the expression for the
gravitational fourforce.
In classical mechanics instead of the total 4potential its
scalar component is used in the form of the gravitational potential. This
allows us to find the potential gravitational energy of bodies and the
equations of their motion. In order to calculate the scalar gravitational
potential the Poisson equation is used of the form: , where is
the Laplace operator, is
the volume density of the mass distribution at the considered point. However,
the expressions obtained for the potential, forces and energies are not Lorentz
covariant, that is, the problem arises during the translation of the results
from one inertial frame to another.
The gravitational 4potential practically is not
considered in the general relativity (GR). This is due to the fact that in GR
the gravitational field is considered identical to the metric field, and the
components of the metric tensor are used as the gravitational potentials, and
the Christoffel symbols are used instead of the field strength. In the weak
field the relation can be established between the component of
the metric tensor of the spacetime and the gravitational scalar potential in
classical mechanics: , where is
the speed of light. The vector potential of gravitational field that is used in the Lorentzinvariant theory
of gravitation can also be expressed in terms of the components of the metric
tensor of GR.
On the other hand, in the axiomatic construction of the Lorentzinvariant theory of gravitation
(LITG) it is the 4potential that represents the gravitational field,
while as for the substance it is represented by the mass 4current . The fifth axiom of LITG states that
d'Alembertian of the 4potential equals the mass 4current with a corresponding
constant factor. ^{[2]} It is sufficient to derive all of the
relations of the Lorentzinvariant theory of gravitation. The axiomatics of
LITG is the same for the covariant theory
of gravitation (CTG), since CTG is the generalization of LITG to the curved
spacetime, in which the metric tensor is dependent on the time and coordinates.
The gravitational 4potential as well as the
electromagnetic 4potential, acting on test bodies, influences the rate of time
in these bodies. ^{[10]} This leads to the fact that
the same processes that occur in bodies, which are located in different
4potentials, get out of phase. For the phase shift between two identical
particles with the mass and the charge , one of which is in a certain
gravitational (electromagnetic) field, we obtain:
here is
the Dirac constant, is the electromagnetic 4potential, is
the 4displacement of the particle in space and time.
The latter relation for the phase shift in
electromagnetic field is confirmed by the AharonovBohm
effect.
The field
energy theorem, which has the same meaning for fields as
the virial theorem for particles, is applicable to a vector gravitational field
in a curved spacetime. In the formulation of the theorem there is the gravitational
4potential: ^{[11]}