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In the Jefimenko’s generalized theory of gravitation, it is proposed the existence of certain potentials to help us to calculate the gravitational and cogravitational fields, such potentials are also presumed non-invariant under certain gauge transformations. In return, we propose that there is a way to perform the calculation of certain potentials that can be derived without using some kind of gauge transformation, and to achieve this we apply the Helmholtz’s theorem. This procedure leads to the conclusion that both gravitational and cogravitational fields propagate simultaneously in a delayed and in an instant manner. On the other hand, it is also concluded that these potentials thus obtained can be real physical quantities, unlike potentials obtained by Jefimenko, which are only used as a mathematical tool for calculating gravitational and cogravitational fields.

Jefimenko’s generalization of Newton’s gravitational theory [

Newton’s theory does not include inductive phenomena, but a relativistic theory of gravitation should include them. Indeed, under the relativistic mass-energy equivalence, not only the mass is a source of gravitational field but any kind of energy also is. Therefore, a body creates gravitational field not only by mass but also by their kinetic energy, i.e. by their movement. And this, ultimately, is what it means induction: the production of forces by moving bodies [

In the general relativity theory, Einstein predicted the existence of gravitational induction phenomena, such phenomena are appointed by Einstein as gravitomagnetism. It can be showed that Jefimenko equations are also derived from linearized Einstein equations (see, for example, pp 47 and 48 in [

Since in 2004, NASA has orbited the “Gravity Probe B”, whose purpose was to prove the existence of gravitomagnetism, (see [

In order to describe the time-dependent gravitational systems, the Jefimenko’s generalized theory of gravitation is based on postulating of retarded expressions for the accustomed gravitational field g and the Heaviside’s or cogravitational field K (Heaviside [

First of all, we write the equations describing time-dependent gravitational systems [

where c is the velocity of propagation of the fields, which is supposed equal to the velocity of light for the retarded component, G is the constant of gravitation,

There are some differences between Maxwell’s equations of electrodynamics and the Jefimenko’s equations of gravitation, i.e. the analogy is not perfect. For example, we have two kinds of electric charges, positives and negatives, which repel each other if the charges are equal and attract each other if they are different, whereas while we have only one type of mass, and if we have a system of two masses in repose, they always attract each other. While the electric field is directed from positive charges generating this field and is directed to the negative charges, the gravitational field is always directed to the masses by which is created. Another difference is that the magnetic field is always right- handed relative to the electric current by which is created, while the cogravitational field is always left-handed relative to the mass current by which is created.

In the analogy between electrodynamics and the so-called gravitodynamics, following the Jefimenko’s book [

Here, we introduce as is made in electrodynamics, the gravitodynamical potentials. If

Electromagnetic | Gravitational |
---|---|

E (electric field) | g (gravitational field) |

B (magnetic field) | K (cogravitational field) |

G (gravitational constant) |

the cogravitational field K satisfies Equation (2), we can always write it as the curl of some other vector quantity

where

The quantity within the parentheses can be written as the gradient of a gravitody- namical scalar potential

therefore,

Substituting the expressions (5) and (8) for the fields

and

Equations (9) and (10) can be decoupled choosing the appropriate form of the potentials

and

in (5) and (8), we get the same original fields

which allows us to separate Equations (9) and (10) for the potentials

Following the ideas of the work of Chubykalo et al. [

The Helmholtz’s theorem claims that under certain conditions all vector fields can be represented as the sum of an irrotational and a solenoidal components. We will use this theorem to separate the fields

Therefore, here we state the Helmholtz’s theorem as [

If the divergence

where

and

^{1}For systems localized in a finite region of space, it is evident that the fields g y K depend on r as 1/r^{2}.

We are going to suppose that all conditions of this theorem are satisfied by the fields

where the indices “i” and “s” mean irrotational and solenoidal components of the vectors, respectively.

For example:

We are going to substitute

and the next equations for the solenoidal part:

By definition, for the irrotational component of the gravitational field

and if we substitute this relation into Equation (22), we obtain the Poisson’s equation

Apparently, we need to take into account that

We show now that Equation (30) is equivalent to the law of conservation of mass. Indeed, let us take the divergence of the Equation (23), then we obtain as the result

But from Equation (22) and because

Now, we will demonstrate that the solution of Equation (30), indeed, is the same solution of the Poisson’s Equation (29). To do this, we note that the irrotational component of

where the potential

or

and where, if we relate Equations (32), (23), (28) y (33) and the fact that

And from (36) and (34), we obtain

or

which is the solution of the Poisson’s Equation (29). So we have found that the Poisson’s equation given by Equation (29), completely defines the potential

Since by definition

Let us now apply the Helmholtz’s theorem to the vector potential

and taking into account (26) we obtain

One can substitute Equations (39) and (40) into (27), and we obtain

where we used the vector identity

We have found that system of Equations (1)-(4) reduces to Equations (29) and (41), applying the Helmholtz’s theorem. Therefore, we obtain separated equations for vector and scalar potentials, namely,

and

Now, we will show that the potentials

If we apply the Helmholtz theorem to the gravitational and cogravitational fields in terms of the ordinary potentials given by (5) and (8) without taking into account any gauge condition, we have

then, by definition,

where

and from Equations (49) and (22) we have the relation between

Now, we apply the Helmholtz theorem and the gauge transformations (11) and (12) and from

comparing the solenoidal parts we obtain

If we seek the transformation law for

From Equation (51), we have the irrotational part of

and including Equation (48) in (53) we get

or

At last, we can use Equation (50) for and, if we consider Equations (11), (12) and (55), we obtain

Hence, we have checked that

It is convenient to remark that the fields g and K are generated only by

And so, we have shown that it is possible to define vector as well as scalar gravitody- namical potentials, which are invariant under gauge transformation. These potentials are defined uniquely from their differential Equations (42) and (43). For this reason, we have arguments for supposing the physical reality of these potentials, similarly to the fields g and K and unlike the gravitational potentials introduced by Jefimenko in [

Our scalar potential T is a generator of the so-called instantaneous action at a distance in gravitation, and the vector potential

Espinoza, A., Chubykalo, A. and Carlos, D.P. (2016) Gauge Invariance of Gravitodynamical Potentials in the Jefimenko’s Generalized Theory of Gravitation. Journal of Modern Physics, 7, 1617-1626. http://dx.doi.org/10.4236/jmp.2016.713146