1 Gravito-electromagnetism
Is general relativity (GR) a final theory or it will be superseded by another theory in the future? We expect that it will survive while its explanatory power is strong enough to describe the available experimental data. However, even if its explanatory power was not strong enough to understand every known phenomenon, we would keep it in the absence of an alternative theory. At the present time, general relativity is a very successful gravitational theory, but we also know that there are several open questions about it, particularly related to their quantization and to to their cosmological applications. Furthermore, we do not know whether GR is suitable to solve these open questions, or whether a different theory is needed. In this situation, it can be interesting to modify the old theory in order to explain singular data effectively or to introduce a different conceptual idea [1]. From a theoretical point of view, it is interesting to study what kind of modification can possibly be done to a theory and keep its mathematical and physical consistency.
In this article, we propose an exercise concerning a recently published gravitation theory that modifies GR within the gravito-electromagnetic precision order [2]. In summary, we will analyze an alternative to this previously proposed modified theory in order to exhaust the possible alterations that are coherent to the original idea. Before considering the new formulation, we give a brief explanation of the origin of gravito-electromagnetism (GEM) using the Chapter 3 of [3]. As the name announces, in this theory an analogy between GR and electromagnetism is established. GEM comes from the weak field approximation to GR, where the
is a constant in cgs units and
describes the motion of a massive particle of velocity v in a gravitational field g and whose gravito-magnetic field is b The boldface characters denote vector quantities in a time-like surface, and the vector product satisfies the usual definition. In terms of components, we have
where
where the space-like components of the space-time index are i, j and k.Hence, one can establish an analogy between the covariant electrodynamics and the field vectors in a tensor formula, so that
However, the quantities of Eq. (4) were obtained from on the zeroth component of
In Ref. [2], the gravitational field g was decomposed as a sum of two auxiliary fields, the gravito-electric field gE and the gravito-magnetic field gBwhere
This decomposition is not usual in gravito-electromagnetism (GEM), and the field equations are also different from the previous formulations. This discussion is already been done in the previous article. However, the previous article does not exhaust the possible formulations, and this paper intends to fill this blank. However, we shall see that this task in not a bureaucratic one. The formulations have a diverse physical content, and the second formulation is necessary for the theoretical comprehension, and for future applications as well.
2 Modified Newtonian gravitation
Modified theories of Newton’s gravitation are not a novelty, and we mention [12-14] as a recent conjectures of such kind. In our proposal, the field equations are such as
Where g is the gravity field vector, ρ is the density of mass, p = ρv is the matter flux density vector. Accordingly, the gravity force F acts over a particle of mass m according to the physical law
Equations (7) are identical to that proposed in Ref. [2], while (8) has a single difference, a flipped sign on of the second term. The ultimate proposal of the present article is to determine the differences concerning this single difference. Additionally, we will confirm that the gravitational field given by (7) has a physical content comparable to that achieved after the truncation of Eintein’s field equations. We remember that truncation of Einstein’s equations generates the Newtonian theory at its first approximation, while higher order terms produce (2-4), and this prevision of GR will be recovered from (7) using a covariant scheme. As first consequence, the continuity equation and the conservation of the mass is obtained from (7)
The energy balance is given by
and Eqs. (9-10) are identical to that obtained in Ref. [2]. We observe the self-interacting terms
Equation (10) is the gravitational equivalent of the Poynting theorem, but a gravitational Poynting vector cannot be obtained. It is interesting to note that every contribution to the energy balance comes from self-interaction. Differently from [2], an expression for the conservation of the linear momentum is not possible in this formulation. Therfore, the field equations, the gravitational force law, the continuity equation and the energy balance encompass all the results that one can obtain from this model. In the following section, the tensor approach will illuminate this physical model from a different standpoint.
3 The gravitational field in the tensor formalism
In this section, we will observe many differences between the model of (7-8) and the previous article. Let us then intoduce the gravitational field tensor
The Minkowskian indices are μ and v, and the metric tensor
where
We also define the contravariant momentum density 4—vector
The gravitational force can as well be obtained in the covariant expression,
The spacelike μ = I components of (16) furnish the gravity force, and the timelike μ = 0 component reveals that the energy density of the model obeys
On the other hand, using the anti-symmetric feature of the field tensor, we obtain
and therefore
In order to obtain the energy-momentum tensor of this self-interacting gravitational theory, we define the
which has been introduced in [2] and that satisfies
where the anti-symmetric Levi-Cività symbol is
Additionally, combining (21-22) produces an equation satisfied by the
where
At this moment, we point out the major difference of the model presented in this article. This approach is more complicated than the former model [2] where
Explicitly, the energy-momentum components are
which generate the scalar quantities
Different from electromagnetism, the gravity energy-momentum tensor is not traceless. This result is in fact expected from general relativity, and thus a consistency condition is fulfilled. Furthermore, using the field (7), we obtain
Using Eqs. (26) and (28) in (23), the energy conservation and the gravitational force components are recovered, and the physical consistency of the model is assured. We have shown in this section that the gravitation model that (7-8) comprise can be consistently described using a tensor language. However, such a formulation seems unsatisfactory, particularly because the conservation of the energy is not clear in Eq. (10). For the sake of clarity, we develop a potential formulation in the next section.
4 The gravitational potentials in the tensor formalism
Introducing the gravitational scalar potential Φ and the gravitational vector potential Ψ, the gravitational field is proposed to be
and the field Eqs. (7) consequently become
Nonetheless, we obtain a simpler formulation after defining auxiliary gravito-electric and gravito-magnetic vector fields, respectively gE and gB Therefore,
Comparing to the the previous formulation [2], the signs of the third term in Eq. (29) and, consequently, of gB in (31) are flipped, and the second equation of (30) is simpler than in the previous paper. In consequence, using (31) in (7) we obtain the gravitational field equations in potential formulation,
that is similar to previous formulations of GEM [2,4-8], and also similar to the Maxwell electromagnetic field equations. Defining the gravitational potential second rank tensor
is the gravitational potential 4—vector, we directly have
The potential tensor (33-34) enables us to regain the equations of motion (30) using
Equation (35) contains the non-homogeneous components of (32), and the homogeneous terms come from
Manipulating the 4—vector momentum density, we consequently have
whose components give
Analyzing (38) in comparison to Eqs. (8) and (17), two constraints emerge, namely,
Therefore, the linear momentum p, the gravito-electric field gE and the gravitational force vector dp/dt are coplanar and the force law (39) conforms perfectly to (2), and the alternated signs in Eq. (4) may be obtained by a redefinition of b. At this moment, we point out the more important drawback of the model. Differently from the previous formulation [2], we cannot obtain a relation expressing the conservation of momentum in the same fashion as the electromagnetic formulation. This does not mean that the momentum is necessarily not conserved, but it may have a more subtle formulation. We may further explain the conservation of momentum by considering the tensor expression of the force law obtained from (35-37), so that
The energy-momentum tensor is
and the interaction term reads
The self-interaction term
Explicitly written, the components of (41) are
Accordingly, we obtain the scalar quantities
By comparing the scalar quantities (44) and (26), the nullity of
Using (38) we generate the energy conservation law that is directly obtained from the field equations (32) and that does not produce additional constraints. Finally, following a formulation of quantum electrodynamics, we use
and (35) is immediately obtained from Eq. (46). As a final remark, the field equations (32) can also be obtained using
However, this formulation flips the sign of
5 The second gravity force law
Let us consider the force law
the field equations
and the field tensor
where (14-17) hold. On the other hand, Eq. (21) changes to
The energy-momentum tensor
Hence, the second formulation is also consistent, and the proper physical content demands experimental investigation of Eqs. (8) and (47). Additionally, the potential formulation is
and finally the field equations are
Additionally,
leads to,
and (35-36) are immediately recovered. From (37), we produce
The constraints are
Essentially, both of the formulations are related by the symmetry transformation
Thus, under the alternative gravity law, the equivalents of Eqs. (40-45) are immediately obtained using Eq. (59), and the difference is the alternate sign in the “Pointing vector” of (45), meaning the reversal of the momentum flux in each formulation.
6 Concluding remarks
We examined several formal questions concerning gravito-electromagnetism, and proposed two gravity force laws, namely (8) and (48), and consistent covariant tensor formulations have been built for both of them. It was also verified that both of the formulations are related through a symmetry operation. The results complement the former article [2], where the force law is identical, but the field equations are different different. The results indicate that the energy is conserved in the present formulation, but the momentum is not conserved. Although this seems a negative result, it is in fact a very important piece of information. The force laws (8) and (48) were obtained using a different set of field equations in Ref. [2], and the choices of the present article introduce the self-interaction terms