Massive Electrodynamic Gravity

Lukasz Andrzej Glinka

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Massive Electrodynamic Gravity

Lukasz Andrzej Glinka

B.M. Birla Science Centre, Hyderabad, India


In this paper, an effcient combination of the diverse theoretical approaches, such like the Einstein gravitational waves, the Lifshitz cosmological perturbation theory, the Veltman perturbative quantum gravity, and the Maxwell electrodynamics, leads to an essentially new discussion of either massless and massive gravitons. The new force law of gravitation is established. The case of the Planckian particle is considered in the context of the Markov hypothesis.

Cite this article:

  • Glinka, Lukasz Andrzej. "Massive Electrodynamic Gravity." Applied Mathematics and Physics 2.3 (2014): 112-118.
  • Glinka, L. A. (2014). Massive Electrodynamic Gravity. Applied Mathematics and Physics, 2(3), 112-118.
  • Glinka, Lukasz Andrzej. "Massive Electrodynamic Gravity." Applied Mathematics and Physics 2, no. 3 (2014): 112-118.

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1. Introduction

Existence of gravitational radiation is one of the proposals due to General Relativity, for the first time considered through Albert Einstein about one hundred years ago [1, 2]. Let us present the core basis of this idea, the detailed derivations can be found in handbooks, Cf. the Refs. [3, 4, 5].

Gravitational waves are defined through the weak field condition


where ημν = diag (1,1,1,1) is the Minkowski metric in the relativistic convention. Assuming the coordinate transformation where are the Lorentz matrices, the Lorentz invariance


leads to


The change of coordinates


generates the Lorentz transformation α


which leads to the gauge transformation for Einstein’s gravitational waves α


In such an approximation


where , and the Einstein field equations for gravitational waves are


where Tμν is the energy-momentum tensor of Matter fields. Making use of the tensor


and require the Lorenz-like gauge


the field equations (12) become the wave equation


whose 00-component and the spherical symmetry


for the energy-momentum tensor with


gives the Poisson equation


with the gravitational potential


which establishes the Newton law of universal gravitation. Then, neglecting other components of ,


one has


and, consequently, the space-time is characterized by the interval


Gravitational waves are one of the most intriguing questions of modern physics [6, 7, 8]. In 2011, V.F. Mukhanov presented the lecture on massive gravity [9] following from Einstein’s gravitational waves, cosmological perturbation theory, and perturbative quantum gravity. This paper shows that the situation is really much more sophisticated than Mukhanov indicated.

2. Cosmological Perturbations

In the cosmological perturbation theory [10, 11]


where the scalar and the tensor perturbations are


while Φ is the gravitational potential responsible for the appropriate law of gravitation, and ∂iB is the vector perturbation. Then, the weak field is


while the condition det gives


However, this state of affairs differs from General Relativity, because the cosmological perturbation theory corresponds with fixation of an inertial reference frame in the background Minkowski space-time and the scalar-vector- tensor decomposition of the perturbation metric components is made according to their transformation properties under spatial rotations, and is similar to decomposition of the electromagnetic field into electric and magnetic fields. The (0,2) spatial tensor can be further decomposed into a trace and a trace- free part, which corresponds with looking for an irreducible representations of the rotation group.

For the metric (23), one has






For this reason


and, therefore,


Consequently, the Einstein field equations are


and, amazingly, can be solved. The 00 component and 0i component give


while the 0j-component leads to


Contraction of this solution with δ and application of (24) give


and, consequently, (42) becomes


Noticing that


one can see that the 00-component of the field equations becomes


Making use of the relation (41), one receives


and, consequently, the ij component of the field equations is


and after contraction with δ gives


where . Finally, one obtains


and the condition (27) for non-zero T00 can be rewritten in the following form


In the equation (56), the evolutions of B and Φ can not be separated, but, consistency with the Newton gravitation suggests the Poisson equation


and then


that is the problem is a choice of suitable .

3. Perturbative Graviton

Gauge field theories offer the perturbative point of view, wherein graviton is a spin-2 particle [12]. Then, of graviton is derived from a Lagrangian


or, explicitly


The Lagrangian of Matter fields has the form


where m is a graviton mass. Straightforward calculations gives


and, consequently,


Implementing the massless limit, one has


and, consequently, the metric (23) becomes


while the condition (27) is equivalent to


4. Massless Limit

The Laplace equation (67) can be rewritten in the form


and can be solved by the most general separation of variables Φ = C, ∂0B = C,


where C is a separation constant. In such a situation,


Taking into account the boundary conditions


one receives


Because the constant in the gravitational potential


vanishes when differentiation is performed, it does not play a role for the force law of gravitation Fi = -m∂iΦ, and, consequently, it can be identically vanishing. This condition gives


and, for this reason, one obtains finally i 2




where Ω0 and ω0 are two constant angular frequency parameters, for which


and the force law of gravitation is 2 3


one sees that the force law is non-trivial if and only if


where K0 and k0 are positive constants.

5. Massive Gravitons

Let m be a positive graviton mass parameter. Then, one has

1. The Newton law of universal gravitation, if


where M > 0 is a mass of a spherically-symmetric body of radius r0

2. The oscillatory law of gravitation, if


3. The force law of gravitation , if


where M is as previously, and is a certain constant reference value of the potential . Then,


and, for consistency, one has


where β is a dimensionless constant, which means that the mass of a body generating gravitation is multiplication of the graviton mass.



where the constant can be established from experimental data.

4. The most general force law of gravitation, which is non-trivial in the massless limit,


where and are certain reference constants, is given by




and, consequently,




If , then


and, consequently,


For M = βm and r0 = R, where R is a radius of the spherical body,


Then, a graviton mass is


and suggests existence of the various gravitons emitted by a mass.

6. Electromagnetic Gravity

The resulting force law of gravitation


can be considered as the analogue of the Lorentz force


where the field dk plays a role analogous to magnetic field, gravitational acceleration gi plays a role of electric field, and mass m of a moving particle plays a role of electric charge.

Introducing the analogue of the Faraday tensor of electromagnetic field


for which the dual tensor is


and taking the four-current


one obtains the analogue of the Maxwell equations


or, explicitly, the analogue of the Gauss law for electricity


the analogue of the Amp`ere law


the analogue of the Gauss law for magnetism


and the analogue of the Faraday law


Introducing the analogue of the electromagnetic four-potential


where is the velocity four-vector, one has


and, making use of (108), one receives


or, equivalently,


Also, the stress-energy tensor is the analogue of the electromagnetic one


where is the field energy density


Si is the analogue of the Poynting vector


and σij is the analogue of the Maxwell stress tensor


Making use of the definitions (111) within the equation (104), one receives


and, similarly, the equation (103) leads to




Particularly, for a graviton mass m= 0, one has


or, taking into account (107),


Comparing the relations (118) and (120), one obtains


and, consequently, one receives the analogue of the Proca equation


describing the massive graviton.

The analogy with the Maxwell electrodynamics gives the Lagrangian


Also, on the quantum level, one obtains the analogue of the quantum electrodynamics and the Feynman diagrams are fully analogous. Making use of (119), the Lagrangian (123) becomes


where we have made use of the four-velocity definition , and the constant Lagrangian has been omitted. Noticing that the continuity equation


should be satisfied, the equation (119) gives the Lorentz gauge


which gives the divergence-free velocity field


which implies that and, consequently, the graviton mass identically vanishes m = 0. However, because this is an additional relation which can be broken and which is not needed for consistency of the theory, one must not take this condition into account. When the continuity equation is non-valid, then a graviton mass is existent and fully argued.

The equations (97) and (98) can be used to find the explicit form of the fields gi and di. There are two identifications


For the first one


and, consequently, m = 0, and the Lorentz gauge is consistent. For the second one,


and for a graviton is a tachyon, whereas for , that is an inverted harmonic oscillator, one has


what compared with (96), gives


Therefore, for non-tachyonic massM, one should have that is a repulsive gravitation, and


Interestingly, for a sphere of radius equal to the Planck length and mass equal to the Planck mass M = MP, one obtains


where ωP is the Planck frequency, while the force law of gravitation becomes


For emission of a graviton through the Planckian particle, M = MP, a hy- pothetical particle existing in the Planck scale energetic regime, a graviton mass exceeds MP, what in the light of the Markov hypothesis [13, 14], that is m ≤ MP, means that this graviton is undetectable.

Moreover, one can see that (111) gives


and when the velocity field vi is known, for example as the solution to the equation (122), both (128) and (129) are the equations for ri


7. Summary

Joining the various approaches to gravity - the Einstein gravitational waves following from General Relativity whose applicability is unclear, the Lifshitz cosmological perturbation theory which is cureently applicable for astrophysical scales, the Veltman perturbative quantum gravity based on the gauge field theories which usually are applicable for high energy physics, and the Maxwell electrodynamics which is applicable at any scales - we have received an essential discussion on either massless and massive gravitons. The analysis of the Planckian particle in the context of the Markov hypothesis has been shown that a massive graviton emitted throughout such a particle is undetectable. The new force law of gravitation has been established. This theory, however, needs a straightforward confrontation with a suitable experimental data. Its should be emphasized that the discussion of this model in more advanced detail was presented in the author’s monograph [15].


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