The conservation of energy–momentum and the mass for the graviton

Abstract

In this work we give special attention to the bimetric theory of gravitation with massive gravitons proposed by Visser in 1998. In his theory, a prior background metric is necessary to take in account the massive term. Although in the great part of the astrophysical studies the Minkowski metric is the best choice to the background metric, it is not possible to consider this metric in cosmology. In order to keep the Minkowski metric as background in this case, we suggest an interpretation of the energy-momentum conservation in Visser's theory, which is in accordance with the equivalence principle and recovers naturally the special relativity in the absence of gravitational sources. Although we do not present a general proof of our hypothesis we show its validity in the simple case of a plane and dust-dominated universe, in which the 'massive term' appears like an extra contribution for the energy density.

Visser’s massive graviton bimetric theory revisited

“…αβ (27) Notice that the above equation satisfies consistently eq. 7 and indicates that all GR solutions including a cosmological constant Λ are also solutions of Visser's massive gravity with a background metric tensor…”

“…In the aforementioned investigations, a flat background (Minkowski space-time) had been adopted despite the fact that according to Visser, "in a cosmological setting it is no longer obvious that we should use the Minkowski metric as a background". If a flat background is adopted, then the only issue resulting in an expanding dynamic metric is to assume that the divergence of the massive graviton tensor is a source for the divergence of the matter stress-energy tensor [27]. In Visser's theory the divergence of the graviton tensor is set to zero and, consequently, when the dynamical equations are linearized around the background metric, the Hilbert-Lorentz condition appears naturally and not as a gauge one.…”

Quintessence cosmology with a traversable wormhole, massive decaying gravitons and with varying G and Λ

“…Several studies have occurred in order to bound the mass of the graviton sorting from e.g. the planetary motions of the solar system (m g < 7.8 × 10 −55 g), the studies of galaxy clusters (m g 2 × 10 −62 g), rotation curves of galactic disks (m g 10 −59 g) and the orbital decay of the binary pulsars PSRB1913+16 (Hulse-Taylor pulsar) and PSR B1534+12 (m g < 1.4 × 10 −52 g) (Alves et al 2009). Additionally, gravitons may acquire effective mass in a wormhole medium dominated by phantom energy with equation of state parameter w = p/ρ < −1 (Sushkov 2005), where the null energy condition (NEC) that requires the stress-energy tensor T αβ to obey for all null vectors is violated, ignoring the back reaction of matter traversing the wormhole.…”

“…From the theoretical point of view, a traversable wormhole is achieved if the shape function obeys the flare-out conditions at the throat: B(r 0 ) = r 0 , B (r 0 ) < 1 and B(r) < r in addition to the asymptotic flatness condition, that is, B(r)/r → 0 as r → ∞ ; . In our approach, we will follow Visser's arguments (Visser 1998) and we take into account massive gravity by means of an additional bimetric tensor in Einstein's field equations, which read (in units = c = 1) (Alves et al 2009)…”

“…Of interest for us for instance are the (0, 0) components of the Friedman equation for the flat case (Alves et al 2009; El-Nabulsi 2009):…”