Abstract
We consider the mass-radius bounds for spherically symmetric static compact objects in the de Rham-Gabadadze-Tolley (dRGT) massive gravity theories, free of ghosts. In this type of gravitational theories the graviton, the quantum of gravity, may have a small, but non-vanishing mass. We derive the hydrostatic equilibrium and mass continuity equations in the Lorentz-violating massive gravity in the presence of a cosmological constant and for a non-zero graviton mass. The case of the constant density stars is also investigated by numerically solving the equilibrium equations. The influence of the graviton mass on the global parameters (mass and radius) of these stellar configurations is also considered. The generalized Buchdahl relations, giving the upper and lower bounds of the mass-radius ratio are obtained, and discussed in detail. As an application of our results we obtain gravitational redshift bounds for compact stellar type objects in the Lorentz-violating dRGT massive gravity, which may (at least in principle) be used for observationally testing this theory in an astrophysical context.
Highlights
An important moment in the development of the massive gravity theory was represented by the paper [3], where it was found that there exists a discrete difference between the zeromass theories and the very small, but non-zero mass theories
In this context it is important to mention that massive gravity is a classical field theory that does not need to be formulated in terms of the graviton, a particle mediating the gravitational interaction in a way similar to the electromagnetic or nuclear interactions
The boundary conditions that must be imposed at the center and on the surface of the star are ρ(0) = ρc, and P(R) = 0, where ρc is the central density, and R is the radius of the compact object
Summary
In four space-time dimensions, we consider a static and spherically symmetric metric of the following form, ds2 = −n(r )d(ct)2 +. We will assume that the energy-momentum tensor of the matter is given by, Tνμ = (ρc2 + P)uμuν + Pδνμ,. I.e., by a perfect fluid, characterized by only two thermodynamic parameters, the matter density ρ, and the thermodynamic pressure P, respectively, as well as by its four-velocity uμ, satisfying the normalization condition uμuμ = −1. In the following we adopt the comoving reference frame, in which the components of the four velocity are given by uμ = −n(r )−1/2, 0, 0, 0. The components of the effective energy-momentum tensor of the massive graviton are given by, X α(3r. Substitute all components in Eq (16), the modified Einstein field equations become f r
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