Abstract
Different theories of gravity can admit the same black hole solution, but the parameters usually have different physical interpretations. In this work we study in depth the linear term beta r in the redshift function of black holes, which arises in conformal gravity, de Rham–Gabadadze–Tolley (dRGT) massive gravity, f(R) gravity (as approximate solution) and general relativity. Geometrically we quantify the parameter beta in terms of the curvature invariants. Astrophysically we found that beta can be expressed in terms of the cosmological constant, the photon orbit radius and the innermost stable circular orbit (ISCO) radius. The metric degeneracy can be broken once black hole thermodynamics is taken into account. Notably, we show that under Hawking evaporation, different physical theories with the same black hole solution (at the level of the metric) can lead to black hole remnants with different values of their physical masses with direct consequences on their viability as dark matter candidates. In particular, the mass of the graviton in massive gravity can be expressed in terms of the cosmological constant and of the formation epoch of the remnant. Furthermore the upper bound of remnant mass can be estimated to be around 0.5 times 10^{27} kg.
Highlights
A static and spherically symmetric black hole solution with a linearly increasing term in the redshift function is an example of degeneracy between four different gravitational 684 Page 2 of 13Eur
Of the Weyl curvature; this can be related to searching for local extrema in the radial evolution of the tidal force generated by the black hole, as considered in a number of specific solutions [57,58], but our approach is grounded on the geometric horizon conjecture rather than on the solution of the geodesic deviation equation
The Novikov–Thorne model, which is widely adopted for the description of the formation of an accretion disk around a black hole, states that the particles constituting the accretion disk follows nearly geodesics paths well approximated by the innermost stable circular orbit (ISCO) falling into the central massive object due to the effects of its gravitational field [17,18]
Summary
A static and spherically symmetric black hole solution with a linearly increasing term (as function of the coordinate radial distance) in the redshift function is an example of degeneracy between (at least1) four different gravitational. R is a solution in conformal gravity [20,21,22], de Rham– Gabadadze–Tolley (dRGT) massive gravity [23], (approximate) f (R) gravity [24,25], and general relativity (in this latter case this spacetime is known under the name of generalized Kiselev black hole) [26] In each of these theories, the parameters M, β, and , which describe the same manifold at a geometrical level, carry different physical meanings. We will illustrate the physical importance of the parameter β in the modeling of the innermost stable circular orbit (which can be taken roughly to correspond to the location of the accretion disk), and of the photon sphere (which may be used as an approximation of the shadow size) In this case there is a degeneracy between the different physical theories because the analysis is based on the study of the geodesic motion which is a purely geometrical concept. We remark that the curvature objects are a geometrical property of the manifold which are fully determined once the metric tensor (1) is provided without the need of knowing the gravitational theory in which it was discovered
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