New perspectives on the TOV equilibrium from a dual null approach
Abstract
The TOV equation appears as the relativistic counterpart of the classical condition for hydrostatic equilibrium. In the present work we aim at showing that a generalised TOV equation also characterises the equilibrium of models endowed with other symmetries besides spherical. We apply the dual null formalism to spacetimes with two dimensional spherical, planar and hyperbolic symmetries with a perfect fluid as the source. We also assume a Killing vector field orthogonal to the surfaces of symmetry, which gives us static solutions, in the timelike Killing field case, and homogeneous dynamical solutions in the case the Killing field is spacelike. In order to treat equally all the aforementioned cases, we discuss the definition of a quasilocal energy for the spacetimes with planar and hyperbolic foliations, since the HawkingHayward definition only applies to compact foliations. After this procedure, we are able to translate our geometrical formalism to the fluid dynamics language in a unified way, to find the generalised TOV equation, for the three cases when the solution is static, and to obtain the evolution equation, for the homogeneous spacetime cases. Remarkably, we show that the static solutions which are not spherically symmetric violate the weak energy condition (WEC). We have also shown that the counterpart of the TOV equation ρ + P = 0, defining a cosmological constanttype behaviour, both in the hyperbolic and spherical cases. This implies a violation of the strong energy condition in both cases, added to the above mentioned violation of the weak energy condition in the hyperbolic case. We illustrate our unified treatment obtaining analogs of Schwarzschild interior solution, for an incompressible fluid ρ = ρ_{0} constant.
 Publication:

Classical and Quantum Gravity
 Pub Date:
 June 2020
 DOI:
 10.1088/13616382/ab8759
 arXiv:
 arXiv:1910.13225
 Bibcode:
 2020CQGra..37l5005M
 Keywords:

 dual null formalism;
 energy conditions;
 spherical;
 planar and hyperbolic geometries;
 generalised TOV equation;
 TOVKilling vector connection;
 General Relativity and Quantum Cosmology;
 Astrophysics  High Energy Astrophysical Phenomena
 EPrint:
 10 pp, 1 fig