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 quasi-local energy for the spacetimes with planar and hyperbolic foliations, since the Hawking–Hayward 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 constant-type 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 = ρ0 constant.

A Comprehensive Analysis of Hyperbolical Fluids in Modified Gravity

“…The distribution of hyperbolically symmetric fluid has mass function as After substituting Equation ( 11 ) into Equation ( 7 ), the value of becomes The mass m and the effective density should be regarded as positive and negative quantities, respectively, according to Equations ( 11 ) and ( 12 ). As previously discovered in [ 33 ], the weak energy requirement is therefore disobeyed. Some intriguing statements on the physical significance of Equation ( 12 ) have been mentioned in [ 35 ].…”

“…The mass m and the effective density µ (e f f ) should be regarded as positive and negative quantities, respectively, according to Equations ( 11) and (12). As previously discovered in [33], the weak energy requirement is therefore disobeyed. Some intriguing statements on the physical significance of Equation ( 12) have been mentioned in [35].…”

“…Harrison [ 25 ] was the first to examine a solution to the Einstein equations of the particular form, which was determined by the hyperbolic symmetry and it has since been the topic of investigation in several contexts [ 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 ]. Herrera et al [ 35 ] have done a detailed investigation on static fluid distributions with hyperbolical symmetry in the framework of GR.…”

Hyperbolically Symmetric Versions of Lemaitre–Tolman–Bondi Spacetimes

“…The above Killing vectors ( 3 ) define the hyperbolic symmetry. Solutions to the Einstein equations endowed with this type of symmetry have been the subject of research in different contexts (see [ 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 ] and the references therein).…”

Hyperbolically symmetric sources in Palatini f(R) gravity

“…The region interior to the horizon had no appropriate mathematical formulation to explicate it [3]. In fact, it was propounded at first by Harrison [4] and later became the subject of interest of many researchers [5][6][7][8][9][10].…”