New perspectives on the TOV equilibrium from a dual null approach
arXiv:1910.13225 · doi:10.1088/1361-6382/ab8759
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 generalized 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 for the spatially homogeneous models is just the familiar equation \r{ho} + 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 constant.
arXiv comment10 pp, 1 fig
References in corpus (42)
- Masses, Radii, and Equation of State of Neutron Stars
- General Laws of Black-Hole Dynamics
- Physical Acceptability of Isolated, Static, Spherically Symmetric, Perfect Fluid Solutions of Einstein's Equations
- Unified first law of black-hole dynamics and relativistic thermodynamics
- Gravitational Energy in Spherical Symmetry
- All static spherically symmetric perfect fluid solutions of Einstein's Equations
- Quasi-Local Gravitational Energy
- Twilight for the energy conditions?
- General relativistic polytropes for anisotropic matter: The general formalism and applications
- Generalized energy conditions in Extended Theories of Gravity
- The realistic models of relativistic stars in f(R) = R + alpha R^2 gravity
- Energy conditions in modified gravity
- Quasilocal mass in general relativity
- Necessary and sufficient conditions for big bangs, bounces, crunches, rips, sudden singularities, and extremality events
- Covariant Perturbations of Schwarzschild Black Holes
- Tolman-Oppenheimer-Volkoff equations in presence of the Chaplygin gas: stars and wormhole-like solutions
- Spacetime geometry of static fluid spheres
- Relativistic hydrodynamics on spacelike and null surfaces: Formalism and computations of spherically symmetric spacetimes
- Polytropic spheres with electric charge: compact stars, the Oppenheimer-Volkoff and Buchdahl limits, and quasiblack holes
- Static, spherically symmetric solutions with a scalar field in Rastall gravity
- Trapped surfaces
- Solution generating theorems for the TOV equation
- Equilibrium and stability of relativistic stars in extended theories of gravity
- Firewalls, black-hole thermodynamics, and singular solutions of the Tolman-Oppenheimer-Volkoff equation
- Post-Tolman-Oppenheimer-Volkoff formalism for relativistic stars
- Incompressible relativistic spheres: Electrically charged stars, compactness bounds, and quasiblack hole configurations
- The covariant Tolman-Oppenheimer-Volkoff equations II: The anisotropic case
- Hawking-Hayward quasi-local energy under conformal transformations
- Possibility of hyperbolic tunneling
- Cosmological black holes from self-gravitating fields
- Quasilocal energy in modified gravity
- Stellar Structure Equations in Extended Palatini Gravity
- A dual null formalism for the collapse of fluids in a cosmological background
- Charged cosmological black holes: a thorough study of a family of solutions
- Quasilocal approach to general universal horizons
- Energy in higher-dimensional spacetimes
- Self-similarity in static axially symmetric relativistic fluids
- Cylindrically symmetric models of gravitational collapse to black holes: a short review
- Revisiting the Birkhoff theorem from a dual null point of view
- Comment on "Covariant Tolman-Oppenheimer-Volkoff equations. II. The anisotropic case"
- The Jeans theorem and the "Tolman-Oppenheimer-Volkov equation" in an exact wave solution of R^2 gravity
- Existence and Classification of Pseudo-Asymptotic Solutions for Tolman-Oppenheimer-Volkoff Systems