The Tolman VII Space-time in the Presence of a Charge and a Cosmological Constant

TL;DR

This paper extends the Tolman VII solution in general relativity to include charge and a cosmological constant, analyzing its geometry, matching conditions, and physical properties, revealing new classes of solutions with potential astrophysical relevance.

Contribution

It derives a generalized charged Tolman VII solution with a cosmological constant, including new interior Nariai solutions and detailed physical analysis.

Findings

Allows for trapped null geodesics over a broad parameter range.

Provides analytic expressions for metric functions and pressure.

Identifies a physically acceptable subclass of solutions.

Abstract

The Tolman VII space-time is one of the few physically acceptable exact solutions in general relativity. In this paper, we derive a generalised Tolman VII solution which includes a charge and a cosmological constant. We analyse the spatial geometry of the solution and present conditions for zero and non-zero spatial curvature. We show that for a particular value of the boundary, the Tolman VII space-time can be matched to the charged Nariai space-time. This represents a new class of interior Nariai solutions. Matching with the Reissner-Nordstr\"om-de Sitter space-time, we derive analytic expressions for the metric functions and the pressure. Using this, we show that the solution allows for trapped null geodesics for a broad range of values for the total charge, central density, and the cosmological constant. We investigate the physical properties and derive an equation of state for the fluid. We show that the fluid can be considered a polytrope with, $\Gamma \sim 2.5$. Finally, we analyse the sound speed and energy conditions to conclude that only a subclass of the solution follows all the basic physical acceptability criteria.