The Tolman VII space-time in the presence of charge and a cosmological constant

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–Nordström-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, $$\Gamma \sim 2.5$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Γ</mml:mi> <mml:mo>∼</mml:mo> <mml:mn>2.5</mml:mn> </mml:mrow> </mml:math> . 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.