Higher dimensional generalization of Buchdahl-Vaidya-Tikekar model for super compact star

TL;DR

This paper extends the Buchdahl-Vaidya-Tikekar super compact star model to higher dimensions, revealing how the key parameters and solutions adapt with increasing spacetime dimensions.

Contribution

It provides a higher dimensional generalization of the Buchdahl-Vaidya-Tikekar metric ansatz, including the relation of the parameter K across dimensions and conditions for constant density stars.

Findings

Higher dimensional solutions are derived for super compact stars.

The parameter K relates to the dimension via a specific formula.

Maximum dimension for a given K_4 is K_4 + 4.

Abstract

We obtain higher dimensional solutions for super compact star for the Buchdahl-Vaidya-Tikekar metric ansatz. In particular, Vaidya and Tikekar characterized the $3$-geometry by a parameter, $K$ which is related to the sign of density gradient. It turns out that the key pressure isotropy equation continues to have the same Gauss form, and hence $4$-dimensional solutions can be taken over to higher dimensions with $K$ satisfying the relation, $K_n = (K_4-n+4)/(n-3)$ where subscript refers to dimension of spacetime. Further $K\geq0$ is required else density would have undesirable feature of increasing with radius, and the equality indicates a constant density star described by the Schwarzschild interior solution. This means for a given $K_4$, maximum dimension could only be $n=K_4+4$, else $K_n$ will turn negative.