No Go Theorem for Kinematic Self-Similarity with A Polytropic Equation of State

TL;DR

This paper proves that in general relativity, spherically symmetric spacetimes with a perfect fluid obeying certain polytropic equations of state cannot admit a specific type of kinematic self-similarity unless they are vacuum, indicating a no-go theorem.

Contribution

It establishes a no-go theorem showing the incompatibility of certain kinematic self-similar symmetries with polytropic fluids in spherically symmetric spacetimes.

Findings

Such spacetimes must be vacuum under the given conditions

Kinematic self-similarity of the second kind is incompatible with polytropic fluids

The result constrains possible models of self-similar gravitational collapse

Abstract

We have investigated spherically symmetric spacetimes which contain a perfect fluid obeying the polytropic equation of state and admit a kinematic self-similar vector of the second kind which is neither parallel nor orthogonal to the fluid flow. We have assumed two kinds of polytropic equations of state and shown in general relativity that such spacetimes must be vacuum.