Embedding the Schwarzschild Ideal Fluid Metric

TL;DR

This paper presents an isochronal, isometric embedding of the Schwarzschild ideal fluid metric to better understand the behavior of the metric as pressure approaches singularity, providing a coordinate-free visualization method.

Contribution

It introduces a novel embedding technique for the Schwarzschild ideal fluid metric, enabling clearer visualization of its properties near pressure singularities.

Findings

Successful embedding of the Schwarzschild ideal fluid metric

Enhanced understanding of metric behavior at pressure singularities

Coordinate-free visualization of the metric's properties

Abstract

Certain semi-Riemannian metrics can be decomposed into a Riemannian part and an isochronal part. The properties of such metrics are particularly easy to visualize in a coordinate-free way, using isometric embedding. We present such an isochronal, isometric embedding of the well known Schwarzschild ideal fluid metric in an attempt to see what is happening when the pressure becomes singular.