Buchdahl-like transformations for perfect fluid spheres

TL;DR

This paper extends solution-generating techniques for static perfect fluid spheres in general relativity across various coordinate systems, introducing Buchdahl-like transformations and classifying their properties for broader applications.

Contribution

It generalizes Buchdahl transformations and solution-generating theorems to multiple coordinate systems, enhancing the toolkit for analyzing perfect fluid spheres in relativity.

Findings

Derived Buchdahl-like transformations in isotropic coordinates.

Generalized solution-generating theorems in diagonal and other coordinates.

Placed Buchdahl transformation in the most general setting with a specific metric ansatz.

Abstract

In two previous articles [Phys. Rev. D71 (2005) 124307 (gr-qc/0503007), and gr-qc/0607001] we have discussed several "algorithmic" techniques that permit one (in a purely mechanical way) to generate large classes of general relativistic static perfect fluid spheres. Working in Schwarzschild curvature coordinates, we used these algorithmic ideas to prove several "solution-generating theorems" of varying levels of complexity. In the present article we consider the situation in other coordinate systems: In particular, in general diagonal coordinates we shall generalize our previous theorems, in isotropic coordinates we shall encounter a variant of the so-called "Buchdahl transformation", while in other coordinate systems (such as Gaussian polar coordinates, Synge isothermal coordinates, and Buchdahl coordinates) we shall find a number of more complex "Buchdahl-like transformations" and "solution-generating theorems" that may be used to investigate and classify the general relativistic static perfect fluid sphere. Finally by returning to general diagonal coordinates and making a suitable ansatz for the functional form of the metric components we place the Buchdahl transformation in its most general possible setting.