Kottler Spacetime in Isotropic Static Coordinates

TL;DR

This paper derives the Kottler spacetime metric in isotropic static coordinates using Jacobian elliptic functions, revealing differences in null geodesics compared to Schwarzschild spacetime.

Contribution

It provides a new static isotropic coordinate form of the Kottler spacetime and analyzes the differences in null geodesics from Schwarzschild spacetime.

Findings

Kottler spacetime expressed in static isotropic coordinates.

Refractive indices of Kottler and Schwarzschild are not proportional in these coordinates.

Null geodesics differ between Kottler and Schwarzschild spacetimes in isotropic static coordinates.

Abstract

The Kottler spacetime in isotropic coordinates is known where the metric is time-dependent. In this paper, the Kottler spacetime is given in isotropic static coordinates (i.e., the metric components are time-independent). The metric is found in terms of the Jacobian elliptic functions through coordinate transformations from the Schwarzschild-(anti-)de Sitter metric. In canonical coordinates, it is known that the unparameterized spatially projected null geodesics of the Kottler and Schwarzschild spacetimes coincide. We show that in isotropic static coordinates, the refractive indices of Kottler and Schwarzschild are not proportional, yielding spatially projected null geodesics that are different.