Topology of the Causal Boundary for Standard Static Spacetimes

TL;DR

This paper investigates the topology of the causal boundary in standard static spacetimes, comparing function-space and chronological topologies, and identifies conditions under which they coincide or form simple product structures, including classical spacetimes.

Contribution

It provides a detailed analysis of when the chronological topology matches the function-space topology in standard static spacetimes, including classical examples like Schwarzschild.

Findings

Chronological topology often coincides with the function-space topology.

Certain static spacetimes have a simple product structure for their causal boundary.

Classical spacetimes like Schwarzschild are included in the analyzed class.

Abstract

The topology of the causal boundary for standard static spacetimes--spacetimes time-invariantly conformal to a metric product of the Lorentz line and a Riemannian manifold--is studied in depth. As this is given in terms of a set of real-valued functions on the Riemannian factor, one could use a function-space topology, but physical reasons recommend a chronological topology instead. The function-space topology has a simple product structure, while the chronological topology might not. This paper examines when the chronological topology coincides with the function-space topology and when it has a simple product structure. A class of standard static spacetimes is examined, all of which yield a simple product structure for the causal boundary; the conformal class of these spacetimes includes classical spacetimes such as external Schwarzschild or Reissner-Nordstrom.