Maximally symmetric subspace decomposition of the Schwarzschild black hole

TL;DR

This paper introduces a comoving coordinate system for Schwarzschild black holes, revealing that the spatial sections along freely falling observers' world lines are maximally symmetric, providing new insights into the black hole's internal structure.

Contribution

It presents a novel comoving coordinate system that decomposes the Schwarzschild black hole into maximally symmetric subspaces along free-fall trajectories.

Findings

Spatial part in the new coordinates is maximally symmetric.

Black hole interior can be viewed as a family of maximally symmetric subspaces.

Provides a different perspective from Kruskal extension.

Abstract

The well-known Schwarzschild black hole was first obtained as a stationary, spherically symmetric solution of the Einstein's vacuum field equations. But until thirty years later, efforts were made for the analytic extension from the exterior area $(r>2GM)$ to the interior one $(r<2GM)$. As a contrast to its maximally extension in the Kruskal coordinates, we provide a comoving coordinate system from the view of the observers freely falling into the black hole in the radial direction. We find an interesting fact that the spatial part in this coordinate system is maximally symmetric $(E_3)$, i.e., along the world lines of these observers, the Schwarzschild black hole can be decomposed into a family of maximally symmetric subspaces.