Maximally symmetric subspace decomposition of the Schwarzschild black hole

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.