An alternative well-posedness property and static spacetimes with naked singularities

TL;DR

This paper introduces an alternative well-posedness property for wave equations in certain static spacetimes with singular boundaries, showing finite energy solutions without boundary conditions due to metric degeneracy.

Contribution

It identifies a new well-posedness mechanism arising from metric degeneracy, distinct from essential self-adjointness, and characterizes the degeneracy types causing this phenomenon.

Findings

Wave propagation is well-posed with finite energy solutions in some singular spacetimes.

No boundary conditions are needed despite the lack of essential self-adjointness.

Degeneracy of metric components near the boundary causes the alternative well-posedness.

Abstract

In the first part of this paper, we show that the Cauchy problem for wave propagation in some static spacetimes presenting a singular time-like boundary is well posed, if we only demand the waves to have finite energy, although no boundary condition is required. This feature does not come from essential self-adjointness, which is false in these cases, but from a different phenomenon that we call the alternative well-posedness property, whose origin is due to the degeneracy of the metric components near the boundary. Beyond these examples, in the second part, we characterize the type of degeneracy which leads to this phenomenon.