On spherically symmetric metric satisfying the positive kinetic energy coordinate condition

TL;DR

This paper investigates coordinate systems in spherically symmetric spacetimes that eliminate negative kinetic energy terms in the Einstein-Hilbert action, presenting solutions including a Reissner-Nordstrom metric variant and discussing Tolman metrics.

Contribution

It derives a linear PDE to find coordinate systems satisfying the positive kinetic energy condition and provides explicit metric solutions, including a Reissner-Nordstrom case.

Findings

Derived a PDE for positive kinetic energy coordinate condition.

Presented a Reissner-Nordstrom metric satisfying the condition.

Discussed the application to Tolman metrics.

Abstract

Generally speaking, there is a negative kinetic energy term in the Lagrangian of the Einstein-Hilbert action of general relativity; On the other hand, the negative kinetic energy term can be vanished by designating a special coordinate system. For general spherically symmetric metric, the question that seeking special coordinate system that satisfies the positive kinetic energy coordinate condition is referred to solving a linear first-order partial differential equation. And then, we present a metric corresponding to the Reissner-Nordstrom solution that satisfies the positive kinetic energy coordinate condition. Finally, we discuss simply the case of the Tolman metric.