On a regular modified C-metric

TL;DR

This paper explores a modified C-metric with a non-singular core, analyzing its properties in the weak field limit, and proposes a link between the cosmological constant and inertial forces in accelerated frames.

Contribution

It introduces a regularized form of the C-metric with a non-standard interpretation and connects the cosmological constant to inertial effects in accelerated systems.

Findings

Proper acceleration is constant in the weak field limit.

The spacetime becomes conformally-flat and anti-de Sitter.

Stress tensor components are regulated by exponential factors.

Abstract

A particular form of the C-metric is investigated, giving it a non-standard interpretation and removing any singularity at $r = 0$. In the weak field limit of the accelerating black hole, the proper acceleration $A$ of a static observer is constant and the geometry becomes conformally-flat (anti de Sitter). The stress tensor is of $\Lambda$-type ($\Lambda = -3a^{2}/8\pi G$) and its energy density is negative. We propose that $\Lambda$ is responsible of inertial forces that appear in uniformly accelerated systems (far from the accelerating source $m$ and for $r << 1/a$ the dominant term in the expression of $a^{r}$ is $-a cos\theta$). The components of the stress tensor and all invariants of the conformally-flat Schwarzschild spacetime are regulated by means of the exponential factor $exp(-k/r), k > 0$.