On a regular modified C-metric
  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 $exp(-k/r), k > 0$. 
Hristu Culetu
gr-qc hep-th