A new form of the C-metric

TL;DR

The paper introduces a new, explicitly factorisable form of the C-metric's structure function, simplifying calculations and root determination, with extensions to charged and extremal cases.

Contribution

A new factorisable form of the C-metric's structure function that simplifies root calculations and coordinate transformations, including charged and extremal cases.

Findings

Simplified roots of the structure function G(ξ)

Easier casting of the C-metric in Weyl coordinates

Explicit form of the extremally charged C-metric in Weyl coordinates

Abstract

The usual form of the C-metric has the structure function G(\xi)=1-\xi^2-2mA\xi^3, whose cubic nature can make calculations cumbersome, especially when explicit expressions for its roots are required. In this paper, we propose a new form of the C-metric, with the explicitly factorisable structure function G(\xi)=(1-\xi^2)(1+2mA\xi). Although this form is related to the usual one by a coordinate transformation, it has the advantage that its roots are now trivial to write down. We show that this leads to potential simplifications, for example, when casting the C-metric in Weyl coordinates. These results also extend to the charged C-metric, whose structure function can be written in the new form G(\xi)=(1-\xi^2)(1+r_{+}A\xi)(1+r_{-}A\xi), where r_{\pm} are the usual locations of the horizons in the Reissner-Nordstrom solution. As a by-product, we explicitly cast the extremally charged C-metric in Weyl coordinates.