Spacetime Structure of a Regular Accelerating Black Hole Pair in General Relativity

TL;DR

This work constructs and analyzes Ernst's one-parameter generalization of the C-metric, identifying the unique regularization values $k_0$ and $C$ that remove axis line sources and yield a regularized, vacuum spacetime describing a pair of accelerating black holes. The generalized metric transitions from Petrov type D to I, and a Gauss-Bonnet analysis clarifies the nodal singularities and horizon topology, with both horizons embeddable in $\mathbb{E}^3$. However, the exponential factors make the spacetime not asymptotically flat, and the conformal infinity $\mathscr{I}$ acquires a dipole structure that alters its topology and prevents a standard Bondi form from being obtained via Weyl coordinates. The paper also develops Weyl and boost-rotation formulations, demonstrates the line sources vanish for the regularized metric, and constructs comprehensive 2D/3D conformal diagrams to illustrate the causal architecture and the wormhole-compatible topologies. Overall, the regularized C-metric provides a rigorous, purely gravitational model of accelerating black holes without a line source and clarifies the geometric and asymptotic properties of this extended solution.

Abstract

We revisit the one-parameter generalization of the C-metric derived by Ernst, which solves the vacuum Einstein equations. Resolving conflicting claims in the literature, we determine the correct value of the parameter that ensures the regularity of the metric on the axis. This "regularized C-metric" describes a pair of accelerating black holes without the line source present in the original C-metric. Additionally, this generalization changes the Petrov type from D to I. We use the Gauss-Bonnet theorem to analyze the nodal singularities, the line source, and their relation to the horizon topology. Both the black hole and acceleration horizons are found to be embeddable in $\mathrm{E}^3$. We examine various geometric and asymptotic properties in detail using several coordinate systems and construct the corresponding 2D and 3D conformal diagrams. This process is more involved than for the original C-metric due to the presence of the exponential factors. These exponential factors also introduce curvature singularities at infinity, which obstructs asymptotic flatness. Contrary to Bonnor's expectation, we demonstrate why Bondi's algorithm for obtaining the standard Bondi form fails for the C-metric, despite its asymptotic flatness. We also show that Ernst's solution-generating prescription in boost-rotation symmetric coordinates is a symmetry of the wave equation.

Figures (16)

• Figure 1: Representation $t,r=\text{constant}$ surfaces with various choices for the range parameter $C$. Dotted lines represent strings and thick lines represent struts.
• Figure 2: Coordinate diagram representing the three new singularities (large colored dots) of the distorted C-metric. These points are regular for the C-metric. Note that the singularity at $(x,y) = (-1,1)$ is $(r,\theta)=(1/\alpha, 0)$ and the one at $(x,y)=(1,-1)$ is not covered by the $r$ coordinate. The singularity at $y=1/2\alpha m$ is outside the physical spacetime.
• Figure 3: Representation of $r$-constant, $t$-constant surfaces, for $r\geq 1/\alpha$ for the generalized C-metric (same for the C-metric). The red lines (or dots) represent the boundary of the surfaces.
• Figure 4: Figure \ref{['fig:surface-area']} shows the comparison of the surface areas of the regularized C-metric $(\mathrm{i.e.,~} k=k_0)$ vs. the original C-metric $(\mathrm{i.e.,~} k=0)$ for $(m,\alpha) = (1,0.1)$, in logarithmic scale. Red (dash) represents the regularized C-metric and the blue (line) represents the original C-metric. Figures \ref{['fig:ceq']} and \ref{['fig:cpol']} plot the equatorial and polar circumferences, respectively, with $(m,\alpha) = (1, 0.1)$. Figure \ref{['radial-distance']} plots the radial distance $R$ from $r=2m$ to $1/\alpha$ vs. $\theta$ for $(m,\alpha) = (1, 0.1)$.
• Figure 5: C-metric black hole horizon showing conical singularities, embeddability issues and regularizing either side of the axis with appropriate choice of the range parameter $C$. Pictures are for $(m,\alpha) = (1,0.2)$. Horizons in Figures \ref{['fig:c=1']} (\ref{['fig:c=1-bottom']}) and \ref{['fig:c-top-reg']} are not entirely embeddable and will have a hole in the bottom. The horizon in Figure \ref{['fig:c-bottom-reg']} is completely embeddable, and is the diagram that is usually presented in the literature (for example, Figure 3 in Kinnersley_Walker and Figure 3 in griffiths_krtous_podolsky). These figures can also be compared with the line source diagrams in Figure \ref{['conic_figure']}.