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

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 E³. 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.