C0L - pairs of black holes accelerated in asymptotically flat spacetime

In the case of a vanishing cosmological constant Λ, C-metric describes two accelerated black holes which enter asymptotically flat spacetime with the velocity of light, and move with a deceleration toward each other until they stop. Then they accelerate away from each other and move back to the infinity where they leave the spacetime.

Three-dimensional diagrams on the following pages represent a causal structure of the C-metric universe outside of the black holes. Angular coordinate φ around the axis of motion is suppressed (a rotation around this axis is a symmetry of the spacetime).

The boundary of the diagram corresponds to the conformal infinity. Horizons of the black holes are indicated by dark surfaces. Blue surfaces indicate acceleration horizon. Horizons are null surfaces and thus they are only one way traversable for a physical observer.

In one set of diagrams, the black hole horizons are depicted by two joined cone-like surfaces. It reflects the null character of the horizons with null generators running from the neck of the horizons to the infinity. An alternative set of diagrams represents the horizons by two joined drop-like surfaces. It suggests that the black holes are localized objects. This representation is more useful in a limit of a negligible mass when the black hole turns to a test particle – compare with the graphical representation of uniformly accelerated observeres in Minkowski spacetime.

Standard two-dimensional conformal diagrams for C-metric depict surfaces ξ,φ=constant. Embedding of these surfaces into the whole spacetime is shown.