For a negative cosmological constant Λ, C-metric with acceleration parameter larger than the critical value, a>1/lΛ, (where lΛ2=–3/Λ) describes a sequence of pairs of accelerated black holes. Black holes of each pair enter asymptotically anti-de Sitter spacetime and move toward each other until they stop. Then, they move away from each other back to the infinity where they leave the spacetime. It starts a cosmological phase when the universe does not contain any black holes. After it, a new pair of holes enters again the universe.
Three-dimensional diagrams on the following pages represent a causal structure of the C-metric universe outside of the black holes. Only a part of the exterior of black holes is depicted – the diagram should repeat in a time (vertical) direction again and again. 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. Outer horizons of the black holes are indicated by dark surfaces. Blue surfaces indicate acceleration horizons and red surfaces cosmological horizons. All 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. Such a representation is more useful in a limit of a negligible mass when the black holes turn to test particles.
Standard two-dimensional conformal diagrams for C-metric depict surfaces ξ,φ=constant. They can have a qualitativelly different shape depending on a value of ξ. Embedding of these surfaces into the whole spacetime is shown.
|black holes visualized with cone-like horizon||black holes visualized with drop-like horizon|
Interactive diagrams needs a browser supporting Java applets. It uses LiveGraphics3D by Martin Kraus. It can take some time to download and to initiate these diagrams.
© 2005-12-31; Pavel Krtouš <Pavel.Krtous@mff.cuni.cz>