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The basic properties of the C-metric are well known. It describes a pair of causally
separated black holes which accelerate in opposite directions under the action of
forces represented by conical singularities. However, these properties can be demonstrated
much more transparently by making use of recently developed coordinate
systems for which the metric functions have a simple factor structure. These enable
us to obtain explicit Kruskal–Szekeres-type extensions through the horizons and
construct two-dimensional conformal Penrose diagrams. We then combine these
into a three-dimensional picture which illustrates the global causal structure of the
space-time outside the black hole horizons. Using both the weak field limit and
some invariant quantities, we give a direct physical interpretation of the parameters
which appear in the new form of the metric. For completeness, relations to other
familiar coordinate systems are also discussed.