On the Construction of Time-Symmetric Black Hole Initial Data⋆

We review the 3+1 split which serves to put Einstein’s equations into the form of a dynamical system with constraints. We then discuss the constraint equations under the simplifying assumption of time-symmetry. Multi-Black-Hole data are presented and more explicitly described in the case of two holes. The effect of different topologies is emphasized. Notation. Space-time is a manifold M with Lorentzian metric g of signature (−,+,+,+). Greek indices are ∈ {0, 1, 2, 3} and latin indices are ∈ {1, 2, 3}. Indices from the beginning of the alphabet, like α, β, . . . and a, b, . . ., refer to orthonormal frames and indices from the middle, like λ, μ, . . . and l,m, . . . to coordinate frames. The symbol ◦ denotes the composition of maps. The relation := (=:) defines the left (right) hand side. The torsion and curvature tensors for the connection ∇ are defined by T (X,Y ) := ∇XY −∇Y X−[X,Y ] and R(X,Y )Z := ∇X∇Y Z − ∇Y ∇XZ − ∇[X,Y ]Z respectively. The covariant components of the Riemann and Ricci tensors are defined by Rαβγδ := g(eα, R(eγ , eδ)eβ) and Rαγ := g Rαβγδ respectively.