100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

156 P. Lagunawas originally introduced by Shibata and Nakamura, 54 and later reintroducedby Baumgarte and Shapiro. 10 This formulation is commonly knownas the BSSN formulation. In general terms, the essence of the BSSN formulationis to work not with the pair (h ij , K ij ) used by the ADM formulationbut instead with (Φ, ĥij, K, Âij, ̂Γ i ). The relation between ADM and BSSNvariables are:Φ = ln h 1/12 (2)ĥ ij = e −4Φ h ij (3)K = K i i (4)Â ij = e −4Φ A ij (5)̂Γ i = −∂ j ĝ ij , (6)where A ij = K ij − h ij K/3. Working explicitly with the scalars Φ and Kis only one of the crucial steps in the BSSN equations. The other is theintroduction of the connection variable ˜Γ i . It is this variable that is mostlyresponsible for improving the hyperbolic properties of the BSSN systemover those of the ADM system. 34The BSSN formulation has been quite successful. Codes based on theseequations have produced simulations of binary neutron stars, 55,43,42 wobblingblack holes, 60 boosted black holes, 59 distorted black holes, 3 blackholes head on collisions, 5,27,58 black hole plunges 5 and binary black holeevolutions with angular momentum up to timescales of a single orbit. 21The second popular 3+1 formulation of the Einstein equation is the hyperbolicformulation developed by Kidder, Scheel and Teukolsky (KST). 39Strictly speaking the KST system is a parameterized family of hyperbolicformulations. This formulation has had remarkable success in evolving singleblack holes. It is expected that in the near future this success will betranslated to evolutions of binary black holes. The starting point in derivingthe KST system is the introduction of a new auxiliary variable d kij ≡ ∂ k h ij ,to eliminate second derivatives of the spatial metric. With this variable, theEinstein equations can be rewritten as:∂ t u = A i ∂ i u + ρ (7)where u is a vector constructed from the evolution variables, the matricesA i (u) determine the hyperbolic properties of the system and ρ(u) denotesthe lower order terms, namely terms that do not contain derivatives ofu. With a suitable choice of parameters, the KST system can be makesymmetric hyperbolic.