Introduction to Numerical Relativity Erik Schneer Perimeter Ins1tute for Theore1cal Physics CGWAS 2013, Caltech, 2013-07-26 What is Numerical Relativity? • Solving Einstein equaons numerically • Can handle arbitrarily complex systems • Sub-field of Computaonal Astrophysics • Also beginning to be relevant in cosmology • Einstein equaons relevant only for dense, compact objects: • Black holes • Neutron stars Overview • General Relavity: Geometry, Coordinates • Solving the Einstein equaons • Relavis1c Hydrodynamics • Analyzing Space1mes: horizons, gravitaonal waves Some relevant concepts GENERAL RELATIVITY Einstein Equations • Gab = 8π Tab • Gab: Einstein tensor, one measure of curvature • Tab: stress-energy tensor, describes mass/energy/momentum/ pressure/stress densi1es • Gab and Tab are symmetric: 10 independent components in 4D • Loose reading: (some part of) the space-1me curvature equals (“is generated by”) its maer content Spacetime Curvature • Difference between special and general relavity: in GR, space1me is curved • 2D example of a curved manifold: earth’s surface • Can’t use a straight coordinate system for a curved manifold! • E.g. Cartesian coordinate system doesn’t “fit” earth’s surface • In GR, one needs to use curvilinear, 1me-dependent coordinate systems • In fact, if one knows how to use arbitrary coordinate systems for a theory (e.g. hydrodynamics or electrodynamics), then working with this theory in GR becomes trivial Riemann, Ricci, Weyl • Curvature is measured by Riemann tensor Rabcd; has 20 independent components in 4D describing curvature completely • (unrelated to “Riemann problem” in hydrodynamics) • [blackboard sketch explaining how to measure curvature] • Can split Riemann tensor into two pieces: Ricci tensor Rab and Weyl tensor Cabcd • Ricci tensor directly (locally) determined by maer content (similar to div E and div B in electrodynamics); does not evolve • Weyl tensor evolves freely, determined by ini1al and boundary condi1ons (describing gravitaonal interac1ons) Metric • Metric gab is fundamental concept in GR • “Solving” a space1me usually means calculang its metric • Everything (curvature, gravitaonal waves, horizons, …) can be calculated from the metric • Conceptually (within theory of GR) similar to gravitaonal poten1al in Newtonian gravity, or poten1als Φ/Ai in electrodynamics • [blackboard sketch explaining meaning of metric] • Metric gab is symmetric, has 10 components in 4D • Metric defines geometry: lengths and angles Covariant Derivatives • Par1al derivaves ∂a are coordinate dependent a a • In other words, given e.g. a vector field Φ , the derivave ∂b Φ does not transform like a tensor under a coordinate transformaon • Note: This is unrelated to curvature – just a property when using non-uniform coordinate systems • Covariant derivaves Da are coordinate independent (“covariant”) a a a c • Db Φ = ∂b Φ + Γ bc Φ a • Christoffel symbol Γ bc contains first derivaves of the metric • A physically meaningful theory must be covariant (coordinate independent) The Einstein Equations as Wave-Type Equations • Can express the Riemann tensor, Ricci tensor etc. in terms of the metric: cd • Rab = -½ g ∂c ∂d gab + (many other terms involving first and second derivaves of the metric) • That is: • Einstein equaons are 10 wave-type equaons (coupled, non- linear) for the metric components • It is (in principle) well known how to solve such equaons (compare Maxwell equaons when wri+en in terms of poten1als) • However, the Einstein equaons have many terms and are thus technically complex 3+1 Decomposition • 4-vectors and 4-tensors are very elegant… • …but our everyday understanding, our astrophysical experience, and most numerical methods are based on space and 1me being separate… • …we thus split 4D space1me into 3D space and 1D 1me, so that the Einstein equaons look like a “conven1onal” theory. • Procedure: 1. Choose some 1me coordinate t for the space1me 2. Hypersurfaces t=const define a 3D space each (must be spacelike) 3. Split 4-vectors into scalar and 3-vector, 4-tensors into scalar, 3-vector, and 3-tensor Lapse and Shift i • The 4-metric gab is split into lapse α, ship β , and 3-metric γij (with simple geometric meanings) • Introduce extrinsic curvature Kij of t=const hypersurfaces (essen1ally 1me derivave of 3-metric) • [blackboard sketch explaining lapse and ship] • [blackboard sketch explaining extrinsic curvature] 2 i j • g00 = -α + γij β β j • g0i = γij β • gij = γij • ∂t γij = -2 α Kij + Di βj + Dj βi [Di is covariant derivave] Gauge Freedom • Certain parts of the 3-metric γij and extrinsic curvature Kij do not influence curvature • They can be chosen freely, and only influence the choice of coordinate system (e.g. div γij and trace Kij) • Called gauge degrees of freedom • Similarly, lapse α and ship βi are not determined by Einstein equaons • They select how the gauge degree of freedoms evolve in 1me • Can thus be chosen freely as well • 4 gauge degrees of freedom in total • Compare to choice of gauge freedom (div A) for vector poten1al A in electrodynamics Constraints • It turns out that some of the Einstein equaons do not involve 1me derivaves, i.e. they are not evolu1on equaons • These have to be sasfied at all 1mes • Compare to div E, div B in electrodynamics • Called Hamiltonian constraint (energy constraint) (scalar) and momentum constraint (3-vector) • Connec1ng curvature and maer content • Can be cast as ellip1c equaons (i.e. boundary value problems) • 4 constraints in total • Constraints need to be sasfied for ini1al condi1on • If so, they remain sasfied during 1me evolu1on • unless there are numerical errors ADM Formulation • A common formulaon that casts the 4D Einstein equaons into a 3+1 1me evolu1on system • ADM: named aer Arnowi+, Deser, Misner (1959) • Developed by James york • Note: several versions of ADM, beware e.g. sign conven1ons for Kij or Rij • ADM system: • 12 evolved variables: γij, Kij • 4 “gauge” variables: α, βi • their 1me derivaves A, Bi • 4 constraints H, Mi • 4 gauge condi1ons Literature • Wald, General Relavity • Gourgoulhon, 3+1 Formalism and Bases of Numerical Relavity, arXiv:gr-qc/0703035 Solving the Einstein Equaons numerically INITIAL VALUE PROBLEMS Time Evolution • Par1al Differen1al Equaons generally fall into one of two classes: • Hyperbolic systems: ini1al value problems (IVP, 1me evolu1on) • Ellip1c systems: boundary problems • Einstein equaons form an IVP • Made explicit by 3+1 split, ADM formulaon • Numerical solvers can be viewed as black boxes: • Input: • Ini1al condi1on at t=0 • Evolu1on equaons: ∂tu = f(u, ∂xu, …) • Output: • Solu1on for all 1mes t≥0 Numerical Stability • In principle, could now: • Write down evolu1on equaons for γij, Kij (ADM system) • Choose some simple gauge condi1ons α, βi (e.g. α=1, βi=0: normal coordinates) • …and solve the equaons on a computer • However, it turns out that this is impossible for two (unrelated) instabili1es: • ADM system amplifies small errors in constraints (which are numerically always present) • Such simple gauge condi1ons lead to coordinate singulari1es (coordinate lines cross aer some 1me) • Numerical relavity community struggled with this for a long 1me: hard mathemacal problem BSSN System • Through trial and error, good evolu1on systems were found (family of BSSN formulaBons) • Through mathemacal analysis, another good class of evolu1on systems was found (harmonic formulaBons) • It turns out that stability crucially depends on the gauge condi1ons, hence lapse and ship cannot really be chosen freely either • Literature: Well-posedness of the BSSN system proven (and BSSN equaons/gauge condi1ons listed in detail) in Brown et al., Phys. Rev. D 79, 044023 (2009) BSSN Variables i i • ADM variables: γij, Kij, α, β , A, B (20 evolved variables) • BSSN uses a different set of variables (25 evolved variables): 1 • φ = /12 log det γij ~ -4φ • γ ij = e γij ij • K = γ Kij ~ -4φ • A ij = e (Kij - ⅓ γij K) ~I ~ij • Γ = - ∂j γ • α, βi remain • A, Bi are essen1ally 1me derivaves of α and βi • Note: there are several slightly different BSSN variants; all work fine, but they have slightly different behaviour near black holes BSSN Gauge Conditions • Time evolu1on equaons of the BSSN variables are readily determined from their defini1ons and the ADM evolu1on equaons • BSSN lapse condi1on: 1+log slicing • Similar to a harmonic 1me coordinate • Also called K driver: drives K (trace of Kij) or ∂t K to zero • BSSN shi condion: Γ driver • Somewhat similar to harmonic spaal coordinates ~I ~I • Drives Γ or ∂t Γ to zero • These gauge condi1ons can drive the coordinate system to being Cartesian (Minkowski) when space1me is flat BSSN Constraints • Since the BSSN system has more variables, it also introduces new constraints (in addi1on to Hamiltonian and momentum constraint): ~ • det γ ij = 1 ~ • trace A ij = 0 ~I ~ij ~I • Γ = - ∂j γ [definion of Γ ] • As with the Hamiltonian and momentum constraint, they need to be sasfied ini1ally, and will then be preserved during evolu1on • BSSN contains divergence cleaning terms ~ • Numerically, trace A ij = 0 needs to be enforced explicitly; the other constraints are well-behaved on their own Initial Conditions • Metric and extrinsic curvature need to sasfy constraints • Thus need to solve constraints before evolu1on – highly non-trivial step! • Typically, ini1al condi1ons are prepared in terms of ADM variables, then converted to BSSN, then evolved • ADM variables are “common language” in numerical relavity (used for ini1al condi1ons, analysis, interac1on with GR hydro, etc.) • York-Lichnerowicz procedure for