Introduction to Numerical Relativity. Erik Schne+er Perimeter Ins1tute for Theore1cal Physics CGWAS 2013, Caltech,

Transcription

1 Introduction to Numerical Relativity Erik Schne+er Perimeter Ins1tute for Theore1cal Physics CGWAS 2013, Caltech,

2 What is Numerical Relativity? Solving Einstein equa1ons numerically Can handle arbitrarily complex systems Sub- field of Computa1onal Astrophysics Also beginning to be relevant in cosmology Einstein equa1ons relevant only for dense, compact objects: Black holes Neutron stars

3

4

5

6

7 Overview General Rela1vity: Geometry, Coordinates Solving the Einstein equa1ons Rela1vis1c Hydrodynamics Analyzing Space1mes: horizons, gravita1onal waves

8 Some relevant concepts GENERAL RELATIVITY

9 Einstein Equations G ab = 8π T ab G ab : Einstein tensor, one measure of curvature T ab : stress- energy tensor, describes mass/energy/momentum/ pressure/stress densi1es G ab and T ab are symmetric: 10 independent components in 4D Loose reading: (some part of) the space- 1me curvature equals ( is generated by ) its ma+er content

10 Spacetime Curvature Difference between special and general rela1vity: 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

11 Riemann, Ricci, Weyl Curvature is measured by Riemann tensor R abcd ; 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 R ab and Weyl tensor C abcd Ricci tensor directly (locally) determined by ma+er content (similar to div E and div B in electrodynamics); does not evolve Weyl tensor evolves freely, determined by ini1al and boundary condi1ons (describing gravita1onal interac1ons)

12 Metric Metric g ab is fundamental concept in GR Solving a space1me usually means calcula1ng its metric Everything (curvature, gravita1onal waves, horizons, ) can be calculated from the metric Conceptually (within theory of GR) similar to gravita1onal poten1al in Newtonian gravity, or poten1als Φ/A i in electrodynamics [blackboard sketch explaining meaning of metric] Metric g ab is symmetric, has 10 components in 4D Metric defines geometry: lengths and angles

13 Covariant Derivatives Par1al deriva1ves a are coordinate dependent In other words, given e.g. a vector field Φ a, the deriva1ve b Φ a does not transform like a tensor under a coordinate transforma1on Note: This is unrelated to curvature just a property when using non- uniform coordinate systems Covariant deriva1ves D a are coordinate independent ( covariant ) D b Φ a = b Φ a + Γ a bc Φc Christoffel symbol Γ a bc contains first deriva1ves of the metric A physically meaningful theory must be covariant (coordinate independent)

14 The Einstein Equations as Wave- Type Equations Can express the Riemann tensor, Ricci tensor etc. in terms of the metric: R ab = - ½ g cd c d g ab + (many other terms involving first and second deriva1ves of the metric) That is: Einstein equa1ons are 10 wave- type equa1ons (coupled, non- linear) for the metric components It is (in principle) well known how to solve such equa1ons (compare Maxwell equa1ons when wri+en in terms of poten1als) However, the Einstein equa1ons have many terms and are thus technically complex

15 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 equa1ons 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

16 Lapse and Shift The 4- metric g ab is split into lapse α, ship β i, and 3- metric γ ij (with simple geometric meanings) Introduce extrinsic curvature K ij of t=const hypersurfaces (essen1ally 1me deriva1ve of 3- metric) [blackboard sketch explaining lapse and ship] [blackboard sketch explaining extrinsic curvature] g 00 = - α 2 + γ ij β i β j g 0i = γ ij β j g ij = γ ij t γ ij = - 2 α K ij + D i β j + D j β i [D i is covariant deriva1ve]

17 Gauge Freedom Certain parts of the 3- metric γ ij and extrinsic curvature K ij do not influence curvature They can be chosen freely, and only influence the choice of coordinate system (e.g. div γ ij and trace K ij ) Called gauge degrees of freedom Similarly, lapse α and ship β i are not determined by Einstein equa1ons 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

18 Constraints It turns out that some of the Einstein equa1ons do not involve 1me deriva1ves, i.e. they are not evolu1on equa1ons These have to be sa1sfied 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 ma+er content Can be cast as ellip1c equa1ons (i.e. boundary value problems) 4 constraints in total Constraints need to be sa1sfied for ini1al condi1on If so, they remain sa1sfied during 1me evolu1on Unless there are numerical errors

19 ADM Formulation A common formula1on that casts the 4D Einstein equa1ons into a 3+1 1me evolu1on system ADM: named aper Arnowi+, Deser, Misner (1959) Developed by James York Note: several versions of ADM, beware e.g. sign conven1ons for K ij or R ij ADM system: 12 evolved variables: γ ij, K ij 4 gauge variables: α, β i their 1me deriva1ves A, B i 4 constraints H, M i 4 gauge condi1ons

20 Literature Wald, General Rela1vity Gourgoulhon, 3+1 Formalism and Bases of Numerical Rela1vity, arxiv:gr- qc/

21 Solving the Einstein Equa1ons numerically INITIAL VALUE PROBLEMS

22 Time Evolution Par1al Differen1al Equa1ons generally fall into one of two classes: Hyperbolic systems: ini1al value problems (IVP, 1me evolu1on) Ellip1c systems: boundary problems Einstein equa1ons form an IVP Made explicit by 3+1 split, ADM formula1on Numerical solvers can be viewed as black boxes: Input: Ini1al condi1on at t=0 Evolu1on equa1ons: t U = f(u, x U, ) Output: Solu1on for all 1mes t 0

23 Numerical Stability In principle, could now: Write down evolu1on equa1ons for γ ij, K ij (ADM system) Choose some simple gauge condi1ons α, β i (e.g. α=1, β i =0: normal coordinates) and solve the equa1ons 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 aper some 1me) Numerical rela1vity community struggled with this for a long 1me: hard mathema1cal problem

24 BSSN System Through trial and error, good evolu1on systems were found (family of BSSN formulabons) Through mathema1cal 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 equa1ons/gauge condi1ons listed in detail) in Brown et al., Phys. Rev. D 79, (2009)

25 BSSN Variables ADM variables: γ ij, K ij, α, β i, A, B i (20 evolved variables) BSSN uses a different set of variables (25 evolved variables): φ = 1 / 12 log det γ ij γ ~ ij = e- 4φ γ ij K = γ ij K ij A ~ ij = e- 4φ (K ij - ⅓ γ ij K) Γ ~I = - j γ ~ij α, β i remain A, B i are essen1ally 1me deriva1ves of α and β i Note: there are several slightly different BSSN variants; all work fine, but they have slightly different behaviour near black holes

26 BSSN Gauge Conditions Time evolu1on equa1ons of the BSSN variables are readily determined from their defini1ons and the ADM evolu1on equa1ons BSSN lapse condi1on: 1+log slicing Similar to a harmonic 1me coordinate Also called K driver: drives K (trace of K ij ) or t K to zero BSSN ship condi1on: Γ driver Somewhat similar to harmonic spa1al coordinates Drives Γ ~I or t Γ ~I to zero These gauge condi1ons can drive the coordinate system to being Cartesian (Minkowski) when space1me is flat

27 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 Γ ~I = - j γ ~ij [defini1on of Γ ~I ] As with the Hamiltonian and momentum constraint, they need to be sa1sfied 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

28 Initial Conditions Metric and extrinsic curvature need to sa1sfy 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 rela1vity (used for ini1al condi1ons, analysis, interac1on with GR hydro, etc.) York- Lichnerowicz procedure for constraint solving: Introduce certain new variables that render constraints ellip1c equa1ons Constraints can then be solved with standard methods Most generic way to solve constraints many other ways exist See e.g. Cook, LRR

29 Singularities Black hole space1mes contain singulari1es Loosely speaking, things become infinite at a singularity Mathema1cally, a singularity is a boundary of the domain, and certain field values may diverge near that boundary Physically, a singularity indicates that a theory is invalid/ inconsistent in this regime E.g. general rela1vity has not been tested for very high curvature, and one expects that quantum gravity will be necessary to describe this consistently Prac1cally, we have to live with singulari1es in our space1mes Cosmic censorship conjecture states that all singulari1es will be hidden behind an event horizon, i.e. will not affect anything that can be observed from far away

30 Singularities Method 1: Excision Nothing escapes an event horizon, i.e. event horizon is ou low boundary Cut out a part of the simula1on domain inside the event horizon Problem: Black holes move, handling irregularly shaped boundaries is complex Method 2: Punctures Separate solu1on into divergent and non- divergent parts E.g. conformal factor Handle divergent part analy1cally Problem: Black holes move, handling a divergent func1on analy1cally imposes inconvenient condi1ons onto simula1on

31 Singularities Method 3: Numerical dissipabon / moving punctures / Turduckening Nothing escapes an event horizon, i.e. can violate Einstein equa1ons inside Can e.g. smooth out solu1on inside (for ini1al data), or can prevent singularity from forming via dissipa1on (during 1me evolu1on) Works beau1fully in prac1ce, proven to be physically correct (since inside EH), was key contribu1on to successful binary black hole simula1ons Doesn t seem to work with harmonic formula1on

32 BSSN Summary BSSN is a family of evolu1on systems for the Einstein equa1ons Other formula1ons exist, e.g. harmonic formula1ons Good gauge condi1ons known Can use arbitrary ini1al condi1ons(if e.g. available in ADM variables) Can also easily calculate ADM variables, if needed by others BSSN includes simple way of handling singulari1es Good gauge condi1ons and singularity handling are probably the largest advantages of BSSN

33 RELATIVISTIC HYDRODYNAMICS

34 Relativistic Hydrodynamics So far, only discussed space1me geometry, i.e. vacuum (including black holes and gravita1onal waves) 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 Hydrodynamics derived from conserva1on laws Baryon number/fermion number/rest mass conserva1on Energy- momentum conserva1on [blackboard sketch describing conserva1on in 4D]

35 Stress- Energy Tensor Energy- momentum conserva1on: a T ab = 0 For a perfect fluid: T ab = ρ u a u b + p (g ab +u a u b ) ρ: density, p: pressure, u a : 4- velocity, g ab : 4- metric Requires also equa1ons of state for p(ρ,ε) ε: internal energy Baryon number/fermion number conserva1on yields equa1on for ρ Energy- momentum conserva1on yields equa1ons for u a and ε

36 Tensor Densities Conserva1on law: a nu a = 0 n: (Fermion, Baryon) number density; u a : 4- velocity Note that conserva1on does not depend on geometry Need to dis1nguish between Density- per- coordinate- volume (independent of geometry) Density- per- physical- volume (depends on geometry) g: volume element, physical volume per coordinate volume Mul1plying with g is called densibzing, turns tensors into tensor densi1es

37 Tensor Densities Conserva1on laws need to be wri+en in terms of tensor densi1es Philosophically: Want to describe hydrodynamics (conserva1on laws) as simple as possible, without referring to the space1me geometry Prac1cally: GR hydro equa1ons become much simpler if expressed via densi1zed quan11es Absorbing factor g into defini1on of variables E.g. total mass: M = ρ g= D ρ: physical density D: conserved ( densi1zed ) density

38 The equations of general relativistic hydrodynamics can be written in the flux conservative t q x if (i) (q) =s(q), (Q1.1) where q is a set of conserved variables, f (i) (q) the fluxes and s(q) the source terms. The five conserved variables are labeled D, S i, and, where D is the generalized particle number density, S i are the generalized momenta in each direction, and is an internal energy term. These conserved variables are composed from a set of primitive variables, which are, the rest-mass density, p, thepressure,v i,the fluid 3-velocities,, the specific internal energy, and W, the Lorentz factor, via the following relations D = p W S i = p hw 2 v i = p hw 2 p D, (Q1.2) where is the determinant of the spatial 3-metric ij and h p/. Only five of the primitive variables are independent. Usually the Lorentz factor is defined in terms of the velocities and the metric as W =(1 ijv i v j ) 1/2. Also one of the pressure, rest-mass density or specific internal energy terms is given in terms of the other two by an equation of state. The fluxes are usually defined in terms of both the conserved variables and the primitive variables: F i (U) = [D( v i i ),S j ( v i i )+p i j, ( v i i )+pv i ] T. (Q1.3) The source terms are s(u) = h 0,T µ g j + µ g j, T µ ln T µ 0 µ i. (Q1.4)

39 Conserved and Primitive Variables Conserved variables: D, S i, τ Necessary to express conserva1on laws, 1me evolu1on Primi1ve variables: ρ, v i, ε, p Necessary to describe microphysics, e.g. equa1on of state p(ρ,ε) Both conserved and primi1ve variables completely describe the hydrodynamic state Conver1ng from conserved to primi1ve variables ( con2prim ) is difficult and involves non- linear root- finding

40 Note on Momentum Conservation In a curved space, cannot define vector- valued conserva1on laws (momentum conserva1on, centre of mass velocity) [blackboard sketch calcula1ng centre of mass in polar coordinates]

41 Literature Gourgoulhon, An Introduc1on to Rela1vis1c Hydrodynamics, h+p://luth.obspm.fr/~luthier/gourgoulhon/pdf/gourg06.pdf Font, Numerical Hydrodynamics and Magnetohydrodynamics in General Rela1vity, lrr

42

43 Black holes, global quan11es, gravita1onal waves ANALYSIS METHODS

44 Event Horizons It is very difficult to find out where there are black holes in the simula1on domain! All one has in the metric (and extrinsic curvature) Event horizons are a teleological concept and hence open not useful The event horizon is the boundary of the region that can be observed from far away (future null infinity) The loca1on of an event horizon depends only on the future development of the space1me, not on the past There is no mathema1cal procedure (nor any kind of measurement) that one could perform to locate an event horizon There could be an event horizon passing through this room right now

45 Event Horizons The loca1on of an event horizon is known iff one knows the future of the space1me: 1. In sta1onary systems 2. Aper a numerical 1me evolu1on is complete Cannot be used to define ini1al data, cannot be used during 1me evolu1on Open, 1me evolu1on ends in sta1onary state Procedure for finding event horizons: Event horizon is null surface, made of light rays moving radially outwards Start in the future, then trace these light rays backwards in 1me

46 Thornburg, LRR Figure 2: This figure shows part of a simulation of the spherically symmetric collapse of a model stellar core (a = 5 3 polytrope) to a black hole. The event horizon (shown by the dashed line) was computed using the integrate null geodesics forwards algorithm described in Section 5.1; solid lines show outgoing null geodesics. The apparent horizon (the boundary of the trapped region, shown shaded) was computed using the zero-finding algorithm discussed in Section 8.1. The dotted lines show the world lines of Lagrangian matter tracers and are labeled by the fraction of baryons interior to them. Figure reprinted with permission from [142]. c 1980 by the American Astronomical Society.

47 Apparent Horizons There are many defini1ons for various kinds of horizons in GR Most useful probably apparent horizon (AH), or marginally trapped surface (MTS) Choose a closed 2- surface (distorted sphere) Send light rays outwards Observe how this sphere of light rays grows If its area stays constant (doesn t grow), it is an MTS Technicality: If there are several MTS, then the outermost is called AH, but terminology is loose [blackboard sketch describing MTS]

48 Apparent Horizons Event horizons are a property of the space1me; they are a global, powerful concept EH is 3D null surface EH begins at point/line, always grows, never ends Apparent horizons depend on a choice of folia1on (1me coordinate), and are thus coordinate dependent AH are 2D surfaces, each defined independently at its own 1me t, forming a 3D world tube AH world tube can begin/end any1me AH cannot be arbitrarily small We observe that AH world tubes form and annihilate in pairs Numerically, finding an AH means solving an ellip1c equa1on, and requires a good ini1al guess (horizon tracking)

49 Horizon Mass and Angular Momentum Can calculate mass and spin of AH, observe how black hole grows (Irreducible) Mass: essen1ally given by area of AH Will not decrease Angular momentum: more involved, need to find/define approximate Killing vector field (approximate axial symmetry) on horizon Several slightly differing defini1ons Angular momentum can increase or decrease Will only be conserved if horizon is axially symmetric Total mass: depends on both irreducible mass and angular momentum Can decrease if angular momentum decreases

50 ADM Quantities Can define total mass and angular momentum Cannot just integrate over space: singulari1es, equivalence principle! Instead, examine metric far away (near infinity) Far away, space1me will be flat Look closer, will see a perturba1on depending on total mass Look closer, will see a perturba1on depending on total angular momentum Difficult to calculate numerically, since need to evaluate (a) far away from source, and (b) have high accuracy Mostly used for ini1al condi1ons

51 Gravitational Waves Gravita1onal wave concept not well defined in strong field regime Need to have a background (possibly flat) and a perturbabon to define waves Gravita1onal waves are transverse and have two modes: h + and h Numerically, two possible defini1ons: Perturba1ve Curvature based

52 Figure 1: In Einstein s theory, gravitational waves have two independent polarizations. The e ect on proper separations of particles in a circular ring in the (x, y)-plane due to a plus-polarized wave traveling in the z-direction is shown in (a) and due to a cross-polarized wave is shown in (b). The ring continuously gets deformed into one of the ellipses and back during the first half of a gravitational wave period and gets deformed into the other ellipse and back during the next half. Sathyaprakash & Schutz, LRR

53 Gravitational Waves: Perturbative Calculation At large distances from a compact object, space1me will be simple will look like a black hole with linear perturba1ons: gravita1onal waves Procedure: Examine metric far away, decompose into background and perturba1on, read off wave content directly Need to know mass, angular momentum of background Need high accuracy

54 Gravitational Waves: Curvature Based Calculation Near infinity, most curvature components will decay away Those decaying most slowly are gravita1onal waves Fall off with 1/r, i.e. can carry away energy Procedure: Examine Weyl tensor far away Decompose it into 5 complex Weyl scalars Ψ n with different fall- off proper1es Ψ 4 falls off with 1/r Need to integrate to obtain wave content Mathema1cally elegant Weyl scalars are coordinate independent - - but Weyl scalars depend on choice of tetrad instead

55 Questions? Geometry: describe space1me via metric 3+1 decomposi1on to obtain ini1al value problem Gauge condi1ons, constraints Hydrodynamics: conserva1on laws Conserved and primi1ve variables Horizons, wave extrac1on Did not discuss: numerical methods, computa1onal issues