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Numerical Relativity: (Almost) All You Need to Know

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Numerical Relativity: (Almost) All You Need to Know Numerical relativity: (Almost) all you need to know Seth Hopper Vitor Cardoso Fig: Ian Hinder CERN - November 16, 2016 Binary black holes come in a few different varieties 396 Chapter 12 Binary black hole initial data quasi-circular plunge & ringdown inspiral merger 4 BHPT + 3 Post-Newtonian Theory PN ) h(t) r/M 2 Black Hole log( Perturbation 1 Numerical Theory t Relativity 0 01234 post-Newtonian numerical perturbation log(M/µ) techniques relativity methods Figure 12.1 The different phases of compact binary inspiral and coalescence. The gravitational wave amplitude Fig: Forseth et al. h(t) is sketched schematically and the analysisFig: Baumgarte technique is identified& Shapiro for each phase. Here ij is the reduced quadrupole moment, I 1 2 3 1 2 jk ρ x j xk δ jkr d x m A x j xk δ jkr , (12.7) I ≡ − 3 = A − 3 ! " # $ " # the bracket denotes an average over several orbital periods, and the triple dot denotes ⟨⟩ the third time derivative. For a binary at large separation we may treat the stars as point i masses and insert their stellar masses m A and Newtonian trajectories x (t) into equa- tion (12.7) to evaluate equations (12.5) and (12.6). For a binary orbit with eccentricity e the emission of gravitational radiation always leads to a decrease in the eccentricity, e˙ < 0.3 Put differently, gravitational radiation circularizes elliptical orbits.Thisresultimplies that during the late stages of compact binary inspiral, we may approximate the orbit as circular.4 Exercise 12.1 Consider a Newtonian binary consisting of two point masses m1 and m2 at a binary separation r. Write the binary’s Hamiltonian H(r, φ, Pr , Pφ), which is equal to its conserved energy E,as 1 P2 1 P2 µM E H r φ , (12.8) = = 2 µ + 2 µr 2 − r where M m1 m2 is the total mass and µ m1m2/M is the reduced mass. Define = + = $orb φ˙ to be the orbital angular velocity, and J Pφ to be the orbital angular momentum.≡ ≡ 3 Peters (1964); see also Lightman et al. (1975), Problem 18.7. 4 See also exercise 12.2,whichshowsthatoncetheybecomecircular,theorbitsremaincircularastheyshrinkinradius. This is a video simulating the second detection, GW151226 M1 = 14.2 M⊙ M2 = 7.5 M⊙ • Simulation: Simulating eXtreme Spacetime project MFinal = 20.8 M⊙ MGWs = 1 M⊙ Outline History and formalism Initial data Evolution of field equations Wave extraction The tools we use There is a long history of successes and failures in numerical relativity Darmois Shibata & Nakamura Arnowitt, Deser, & Lichnerowicz Pretorius Misner Baumgarte & Shapiro Hahn & Fourès-Bruhat Lindquist 1927 1944 1956 1962 1964 Long, long 1995 1998 2005 years pass “Grand Challenge” Here are two good resources 330 HAHN AND LINDQUIST This time is to be compared with the time torit at which the throat collapses to Hahn & Lindquistzero circumference did the : first work in 1964 terit W ?rm M 2$. We have used here a limiting form for Eq. (2.20), m m 7r2/2p:,TWO-BODY PROBLEM which IN GEOMETRODYNAMICS is valid 307 ANNALS OF PHYSICS: forf8, 304-331small (1964)~0 . By choosing ~0 sufficiently small, one can evidently make the ratio tl/tcrit as small as one pleases. The Two-BodyThis ratherProblem surprisingin Geometrodynamics result does not contradict the analysis of Fuller and Wheeler, SUSANfor it G.applies HAHN only to the case of strongly interacting masses. There are good reasons for regarding the latter system as a single compositeI / particle\ , with an (a) (b) International internalBusiness Machinesstructure”; Corporation, violationsNew York, Newof causalityYork within this super-particle would not be at variance ANDwith commonly accepted physical principles. Instead one should ask whether one can send a light ray from the asymptotic region through the RICHARD W. LINDQUIST wormhole and back out to infinity without encountering a singularity in the Adelphi University, Garden City, New York metric. We believe this to be impossible, although we have not been able(C) to prove FIG. 2. Three models for the two-body problems are displayed as two-dimensional sur- The problem itof twoconclusively. interacting masses is investigated within the framework faces imbedded in flat a-space. The first, (a), a direct generalization of Fig. 1, shows a pair of Einstein-Rosen bridges whose upper sheets are joined together. In (b) their lower sheets of geometrodynamics. In It summary,is assumed that thethe space-timenumerical continuum solutionis free of of the Einsteinhave been joined fieldas well, toequations form a multiply connectedpresents space whose upper and lower sheets all real sources of mass or charge; particles are identified with multiply con- are equivalent (i.e. isometric). If the two masses represented by the two mouths are equal, the figure will also be symmetric under reflections in a vertical plane midway between them. nected regions noof empty insurmountable space. Particular attentiondifficulties. is focused onMuch an asymp- still remains These tosymmetries be done,then allow onehowever, to identify corresponding in thepoints on the two sheets. and totically flat space containing a “handle” or “wormhole.” When the two also to identify corresponding points at the two mout,hs; the space which results is shown investigation both of stable difference schemes (a inproof (c). of stability being one of “mouths” of the wormhole are well separated, they seem to appear as two cen- ters of gravitational the attractionoutstanding of equal mass.unsolved To simplify problems)the problem, it andis of coordinatemodel was firstconditions proposed by Wheeler that (1) andare later calledwell by him a “wormhole” assumed that the metric is invariant under rotations about the axis of sym- manifold (2). In order that the two mouths shall match up sufficiently smoothly metry, and symmetricsuited with torespect numerical to the time work.t = 0 of maximumThe practicalseparation impossibility(i.e., analytically),of carrying they must bethe identical; numerical thus this model describes a universe of the two mouths. Analytic initial value data for this case have been ob- containing two particles of equal mass.5 tained by Misner; solutionthese contain sufficientlytwo arbitrary parameters,far into whichthe arefuture uniquely limits the conclusionsWhile there is no compellingwhich reasoncan for bepreferring drawn one model over another, we have focused attention exclusively on the wormhole picture. As far as the purely determined when aboutthe mass theof the dynamicaltwo mouths and theirbehavior initial separation of thehave wormhole system. Nevertheless, one sees been specified. We treat a particular case in which the ratio of mass to initial 6 The present investigation has been restricted to the case of pure gravitation, for which the appropriate field equations are those given in Eq. (l.la) with the right-hand side set separation is approximatelyevidence one-half.for a Togravitational determine a unique collapsesolution of theof each mouth,equal to zero analogous(i.e., with F,, = 0). Oneto can, thathowever, alsoof investigate the solutions to the corn remaining (dynamic) field equations, the coordinate conditions go- = -& are plete set of equations (1.1) on this manifold. In this more general approach it is also possible Schwarzschild metric, together with an interactionto trap electricbetween lines of force inthe the “wormhole,”two givingof them.the appearance of two particles with imposed; then the set of second order equations is transformed into a quasi- equal mass and equal but opposite charge. The time-symmetric initial value problem for this linear first order Thesesystem andtwo the effectsdifference schemecan onlyof Friedrichs be properlyused to ob- disentangledcase has beenthrough solved by Lindquistmeasurements (13). in tain a numerical solution. Its behavior agrees qualitatively with that of the one-body problem, theand asymptoticcan be interpreted region;as a mutual attractionwith theand pinching-limited data at our disposal, such an analysis off of the two mouthshas notof the beenwormhole. possible. I. INTRODUCTION Wheeler (1, 2) has used the term “geometrodynamics” to characterizeACKNOWLEDGMENTS those solutions of the field equations for gravitation and electromagnetism’ We are grateful to Professor J. A. Wheeler and Professor C. W. Misner for many valuable 41 =discussions R,v - ?4 g&about = 2(F,,FPthe formulation- Pi gj,.F,sF=B) of the problem(l.la) and interpretation of the results, and to Professor FPu;vR. 0. = Friedrichs0 for much useful (l.lb)advice on the mathematical approach. We also 1 Throughout this paperthank Greek Professorsubscripts andM. superscriptsFerentz rangefor hisfrom assistance0 to 3 and Latin with the analysis and programming during ones from 1 to 3. Also, theunits earlierare chosen stagesso that Gof (universalthe work.gravitation constant) = c = 1. 304 RECEIVED: December 9, 1963. REFERENCES 1. J. A. WHEELER, Phys. Rev. 97,511 (1955). 2. C. W. MISNER AND J. A. WHEELER, Ann. Phys. (N.Y.) 2, 525 (1957). 15 See, e.g., Brill and Lindquist (18). 2.2 Maxwell’s equations in Minkowski spacetime 27 We denote the 4-dimensional spacetime metric by gab, the 3-dimensional spatial metric by γij, and its conformally related metric byγ ¯ij. All of these are objects that we will encounter in later chapters. Four-dimensional objects associated with gab are denoted with (4) asuperscript2.2in Maxwell’s front of equations the symbol, in Minkowski objects spacetime associated with27 γ ¯ij carry a bar, and objects i ¯ i related to γij carry no decorations. For example, " jk is associated with γij, " jk withγ ¯ij, We denote the 4-dimensional(4) i spacetime metric by gab, the 3-dimensional spatial metric ¯ and " jk with gab.ThecovariantderivativeoperatorisdenotedwithDi and Di when by γij, and its conformallyassociated related with metric the spatial byγ ¯ij. metric All of and these the are conformally objects that related we will metric, respectively, but with encounter in later chapters. Four-dimensional objects associated with gab are denotedThe with 3+1 decompostion of (4) the nabla symbol a when associated with the 4-dimensional metric gab.Weoccasionally asuperscript in front of the symbol, objects∇flat associated withγ ¯ij carry a bar, and objects use the symbol # for thei flat scalar Laplace2 operator.¯ i related to γij carry no decorations.
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