Gravitational Wavesand Their Mathematics

Lydia Bieri, David Garfinkle,Nicolas Yunes

A version of this article was published in

the AMS Notices, Vol. 64, Issue 07, 2017. 1

1 Introduction

In 2015 gravitational waves were detected forthe first time by the LIGO-Virgo collaboration[1]. This triumph happened 100 years after Al-bert Einstein’s formulation of the Theory of Gen-eral Relativity [38, 39, 40] and 99 years after hisprediction of gravitational waves [41]. This ar-ticle focuses on the mathematics of Einstein’sgravitational waves, from the properties of theEinstein vacuum equations and the initial valueproblem (Cauchy problem), to the various ap-proximations used to obtain quantitative predic-tions from these equations, and eventually an ex-perimental detection.

General Relativity is studied as a branch of as-tronomy, physics, and mathematics. At its coreare the Einstein equations, which link the physi-cal content of our universe to geometry. By solv-ing these equations, we construct the spacetimeitself, a continuum that relates space, time, ge-ometry, and matter (including energy). The dy-namics of the gravitational field are studied in theCauchy problem for the Einstein equations, rely-ing on the theory of nonlinear partial differentialequations (pde) and geometric analysis. The con-nections between astronomy, physics, and math-ematics are richly illustrated by the story of grav-itational radiation.

In General Relativity, the universe is describedas a spacetime manifold with a curved metricwhose curvature encodes the properties of thegravitational field. While sometimes one wantsto use General Relativity to describe the wholeuniverse, often we just want to know how a singleobject or small collection of objects behaves. To

1A version of this article (with more graphicsand photos) was published in the AMS Notices,Vol. 64, Issue 07, 2017, (August issue 2017). Itis available for free to download from the jour-nal’s website under http://www.ams.org/publications/

journals/notices/201707/rnoti-p693.pdf

The full August 2017 issue is available under http://www.ams.org/journals/notices/201707/index.html

Figure 1: Albert Einstein predicted gravitationalwaves in 1916.

address that kind of problem, we use the ideal-ization of the isolated system: a spacetime con-sisting of just the objects we want to study andnothing else. We might consider the solar sys-tem as an isolated object, or a pair of black holesspiraling into one another until they collide. Weask how those objects look to a distant, far awayobserver in a region where presumably the cur-vature of spacetime is very small. Gravitationalwaves are vibrations in spacetime that propa-gate at the speed of light away from their source.They may be produced, for example, when blackholes merge. This is what was first detected byAdvanced LIGO (aLIGO) and this is the focusof this article.

First we describe the basic differential geom-etry used to define the universe as a geometricobject. Next we describe the mathematicalproperties of the Einstein vacuum equations,including a discussion of the Cauchy problemand gravitational radiation. Then we turn to thevarious approximation schemes used to obtainquantitative predictions from these equations.We conclude with the experimental detectionof gravitational waves and the astrophysicalimplications of this detection. This detection isnot only a spectacular confirmation of Einstein’stheory, but also the beginning of the era of gravi-tational wave astronomy, the use of gravitational

1

arX

iv:1

710.

0327

2v1

[gr

-qc]

9 O

ct 2

017

waves to investigate aspects of our universe thathave been inaccessible to telescopes.

2 The Universe as a Geomet-ric Object

A spacetime manifold is defined to be a 4-dimensional, oriented, differentiable manifold Mwith a Lorentzian metric tensor, g, which is anon-degenerate quadratic form of index one,

g =

3∑µ,ν=0

gµνdxµ ⊗ dxν ,

defined in TqM for every q in M varyingsmoothly in q. The trivial example, theMinkowski spacetime as defined in Einstein’sSpecial Relativity, is R4 endowed with the flatMinkowski metric:

g = η = −c2dt2 + dx2 + dy2 + dz2. (1)

Taking x0 = t, x1 = x, x2 = y and x3 = z, wehave η00 = −c2, ηii = 1 for i = 1, 2, 3 and ηµν = 0for µ 6= ν. In mathematical General Relativitywe often normalize the speed of light, c = 1.

The family of Schwarzschild metrics are solu-tions of the Einstein vacuum equations that de-scribe spacetimes containing a black hole, wherethe parameter values are M > 0. Taking rs =2GM/c2, it has the metric:

g = −c2

(1− rs

4ρ

)2(

1 + rs4ρ

)2 dt2 +(

1 + rs4ρ

)4h, (2)

where ρ2 = x2+y2+z2, h = dx2+dy2+dz2, andG denotes the Newtonian gravitational constant.This space is asymptotically flat as ρ→∞.

The Friedmann-Lemaıtre-Robertson-Walkerspacetimes describe homogeneous and isotropicuniverses through the metric

g = −c2dt2 + a2(t)gχ, (3)

where gχ is a Riemannian metric with constantsectional curvature, χ, (e.g. a sphere when χ =1) and a(t) describes the expansion of the uni-verse. The function, a(t), is found by solving theEinstein equations as sourced by fluid matter.

In an arbitrary Lorentzian manifold, M , a vec-tor X ∈ TxM is called null or lightlike if

gx(X,X) = 0.

At every point there is a cone of null vectorscalled the null cone. A vector X ∈ TxM is calledtimelike if

gx(X,X) < 0,

and spacelike if

gx(X,X) > 0.

In General Relativity nothing travels faster thanthe speed of light, so the velocities of masslessparticles are null vectors whereas those for mas-sive objects are timelike. A causal curve is adifferentiable curve for which the tangent vectorat each point is either timelike or null.

A hypersurface is called spacelike if its normalvector is timelike, so that the metric tensor re-stricted to the hypersurface is positive definite.A Cauchy hypersurface is a spacelike hypersur-face where each causal curve through any pointx ∈ M intersects H exactly at one point. Aspacetime (M, g) is said to be globally hyperbolicif it has a Cauchy hypersurface. In a globallyhyperbolic spacetime, there is a time function twhose gradient is everywhere timelike or null andwhose level surfaces are Cauchy surfaces. A glob-ally hyperbolic spacetime is causal in the sensethat no object may travel to its own past.

As in Riemannian geometry, curves with 0acceleration are called geodesics. Light travelsalong null geodesics. Geodesics which enter theevent horizon of a black hole never leave. Objectsin free fall travel along timelike geodesics. Theyalso can never leave once they have entered ablack hole. When two black holes fall into one an-other, they merge and form a single larger blackhole.

In curved spacetime, geodesics bend togetheror apart and the relative acceleration betweengeodesics is described by the Jacobi equation,also known as the geodesic deviation equation.In particular, the relative acceleration of nearbygeodesics is given by the Riemann curvature ten-sor times the distance between them. The Riccicurvature tensor, Rµν , measures the average wayin which geodesics curve together or apart. Thescalar curvature, R, is the trace of the Ricci cur-vature.

Einstein’s field equations are:

Rµν − 12Rgµν =

8πG

c4Tµν , (4)

where Tµν denotes the energy-momentum tensor,which encodes the energy density of matter. Notethat for cosmological considerations, one can add

2

Λgµν on the left hand side, where Λ is the cosmo-logical constant. However, nowadays, this termis commonly absorbed into Tµν on the right handside. Here we will consider the non-cosmologicalsetting. One then solves the Einstein equationsfor the metric tensor gµν . If there are no otherfields, then Tµν = 0 and (4) reduce to the Ein-stein vacuum equations:

Rµν = 0 . (5)

Note that the Einstein Equation is a set ofsecond order quasilinear partial differential equa-tions for the metric tensor. In fact, when choos-ing the right coordinate chart (wave coordi-nates), taking c = 1, and writing out the formulafor the curvature tensor, Rµν , in those coordi-nates, the equation becomes:

�ggαβ = Nαβ (6)

where �g is the wave operator and Nαβ =Nαβ(g, ∂g) denote nonlinear terms with quadrat-ics in ∂g.

Quite a few exact solutions to the Einstein vac-uum equations are known. Among the most pop-ular are the trivial solution (Minkowski space-time) as in (1), the Schwarzschild solution, whichdescribes a static black hole, as in (2), and theKerr solution, which describes a black hole withspin angular momentum. Note that the exteriorgravitational field of any spherically symmetricobject takes the form of (2) for r > r0 wherer0 > rs is the radius of the object, so this modelcan be used to study the spacetime around anisolated star or planet. However, in order to un-derstand the dynamics of the gravitational fieldand radiation, we have to investigate large classesof spacetimes. This can only be done by solvingthe initial value problem (Cauchy problem) forthe Einstein equations, which will be discussedin the next section.

If there are matter fields, so that Tµν 6= 0, thenthese fields satisfy their own evolution equations,which have to be solved along with the Einsteinfield equations (4) as a coupled system. Thescale factor, a(t), of the Friedmann-Lemaıtre-Robertson-Walker cosmological spacetimes in (3)can then be found by solving a second order, or-dinary differential equation derived from (4). Ifa solution has a time where the scale factor van-ishes, then the solution is said to describe a cos-mos whose early phase is a “big bang.”

Figure 2: Albert Einstein

3 The Einstein Equations

3.1 Beginnings of Cauchy Problem

In order to study gravitational waves, stabilityproblems, and general questions about the dy-namics of the gravitational field, we have to for-mulate and solve the Cauchy problem. That is weare given initial data: a prescribed Riemannianmanifold H with a complete Riemannian met-ric gij and a symmetric 2-tensor Kij satisfyingcertain consistency conditions called the Einsteinconstraint equations. We then solve for a space-time (M, g) which satisfies the Einstein equationsevolving forward from this initial data set. Thatis, the given Riemannian manifold H is a space-like hypersurface in this spacetime solution M ,where g is the restriction of g. Furthermore, thesymmetric two tensor Kij is the prescribed sec-ond fundamental form.

All the different methods used to describegravitational radiation have to be thought of asembedded into the aim of solving the Cauchyproblem. We solve the Cauchy problem by meth-ods of analysis and geometry. However, for sit-uations where the geometric-analytic techniquesare not (yet) at hand, one uses approximationmethods and numerical algorithms. The goal ofthe latter methods is to produce approximationsto solutions of the Cauchy problem for the Ein-

3

stein equations.

In order to derive the gravitational waves frombinary black hole mergers, binary neutron starmergers, or core-collapse supernovae, we describethese systems by asymptotically flat spacetimes.These are solutions of the Einstein equations thatat infinity tend to Minkowski space with a metricas in (1). Schwarzschild space is a simple exam-ple of such an isolated system containing onlya single stationary black hole (2). There is ahuge literature about specific fall-off rates whichwe will not describe here. The null asymptoticsof these spacetimes contain information on grav-itational radiation (gravitational waves) out toinfinity.

Recall that the Einstein vacuum equations (5)are a system of ten quasilinear, partial differen-tial equations that can be put into hyperbolicform. However, with the Bianchi identity impos-ing four constraints, the Einstein vacuum system(5) constitutes only six independent equationsfor the ten unknowns of the metric gµν . Thiscorresponds to the general covariance of the Ein-stein equations. In fact, uniqueness of solutionsto these equations holds up to equivalence un-der diffeomorphisms. We have just found a corefeature of General Relativity. This mathematicalfact also means that physical laws do not dependon the coordinates used to describe a particularprocess.

The Einstein equations split into a set of evolu-tion equations and a set of constraint equations.As above, t denotes the time coordinate whereasindices i, j = 1, · · · , 3 refer to spatial coordinates.Taking c = 1, the evolution equations read:

∂gij∂t

= −2ΦKij + LX gij (7)

∂Kij∂t

= −∇i∇jΦ + LXKij

+ (Rij + Kij trK − 2KimKmj )Φ (8)

Here Kij is the extrinsic curvature of the t =const. surface H as above. The lapse Φ andshift X are essentially the gtt and gti componentsof the metric, and are given by T = Φn + Xwhere T is the evolution vector field ∂/∂t and nis the unit normal to the constant time hypersur-face. ∇i is the spatial covariant derivative andL is the Lie derivative. However, the initial data(gij ,Kij) cannot be chosen freely: the remain-ing four Einstein vacuum equations become the

following constraint equations:

∇iKij − ∇j trK = 0 (9)

R + (trK)2 − |K|2 = 0 . (10)

An initial data set is a 3-dimensional manifoldH with a complete Riemannian metric gij and asymmetric 2-tensor Kij satisfying the constraintequations ((9), (10)). We will evolve an asymp-totically flat initial data set (H, gij ,Kij), thatoutside a sufficiently large compact set D, H\Dis diffeomorphic to the complement of a closedball in R3 and admits a system of coordinateswhere gij → δij and Kij → 0 sufficiently fast.

It took a long time before the Cauchy problemfor the Einstein equations was formulated cor-rectly and understood. Geometry and pde the-ory were not as developed as they are today, andthe pioneers of General Relativity had to strugglewith problems that have elegant solutions nowa-days. The beauty and challenges of General Rel-ativity attracted many mathematicians, as for in-stance D. Hilbert or H. Weyl, to work on GeneralRelativity’s fundamental questions. Weyl in 1923talked about a “causally connected” world, whichhints at issues that the domain of dependencetheorem much later would solve. G. Darmois inthe 1920s studied the analytic case, which is notphysical but a step in the right direction. He rec-ognized that the analyticity hypothesis is physi-cally unsatisfactory, because it hides the propa-gation properties of the gravitational field. With-out going into details, important work followedby K. Stellmacher, K. Friedrichs, T. de Donder,and C. Lanczos. The latter two introduced wavecoordinates, which Darmois later used. In 1939,A. Lichnerowicz extended Darmois’ work. Healso suggested the extension of the 3+1 decompo-sition with non-zero shift to his student YvonneChoquet-Bruhat, which she carried out.

Choquet-Bruhat, encouraged by Jean Leray in1947, searched for a solution to the non-analyticCauchy problem of the Einstein equations, whichturned into her famous result of 1952. There aremany more players in this game that should bementioned, but there is not enough space to dojustice to their work. These works also built onprogress in analysis and pde theory by H. Lewy,J. Hadamard, J. Schauder, and S. Sobolev amongmany others. Details on the history of the proofcan be found in Choquet-Bruhat’s survey articlepublished in [27] Surveys in Differential Geome-try 2015: One hundred years of general relativity,and more historical background (including a dis-cussion between Choquet-Bruhat and Einstein)

4

is given in Choquet-Bruhat’s forthcoming auto-biography.

In 1952 Choquet-Bruhat [26] proved a localexistence and uniqueness theorem for the Ein-stein equations, and in 1969 Choquet-Bruhat andR. Geroch [28] proved the global existence ofa unique maximal future development for everygiven initial data set.

Theorem 1 (Choquet-Bruhat, 1952) Let(H, g,K) be an initial data set satisfying thevacuum constraint equations. Then there exists aspacetime (M, g) satisfying the Einstein vacuumequations with H ↪→M being a spacelike surfacewith induced metric g and second fundamentalform K.

This was proven by finding a useful coordi-nate system, called wave coordinates, in whichEinstein’s vacuum equations appear clearly asa hyperbolic system of partial differential equa-tions. The pioneering result by Choquet-Bruhatwas improved by Dionne (1962), Fisher-Marsden(1970), and Hughes-Kato-Marsden (1977) usingthe energy method.

3.2 Global Cauchy Problem

Choquet-Bruhat’s local theorem 1 of 1952 wasa breakthrough and has since been fundamen-tal for further investigations of the Cauchy prob-lem. Once we have local solutions of the Einsteinequations, do they exist for all time, or do theyform singularities? And of what type would thelatter be? In 1969 Choquet-Bruhat and Gerochproved there exists a unique, globally hyperbolic,maximal spacetime (M, g) satisfying the Einsteinvacuum equations with H ↪→M being a Cauchysurface with induced metric g and second funda-mental form K. This unique solution is called themaximal future development of the initial dataset.

However, there is no information about thebehavior of the solution. Will singularities oc-cur or will it be complete? One would expectthat sufficiently small initial data evolves foreverwithout producing any singularities, whereas suf-ficiently large data evolves to form spacetime sin-gularities such as black holes. From a mathe-matical point of view the question is whethertheorems can be proven that establish this be-havior. A breakthrough occurred in 2008 withChristodoulou’s proof [31], building on an earlierresult due to Penrose [71], that black hole singu-larities form in the Cauchy development of ini-

tial data, which do not contain any singularities,provided that the incoming energy per unit solidangle in each direction in a suitably small timeinterval is sufficiently large. This means that ablack hole forms through the focussing of gravi-tational waves. This result has since been gener-alized by various authors, and the main methodshave been applied to other nonlinear pdes.

The next burning question to ask is whetherthere is any asymptotically flat (and non-trivial)initial data with complete maximal development.This can be thought of as a question about theglobal stability of Minkowski space. In their cel-ebrated work [32] of 1993 D. Christodoulou andS. Klainerman proved the following result, whichhere we state in a very general way. The detailsare intricate and the smallness assumptions arestated for weighted Sobolev norms of the geomet-ric quantities.

Theorem 2 (Christodoulou and Klainerman,1993, [32]) Given strongly asymptotically flat ini-tial data for the Einstein vacuum equations (5),which is sufficiently small, there exists a unique,causally geodesically complete and globally hy-perbolic solution (M, g), which itself is globallyasymptotically flat.

The proof relies on geometric analysis and isindependent of coordinates. First, energies areidentified with the help of the Bel-Robinson ten-sor, which basically is a quadratic of the Weylcurvature. Then, the curvature components areestimated in a comparison argument using theenergies. Finally, in a large bootstrap argumentwith assumptions on the curvature, the remain-ing geometric quantities are proven to be con-trolled. The proof comprises various new ideasand features that became important not only forfurther studies of relativistic problems but alsoin other nonlinear hyperbolic pdes.

The Christodoulou-Klainerman result of the-orem 2 was generalized in 2000 by Nina Zipser[98], [99] for the Einstein-Maxwell equations andin 2007 by Lydia Bieri [11], [12] for the Einsteinvacuum equations assuming less on the decay atinfinity and less regularity. Thus, the latter re-sult establishes the borderline case for decay ofinitial data in the Einstein vacuum case. Bothworks use geometric analysis in a way that is in-dependent of any coordinates.

Next, let us go back to the pioneering resultsby Choquet-Bruhat and Geroch, and say a fewwords about further extensions of these works.A standard result ensures that for an Einstein

5

vacuum initial data set (H0, g,K) with H0 al-lowing to be covered by a locally finite systemof coordinate charts with transformations beingC1-diffeomorphisms, and

gmn|H0 ∈ Hkloc , ∂0gmn|H0 ∈ Hk−1

loc , k >5

2,

(11)there exists a unique globally hyperbolic solu-tion with H0 being a Cauchy hypersurface. Sev-eral improvements followed, including those byBahouri-Chemin [8], Tataru [85], Smith-Tataru[80], and then by Klainerman-Rodnianski [57],[58]. The latter proved that for the same prob-lem but with k > 2 there exists a time in-terval [0, T ] and a unique solution g such thatgmn ∈ C0([0, T ], Hk) where T depends only on||gmn|H0 ||Hk +||∂0gmn|H0 ||Hk−1 . Recently the L2

curvature conjecture was proven by Klainerman-Rodnianski-Szeftel [59], [60], [81], [82], [83], [84]:under certain assumptions they relax the regular-ity condition such that the time of existence ofthe solution depends only on the L2-norms of theRiemann curvature tensor and on the gradient ofthe second fundamental form.

For our purposes, we want to know what theglobal existence theorem says about the proper-ties of radiation, i.e. the behavior of curvatureat large distances. In particular, because we ex-pect gravitational radiation to propagate at thespeed of light, we would like to study the behav-ior at large distances along outgoing light rays.This sort of question was addressed long beforeChristodoulou and Klainerman. However, theseworks assume a lot on the spacetimes considered.As a consequence, components of the Riemanncurvature tensor show a specific hierarchy of de-cay in r. The spacetimes of the Christodoulou-Klainerman theorem do not fully satisfy theseproperties, showing only some of the fall-off butnot all. In fact, Christodoulou showed that phys-ical spacetimes cannot fulfill the stronger decay.The results by Christodoulou-Klainerman pro-vide a precise description of null infinity for phys-ically interesting situations.

3.3 Gravitational Radiation

In this section, we consider radiative space-times with asymptotic structures as derived byChristodoulou-Klainerman [32]. The asymp-totic behavior of gravitational waves near infin-ity [17, 78, 70, 47, 48] approximates how gravita-tional radiation emanating from a distant blackhole merger would appear when observed by

aLIGO. Asymptotically the gravitational wavesappear to be planar, stretching and shrinking di-rections perpendicular to the wave’s travel direc-tion.

As an example, let us consider the merger oftwo black holes. Long before the merger, the to-tal energy of the two-black-hole spacetime, theso-called ADM energy or “mass,” named for itscreators Arnowitt-Deser-Misner [6], is essentiallythe sum of the masses of the individual blackholes. During the merger, energy and momen-tum are radiated away in the form of gravita-tional waves. After the merger, once the waveshave propagated away from the system, the en-ergy left in the system, the so-called Bondi mass,decreases and this can be calculated throughthe formalism introduced by Bondi, Sachs, andTrautman.

Gravitational radiation travels along null hy-persurfaces in the spacetime. As the source isvery far away from us, we can think of thesewaves as reaching us (the experiment) at null in-finity, which is defined as follows.

Definition 1 Future null infinity I+ is de-fined to be the endpoints of all future-directed nullgeodesics along which r →∞. It has the topologyof R× S2 with the function u taking values in R.

A null hypersurface Cu intersects I+ at infinity ina 2-sphere. To each Cu at null infinity is assigneda Trautman-Bondi mass M(u), as introduced byBondi, Trautman, and Sachs in the middle ofthe last century. This quantity measures theamount of mass that remains in an isolated grav-itational system at a given retarded time, i.e. theTrautman-Bondi mass measures the remainingmass after radiation through I+ up to u. TheBondi mass-loss formula reads for u1 ≤ u2

M (u2) = M (u1)− C∫ u2

u1

∫S2

|Ξ|2 dµ◦γdu (12)

with |Ξ|2 being the norm of the shear tensorat I+ and dµ◦

γthe canonical measure on S2.

If other fields are present, like electromagneticfields, then the formula contains a correspondingterm for that field. In the situations consideredhere, it has been proven that limu→−∞M(u) =MADM .

The effects of gravitational waves on neigh-boring geodesics is encoded in the Jacobi equa-tion. This very fact is at the heart of the detec-tion by aLIGO and is discussed in Sec. 6. Fromthis, we derive a formula for the displacement of

6

test masses, while the wave packet is travelingthrough the apparatus. This is what was mea-sured by the aLIGO detectors.

Now, there is more to the story. From the anal-ysis of the spacetime at I+ one can prove that thetest masses will go to rest after the gravitationalwave has passed, meaning that the geodesics willnot be deviated anymore. However, will the testmasses be at the “same” position as before thewave train passed or will they be dislocated? Inmathematical language, will the spacetime geom-etry have changed permanently? If so, then thisis called the memory effect of gravitational waves.This effect was first computed in 1974 by Ya.B.Zel’dovich and A.G. Polnarev in the linearizedtheory [96], where it was found to be very smalland considered not detectable at that time.

In 1991 D. Christodoulou, studying the fullnonlinear problem [30], showed that this effectis larger than expected and could in principle bemeasured. Bieri and Garfinkle showed [13] thatthe formerly called “linear” (now ordinary) and“nonlinear” (now null) memories are two differ-ent effects, the former sourced by the difference ofa specific component of the Weyl tensor, and thelatter due to fields that do reach null infinity I+.In the case of the Einstein vacuum equations, thisis the shear appearing in (12). In particular, thepermanent displacement (memory) is related to

F = C

∫ +∞

−∞|Ξ(u)|2 du (13)

where F/4π denotes the total energy radiated ina given direction per unit solid angle. A very re-cent paper by P.D. Lasky, E. Thrane, Y. Levin, J.Blackman, and Y. Chen [61] suggests a methodfor detecting gravitational wave memory withaLIGO.

4 Approximation Methods

To compare gravitational wave experimentaldata to the predictions of the theory, one needs acalculation of the predictions of the theory. It isnot enough to know that solutions of the Einsteinfield equations exist; rather, one needs quantita-tive solutions of those equations to at least theaccuracy needed to compare to experiments. Inaddition, sometimes the gravitational wave sig-nal is so weak that to keep it from being over-whelmed by noise one must use the techniqueof matched filtering [52] in which one looks formatches between the signal and a set of templates

of possible expected waveforms. These quantita-tive solutions are provided by a set of overlappingapproximation techniques, and by numerical sim-ulations. We will discuss the approximation tech-niques in this section and the numerical methodsin section 5.

4.0.1 Linearized Theoryand Gravitational Waves

Since gravitational waves become weaker as theypropagate away from their sources, one mighthope to neglect the nonlinearities of the Einsteinfield equations and focus instead on the linearizedequations, which are easier to work with. Onemay hope that these equations would provide anapproximate description of the gravitational ra-diation for much of its propagation and for itsinteraction with the detector. In linearized grav-ity, one then writes the spacetime metric as

gµν = ηµν + hµν (14)

where ηµν is the Minkowski metric as in (1) andhµν is assumed to be small. One then keeps termsin the Einstein field equations only to linear or-der in hµν . The coordinate invariance of GeneralRelativity gives rise to what is called gauge in-variance in linearized gravity. In particular, con-sider any quantity F written as F = F + δFwhere F is the value of the quantity in the back-ground and δF is the first order perturbation ofthat quantity. Then for an infinitesimal diffeo-morphism along the vector field ξ, the quantityδF changes by

δF → δF + LξF ,

where recall that L stands for the Lie deriva-tive. Recall also that harmonic coordinates madethe Einstein vacuum equations look like the waveequation in (6). We would like to do somethingsimilar in linearized gravity. To this end wechoose ξ to impose the Lorenz gauge condition(not Lorentz!)

∂µhµν = 0,

wherehµν = hµν − (1/2)ηµνh.

The linearized Einstein field equations then be-come

�hµν = −16πGTµν , (15)

where � is the wave operator in Minkowskispacetime.

In a vacuum one can use the remaining free-dom to choose ξ to impose the conditions that

7

hµν has only spatial components and is trace-free, while remaining in Lorenz gauge. This re-finement of the Lorenz gauge is called the TTgauge, since it guarantees that the only two prop-agating degrees of freedom of the metric pertur-bation are transverse ∂ihij = 0 and (spatially)traceless ηijhij = 0. The metric in TT gaugehas a direct physical interpretation given by thefollowing formula for the linearized Riemanniancurvature tensor

Ritjt = −1

2hTTij , (16)

which sources the geodesic deviation equation,and thus encapsulates how matter behaves inthe presence of gravitational waves. Combin-ing Eq. (16) and the Jacobi equation, one cancompute the change in distance between two testmasses in free fall [73]:

4di(t) =1

2hTTij (t)dj0

where dj0 is the initial distance between the testmasses.

The TT nature of gravitational wave pertur-bations allows us to immediately infer that theyonly have two polarizations. Consider a wavetraveling along the z-direction, such that hTTij (t−z) is a solution of �hTTij = 0. The Lorenz con-dition, the assumption that the metric perturba-tion vanishes for large r, the trace-free condition,and symmetries imply that there are only two in-dependent propagating degrees of freedom:

h+(t− z) = hTTxx = −hTTyy

andh×(t− z) = hTTxy = hTTyx .

The h+ gravitational wave stretches the x direc-tion in space while it squeezes the y direction,and vice-versa. The interferometer used to de-tect gravitational waves has two long perpendic-ular arms that measure this distortion. There-fore, one must approximate these displacementsin order to predict what the interferometer willsee under various scenarios.

4.1 The post-Newtonian Approxi-mation

The post-Newtonian (PN) approximation forgravitational waves [14] extends the linearizedstudy presented above to higher orders in themetric perturbation, while also assuming that

the bodies generating the gravitational field moveslowly compared to the speed of light [74].The PN approach was developed by Einstein,Infeld, Hoffman, Damour, Deruelle, Blanchet,Will, Schaefer, and many others (see [14, 74]and references therein). In the harmonic gauge∂α(√−ggαβ) = 0 commonly employed in PN the-

ory, the expanded equations take the form

�hαβ = −16πG

c4ταβ , (17)

where � is the wave operator and

ταβ = −(g)Tαβ + (16π)−1Nαβ ,

with Nαβ composed of quadratic forms of themetric perturbation.

These expanded equations can then be solvedorder by order in the perturbation through Greenfunction methods, where the integral is over thepast lightcone of Minkowski space for x ∈ M .When working at sufficiently high PN order,the resulting integrals can be formally diver-gent, but these pathologies can be bypassed orcured through asymptotic matching methods (asin Will’s method of the direct integration ofthe relaxed Einstein equations [92, 91, 69]) orthrough regularization techniques (as in Blanchetand Damour’s Hadamard and dimensional regu-larization approach [16, 34]). All approaches tocure these pathologies have been shown to leadto exactly the same end result for the metric per-turbation.

The metric perturbation is solved for order byorder, where at each order one uses the previ-ously calculated information in the expression forNαβ and also to find the motion of the mattersources, thus leading to an improved expressionfor Tαβ at each order. In particular, the emis-sion of gravitational waves by a binary systemcauses a change in the period of that system, andthis change was used by Hulse and Taylor to in-directly detect gravitational waves through theirobservations of the binary pulsar [51]. In thisway, the PN iterative procedure provides a per-turbative approximation to the solution to theEinstein equations to a given order in the fee-bleness of the gravitational interaction and thespeed of the bodies.

Little work has gone into studying the math-ematical properties of the resulting perturbativeseries. Clearly, the PN approximation should notbe valid when the speed of the bodies becomescomparable to the speed of light or when the ob-jects described are black holes or neutron stars

8

with significant self-gravity. Damour, however,has shown that the latter can still be describedby the PN approximation up to a given order inperturbation theory [33]. Moreover, recent nu-merical simulations of the merger of binary blackholes and neutron stars have shown that the PNapproximation is accurate even quite late in theinspiral, when the objects are moving at close toa third of the speed of light [15, 18, 93, 20, 19, 97].

4.2 Resummations of thePN Approximation

The accuracy of the approximate solutions can beimproved by applying resummation techniques:the rewriting of the perturbative expansion in anew form (e.g. a Chebyshev decomposition or aPade series) that makes use of some physical fea-ture one knows should be present in the exactsolution. For example, one may know (throughsymmetry arguments or by taking certain lim-its) that some exact result contains a first-orderpole at a certain spacetime position, so one couldrewrite the approximate solution as a Pade ap-proximant that makes this pole explicit [50].

A particular resummation of the PN approx-imation that has been highly successful at ap-proximating numerical solutions is the effectiveone-body approach introduced by Buonanno andDamour [21, 22]. Recall that in Newtonian grav-ity, the motion of masses m1 and m2 under theirmutual gravitational attraction is mathemati-cally equivalent to the motion of a single mass µin the gravitational field of a stationary mass M ,where M = m1 + m2 and µ = (m1m2)/M . Theeffective one body approach similarly attempts torecast the motion of two black holes under theirmutual gravitational attraction as the motion ofa single object in a given spacetime metric.

More precisely, one recasts the two-body prob-lem onto the problem of an effective body thatmoves on an effective external metric throughan energy map and a canonical transforma-tion. The dynamics of the effective body arethen described through a (conservative) im-proved Hamiltonian and a (dissipative) improvedradiation-reaction force. The improved Hamil-tonian is resummed through two sets of square-roots of PN series, in such a way so as to repro-duce the standard PN Hamiltonian when Taylorexpanded about weak-field and slow-velocities.The improved radiation-reaction force is con-structed from quadratic first-derivatives of thegravitational waves, which in turn are product-

resummed using the Hamiltonian (from knowl-edge of the extreme mass-ratio limit of the PN ex-pansion) and a field-theory resummation of cer-tain tail-effects.

Once the two-body problem has been refor-mulated, the Hamilton equations associated withthe improved Hamiltonian and radiation-reactionforce are solved numerically, a significantly easierproblem than solving the full Einstein equations.This resummation, however, is not enough be-cause the improved Hamiltonian and radiation-reaction force are built from finite PN expan-sions. The very late inspiral behavior of the so-lution can be corrected by adding calibration co-efficients (consistent with PN terms not yet cal-culated) to the Hamiltonian and the radiation-reaction force, which are then determined by fit-ting to a set of full, numerical relativity simula-tions (see e.g. [23, 66]).

The calibrated effective-one-body waveformsdescribed above are incredibly accurate represen-tations of the gravitational waves emitted in theinspiral of compact objects, up to the momentwhen the black holes merge [7]. They becomeaccurate after the merger by adding on informa-tion from black hole perturbation theory that wedescribe next [67, 68].

4.3 Perturbations about a blackhole background

After black holes merge, they form a single dis-torted black hole that sheds its distortions byemitting gravitational waves and eventually set-tling down to a Kerr black hole. This “ringdown”phase is described using perturbation theory withthe Einstein vacuum equations linearized arounda Kerr black hole background. Teukolsky showedhow to obtain a wave type equation for these per-turbed Weyl tensor components from the Ein-stein vacuum equations [86]. The result of theTeukolsky method is that the distortions canbe expanded in modes, each of which has acharacteristic frequency and exponential decaytime [25]. The ringdown is well approximatedby the most slowly decaying of these modes.This ringdown waveform can be stitched to theeffective-one-body inspiral waveforms to obtaina complete description of the gravitational wavesemitted in the coalescence of black holes.

9

5 Mathematics andNumerics

In numerical relativity, one creates simulationsof the Einstein field equations using a computer.This is needed when no other method will work,in particular when gravity is very strong andhighly dynamical (as it is when two black holesmerge).

The Einstein field equations, like most of theequations of physics, are differential equations,and the most straightforward of the techniquesfor simulating differential equations are finite dif-ference equations [75]. In the one dimensionalsetting, one approximates a function f(x) by itsvalues on equally spaced points

fi = f(iδ) for i ∈ N.

One then approximates derivatives of f using dif-ferences

f ′ ≈ (fi+1 − fi−1)/(2δ)

andf ′′(i) ≈ (fi+1 + fi−1 − 2fi)/δ

2.

For any pde with an initial value formulation onereplaces the fields by their values on a spacetimelattice, and the field equations by finite differenceequations that determine the fields at time stepn+ 1 from their values at time step n. Thus theEinstein vacuum equations are written as differ-ence equations where the step 0 information isthe initial data set.

One then writes a computer program that im-plements this determination and runs the pro-gram. Sounds simple, right? So what could gowrong? Quite a lot, actually. It is best to thinkof the solution of the finite difference equation assomething that is supposed to converge to a so-lution of the differential equation in the limit asthe step size δ between the lattice points goes tozero. But it is entirely possible that the solutiondoes not converge to anything at all in this limit.In particular, the coordinate invariance of gen-eral relativity allows one to express the Einsteinfield equations in many different forms, some ofwhich are not strongly hyperbolic. Computersimulations of these forms of the Einstein fieldequations generally do not converge.

Another problem has to do with the constraintequations. Recall that initial data have to sat-isfy constraint equations. It is a consequence ofthe theorem of Choquet-Bruhat that if the initialdata satisfy those constraints then the results ofevolving those initial data continues to satisfy the

constraints. However, in a computer simulationthe initial data only satisfy the finite differenceversion of the constraints and therefore have asmall amount of constraint violation. The fieldequations say that data with zero constraint vi-olation evolve to data with zero constraint vio-lation. But that still leaves open the possibility(usually realized in practice) that data with smallconstraint violation evolve in such a way that theconstraint violation grows rapidly (perhaps evenexponentially) and thus destroys the accuracy ofthe simulation.

Finally there is the problem that these simula-tions deal with black holes, which contain space-time singularities. A computer simulation can-not be continued past a time where a slice ofconstant time encounters a spacetime singular-ity. Thus either the simulations must only berun for a short amount of time, or the time slicesinside the black hole must somehow be “sloweddown” so that they do not encounter the singu-larity. But then if the time slice advances slowlyinside the black hole and rapidly outside it, thiswill lead to the slice being stretched in such a wayas to lead to inaccuracies in the finite differenceapproximation.

Before 2005 these three difficulties were insur-mountable, and none of the computer simulationsof colliding black holes gave anything that couldbe used to compare with observations. Then sud-denly in 2005 all of these problems were solved byFrans Pretorius [77] who produced the first fullysuccessful binary black hole simulation. Thenlater that year the problem was solved again (us-ing completely different methods!) by two othergroups: one consisting of Campanelli, Lousto,Marronetti, and Zlochower [24] and the other ofBaker, Centrella, Choi, Koppitz, and van Me-ter [9]. Though the methods are different, bothsets of solutions can be thought of as consistingof the ingredients hyperbolicity, constraint damp-ing, and excision, and we will treat each one inturn.

Hyperbolicity. Since one needs the equationsto be strongly hyperbolic, one could performthe simulations in harmonic coordinates. How-ever, one also needs the time coordinate to re-main timelike, so instead Pretorius used general-ized harmonic coordinates (as first suggested byFriedrich) where the coordinates satisfy a waveequation with a source. The other groups imple-mented hyperbolicity by using the BSSN equa-tions [64, 79, 10] (named for its inventors: Baum-garte, Shapiro, Shibata, and Nakamura). These

10

equations decompose the spatial metric into aconformal factor and a metric of unit determi-nant and then evolve each of these quantitiesseparately, adding appropriate amounts of theconstraint equations to convert the spatial Riccitensor into an elliptic operator.

Constraint damping. Because the constraintsare zero in exact solutions to the theory, onehas the freedom to add any multiples of the con-straints to the right-hand side of the field equa-tions without changing the class of solutions tothe field equations. In particular, with cleverchoices of which multiples of the constraints goon the right-hand side, one can arrange that inthese new versions of the field equations smallviolations of the constraints get smaller underevolution rather than growing. Carsten Gund-lach and his collaborators showed how to do thisfor evolution using harmonic coordinates [49],and their method was implemented by Preto-rius. The BSSN equations already have somerearrangement of the constraint and evolutionequations. The particular choice of lapse andshift (Φ and X from eqns. (7-8)) used by theother groups (called 1+log slicing and Gammadriver shift) were found to have good constraintdamping properties.

Excision. Because nothing can escape from ablack hole, nothing that happens inside can haveany influence on anything that happens outside.Thus in performing computer simulations of col-liding black holes, one is allowed to simply excisethe black hole interior from the computationalgrid and still obtain the answer to the questionof what happens outside the black holes. By ex-cising, one no longer has to worry about singu-larities or grid stretching. Excision was first pro-posed by Unruh and Thornburg [87], and firstimplemented by Seidel and Suen and their col-laborators [5], and used in Pretorius’ simulations.The other groups essentially achieve excision byother methods. They use a “moving puncturemethod” which involves a second asymptoticallyflat end inside each black hole, that is compact-ified to a single point that can move aroundthe computational grid. The region between thepuncture and the black hole event horizon under-goes enormous grid stretching, so that effectivelyonly the exterior of the black hole is covered bythe numerical grid.

Since 2005, many simulations of binary blackhole mergers have been performed, for variousblack hole masses and spins. Some of the mostefficient simulations are done by the SXS collab-

oration using spectral methods instead of finitedifference methods [63]. (SXS stands for “Sim-ulating eXtreme Spacetimes” and the collabora-tion is based at Cornell, Caltech, and elsewhere).Spectral methods use the grid values fi to ap-proximate the function f(x) as an expansion in aparticular basis of orthogonal functions. The ex-pansion coefficients and the derivatives of the ba-sis functions are then used to compute the deriva-tives of f(x). Compared to finite difference meth-ods, spectral methods can achieve a given accu-racy of the derivatives with significantly fewergrid points.

6 Gravitational WaveExperiment

The experimental search for gravitational wavesstarted in the 1960s through the construction ofresonant bar detectors [89]. The latter essen-tially consist of a large (meter-size) cylinder ina vacuum chamber that is isolated from vibra-tions. When a gravitational wave at the rightfrequency interacts with such a bar, it can excitethe latter’s resonant mode, producing a changein length that one can search for. In 1968, JosephWeber announced that he had detected gravita-tional waves with one such resonant bar. Thesensitivity of Weber’s resonant bar to gravita-tional waves was not high enough for this to bepossible and other groups could not reproducehis experiment.

In the late 1960s and early 1970s, the search forgravitational waves with laser interferometers be-gan through the pioneering work of Rainer Weissat MIT [90] and Kip Thorne [76] and RonaldDrever at Caltech [37], among many others. Thebasic idea behind interferometry is to split a laserbeam into two sub-beams that travel down or-thogonal arms, bounce off mirrors, and then re-turn to recombine. If the light travel time is thesame in each sub-beam, then the light recom-bines constructively, but if a gravitational wavegoes through the detector, then the light traveltime is not the same in each arm and interfer-ence occurs. Gravitational wave interferometersare devices that use this interference process tomeasure small changes in light travel time veryaccurately so as to learn about the gravitationalwaves that produced them, and thus, in turn,about the properties of the source of gravita-tional waves.

The initial Laser Interferometer Gravitational-

11

Figure 3: Schematic diagram of a Michelson in-terferometer, which is at the heart of the instru-mental design used by aLIGO. In the diagram, alaser beam is split into two sub-beams that traveldown orthogonal arms, bounce off mirrors, andthen return to recombine.

Wave Observatory (iLIGO) was funded by theNational Science Foundation in the early 1990sand operations started in the early 2000s. Thereare actually two LIGO facilities (one in Hanford,Washington, and one in Livingston, Louisiana) inoperation right now, with an Italian counterpart(Virgo) coming online soon, a Japanese counter-part (KAGRA) coming online by the end of thedecade, and an Indian counterpart (LIGO-India)coming online in the 2020s. The reason for mul-tiple detectors is to achieve redundancy and in-crease the confidence of a detection by observingthe signal by independent detectors with uncor-related noise. Although iLIGO was over four or-ders of magnitude more sensitive than Weber’soriginal instrument in a wide frequency band, nogravitational waves were detected.

In the late 2000s, upgrades to convert iLIGOinto advanced LIGO (aLIGO) commenced.These upgrades included an increase in the laserpower to reduce quantum noise, larger and heav-ier mirrors to reduce thermal and radiation pres-sure noise, better suspension fibers for the mir-rors to reduce suspension thermal noise, amongmany other improvements. aLIGO commencedscience operations in 2015 with a sensitivityroughly 3-4 times greater than that of iLIGO’slast science run.

Within days of the first science run, the aLIGO

Prepublication copy provided to Rachel Rossi.

Not for print or electronic distribution. This file may not be posted electronically.

Figure 15. (from top to bottom) Ronald Drever, KipThorne, and Rainer Weiss pioneered the effort todetect gravitational waves with laser interferometers.

Figure 16. One of the two aLIGO facilities, this one inLivingston, Louisiana, where the interference patternassociated with a gravitational wave produced in themerger of two black holes was recorded within daysof the first science run.

Figure 17. Top: Filtered GW strain as a function oftime detected at the Hanford location of aLIGO.Bottom: Best fit reconstruction of the signal using anumerical relativity simulation (red), an analyticalwaveform template (gray), and a set of Morleywavelets. The latter two are shown as 90 percentconfidence regions, while the simulation is aparticular run with a choice of parameters within thisthe 90 percent confidence region.

are suspended from wires like a pendulum, but thismeans that for short time motion in the horizontaldirection, the motion of each mirror can be treated asa spacetime geodesic. But the interferometer measuresdistance between the mirrors, so what we want to know

22 Notices of the AMS Volume 64, Number 7

Figure 4: (from top to bottom) Ronald Drever,Kip Thorne, and Rainer Weiss pioneered the ef-fort to detect gravitational waves with laser in-terferometers.

detectors recorded the interference pattern asso-ciated with a gravitational wave produced in themerger of two black holes 1.3 billion light yearsaway [2]. The signal was so loud (relative tothe level of the noise) that the probability thatthe recorded event was a gravitational wave wasmuch larger than 5σ, meaning that the probabil-ity of a false alarm was much smaller than 10−7.There is no doubt that this event, recorded on14th September 2015, as well as a second one,detected the day after Christmas of that sameyear [3], were the first direct detections of gravi-tational waves.

In order to understand how gravitationalwaves are detected, we must understand how thewaves affect the motion of the parts of the in-terferometer. The mirrors are suspended fromwires like a pendulum, but this means that forshort time motion in the horizontal direction, themotion of each mirror can be treated as a space-time geodesic. But the interferometer measures

12

Figure 5: One of the two aLIGO facilities in Liv-ingston, Louisiana, where the interference pat-tern associated with a gravitational wave pro-duced in the merger of two black holes wasrecorded within days of the first science run.

distance between the mirrors, so what we wantto know is how does this distance change underthe influence of a gravitational wave. The answerto this question comes from the Jacobi equation:the relative acceleration of nearby geodesics isequal to the Riemann tensor times the separa-tion of those geodesics.

Thus if at any time we want to know theseparation, we need to integrate the Jacobiequation twice with respect to time. However,the Riemann tensor is the second derivativeof the TT gauge metric perturbation. Thus,by using this particular gauge we can say thatLIGO directly measures the metric perturbationby using laser interferometry to keep track ofthe separation of its mirrors.

7 Astrophysics andFundamental Physics

Up until now, we have created a picture of theuniverse from the information we have obtainedfrom amazing telescopes, such as Chandra in theX-rays, Hubble in the optical, Spitzer in the in-frared, WMAP in the microwave, and Arecibo inthe radio frequencies. This information was pro-vided by light that traveled from astrophysicalsources to Earth. Every time humankind built anew telescope that gave us access to a new fre-quency range of the light spectrum, amazing dis-coveries were made; case in point, accretion disksignatures of black holes using X-ray astronomy.This expectation is especially true for gravita-tional wave detectors, which do not just open a

Figure 6: Top: Filtered GW strain as a func-tion of time detected at the Hanford locationof aLIGO. Bottom: Best fit reconstruction ofthe signal using a numerical relativity simulation(red), an analytical waveform template (gray)and a set of Morley wavelets. The latter twoare shown as 90% confidence regions, while thesimulation is a particular run with a choice of pa-rameters within this the 90% confidence region.

new frequency range, but rather aim to listen tothe universe in an entirely new way: with gravityinstead of light.

This new type of astrophysics has an immensepotential to truly revolutionize science becausegravitational waves can provide very clean infor-mation about their sources. Unlike light, gravita-tional waves are very weakly coupled to matter,allowing gravitational waves to go right throughthe intermediate matter (which would absorblight) and provide a clean picture (or soundtrack)of astrophysical sources that until now had re-mained obscure. Of course, this is a double-edged sword because the detection of gravita-tional waves is extremely challenging, requiringthe ability to measure distances that are as smallas 10−3 times the size of a proton over a 4 kmbaseline.

The aLIGO detectors achieved just that, pro-viding humanity with not only the first direct de-tection of gravitational waves, but also the firstdirect evidence of the existence of black hole bi-naries and their coalescence. As of the writingof this article, aLIGO had detected two events,

13

both of which correspond to the coalescence of bi-nary black hole systems in a quasi-circular orbit.Fitting the hybrid analytic and numerical mod-els described in Sections 4 and 5 to the data, theaLIGO collaboration found that the first eventconsisted of two black holes with masses [2]

(m1,m2) ≈ (36.2, 29.1)M�,

where M� is the mass of our sun, colliding atroughly half the speed of light to produce a rem-nant black hole with mass

mf ≈ 62.3M�

and dimensionless spin angular momentum

|~S|/m2f ≈ 0.68,

located 420 mega-parsecs away from Earth(roughly 1.3 billion times the distance light trav-els in one year). The second event consisted oflighter black holes, with masses [3]

(m1,m2) ≈ (14.2, 7.5)M�

that collided to produce a remnant black holewith mass

mf ≈ 20.8M�

and dimensionless spin angular momentum

|~S|/m2f ≈ 0.74,

located 440 mega-parsecs away from Earth. Inboth cases, the peak luminosity radiated was inthe range of 1056 ergs/s with the systems effec-tively losing 3M� and 1M� respectively in lessthan 0.1 seconds. Thus, for a very brief moment,these events produced more energy than all ofthe stars in the observable universe put together.

Perhaps one of the most interesting inferencesone can draw from such events is that blackholes (or at the very least, objects that look and“smell” a lot like black holes) truly do form bi-naries and truly do merge in nature within anamount of time smaller than the age of the uni-verse. Until now, we had inferred the existenceof black holes by either observing how other starsorbit around supermassive ones at the center ofgalaxies or by observing enormous disks of gasorbit around stellar mass black holes and the X-rays emitted as some of that gas falls into theblack hole. The aLIGO observations are the firstdirect observation of radiation produced by bi-nary black holes themselves through the wave-like excitations of the curvature they generate

when they collide. Not only did the aLIGOobservation prove the existence of binary blackholes, but even the first observation broughtabout a surprise: the existence and merger ofblack holes in a mass range that had never beenobserved before.

The aLIGO observations have demonstratedthat General Relativity is not only highly accu-rate at describing gravitational phenomena in thesolar system, in binary pulsar observations, andin cosmological observations, but also in the lateinspiral, merger and ringdown of black hole bi-naries [94, 4, 95]. Gravity is truly described byEinstein’s theory even in the most extreme grav-ity scenarios: when the gravitational interactionis strong, highly non-linear, and highly dynami-cal. Such consistency with Einstein’s theory hasimportant consequences on theories that modifygravity in hopes of arriving at a quantum grav-itational completion. Future gravitational waveobservations will allow us to verify many otherpillars of Einstein’s theory, such as that the gravi-tational interaction is parity invariant, that grav-itational waves propagate at the speed of light,and that it only possesses two transverse polar-izations.

The detection of gravitational waves is notonly a spectacular confirmation of Einstein’s the-ory, but also the beginning of a new era inastrophysics. Gravitational waves will providethe soundtrack to the movie of our universe, asoundtrack we had so far been missing with tele-scopes. No doubt that they will be a rich sourcefor new questions and inspiration in physics aswell as mathematics. We wait anxiously for theunexpected beauty this music will provide.

Acknowledgments

The authors acknowledge their NSF support.LB is supported by NSF CAREER grant DMS-1253149 to The University of Michigan. DGis supported by NSF grant PHY-1505565 toOakland University. N.Y. acknowledges supportfrom NSF CAREER Grant PHY-1250636.

14

Nobel Prize in Physics 2017

While we were editing the arxiv version of thisarticle, 2 “the Royal Swedish Academy of Sci-ences has decided to award the Nobel Prize inPhysics 2017 with one half to Rainer Weiss,LIGO/VIRGO Collaboration, and the other halfjointly to Barry C. Barish, LIGO/VIRGO Col-laboration, and Kip S. Thorne, LIGO/VIRGOCollaboration, for decisive contributions to theLIGO detector and the observation of gravita-tional waves.”

Prepublication copy provided to Rachel Rossi.

Not for print or electronic distribution. This file may not be posted electronically.

Figure 15. (from top to bottom) Ronald Drever, KipThorne, and Rainer Weiss pioneered the effort todetect gravitational waves with laser interferometers.

Figure 16. One of the two aLIGO facilities, this one inLivingston, Louisiana, where the interference patternassociated with a gravitational wave produced in themerger of two black holes was recorded within daysof the first science run.

Figure 17. Top: Filtered GW strain as a function oftime detected at the Hanford location of aLIGO.Bottom: Best fit reconstruction of the signal using anumerical relativity simulation (red), an analyticalwaveform template (gray), and a set of Morleywavelets. The latter two are shown as 90 percentconfidence regions, while the simulation is aparticular run with a choice of parameters within thisthe 90 percent confidence region.

are suspended from wires like a pendulum, but thismeans that for short time motion in the horizontaldirection, the motion of each mirror can be treated asa spacetime geodesic. But the interferometer measuresdistance between the mirrors, so what we want to know

22 Notices of the AMS Volume 64, Number 7

Figure 7: Rainer Weiss (LIGO/VIRGO Collabo-ration).

Figure 8: Barry Barish (LIGO/VIRGO Collabo-ration).

Prepublication copy provided to Rachel Rossi.

Not for print or electronic distribution. This file may not be posted electronically.

Figure 15. (from top to bottom) Ronald Drever, KipThorne, and Rainer Weiss pioneered the effort todetect gravitational waves with laser interferometers.

Figure 16. One of the two aLIGO facilities, this one inLivingston, Louisiana, where the interference patternassociated with a gravitational wave produced in themerger of two black holes was recorded within daysof the first science run.

Figure 17. Top: Filtered GW strain as a function oftime detected at the Hanford location of aLIGO.Bottom: Best fit reconstruction of the signal using anumerical relativity simulation (red), an analyticalwaveform template (gray), and a set of Morleywavelets. The latter two are shown as 90 percentconfidence regions, while the simulation is aparticular run with a choice of parameters within thisthe 90 percent confidence region.

are suspended from wires like a pendulum, but thismeans that for short time motion in the horizontaldirection, the motion of each mirror can be treated asa spacetime geodesic. But the interferometer measuresdistance between the mirrors, so what we want to know

22 Notices of the AMS Volume 64, Number 7

Figure 9: Kip Thorne (LIGO/VIRGO Collabo-ration).

2See the official announcement on the Nobel Prizewebsite https://www.nobelprize.org/nobel_prizes/

physics/laureates/2017/press.html

References

[1] LIGO Scientific Collaboration and VirgoCollaboration; Abbott, B.P., et al. Ob-servation of Gravitational Waves from Bi-nary Black Hole Merger. PRL. 116. 061102.(2016).

[2] B. P. Abbott et al. [LIGO Scien-tific and Virgo Collaborations], Phys.Rev. Lett. 116, no. 6, 061102 (2016)doi:10.1103/PhysRevLett.116.061102[arXiv:1602.03837 [gr-qc]].

[3] B. P. Abbott et al. [LIGO Scien-tific and Virgo Collaborations], Phys.Rev. Lett. 116, no. 24, 241103 (2016)doi:10.1103/PhysRevLett.116.241103[arXiv:1606.04855 [gr-qc]].

[4] B. P. Abbott et al. [LIGO Scien-tific and Virgo Collaborations], Phys.Rev. Lett. 116, no. 22, 221101 (2016)doi:10.1103/PhysRevLett.116.221101[arXiv:1602.03841 [gr-qc]].

[5] P. Anninos, G. Daues, J. Masso, E. Seideland W.-M. Suen, Phys. Rev. D 51, 5562(1995)

[6] R. Arnowitt, S. Deser, C. Misner The dy-namics of general relativity in Gravitation:an introduction to current research L. Wit-ten (ed) Wiley, New York (1962)

[7] S. Babak, A. Taracchini and A. Buo-nanno, Phys. Rev. D 95, no. 2, 024010(2017) doi:10.1103/PhysRevD.95.024010[arXiv:1607.05661 [gr-qc]].

[8] H. Bahouri, J.Y. Chemin. Equationsd’ondes quasilineaires et effect dispersif.Amer.J.Math. 121. (1999).

[9] J.G. Baker et al. Phys.Rev.Lett. 96. (2006).

[10] T.W. Baumgarte, and S.L. Shapiro, Phys.Rev. D 59, 024007 (1998)

[11] L. Bieri. An Extension of the Stability Theo-rem of the Minkowski Space in General Rel-ativity. ETH Zurich, Ph.D. thesis. 17178.Zurich. (2007).

[12] L. Bieri. Extensions of the Stability Theo-rem of the Minkowski Space in General Rel-ativity. Solutions of the Einstein VacuumEquations. AMS-IP. Studies in AdvancedMathematics. Cambridge. MA. (2009).

15

[13] L. Bieri, D. Garfinkle. Perturbative andgauge invariant treatment of gravitationalwave memory. Phys. Rev. D. 89. 084039.(2014).

[14] L. Blanchet, Living Rev. Rel. 9, 4 (2006).

[15] L. Blanchet, doi:10.1142/9789812791368-0022 gr-qc/0207037.

[16] L. Blanchet and G. Faye, J. Math. Phys.41, 7675 (2000) doi:10.1063/1.1308506 [gr-qc/0004008].

[17] H. Bondi, M. G. J. van der Burg and A.W. K. Metzner. Gravitational Waves inGeneral Relativity. VII. Waves from Axi-Symmetric Isolated Systems. Proc. Roy.Soc. A. 269 (1962). 21-52

[18] M. Boyle, D. A. Brown, L. E. Kid-der, A. H. Mroue, H. P. Pfeif-fer, M. A. Scheel, G. B. Cook andS. A. Teukolsky, Phys. Rev. D 76, 124038(2007) doi:10.1103/PhysRevD.76.124038[arXiv:0710.0158 [gr-qc]].

[19] M. Boyle, D. A. Brown and L. Pekowsky,Class. Quant. Grav. 26, 114006 (2009)doi:10.1088/0264-9381/26/11/114006[arXiv:0901.1628 [gr-qc]].

[20] M. Boyle, A. Buonanno, L. E. Kidder,A. H. Mroue, Y. Pan, H. P. Pfeiffer andM. A. Scheel, Phys. Rev. D 78, 104020(2008) doi:10.1103/PhysRevD.78.104020[arXiv:0804.4184 [gr-qc]].

[21] A. Buonanno, T. Damour. Phys.Rev.D. 59.(1999).

[22] A. Buonanno, T. Damour. Phys.Rev.D. 62.(2000).

[23] A. Buonanno, Y. Pan, H. P. Pfeif-fer, M. A. Scheel, L. T. Buchman andL. E. Kidder, Phys. Rev. D 79, 124028(2009) doi:10.1103/PhysRevD.79.124028[arXiv:0902.0790 [gr-qc]].

[24] M. Campanelli, C.O. Lousto, P. Marronetti,Y. Zlochower. Phys.Rev.Lett. 96. (2006).

[25] S. Chandrasekhar, The Mathematical The-ory of Black Holes Oxford University Press,Oxford, 1992

[26] Y. Choquet-Bruhat. Theoreme d’existencepour certain systemes d’equations auxderivees partielles nonlineaires. Acta Math.88. (1952). 141-225.

[27] Y. Choquet-Bruhat. Beginnings of theCauchy problem for Einstein’s field equa-tions. Surveys in Differential Geometry.Volume 20. One hundred years of generalrelativity. (2015).

[28] Y. Choquet-Bruhat, R. Geroch. Global As-pects of the Cauchy Problem in General Rel-ativity. Comm.Math.Phys. 14. (1969). 329-335.

[29] D. Christodoulou. The Global Initial ValueProblem in General Relativity. The NinthMarcel Grossmann Meeting (Rome, 2000).ed. by V. G. Gurzadyan et al., World Scien-tific. Singapore (2002).

[30] D. Christodoulou. Nonlinear Nature ofGravitation and Gravitational-Wave Experi-ments. Phys.Rev.Letters. 67. (1991). no.12.1486-1489.

[31] D. Christodoulou. The Formation of BlackHoles in General Relativity. EMS publish-ing house ETH Zurich. (2009).

[32] D. Christodoulou, S. Klainerman. Theglobal nonlinear stability of the Minkowskispace. Princeton Math.Series 41. PrincetonUniversity Press. Princeton. NJ. (1993).

[33] T. Damour, Fundam. Theor. Phys. 9, 89(1984).

[34] T. Damour, P. Jaranowski and G. Schae-fer, Phys. Lett. B 513, 147 (2001)doi:10.1016/S0370-2693(01)00642-6 [gr-qc/0105038].

[35] T. Damour, A. Nagar. Phys.Rev.D. 90.4.(2014).

[36] T. De Donder. La Gravifique Einsteinienne.Annales de l’Observatoire Royal de Bel-gique. Brussels. (1921).

[37] R.W.P. Drever, F.J. Raab, K.S. Thorne,R. Vogt, R. Weiss. Laser Interferome-ter Gravitational-wave Observatory (LIGO)Technical Report. (1989).

[38] A. Einstein. Zur Allgemeinen Rela-tivitatstheorie. SAW (Sitzungsber. K.Preuss. Akad. Wiss. Berlin.) (1915). 778-786.

16

[39] A. Einstein. Die Feldgleichungen der Gravi-tation. SAW (Sitzungsber. K. Preuss. Akad.Wiss. Berlin.) (1915). 844-847.

[40] A. Einstein. Die Grundlage der allgemeinenRelativitatstheorie. Annalen der Physik. 49.(1916). 769-822.

[41] A. Einstein. Naherungsweise Integra-tion der Feldgleichungen der Gravitation.SAW (Sitzungsber. K. Preuss. Akad. Wiss.Berlin.) (1916). 688-696.

[42] A. Einstein. Kosmologische Betrachtun-gen zur allgemeinen Relativitatstheorie.SAW (Sitzungsber. K. Preuss. Akad. Wiss.Berlin.) (1917). 142-152.

[43] A. Einstein. Uber Gravitationswellen.SAW (Sitzungsber. K. Preuss. Akad. Wiss.Berlin.) (1918). 154-167.

[44] H. Friedrich. Comm.Math.Phys. 100.(1985).

[45] H. Friedrich. On the Existence of n-Geodesically Complete or Future Com-plete Solutions of Einstein’s Field Equa-tions with Smooth Asymptotic Structure.Comm.Math.Phys. 107. (1986). 587-609.

[46] D. Garfinkle. Phys.Rev.D. 65. (2002).

[47] R. Geroch, Asymptotic Structure of Space-time in P. Esposito and L. Witten(eds.) Asymptotic Structure of Spacetime,Plenum, New York (1977)

[48] R. Geroch and B. Xanthopoulos, J. Math.Phys. 19, 714 (1978)

[49] C. Gundlach, J.M. Martin-Garcia, G. Cal-abrese, I. Hinder. Class.Quant.Grav. 22.(2005).

[50] A. Gupta, A. Gopakumar, B. R. Iyerand S. Iyer, Phys. Rev. D 62, 044038(2000) doi:10.1103/PhysRevD.62.044038[gr-qc/0002094].

[51] R. A. Hulse and J. H. Taylor, Astrophys. J.195, L51 (1975). doi:10.1086/181708

[52] P. Jaranowski and A. Krolak, Living Rev.Rel. 8, 3 (2005) [Living Rev. Rel. 15, 4(2012)] [arXiv:0711.1115 [gr-qc]].

[53] S. Klainerman, J. Luk, I. Rodnianski. AFully Anisotropic Mechanism for Formationof Trapped Surfaces. Inventiones Mathe-maticae. 195. (2014)

[54] S. Klainerman, F. Nicolo. The EvolutionProblem in General Relativity. Progress inMath.Phys. 25. Birkhauser. Boston. (2003).

[55] S. Klainerman, F. Nicolo. Peeling proper-ties of asymptotically flat solutions to theEinstein vacuum equations. Class. Quan-tum Gravity. 20. no. 14. (2003). 3215-3257.

[56] S. Klainerman, I. Rodnianski. On the for-mation of trapped surfaces. Acta Mathe-matica. 208.2. (2012).

[57] S. Klainerman, I. Rodnianski. Improved lo-cal well posedness for quasilinear wave equa-tions in dimension three. Duke Math.J.,117. 1. (2003).

[58] S. Klainerman, I. Rodnianski. Improved lo-cal well posedness for the Einstein equationsin wave coordinates. Annals of Math. 161.(2005).

[59] S. Klainerman, I. Rodnianski, J. Szeftel.The bounded L2 curvature conjecture. In-vent.Math. 202.1. (2015).

[60] S. Klainerman, I. Rodnianski, J. Szef-tel. Overview of the proof of thebounded L2 curvature conjecture. (2012)arXiv:1204.1772v1.

[61] P.D. Lasky, E. Thrane, Y. Levin, J. Black-man, Y. Chen, Detecting gravitational-wave memory with LIGO: implications ofGW150914. Phys.Rev.Lett. 117, no. 6,(2016)

[62] J. Leray. Hyperbolic differential equations.The Institute for Advanced Study. Prince-ton. N.J. (1953).

[63] A. Mroue, M. Scheel, B. Szilagyi, H. Pfeif-fer, M. Boyle, D. Hemberger, L. Kidder,G. Lovelace, S. Ossokine, N. Taylor, A.Zenginoglu, L. Buchman, T. Chu, E. Foley,M. Giesler, R. Owen, S. Teukolsky, Phys.Rev. Lett. 111, 241104 (2013)

[64] T. Nakamura, K. Oohara, and Y. Kojima,Prog. Theor. Phys. Suppl. 90, 1 (1987)

[65] E.T. Newman and R. Penrose. An approachto gravitational radiation by a method ofspin coefficients. Journal of Math.Phys. 3(1962). 566-578.

17

[66] Y. Pan, A. Buonanno, L. T. Buchman,T. Chu, L. E. Kidder, H. P. Pfeiffer andM. A. Scheel, Phys. Rev. D 81, 084041(2010) doi:10.1103/PhysRevD.81.084041[arXiv:0912.3466 [gr-qc]].

[67] Y. Pan, A. Buonanno, M. Boyle, L. T. Buch-man, L. E. Kidder, H. P. Pfeiffer andM. A. Scheel, Phys. Rev. D 84, 124052(2011) doi:10.1103/PhysRevD.84.124052[arXiv:1106.1021 [gr-qc]].

[68] Y. Pan, A. Buonanno, A. Tarac-chini, L. E. Kidder, A. H. Mroue,H. P. Pfeiffer, M. A. Scheel and B. Szi-lagyi, Phys. Rev. D 89, no. 8, 084006(2014) doi:10.1103/PhysRevD.89.084006[arXiv:1307.6232 [gr-qc]].

[69] M. E. Pati and C. M. Will,Phys. Rev. D 62, 124015 (2000)doi:10.1103/PhysRevD.62.124015 [gr-qc/0007087].

[70] R. Penrose. Zero Rest-Mass Fields In-cluding Gravitation: Asymptotic Behaviour.Proc.Roy.Soc. of London. A284 (1965). 159.

[71] R. Penrose. Gravitational collapse andspace-time singularities. Phys.Rev.Lett. 14.(1965).

[72] F. Pirani. Invariant formulation of grav-itational radiation theory. Phys.Rev. 105(1957). 1089-1099.

[73] M. Pitkin, S. Reid, S. Rowan andJ. Hough, Living Rev. Rel. 14, 5 (2011)doi:10.12942/lrr-2011-5 [arXiv:1102.3355[astro-ph.IM]].

[74] E. Poisson and C. M. Will Gravity Cam-bridge University Press, Cambridge, U.K.(2014)

[75] W. Press, S. Teukolsky, W. Vetterling, B.Flannery, Numerical Recipes in Fortran(2nd edition) Cambridge University Press,Cambridge (1992)

[76] W. Press, K. Thorne.Annu.Rev.Astron.Astrophys. 10. 335.(1972).

[77] F. Pretorius. Phys.Rev.Lett. 95. (2005).

[78] R. K. Sachs. Gravitational Waves in Gen-eral Relativity. VIII. Waves in Asymptot-ically Flat Space-Time. Proc.Roy.Soc. ofLondon A270. (1962). 103-126.

[79] M. Shibata and T. Nakamura, Phys. Rev. D52, 5428 (1995)

[80] H. Smith, D. Tataru. Sharp local well posed-ness results for the nonlinear wave equation.Annals of Math. 162.1. (2005).

[81] J. Szeftel. Parametrix for wave equa-tions on a rough background I: Regular-ity of the phase at initial time. (2012)arXiv:1204.1768v1.

[82] J. Szeftel. Parametrix for wave equationson a rough background II: Construction ofthe parametrix and control at initial time.(2012) arXiv:1204.1769v1

[83] J. Szeftel. Parametrix for wave equations ona rough background III: Space-time regular-ity of the phase. (2012) arXiv:1204.1770v1

[84] J. Szeftel. Parametrix for wave equations ona rough background IV: Control of the errorterm. (2012) arXiv:1204.1771v1

[85] D. Tataru. Strichartz estimates for oper-ators with nonsmooth coefficients and thenonlinear wave equation. Amer.J.Math.122. (2000).

[86] S. Teukolsky, Phys. Rev. Lett. 29, 1114(1972)

[87] J. Thornburg, Class. Quantum Grav. 4,1119 (1987)

[88] A. Trautman. Radiation and boundaryconditions in the theory of gravitation.Bull.Acad.Pol.Sci.Ser.Math.Astron.Phys. 6(1958). 407.

[89] J. Weber. Phys. Rev. Lett. 24 (1970) 276-279

[90] R. Weiss. Quarterly Report of the ResearchLab. for Electronics. MIT Report No. 105.(1972).

[91] C. M. Will, Prog. Theor. Phys. Suppl. 136,158 (1999) doi:10.1143/PTPS.136.158 [gr-qc/9910057].

[92] C. M. Will and A. G. Wiseman,Phys. Rev. D 54, 4813 (1996)doi:10.1103/PhysRevD.54.4813 [gr-qc/9608012].

18

[93] N. Yunes and E. Berti, Phys. Rev.D 77, 124006 (2008) Erratum:[Phys. Rev. D 83, 109901 (2011)]doi:10.1103/PhysRevD.77.124006,10.1103/PhysRevD.83.109901[arXiv:0803.1853 [gr-qc]].

[94] N. Yunes and X. Siemens, Living Rev.Rel. 16, 9 (2013) doi:10.12942/lrr-2013-9[arXiv:1304.3473 [gr-qc]].

[95] N. Yunes, K. Yagi and F. Preto-rius, Phys. Rev. D 94, no. 8, 084002(2016) doi:10.1103/PhysRevD.94.084002[arXiv:1603.08955 [gr-qc]].

[96] Ya.B. Zel’dovich, A.G. Polnarev. Astron.Zh.51, 30. (1974). [Sov.Astron. 18, 17. (1974)].

[97] Z. Zhang, N. Yunes and E. Berti,Phys. Rev. D 84, 024029 (2011)doi:10.1103/PhysRevD.84.024029[arXiv:1103.6041 [gr-qc]].

[98] N. Zipser. The Global Nonlinear Stabil-ity of the Trivial Solution of the Einstein-Maxwell Equations. Ph.D. thesis. HarvardUniv. Cambridge MA. (2000).

[99] N. Zipser. Extensions of the Stability Theo-rem of the Minkowski Space in General Rel-ativity. - Solutions of the Einstein-MaxwellEquations. AMS-IP. Studies in AdvancedMathematics. Cambridge. MA. (2009).

Photo Credits

Figures 1 and 2 are in the public domain.

Figure 3 is courtesy of Ed Stanner.

Figures 4, 7, and 9 are courtesy of the Gruber

Foundation.

Figure 5 is courtesy of the Caltech/MIT/LIGO

Lab.

Figure 6 is courtesy of [1] B.P. Abbot et al. “Ob-

servation of Gravitational Waves from Binary Black

Hole Merger”. PRL. 116. 061102. (2016).

Figure 8 is in the public domain.

Lydia BieriDepartment of MathematicsUniversity of MichiganAnn Arbor, MI 48109, USAEmail address: lbieri@umich.edu

David GarfinklePhysics DepartmentOakland UniversityRochester, MI 48309, USA,andMichigan Center for Theoret. PhysicsRandall Laboratory of PhysicsUniversity of MichiganAnn Arbor, MI 48109, USAEmail address: garfinkl@oakland.edu

Nicolas YuneseXtreme Gravity InstituteDepartment of PhysicsMontana State UniversityBozeman, MT 59717 USAEmail address: nyunes@physics.montana.edu

19