Gravitational Waves

Credit: NASA/C. Henze

The 2017 Nobel Prize in Physics was awarded to Rainer Weiss, Barry C. Barish, and Kip S. Thorne for their work on the detection of gravitational waves. (See Note 1.)  The physics and engineering that go into this accomplishment are truly impressive.  However, before anyone could imagine setting up the experiment, some mathematical questions needed to be answered.  There are two articles in the August 2017 issue of the AMS Notices that give an overview the mathematics of gravitational waves.  In this post, I crib from those two articles and provide a literature tour of some of the significant papers by relying on MathSciNet. A longer article by Bieri just published in the AMS Bulletin goes into more detail on a selection of the topics.

Introduction

General Relativity (GR) depends heavily on mathematics, in particular, differential geometry and PDEs, since Einstein’s equations are nonlinear PDEs involving the curvature and the metric itself.   The equations hold an inherent beauty that manifests itself not just in the mathematical description of space-time, but also in a rich realm of geometry: Einstein manifolds.  This is digression from the theory of gravitational waves, but it is, in the language of Michelin guides, “Worth a trip.”   As a tour guide for this trip, I heartily recommend the book Einstein manifolds written by a group of geometers under the pseudonym Arthur L. Besse.

There are many places for a mathematician to learn general relativity (GR).  For a first read, Einstein’s own account in The Meaning of Relativity is extremely good.  However it is deceptive: it seems like easy reading, but he is describing very deep ideas that are overturning our view of the world.  You need to pay attention to understand what is going on.  I didn’t pay attention the first time through and had to read it a second time.  Here are three books that are suited to mathematicians for learning about general relativity.

• Hawking and EllisThe large scale structure of space-time. Cambridge Monographs on Mathematical Physics, No. 1. Cambridge University Press, London-New York, 1973. xi+391 pp. MR0424186
• Barrett O’Neill, Semi-Riemannian Geometry, With applications to relativity. Pure and Applied Mathematics, 103. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. xiii+468 pp.  MR0719023
• Misner, Thorne, and WheelerGravitation. W. H. Freeman and Co., San Francisco, Calif., 1973. ii+xxvi+1279+iipp. MR0418833.

The book by Hawking and Ellis assembles the tools you need to understand general relativity, then applies them to discuss black holes, in particular the Hawking and Penrose singularity theorems.  O’Neill’s book is written by a mathematician for mathematicians.  The bulk of it is about differential geometry with a semi-Riemannian metric.  You could give a complete course on differential geometry from O’Neill’s book and not cover relativity.  But why would you leave it out?  He sets everything up for you, then puts it to use.  This is probably not the introduction that a physicist would want, but since it begins on familiar territory, it suits many mathematicians.  Misner, Thorne, and Wheeler is often referred to as “the Bible” for studying general relativity.  It is cited so often, that people often refer to it as MTW.   It is a monster-sized book, but it is really two books in one: a one-semester introduction and a deeper look at the subject.  MTW is the only one of these three books that really treats gravitational waves.

These three books provide a general introduction to GR.  The two Notices papers are describing results that depend on the modern approach of geometric analysis in general relativity.  For that, a good place to start would be the book:

MR2391586
Christodoulou, Demetrios(CH-ETHZ)
Mathematical problems of general relativity. I.
Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2008. x+147 pp. ISBN: 978-3-03719-005-0.

I have not read it – I am relying on the review in MathSciNet!  A short history of the mathematical study of Einstein’s equations can be found in the quite readable and very enjoyable article by Choquet-Bruhat:

MR3467361
Choquet-Bruhat, Yvonne(F-IHES)
Beginnings of the Cauchy problem for Einstein’s field equations.
Surveys in differential geometry 2015. One hundred years of general relativity, 1–16,
Surv. Differ. Geom., 20, Int. Press, Boston, MA, 2015.

The articles in the AMS Notices are:

MR3676432
Hill, C. Denson(1-SUNYS); Nurowski, Paweł(PL-PAN-CTP)
How the green light was given for gravitational wave search.
Notices Amer. Math. Soc. 64 (2017), no. 7, 686–692.

and

MR3676433
Bieri, Lydia(1-MI); Garfinkle, David(1-OAKL-P); Yunes, Nicolás(1-MTS-P)
Gravitational waves and their mathematics.
Notices Amer. Math. Soc. 64 (2017), no. 7, 693–707.

As the Notices  articles point out, there are two big questions that had to be answered before we could have the spectacular results from LIGO.  The paper by Hill and Nurowski addresses the first question:  1. Do gravitational waves exist?  The paper by Bieri, Garfinkle, and Yunes answers the second question: 2. How might one detect gravitational waves?

Some preliminaries

As mentioned above, Einstein’s equations are nonlinear PDEs:

(1)    $R_{i,j} – \frac{1}{2}R g_{i,j} = \kappa T_{i,j}$

where $g$ is the metric, $R_{i,j}$ is the Ricci curvature (which is a bunch of second derivatives of the metric), $R$ is the scalar curvature, and $T_{i,j}$ is the energy-momentum tensor that describes the matter or energy present in the space-time.  There is a linearized form, which was known already to Einstein.  First, think of the metric as a perturbation of the metric $\eta_{i,j}$ on Minkowski space:

$g_{i,j} = \eta_{i,j} + \epsilon h_{i,j}$.

To linearize, develop the left-hand side of (1) in powers of $\epsilon$, neglecting terms of order $2$ and above.   Away from sources, the energy-momentum tensor is zero.  So in these regions, the Einstein equations linearize to a system of decoupled relativistic wave equations in the unknowns $h_{i,j}$.  Thus, at least in linearized GR, there are plane waves traveling at the speed of light. Because these are linear equations, superposition applies and you can add these plane waves to make any wave you want .  These are the gravitational waves first described by Einstein.   Far from sources, the linearized theory should coincide closely with the nonlinear theory.  So one hopes to be able to detect gravitational waves by looking for phenomena that behave like the waves in the linearized model.  But first one has to answer Question 1.

Question 1: Do gravitational waves exist?

The question is whether or not waves exist in the full theory, not just the linearized version.  Hill and Nurowski reduce this question to seven sub-questions.  (Well that’s simpler, isn’t it?)

1. What is the definition of a plane gravitational wave in the full theory?
2. Does the so-defined plane wave exist as a solution to the full Einstein system?
3. Do such waves carry energy?
4. What is a definition of a gravitational wave with nonplanar front in the full theory?
5. What is the energy of such waves?
6. Do there exist solutions to the full Einstein system satisfying this definition?
7. Does the full theory admit solutions corresponding to the gravitational waves emitted by bounded sources?

Sub-questions 1 and 4 look surprising at first – the answers ought to be “Of course!”   But the mathematics of general relativity can be subtle, and simple things like bad choices of coordinates can mask important phenomena, as Einstein and Rosen discovered:

MR3363463
Einstein, A.; Rosen, N.
On gravitational waves.
J. Franklin Inst. 223 (1937), no. 1, 43–54.

(Well, actually, they made the bad choice and Howard Robertson discovered the badness of it when he refereed the paper.)   Ignoring Robertson’s observation led Einstein and Rosen to conclude that plane waves don’t exist in physical space-times, since their coordinate choice indicated that the space-time had singularities.

The positive answer to sub-questions 1, 2, and 3 comes in a paper by Bondi in Nature and the paper:

MR0106747
Bondi, H.; Pirani, F. A. E.; Robinson, I.
Gravitational waves in general relativity. III. Exact plane waves.
Proc. Roy. Soc. London Ser. A 251 1959 519–533.

MR0096537
Pirani, F. A. E.
Invariant formulation of gravitational radiation theory.
Phys. Rev. (2) 105 1957 1089–1099.

Hill and Nurowski point out that there was a parallel discovery much earlier by a mathematician that included the Bondi-Pirani-Robinson waves as a special case, but the gravitational theorists overlooked it:

MR1512246
Brinkmann, H. W.;
Einstein spaces which are mapped conformally on each other.
Math. Ann. 94 (1925), no. 1, 119–145.

Oops.

Now you have a definition of a plane wave, but you still have a problem because full GR is nonlinear, so superposition doesn’t work — you cannot build arbitrary waves out of plane waves.  However, Pirani gives a geometric definition of nonplanar waves by considering the principal null directions, which are the eigendirections of the Weyl tensor (the traceless part of the Riemann tensor).  Specifically, waves occur for space-times that are (asymptotically) algebraically special:

MR0096537
Pirani, F. A. E.
Invariant formulation of gravitational radiation theory.
Phys. Rev. (2) 105 1957 1089–1099.

Positive answers to sub-questions 4 and 5 come from Andrzej Trautman in his two papers:

MR0097265
Trautman, A.
Boundary conditions at infinity for physical theories.
Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 6 1958 403–406.

and

MR0097266
Trautman, A.
Radiation and boundary conditions in the theory of gravitation.
Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 6 1958 407–412.

The basic idea is to say that a radiative spacetime should satisfy certain boundary conditions at infinity, in analogy with radiative fields in Maxwell’s theory of electromagnetism.  (At some point, I probably should have introduced null infinity, but this post is already long enough.)

That leaves us with questions 6 and 7.  These were answered by Robinson and Trautman by actually producing some solutions with the desired property.

MR0135928
Robinson, I.; Trautman, A.
Some spherical gravitational waves in general relativity.
Proc. Roy. Soc. Ser. A 265 1961/1962 463–473.

Review: The authors obtain a class of metrics of which some represent, in their own words, “a very simple kind of spherical radiation”. They begin by establishing

$ds^2=-\rho^2p^{-2}\{(d\xi-ad\sigma)^2+(d\eta-bd\sigma)^2\}+2d\rho\,d\sigma+c\,d\sigma^2$

($a$, $b$, $c$, $p$ functions of the coordinates $\xi$, $\eta$, $\rho$, $\sigma$, but with $\partial p/\partial\rho=0$) as a canonical form for a metric which admits a shear-free, diverging and hypersurface-normal null vector field $\sigma_i$ that satisfies $R_{ik}\sigma^i\sigma^k=0$. They then solve the remaining field-equations for empty space and show that the solutions define two families of $V_2$ and admit a number of local and integral invariants. Algebraic properties of the curvature tensor and its rate of change along propagation-rays are discussed, and several explicit solutions of the Einstein and Maxwell-Einstein equations are obtained. These include the Schwarzschild metric as well as some static degenerate solutions of Levi-Civita.
Reviewed by H. S. Ruse

So, after some hiccups and a lot of hard work, we know what gravitational waves are and even have some information about their existence.

Question 2: How might one detect gravitational waves?

Gravity appears to us as a fairly strong force.  It sticks us to the Earth.  The sun has a massive gravitational field that bends light (one thing that makes eclipses interesting to physicists).  In the event detected by LIGO, about 3 times the mass of the sun was converted into gravitational waves in a fraction of a second.  So if a gravitational wave is going to come our way, especially one caused by the collision of two black holes, you might expect it to arrive like the big waves on the North Shore of Oahu.  But, alas, we are observing the waves far from the source, in an area that we are assuming is essentially flat.  So, in order to observe gravitational waves, LIGO needed to detect a displacement that was on the scale of 1/1000 the charge diameter of a proton over the course of  a 4 km baseline, or a change of about one part in $10^{21}$.  In order to do this, the experimenters need to have a very precise picture of what to look for.

A good way to find solutions to the Einstein equations is to solve a Cauchy problem. The Einstein equations split into a set of evolution equations and a set of constraint equations.   You want to solve this system by specifying initial data.  Typically, you either prescribe data on a space-like hypersurface (the classical case) or on a null hypersurface (the characteristic case).  In the classical case, the hypersurface is known as a Cauchy surface, which, at least locally, you can picture as a slice corresponding to a fixed time that will then evolve under the equation.   For gravitational waves, though, the characteristic case is more suitable, as explained in the Bulletin paper by Bieri.  In any case, a priori, there is no guarantee that if your initial data satisfy the constraints that the solution of the evolution equation also satisfies the constraints.  But in the case of the Einstein equations, Choquet-Bruhat showed exactly that. This is great!  Her local solution is in the paper

MR0053338
Fourès-Bruhat, Y.
Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires. (French)
Acta Math. 88, (1952). 141–225.

The global solution is in her paper with Geroch:

MR0250640
Choquet-Bruhat, Yvonne; Geroch, Robert
Global aspects of the Cauchy problem in general relativity.
Comm. Math. Phys. 14 1969 329–335.

Once we know that solutions exist, we need to know more about them because we need to know what we are looking for.   You can read about the long-term existence of solutions in the book by Christodoulou and Klainerman

MR1316662
Christodoulou, Demetrios(1-PRIN); Klainerman, Sergiu(1-PRIN)
The global nonlinear stability of the Minkowski space.
Princeton Mathematical Series, 41. Princeton University Press, Princeton, NJ, 1993. x+514 pp. ISBN: 0-691-08777-6

where they also investigate the asymptotic structure of these spacetimes.  As stated in the review of the book, “It is shown that the laws of gravitational radiation discovered by Bondi and others more than thirty years ago using formal power series expansions are rigorously true in this class of spacetimes.”

So, now we are getting closer to the question of how you might detect gravitational waves.  There is a good mathematical description of the waves.  What is needed next is a description of what might actually be observable.   The behavior of neighboring geodesics in any geometric setting is governed by the Jacobi equation.  In typical differential geometry courses, the Jacobi equation is used to identify conjugate points.   Here, though, the equation allows for a computation of the (small) displacement of test masses when a gravitational wave passes.  Since gravitational waves move at the speed of light, it is helpful to know that there is a memory effect that lingers and can be computed.  See, for instance,

MR1144215
Thorne, Kip S.(1-CAIT-TA)
Gravitational-wave bursts with memory: the Christodoulou effect.
Phys. Rev. D (3) 45 (1992), no. 2, 520–524

as well as more recent work as found in

MR3467364
Bieri, Lydia(1-MI); Garfinkle, David(1-OAKL-P); Yau, Shing-Tung(1-HRV)
Gravitational waves and their memory in general relativity. (English summary) Surveys in differential geometry 2015. One hundred years of general relativity, 75–97,
Surv. Differ. Geom., 20, Int. Press, Boston, MA, 2015.

What we need, though, is to be able to observe it and to measure it in the real world, thus confirming that it was caused by a gravitational wave. This uses the technique of matched filtering, meaning you have a collection of templates (derived from a solution) and you match the collected data to these templates.

Once again, the nonlinearity of the Einstein equations means that we don’t have many closed-form solutions to use for templates.  Rather, we need simulations.  The Choquet-Bruhat result tells us that for actual solutions, initial data that satisfy the constraints will evolve and still satisfy the constraints.  But in a simulation — an approximation, even a tiny deviation can cause a violation of the constraint and invalidate the simulation.  Fortunately, mathematical physicists are good at numerical solutions and simulations and came up with a suite of three techniques that overcome this:  hyperbolicity, constraint damping, and excision.  Here are some papers that are representative of each:

Hyperbolicity
MR2254287
Pretorius, Frans(3-AB-P)
Simulation of binary black hole spacetimes with a harmonic evolution scheme.
Classical Quantum Gravity 23 (2006), no. 16, S529–S552.

Constraint damping
MR3079062
Gundlach, Carsten(4-SHMP-SM); Martín-García, José M.(F-PARIS6-IAP); Garfinkle, David(1-OAKL-P)
Summation by parts methods for spherical harmonic decompositions of the wave equation in any dimensions.
Classical Quantum Gravity 30 (2013), no. 14, 145003, 31 pp.

Excision
MR2079939
Thornburg, Jonathan(D-MPIGP)
Black-hole excision with multiple grid patches.
Classical Quantum Gravity 21 (2004), no. 15, 3665–3691.

Finally, all the mathematical ingredients are in place:  a detailed picture of gravitational waves in the full, nonlinear theory and a means of computing samples of gravitational waves to the level of accuracy needed for matched filtering.  Performing the experiment requires turning the mathematics into machinery, which is in itself an exquisite feat of science and engineering.

Feynman used to boast that QED (quantum electrodynamics) was the physical theory that was able to make the most accurate predictions and that had been tested to the greatest level of precision, phrasing it as like measuring the distance from New York to LA to within the width of a human hair (one part in $10^{12}$). With the LIGO experiments, the general relativists have verified a theory using an instrument that is sensitive on the order of one part in $10^{21}$.  (See Note 4).

Acknowledgment:  I am grateful to Lydia Bieri for some extra comments on the work involved, in particular for helping me to understand Choquet-Bruhat’s work better.

Notes:

(1) The paper announcing the LIGO results is

MR3707758
Abbott, B. P.(1-CAIT-LIG); et al.;
Observation of gravitational waves from a binary black hole merger.
Authors include B. C. Barish, K. S. Thorne and R. Weiss.
Phys. Rev. Lett. 116 (2016), no. 6, 061102, 16 pp.

(2) Before Choquet-Bruhat, various people made some headway on the existence problem for Cauchy data for the Einstein equations.  One such person was Cornelius Lanczos, who had a remarkable life.  His mixture of bad luck and good luck led to him making contributions to several different areas of mathematics, including general relativity, but also numerical analysis and linear algebra.   The AMS published a nice biography of Lanczos by Barbara Gellai where you can read the details.

(3) The experiments to detect gravitational waves are impressive and expensive.  CERN’s search for the Higgs boson was even more expensive.  These are two major accomplishments in experimental physics that don’t get off the ground without some good mathematics to indicate that there was something interesting out there and giving a clue as to how to look for it.  But there are some mathematically inspired physics experiments that are much less expensive.  For instance, you can build a magnetic monopole detector for about the cost of a cup of coffee at Starbucks.  It is described in

MR0856880
Clifford Henry Taubes
Physical and Mathematical Applications of Gauge Theories,
Notices Amer. Math. Soc., 33 (1986), no. 5, 707–715.

All you need is a battery, a lightbulb, and some wire.  Oh, and the patience to sit unblinking for a very long time. The paper provides a very accessible introduction to gauge theory suitable for a general mathematical audience.  Taubes also has a little fun with the writing.  The article has passages like:

These are the plans, good luck in the chase,
the Nobel is the prize at the end of the race.

Earlier, discussing decay of quarks, Taubes writes: