{"id":1790,"date":"2017-11-07T17:27:37","date_gmt":"2017-11-07T22:27:37","guid":{"rendered":"http:\/\/blogs.ams.org\/beyondreviews\/?p=1790"},"modified":"2018-03-12T15:12:57","modified_gmt":"2018-03-12T20:12:57","slug":"gravitational-waves","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/beyondreviews\/2017\/11\/07\/gravitational-waves\/","title":{"rendered":"Gravitational Waves"},"content":{"rendered":"<div id=\"attachment_1792\" style=\"width: 308px\" class=\"wp-caption alignleft\"><a href=\"https:\/\/blogs.ams.org\/beyondreviews\/files\/2017\/10\/GravWaves-NASA_C.-Henze.073344.jpg\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1792\" class=\" wp-image-1792\" src=\"http:\/\/blogs.ams.org\/beyondreviews\/files\/2017\/10\/GravWaves-NASA_C.-Henze.073344-1024x1024.jpg\" alt=\"A simulation of the merger of two black holes and the resulting emission of gravitational radiation.\" width=\"298\" height=\"298\" srcset=\"https:\/\/blogs.ams.org\/beyondreviews\/files\/2017\/10\/GravWaves-NASA_C.-Henze.073344-1024x1024.jpg 1024w, https:\/\/blogs.ams.org\/beyondreviews\/files\/2017\/10\/GravWaves-NASA_C.-Henze.073344-150x150.jpg 150w, https:\/\/blogs.ams.org\/beyondreviews\/files\/2017\/10\/GravWaves-NASA_C.-Henze.073344-300x300.jpg 300w, https:\/\/blogs.ams.org\/beyondreviews\/files\/2017\/10\/GravWaves-NASA_C.-Henze.073344-768x768.jpg 768w, https:\/\/blogs.ams.org\/beyondreviews\/files\/2017\/10\/GravWaves-NASA_C.-Henze.073344.jpg 2000w\" sizes=\"auto, (max-width: 298px) 100vw, 298px\" \/><\/a><p id=\"caption-attachment-1792\" class=\"wp-caption-text\">Credit: NASA\/C. Henze<\/p><\/div>\n<p>The <a href=\"https:\/\/www.nobelprize.org\/nobel_prizes\/physics\/laureates\/2017\/\">2017 Nobel Prize in Physics<\/a> was awarded to <a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/323370\">Rainer Weiss<\/a>, <a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/777128\">Barry C. Barish<\/a>, and <a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/172390\">Kip S. Thorne<\/a> for their work on the <a href=\"https:\/\/www.ligo.caltech.edu\/news\/ligo20160211\">detection of gravitational waves<\/a>. (See Note 1.)\u00a0 The physics and engineering that go into this accomplishment are truly impressive.\u00a0 However, before anyone could imagine setting up the experiment, some mathematical questions needed to be answered.\u00a0 There are <a href=\"http:\/\/www.ams.org\/journals\/notices\/201707\/rnoti-p684.pdf\">two articles<\/a> in the <a href=\"http:\/\/www.ams.org\/journals\/notices\/201707\/\">August 2017 issue<\/a> of the <em>AMS Notices<\/em> that give an overview the mathematics of gravitational waves.\u00a0 In this post, I crib from those two articles and provide a literature tour of some of the significant papers by relying on MathSciNet.\u00a0A <a href=\"http:\/\/www.ams.org\/journals\/bull\/0000-000-00\/S0273-0979-2017-01592-3\/\">longer article<\/a>\u00a0by Bieri just published in the <a href=\"http:\/\/www.ams.org\/publications\/journals\/journalsframework\/bull\">AMS Bulletin<\/a>\u00a0goes into more detail on a selection of the topics.\u00a0\u00a0<!--more--><\/p>\n<h2>Introduction<\/h2>\n<p>General Relativity (GR) depends heavily on mathematics, in particular, differential geometry and PDEs, since Einstein&#8217;s equations are nonlinear PDEs involving the curvature and the metric itself.\u00a0 \u00a0The 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.\u00a0 This is digression from the theory of gravitational waves, but it is, in the language of Michelin guides, &#8220;Worth a trip.&#8221;\u00a0 \u00a0As a tour guide for this trip, I heartily recommend the book\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet-getitem?mr=867684\"><em>Einstein manifolds<\/em><\/a> written by a group of geometers under the pseudonym\u00a0Arthur L.\u00a0Besse.<\/p>\n<p>There are many places for a mathematician to learn general relativity (GR).\u00a0 For a first read, Einstein&#8217;s own account in <em><a href=\"https:\/\/mathscinet.ams.org\/mathscinet-getitem?mr=12946\">The Meaning of Relativity<\/a><\/em>\u00a0is extremely good.\u00a0 However it is deceptive: it seems like easy reading, but he is describing very deep ideas that are overturning our view of the world.\u00a0 You need to pay attention to understand what is going on.\u00a0 I didn&#8217;t pay attention the first time through and had to read it a second time.\u00a0 Here are three books that are suited to mathematicians for learning about general relativity.<\/p>\n<ul>\n<li><a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/82830\">Hawking<\/a> and <a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/63015\">Ellis<\/a>,\u00a0<em>The large scale structure of space-time<\/em>. Cambridge Monographs on Mathematical Physics, No. 1. Cambridge University Press, London-New York, 1973. xi+391 pp.\u00a0<a href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=424186\">MR0424186<\/a><\/li>\n<li><a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/195638\">Barrett O&#8217;Neill<\/a>, <em>Semi-Riemannian Geometry<\/em>,\u00a0With applications to relativity. Pure and Applied Mathematics, 103. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. xiii+468 pp.\u00a0 <a href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=719023\">MR0719023<\/a><\/li>\n<li><a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/348873\">Misner<\/a>, <a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/172390\">Thorne<\/a>, and <a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/182260\">Wheeler<\/a>,\u00a0<em>Gravitation<\/em>. W. H. Freeman and Co., San Francisco, Calif., 1973. ii+xxvi+1279+iipp. <a href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=418833\">MR0418833<\/a>.<\/li>\n<\/ul>\n<p>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.\u00a0 O&#8217;Neill&#8217;s book is written by a mathematician for mathematicians.\u00a0 The bulk of it is about differential geometry with a semi-Riemannian metric.\u00a0 You could give a complete course on differential geometry from O&#8217;Neill&#8217;s book and not cover relativity.\u00a0 But why would you leave it out?\u00a0 He sets everything up for you, then puts it to use.\u00a0 This is probably not the introduction that a physicist would want, but since it begins on familiar territory, it suits many mathematicians.\u00a0 Misner, Thorne, and Wheeler is often referred to as &#8220;the Bible&#8221; for studying general relativity.\u00a0 It is cited so often, that people often refer to it as MTW.\u00a0 \u00a0It is a monster-sized book, but it is really two books in one: a one-semester introduction and a deeper look at the subject.\u00a0\u00a0MTW is the only one of these three books that really treats gravitational waves.<\/p>\n<p>These three books provide a general introduction to GR.\u00a0 The two <em>Notices<\/em> papers are describing results that depend on the modern approach of geometric analysis in general relativity.\u00a0 For that, a good place to start would be the book:<\/p>\n<p><a href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=2391586\">MR2391586<\/a><br \/>\nChristodoulou, Demetrios(CH-ETHZ)<br \/>\nMathematical problems of general relativity. I.<br \/>\nZurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Z\u00fcrich, 2008. x+147 pp. ISBN: 978-3-03719-005-0.<\/p>\n<p>I have not read it &#8211; I am relying on the review in MathSciNet!\u00a0 A short history of the mathematical study of Einstein&#8217;s equations can be found in the quite readable and very enjoyable article by Choquet-Bruhat:<\/p>\n<p><a href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=3467361\">MR3467361<\/a><br \/>\nChoquet-Bruhat, Yvonne(F-IHES)<br \/>\nBeginnings of the Cauchy problem for Einstein&#8217;s field equations.<br \/>\nSurveys in differential geometry 2015. One hundred years of general relativity, 1\u201316,<br \/>\nSurv. Differ. Geom., 20, Int. Press, Boston, MA, 2015.<\/p>\n<hr \/>\n<p>The <a href=\"http:\/\/www.ams.org\/journals\/notices\/201707\/rnoti-p684.pdf\">articles in the AMS Notices<\/a> are:<\/p>\n<p style=\"padding-left: 30px\"><a href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=3676432\">MR3676432<\/a><br \/>\n<a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/211060\">Hill, C. Denson<\/a>(1-SUNYS); <a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/250343\">Nurowski, Pawe\u0142<\/a>(PL-PAN-CTP)<br \/>\nHow the green light was given for gravitational wave search.<br \/>\n<em>Notices Amer. Math. Soc.<\/em> 64 (2017), no. 7, 686\u2013692.<\/p>\n<p style=\"padding-left: 30px\">and<\/p>\n<p style=\"padding-left: 30px\"><a href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=3676433\">MR3676433<\/a><br \/>\n<a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/877386\">Bieri, Lydia<\/a>(1-MI); <a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/219633\">Garfinkle, David<\/a>(1-OAKL-P); <a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/720789\">Yunes, Nicol\u00e1s<\/a>(1-MTS-P)<br \/>\nGravitational waves and their mathematics.<br \/>\n<em>Notices Amer. Math. Soc.<\/em> 64 (2017), no. 7, 693\u2013707.<\/p>\n<p>As the <em>Notices\u00a0<\/em> articles point out, there are two big questions that had to be answered before we could have the spectacular results from LIGO.\u00a0\u00a0The paper by Hill and Nurowski addresses the first question:\u00a0 <strong><span style=\"color: #008000\">1. Do gravitational waves exist?<\/span><\/strong>\u00a0 The paper by Bieri, Garfinkle, and Yunes answers the second question: <strong><span style=\"color: #008000\">2. How might one detect gravitational waves?<\/span><\/strong><\/p>\n<h2>Some preliminaries<\/h2>\n<p>As mentioned above, Einstein&#8217;s equations are nonlinear PDEs:<\/p>\n<p>(1)\u00a0 \u00a0 $R_{i,j} &#8211; \\frac{1}{2}R g_{i,j} = \\kappa T_{i,j}$<\/p>\n<p>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.\u00a0\u00a0There is a linearized form, which was known already to Einstein.\u00a0 First, think of the metric as a perturbation of the metric $\\eta_{i,j}$ on Minkowski space:<\/p>\n<p>$g_{i,j} = \\eta_{i,j} + \\epsilon h_{i,j}$.<\/p>\n<p>To linearize, develop the left-hand side of (1) in powers of $\\epsilon$, neglecting terms of order $2$ and above.\u00a0 \u00a0Away from sources, the energy-momentum tensor is zero.\u00a0 So in these regions, the Einstein equations linearize to a system of decoupled relativistic wave equations in the unknowns $h_{i,j}$.\u00a0 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 .\u00a0 These are the\u00a0<em>gravitational waves<\/em> first described by Einstein.\u00a0 \u00a0Far from sources, the linearized theory should coincide closely with the nonlinear theory.\u00a0 So one hopes to be able to detect gravitational waves by looking for phenomena that behave like the waves in the linearized model.\u00a0 But first one has to answer Question 1.<\/p>\n<h2>Question 1: Do gravitational waves exist?<\/h2>\n<p>The question is whether or not waves exist <em>in the full theory<\/em>, not just the linearized version.\u00a0 Hill and Nurowski reduce this question to seven sub-questions.\u00a0 (Well that&#8217;s simpler, isn&#8217;t it?)<\/p>\n<ol>\n<li>What is the definition of a\u00a0<em>plane<\/em> gravitational wave in the full theory?<\/li>\n<li>Does the so-defined plane wave exist as a solution to the full Einstein system?<\/li>\n<li>Do such waves carry energy?<\/li>\n<li>What is a definition of a gravitational wave with <em>nonplanar front<\/em> in the full theory?<\/li>\n<li>What is the energy of such waves?<\/li>\n<li>Do there exist solutions to the full Einstein system satisfying this definition?<\/li>\n<li>Does the full theory admit solutions corresponding to the gravitational waves emitted by bounded sources?<\/li>\n<\/ol>\n<p>Sub-questions 1 and 4 look surprising at first &#8211; the answers ought to be &#8220;Of course!&#8221;\u00a0 \u00a0But 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:<\/p>\n<p><a href=\"https:\/\/mathscinet.ams.org\/mathscinet-getitem?mr=3363463\">MR3363463<\/a><br \/>\nEinstein, A.; Rosen, N.<br \/>\nOn gravitational waves.<br \/>\nJ. Franklin Inst. 223 (1937), no. 1, 43\u201354.<\/p>\n<p>(Well, actually, they made the bad choice and Howard Robertson discovered the badness of it when he refereed the paper.)\u00a0 \u00a0Ignoring Robertson&#8217;s observation led Einstein and Rosen to conclude that plane waves don&#8217;t exist in <em>physical<\/em> space-times, since their coordinate choice indicated that the space-time had singularities.<\/p>\n<p>The positive answer to sub-questions 1, 2, and 3 comes in <a href=\"http:\/\/www.nature.com\/nature\/journal\/v179\/n4569\/abs\/1791072a0.html\">a paper by Bondi in\u00a0<em>Nature<\/em><\/a> and the paper:<\/p>\n<p style=\"padding-left: 30px\"><a href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=106747\">MR0106747<\/a><br \/>\n<a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/39140\">Bondi, H.<\/a>; <a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/139915\">Pirani, F. A. E.<\/a>; <a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/192741\">Robinson, I.<\/a><br \/>\nGravitational waves in general relativity. III. Exact plane waves.<br \/>\nProc. Roy. Soc. London Ser. A 251 1959 519\u2013533.<\/p>\n<p>See also the paper:<\/p>\n<p><a href=\"https:\/\/mathscinet.ams.org\/mathscinet-getitem?mr=96537\">MR0096537<\/a><br \/>\nPirani, F. A. E.<br \/>\nInvariant formulation of gravitational radiation theory.<br \/>\nPhys. Rev. (2) 105 1957 1089\u20131099.<\/p>\n<p>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:<\/p>\n<p style=\"padding-left: 30px\"><a href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=1512246\">MR1512246<\/a><br \/>\nBrinkmann, H. W.;<br \/>\nEinstein spaces which are mapped conformally on each other.<br \/>\nMath. Ann. 94 (1925), no. 1, 119\u2013145.<\/p>\n<p>Oops.<\/p>\n<p>Now you have a definition of a plane wave, but you still have a problem because full GR is nonlinear, so superposition doesn&#8217;t work &#8212; you cannot build arbitrary waves out of plane waves.\u00a0 However, Pirani gives a geometric definition of nonplanar waves by considering the principal null directions, which are the eigendirections of the <a href=\"https:\/\/en.m.wikipedia.org\/wiki\/Weyl_tensor\">Weyl tensor<\/a>\u00a0(the traceless part of the Riemann tensor).\u00a0 Specifically, waves occur for space-times that are\u00a0(asymptotically)\u00a0<a href=\"https:\/\/en.m.wikipedia.org\/wiki\/Petrov_classification\"><em>algebraically special:<\/em><\/a><\/p>\n<p style=\"padding-left: 30px\"><a href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=96537\">MR0096537<\/a><br \/>\n<a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/139915\">Pirani, F. A. E.<\/a><br \/>\nInvariant formulation of gravitational radiation theory.<br \/>\nPhys. Rev. (2) 105 1957 1089\u20131099.<\/p>\n<p>Positive answers to sub-questions 4 and 5 come from <a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/174075\">Andrzej Trautman<\/a> in his two papers:<\/p>\n<p style=\"padding-left: 30px\"><a href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=97265\">MR0097265<\/a><br \/>\n<a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/174075\">Trautman, A.<\/a><br \/>\nBoundary conditions at infinity for physical theories.<br \/>\nBull. Acad. Polon. Sci. S\u00e9r. Sci. Math. Astr. Phys. 6 1958 403\u2013406.<\/p>\n<p>and<\/p>\n<p style=\"padding-left: 30px\"><a href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=97266\">MR0097266<\/a><br \/>\n<a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/174075\">Trautman, A.<\/a><br \/>\nRadiation and boundary conditions in the theory of gravitation.<br \/>\nBull. Acad. Polon. Sci. S\u00e9r. Sci. Math. Astr. Phys. 6 1958 407\u2013412.<\/p>\n<p>The basic idea is to say that a radiative spacetime should satisfy certain boundary conditions at infinity, in analogy with radiative fields in Maxwell&#8217;s theory of electromagnetism.\u00a0 (At some point, I probably should have introduced <a href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=3467365\">null infinity<\/a>, but this post is already long enough.)<\/p>\n<p>That leaves us with questions 6 and 7.\u00a0 These were answered by Robinson and Trautman by actually producing some solutions with the desired property.<\/p>\n<p style=\"padding-left: 30px\"><a href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=135928\">MR0135928<\/a><br \/>\n<a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/192741\">Robinson, I.<\/a>; <a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/174075\">Trautman, A.<\/a><br \/>\nSome spherical gravitational waves in general relativity.<br \/>\nProc. Roy. Soc. Ser. A 265 1961\/1962 463\u2013473.<\/p>\n<p style=\"padding-left: 30px\"><strong>Review<\/strong>:\u00a0The authors obtain a class of metrics of which some represent, in their own words, &#8220;a very simple kind of spherical radiation&#8221;. They begin by establishing<\/p>\n<p style=\"padding-left: 30px\">$ds^2=-\\rho^2p^{-2}\\{(d\\xi-ad\\sigma)^2+(d\\eta-bd\\sigma)^2\\}+2d\\rho\\,d\\sigma+c\\,d\\sigma^2$<\/p>\n<p style=\"padding-left: 30px\">($a$, $b$, $c$, $p$ 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.<br \/>\nReviewed by H. S. Ruse<\/p>\n<p>So, after some hiccups and a lot of hard work, we know what gravitational waves are and even have some information about their existence.<\/p>\n<h2>Question 2:\u00a0How might one detect gravitational waves?<\/h2>\n<p>Gravity appears to us as a fairly strong force.\u00a0 It sticks us to the Earth.\u00a0 The sun has a massive gravitational field that bends light (one thing that makes eclipses interesting to physicists).\u00a0 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.\u00a0 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.\u00a0 But, alas, we are observing the waves far from the source, in an area that we are assuming is essentially flat.\u00a0 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\u00a0 a 4 km baseline, or a change of about one part in $10^{21}$.\u00a0 In order to do this, the experimenters need to have a very precise picture of what to look for.<\/p>\n<p>A good way to find solutions to the Einstein equations is to solve a Cauchy problem.\u00a0The Einstein equations split into a set of evolution equations and a set of constraint equations.\u00a0 \u00a0You want to solve this system by specifying initial data.\u00a0 Typically, you either prescribe data on a space-like hypersurface (the classical case) or on a null hypersurface (the characteristic case).\u00a0 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.\u00a0 \u00a0For gravitational waves, though, the characteristic case is more suitable, as explained in the <a href=\"http:\/\/www.ams.org\/journals\/bull\/0000-000-00\/S0273-0979-2017-01592-3\/\"><em>Bulletin\u00a0<\/em>paper<\/a> by Bieri.\u00a0 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.\u00a0 But in the case of the Einstein equations, <a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/48780\">Choquet-Bruhat<\/a> showed exactly that.\u00a0This is great!\u00a0 Her local solution is in the paper<\/p>\n<p><a href=\"https:\/\/mathscinet.ams.org\/mathscinet-getitem?mr=53338\">MR0053338<\/a><br \/>\n<a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/48780\">Four\u00e8s-Bruhat, Y.<\/a><br \/>\nTh\u00e9or\u00e8me d&#8217;existence pour certains syst\u00e8mes d&#8217;\u00e9quations aux d\u00e9riv\u00e9es partielles non lin\u00e9aires. (French)<br \/>\nActa Math. 88, (1952). 141\u2013225.<\/p>\n<p>The global solution is in her paper with Geroch:<\/p>\n<p><a href=\"https:\/\/mathscinet.ams.org\/mathscinet-getitem?mr=250640\">MR0250640<\/a><br \/>\n<a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/48780\">Choquet-Bruhat, Yvonne<\/a>; <a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/225022\">Geroch, Robert<\/a><br \/>\nGlobal aspects of the Cauchy problem in general relativity.<br \/>\nComm. Math. Phys. 14 1969 329\u2013335.<\/p>\n<p>Once we know that solutions exist, we need to know more about them because we need to know what we are looking for.\u00a0 \u00a0You can read about the long-term existence of solutions in the book by\u00a0<a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/49030\">Christodoulou<\/a> and\u00a0<a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/102350\">Klainerman<\/a><\/p>\n<p><a href=\"http:\/\/mathscinet.ams.org\/mathscinet-getitem?mr=1316662\">MR1316662<\/a><br \/>\n<a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/49030\">Christodoulou, Demetrios<\/a>(1-PRIN); <a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/102350\">Klainerman, Sergiu<\/a>(1-PRIN)<br \/>\nThe global nonlinear stability of the Minkowski space.<br \/>\nPrinceton Mathematical Series, 41. Princeton University Press, Princeton, NJ, 1993. x+514 pp. ISBN: 0-691-08777-6<\/p>\n<p>where they also investigate\u00a0the asymptotic structure of these spacetimes.\u00a0 As stated in the review of the book, &#8220;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.&#8221;<\/p>\n<p>So, now we are getting closer to the question of how you might detect gravitational waves.\u00a0 There is a good mathematical description of the waves.\u00a0 What is needed next is a description of what might actually be observable.\u00a0 \u00a0The behavior of neighboring geodesics in any geometric setting is governed by the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Jacobi_field\">Jacobi equation<\/a>.\u00a0 In typical differential geometry courses, the Jacobi equation is used to identify <a href=\"https:\/\/en.wikipedia.org\/wiki\/Conjugate_points\">conjugate points<\/a>.\u00a0 \u00a0Here, though, the equation allows for a computation of the (small) displacement of test masses when a gravitational wave passes.\u00a0 Since gravitational waves move at the speed of light, it is helpful to know that there is a\u00a0<em>memory effect<\/em> that lingers and can be computed.\u00a0 See, for instance,<\/p>\n<p><a href=\"https:\/\/mathscinet.ams.org\/mathscinet-getitem?mr=1144215\">MR1144215<\/a><br \/>\nThorne, Kip S.(1-CAIT-TA)<br \/>\nGravitational-wave bursts with memory: the Christodoulou effect.<br \/>\nPhys. Rev. D (3) 45 (1992), no. 2, 520\u2013524<\/p>\n<p>as well as more recent work as found in<\/p>\n<p><a href=\"https:\/\/mathscinet.ams.org\/mathscinet-getitem?mr=3467364\">MR3467364<\/a><br \/>\nBieri, Lydia(1-MI); Garfinkle, David(1-OAKL-P); Yau, Shing-Tung(1-HRV)<br \/>\nGravitational waves and their memory in general relativity. (English summary) Surveys in differential geometry 2015. One hundred years of general relativity, 75\u201397,<br \/>\nSurv. Differ. Geom., 20, Int. Press, Boston, MA, 2015.<\/p>\n<p>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\u00a0<a href=\"https:\/\/en.wikipedia.org\/wiki\/Matched_filter\"><em>matched filtering<\/em><\/a>, meaning you have a collection of templates (derived from a solution) and you match the collected data to these templates.<\/p>\n<p>Once again, the nonlinearity of the Einstein equations means that we don&#8217;t have many closed-form solutions to use for templates.\u00a0 Rather, we need simulations.\u00a0 The Choquet-Bruhat result tells us that for actual solutions, initial data that satisfy the constraints will evolve and still satisfy the constraints.\u00a0 But in a simulation &#8212; an approximation, even a tiny deviation can cause a violation of the constraint and invalidate the simulation.\u00a0 Fortunately, mathematical physicists are good at numerical solutions and simulations and came up with a suite of three techniques that overcome this:\u00a0 hyperbolicity, constraint damping, and excision.\u00a0 Here are some papers that are representative of each:<\/p>\n<p><strong>Hyperbolicity<\/strong><br \/>\n<a href=\"http:\/\/mathscinet.ams.org\/mathscinet-getitem?mr=2254287\">MR2254287<\/a><br \/>\n<a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/638839\">Pretorius, Frans<\/a>(3-AB-P)<br \/>\nSimulation of binary black hole spacetimes with a harmonic evolution scheme.<br \/>\nClassical Quantum Gravity 23 (2006), no. 16, S529\u2013S552.<\/p>\n<p><strong>Constraint damping<\/strong><br \/>\n<a href=\"http:\/\/mathscinet.ams.org\/mathscinet-getitem?mr=3079062\">MR3079062<\/a><br \/>\n<a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/318815\">Gundlach, Carsten<\/a>(4-SHMP-SM); <a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/645980\">Mart\u00edn-Garc\u00eda, Jos\u00e9 M<\/a>.(F-PARIS6-IAP); <a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/219633\">Garfinkle, David<\/a>(1-OAKL-P)<br \/>\nSummation by parts methods for spherical harmonic decompositions of the wave equation in any dimensions.<br \/>\nClassical Quantum Gravity 30 (2013), no. 14, 145003, 31 pp.<\/p>\n<p><strong>Excision<\/strong><br \/>\n<a href=\"http:\/\/mathscinet.ams.org\/mathscinet-getitem?mr=2079939\">MR2079939<\/a><br \/>\n<a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/248071\">Thornburg, Jonathan<\/a>(D-MPIGP)<br \/>\nBlack-hole excision with multiple grid patches.<br \/>\nClassical Quantum Gravity 21 (2004), no. 15, 3665\u20133691.<\/p>\n<p>Finally, all the mathematical ingredients are in place:\u00a0 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.\u00a0 Performing the experiment requires turning the mathematics into machinery, which is in itself an exquisite feat of science and engineering.<\/p>\n<p>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\u00a0like 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 $10^{21}$.\u00a0 (See Note 4).<\/p>\n<hr \/>\n<p><strong>Acknowledgment<\/strong>:\u00a0 I am grateful to <a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/877386\">Lydia Bieri<\/a> for some extra comments on the work involved, in particular for helping me to understand Choquet-Bruhat&#8217;s work better.<\/p>\n<hr \/>\n<p><strong>Notes<\/strong>:<\/p>\n<p>(1) The paper announcing the LIGO results is<\/p>\n<p><a href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=3707758\">MR3707758<\/a><br \/>\nAbbott, B. P.(1-CAIT-LIG); et al.;<br \/>\nObservation of gravitational waves from a binary black hole merger.<br \/>\nAuthors include B. C. Barish, K. S. Thorne and R. Weiss.<br \/>\nPhys. Rev. Lett. 116 (2016), no. 6, 061102, 16 pp.<\/p>\n<p>(2) Before Choquet-Bruhat, various people made some headway on the existence problem for Cauchy data for the Einstein equations.\u00a0 One such person was <a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/109775\">Cornelius Lanczos<\/a>, who had a remarkable life.\u00a0 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.\u00a0 \u00a0The AMS published a nice <a href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=2722916\">biography of Lanczos<\/a> by\u00a0Barbara Gellai where you can read the details.<\/p>\n<p>(3) The experiments to detect gravitational waves are impressive and <em>expensive<\/em>.\u00a0 CERN&#8217;s search for the Higgs boson was even more expensive.\u00a0 These are two major accomplishments in experimental physics that don&#8217;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.\u00a0 But there are some mathematically inspired physics experiments that are much less expensive.\u00a0 For instance, you can build a magnetic monopole detector for about the cost of a cup of coffee at <a href=\"https:\/\/en.wikipedia.org\/wiki\/Lieutenant_Starbuck\">Starbucks<\/a>.\u00a0 It is described in<\/p>\n<p><a href=\"https:\/\/mathscinet.ams.org\/mathscinet-getitem?mr=856880\">MR0856880<\/a><br \/>\n<a href=\"http:\/\/www.ams.org\/mathscinet\/MRAuthorID\/171070\">Clifford Henry Taubes<\/a><br \/>\nPhysical and Mathematical Applications of Gauge Theories,<br \/>\n<em>Notices Amer. Math. Soc.<\/em>,\u00a033 (1986), no. 5, 707&#8211;715.<\/p>\n<p>All you need is a battery, a lightbulb, and some wire.\u00a0 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.\u00a0 Taubes also has a little fun with the writing.\u00a0 The article has passages like:<\/p>\n<p><em>These are the plans, good luck in the chase,<br \/>\n<\/em><em>the Nobel is the prize at the end of the race.<\/em><\/p>\n<p>Earlier, discussing decay of quarks, Taubes writes:<\/p>\n<p><em>Please, don&#8217;t blink, as you might miss the prize<\/em><br \/>\n<em>of seeing the lash of this year&#8217;s lone quark&#8217;s predicted demise.<\/em><br \/>\n<em>After five years of patience, statistics are small;<\/em><br \/>\n<em>no announcement&#8217;s been made of matter&#8217;s downfall.<\/em><\/p>\n<p>Other passages are in <a href=\"https:\/\/www.poetryfoundation.org\/learn\/glossary-terms\/blank-verse\">blank verse<\/a>.\u00a0 That era of the AMS\u00a0<em>Notices<\/em> is not (yet) available online.\u00a0 If you want to read the article, I am afraid you will have to go to the library.\u00a0 It is at least worth a detour as you walk from <a href=\"http:\/\/www.gutenberg.org\/ebooks\/2701\">Starbucks<\/a> to your office.<\/p>\n<p>(4) I&#8217;ve skipped over something important in comparing the accuracy of QED to the sensitivity of the LIGO experiments.\u00a0 I refer you to a <a href=\"https:\/\/physics.stackexchange.com\/questions\/235865\/how-are-the-4-km-arms-of-ligo-measured-so-accurately\">discussion<\/a> on <a href=\"https:\/\/physics.stackexchange.com\/\">physics.stackexhange.com<\/a> to address that.<\/p>\n<div style=\"margin-top: 0px; margin-bottom: 0px;\" class=\"sharethis-inline-share-buttons\" ><\/div>","protected":false},"excerpt":{"rendered":"<p>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.)\u00a0 The physics and engineering that go into this accomplishment are &hellip; <a href=\"https:\/\/blogs.ams.org\/beyondreviews\/2017\/11\/07\/gravitational-waves\/\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n<div style=\"margin-top: 0px; margin-bottom: 0px;\" class=\"sharethis-inline-share-buttons\" data-url=https:\/\/blogs.ams.org\/beyondreviews\/2017\/11\/07\/gravitational-waves\/><\/div>\n","protected":false},"author":86,"featured_media":1792,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[40],"tags":[],"class_list":["post-1790","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-announcements"],"jetpack_featured_media_url":"https:\/\/blogs.ams.org\/beyondreviews\/files\/2017\/10\/GravWaves-NASA_C.-Henze.073344.jpg","jetpack_shortlink":"https:\/\/wp.me\/p6C2KK-sS","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/blogs.ams.org\/beyondreviews\/wp-json\/wp\/v2\/posts\/1790","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ams.org\/beyondreviews\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ams.org\/beyondreviews\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/beyondreviews\/wp-json\/wp\/v2\/users\/86"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/beyondreviews\/wp-json\/wp\/v2\/comments?post=1790"}],"version-history":[{"count":61,"href":"https:\/\/blogs.ams.org\/beyondreviews\/wp-json\/wp\/v2\/posts\/1790\/revisions"}],"predecessor-version":[{"id":1965,"href":"https:\/\/blogs.ams.org\/beyondreviews\/wp-json\/wp\/v2\/posts\/1790\/revisions\/1965"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/beyondreviews\/wp-json\/wp\/v2\/media\/1792"}],"wp:attachment":[{"href":"https:\/\/blogs.ams.org\/beyondreviews\/wp-json\/wp\/v2\/media?parent=1790"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ams.org\/beyondreviews\/wp-json\/wp\/v2\/categories?post=1790"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ams.org\/beyondreviews\/wp-json\/wp\/v2\/tags?post=1790"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}