Gravitational Wave Memory in Cosmological Spacetimes Lydia Bieri University of Michigan Department of Mathematics Ann Arbor Black Hole Initiative Conference Harvard University, May 8-9, 2017 Overview Spacetimes and Radiation Gravitational Radiation with Memory A \Footprint in Spacetime" ) Isolated Systems • Cosmological Setting • Photos: Courtesy of ETH-Bibliothek Z¨urich In 1915, Albert Einstein completed the Theory of General Relativity. In a letter of A. Einstein to A. Sommerfeld from November 1915, he mentions: "At present I occupy myself exclusively with the problem of gravitation and now believe that I shall master all difficulties with the help of a friendly mathematician (Marcel Grossmann). But one thing is certain, in all my life I have never labored nearly as hard, and I have become imbued with great respect for mathematics, the subtler part of which I had in my simple-mindedness regarded as pure luxury until now. Compared with this problem, the original relativity is child's play." C:/ITOOLS/WMS/CUP/151317/WORKINGFOLDER/NSS/9780521514842C06.3D 83 [63–87] 31.10.2008 11:47AM Postcard from Albert EinsteinThe toearly Hermann cosmology of Weyl, Einstein and 1923 de Sitter 83 Fig. 6.5 Einstein’s postcard to Weyl. Written on Tuesday before Whitsun, which Courtesy of ETH-Bibliothek Z¨urich corresponds to 23 May 1923. Note the postage of 180 Mark; this is a witness of the beginning of the proper German hyperinflation. (ETH-Bibliothek, Zurich, Einstein Archiv.) Einstein was not the only one to feel uncomfortable about de Sitter’s Pandora’s box. Eddington’s comments show that he was not happy with the situation in cosmology, and several years later Robertson expressed the hope to have found a mathematical solution in which many of de Sitter’s apparent paradoxes were eliminated (Robertson 1928). It is all the more astonishing that Einstein, in 1922, and again in 1923, brushed aside Friedmann’s dynamical solution of the Einstein equations. He did the same in 1927 when Lemaıˆtre showed him his discovery of the expanding universe, corroborated with theore- tical and observational evidence. We will come back to these two episodes. The interpretation of the de Sitter type models was never easy, having to do with the concept of time, and at least as much with the freedom to play with coordinate systems. When discussing Einstein’s and his models (models A and B respectively in his terminology), de Sitter said: ‘In both systems A and B it is always possible, at every point of the four-dimensional space-time, to find systems of reference in which the gμν depend only on one space-variable (the “radius-vector”), and not on “time”. In the system A the “time” of these systems of reference is the same always and everywhere, in B it is not. In B there is no Spacetimes in General Relativity Definition Spacetimes (M; g), where M a 4-dimensional manifold with Lorentzian metric g solving Einstein's equations: 1 Gµν := Rµν gµν R = 8π Tµν ; − 2 where Gµν is the Einstein tensor, Rµν is the Ricci curvature tensor, R the scalar curvature tensor, g the metric tensor and Tµν denotes the energy-momentum tensor. For Tµν 0 these equations reduce to the Einstein-Vacuum equations: ≡ Rµν = 0 : (1) Solutions of (1): Spacetimes (M; g), where M is a four-dimensional, oriented, differentiable manifold and g is a Lorentzian metric obeying (1). Asymptotically Flat versus Cosmological Spacetimes In the cosmological case, we add to the original Einstein equations the term containing Λ, the positive cosmological constant: 1 Rµν gµν R + Λgµν = 8π Tµν ; (2) − 2 Asymptotically Flat Spacetimes: Fall-off (in particular of metric and curvature components) towards Minkowski spacetime at infinity. Natural definition of \null infinity" understand gravitational radiation. ) These are solutions of the original Einstein equations with asymptotically flat initial data. Cosmological Spacetimes: Solutions of the cosmological Einstein equations (2). \Null infinity is spacelike". no \natural" way to discuss radiation. ) < H > 03$"1F Foliations of the Spacetime012 O O ¦ ¦ O P L O P L N ¡¦ ¥¤ ¨ ¡ O P O P ¡ ¡£ ¥¤ ¡£ §¤ N L N ¡£¢§¤ ©¨ O P ¡£¢¥¤ O P N L ¡¦¢§¤ KML Foliation by a time function t spacelike, complete Riemannian hypersurfaces Ht. ) >/> Foliation by a function@ u null hypersurfaces Cu. ) St;u = Ht Cu \ Foliation of Null Infinity Future null infinity + is defined to be the endpoints of all future-directed nullI geodesics along which r . It has the topology of ! 1 R S2 with the function u taking values in R. × + Thus a null hypersurface Cu intersects at infinity in a 2-sphere S1;u. I Consider a null hypersurface Cu in the spacetime M. Let t and explore limits of \local" quantities. ! 1 For instance: Hawking mass tends to Bondi mass along any Cu as t . ! 1 Shears and Expansion Scalars Viewing S as a hypersurface in C, respectively C: 4.6 The Characteristic Initial Value Problem Denote the second fundamental form of in by , and the In Section 3.3 we discussed about the CauchyS problemC χ for the Einstein equations. In par- second fundamental form of in by . ticular, we saw that the initial dataS setC consistsχ of the triplet ( , g, k), where is a H0 H0 three-dimensionalTheir traceless Riemannian parts are called manifold, the shearsg is the and metric denoted on by χ^,andχ^ k is a symmetric (0,2) H0 tensorrespectively. field on and such that g, k satisfy the constraint equations. Recall that g, k are to H0 be theThe first traces and secondtrχ and fundamentaltrχ are the formsexpansion of scalarsin . , respectively. H0 M In this section, we will discuss in detail the2 formulation of the characteristic initial value Null Limits of the Shears: limCu;t!1 r χ^ = Σ(u) and problem, i.e. the case where the initial Riemannian (spacelike) Cauchy hypesurface 0 is limCu;t!1 rχ^ = Ξ(u). H replaced by two degenerate (null) hypersurfaces C C intersecting at a two-dimensional ∪ surface S. Motivation Let us first motivate the formulation of the characteristic initial value problem. Let us assume that g/ is a given degenerate metric on C C and let be the arising spacetime ∪ M manifold and g the Lorentzian metric which satisfies the Einstein equations extending g/ on C C. Let us consider the double null foliation of ( , g) such that Ω = 1 on C C. Let L be ∪ M ∪ the geodesic vector field on C, which coincides with the normalized and equivariant vector field, and let u be its affine parameter such that u = 0 on S. Then, we obtain a foliation of C which consists of the (spacelike) surfaces S = u = τ . The crucial observation is that τ { } the null second fundamental form χ on C, which recall that is defined to be the following (0,2) tensor field on C χ(X, Y ) = g( e ,Y ), ∇X 4 where X, Y T C, is in fact, an tensor field which depends only on the intrinsic geometry of ∈ p C (although L depends on the spacetime metric g). Indeed, the first variational formula ∇X gives us 1 χ = / g/, 2L4 and since the Lie derivative / is intrinsic to the hypersurface C, we deduce that g/ com- LL pletely determines χ on C. On the other hand, by the Raychaudhuri equation we have e (trχ) = χ 2 trα, 4 −| | − and since χ and trχ (and ω = L(log Ω) = 0) are determined from g/ , we deduce that trα is also determined. However, in view of the Einstein equations (see Section 4.3) we have trα = Ric(e4, e4) = 0. This shows that one cannot arbitrarily prescribe a degenerate metric g/ on C C, since ∪ otherwise trα would in general be non-zero. 69 Gravitational Waves - Energy Radiated Fluctuation of curvature of the spacetime propagating as a wave. Gravitational waves: Localized disturbances in the geometry propagate at the speed of light, along outgoing null hypersurfaces. I+ I+ observe gravitational waves source H Gravitational radiation: gravitational waves traveling from source along outgoing null hypersurfaces. Picture: Courtesy of NASA. Memory Effect of Gravitational Waves Gravitational waves traveling from their source to our experiment. Three test masses in a plane as follows. The test masses will experience 1 Instantaneous displacements (while the wave packet is traveling through) 2 Permanent displacements (cumulative, stays after wave packet passed). This is called the memory effect of gravitational waves. Two types of this memory. Class. Quantum Grav. 29 (2012) 000000 L Bieri et al Figure 2. Permanent displacement of test masses caused by Christodoulou memory effect. Test masses m1andm2 are displaced permanently after the passage of a gravitational wave train. p 1488: ‘When matter (i.e. electromagnetic or neutrino) radiation is present then if T is the φ∗( 2 1 ( , )) ∗ →∞ | |2 energy tensor of matter, u r 4 T l l tends to a limit E as r0 and in (7)–(9) is replaced by | |2 +32πE.’ This is a suggestion, in which direction one would have to search to find other contributions to the nonlinear memory effect. It was not known, what the limit E would be. This limit E depending on u could behave in such a way that there were no additive contribution from E to the memory, or that it was negligible. Studying the adapted formulas (7)–(9) in Christodoulou (1991), one has to keep in mind that formula (9) governs the nonlinear memory effect. It is an additive effect. How do we know that E is in fact contributing? What is the structure of this limit? We give the answer in our formulas (15) and (6) based on Bieri et al (2010) and on (2) from Zipser (2009).