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 Zu rich
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.
Postcard from Albert Einstein to Hermann Weyl, 1923 The early cosmology of Einstein and de Sitter 83 Courtesy of ETH-Bibliothek Zürich Fig. 6.5 Einstein s postcard to Weyl. Written on Tuesday before Whitsun, which
Spacetimes in General Relativity Definition Spacetimes (M, g), where M a 4-dimensional manifold with Lorentzian metric g solving Einstein s equations: G µν := R µν 1 2 g µν R = 8π T µν, where G µν is the Einstein tensor, R µν is the Ricci curvature tensor, R the scalar curvature tensor, g the metric tensor and T µν denotes the energy-momentum tensor. For T µν 0 these equations reduce to the Einstein-Vacuum equations: R µν = 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: R µν 1 2 g µν R + Λg µν = 8π T µν, (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 01203$"1F > < Foliations of the Spacetime O L P O P O P O P O P O L P O O N N L N N L KML Foliation by a time function t spacelike, complete Riemannian hypersurfaces H t. >/> @ Foliation by a function u null hypersurfaces C u. S t,u = H t C u
Foliation of Null Infinity Future null infinity I + is defined to be the endpoints of all future-directed null geodesics along which r. It has the topology of R S 2 with the function u taking values in R. Thus a null hypersurface C u intersects I + at infinity in a 2-sphere S,u. Consider a null hypersurface C u in the spacetime M. Let t and explore limits of local quantities. For instance: Hawking mass tends to Bondi mass along any C u as t.
Shears and Expansion Scalars Viewing S as a hypersurface in C, respectively C: 4.6 The Characteristic Initial Value Problem In Section Denote 3.3 the we second discussed fundamental about the form Cauchy of S problem in C by χ, for and the the Einstein equatio ticular, second we saw fundamental that the initial form of data S in set C by consists χ. of the triplet (H 0, g, k), wh three-dimensional Their traceless Riemannian parts are called manifold, the shears g is the andmetric denoted onby H 0 ˆχ, and ˆχ k is a symm tensorrespectively. field on H 0 and such that g, k satisfy the constraint equations. Recall tha be thethe firsttraces and second trχ and fundamental trχ are theforms expansion of Hscalars. 0 in M, respectively. In Null this section, Limits of we the will Shears: discuss lim in detail the Cu,t r 2 ˆχ formulation = Σ(u) and of the characteristic problem, lim i.e. the Cu,t rˆχ case = where Ξ(u). the initial Riemannian (spacelike) Cauchy hypesu replaced by two degenerate (null) hypersurfaces C C intersecting at a twosurface S.
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 Picture: Courtesy of NASA. Gravitational radiation: gravitational waves traveling from sour
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. lass. Quantum Grav. 29 (2012) 000000 L Bieri et al
Memory - Continued - Isolated Systems Ordinary (formerly called linear ) effect => was known for a long time in the slow motion limit [Ya.B. Zel dovich, A.G. Polnarev 1974] Null (formerly called nonlinear ) effect => was found by [D. Christodoulou 1991]. Contribution from electro-magnetic field to nonlinear effect => was found by [L. Bieri, P. Chen, S.-T. Yau 2010 and 2011]. Contribution from neutrino radiation to nonlinear effect => was found by [L. Bieri, D. Garfinkle 2012 and 2013]. See also works by Braginsky, Grishchuk, Thorne, Blanchet, Damour, Wiseman, Will. Recent works on memory include Wald, Tolish, Favata, Flanagan, Strominger, Winicour, Loutrel, Yunes, Hawking, Perry, Zhiboedov, Pasterski and more.
Contribution to the null memory (Christodoulou memory) for a fairly general stress-energy tensor with decay r 2 in the outgoing null direction (L. Bieri, D. Garfinkle). 2 Types of Memory Due to: Fields that do and Fields that do not go out to null infinity! We find an electromagnetic analog of gravitational wave memory. [L. Bieri, D. Garfinkle 2013] charged test masses observe a residual kick.
Detection Detectors of electromagnetic radiation absorb energy from the wave. Flux of energy in the wave: goes as r 2. Sensitivity of the detector falls off like r 2. Detectors of gravitational waves sensitivity falls off like r 1. Gravitational wave detector works not by measuring power absorbed from the wave but rather by following the motion induced in the detector by the wave. What permanent changes occur? Gravitational: Permanent displacement. Electromagnetic: Residual velocity (kick).
Gravitational Wave Experiment For a situation where the geodesics are not too far away from each other, one can replace the geodesic equation for γ 1 and γ 2 by the Jacobi equation (geodesic deviation from γ 0 ). with where k, l = 1, 2, 3. d 2 x k dt 2 = R kt lt x l (3) R kt lt = R (E k, T, E l, T )!!!Information about the curvature and null structures required!!! Analyze the spacetimes!
Memory - Permanent Displacement Asymptotically Flat Spacetimes The permanent displacement of test masses is related to the difference (Σ + Σ ) in the asymptotic shears, which themselves depend on the radiated energy in a nonlinear way. x = ( d 0 r ) (Σ+ Σ ). (4) There are the following contributions to the permanent displacement x: The ordinary memory is sourced by P, that is the change in the radial component of the electric part of the Weyl tensor. The null memory is sourced by F, the energy radiated to infinity (including shear and component of energy-momentum tensor).
The Christodoulou Memory Effect Christodoulou derived the null memory effect of gravitational waves in a fully nonlinear setting with exact solutions (no approximations used). It is based on the precise description of the null asymptotic behavior of relevant spacetimes, as these was established in the Christodoulou-Klainerman work on the nonlinear stability of the Minkwoski space. Other methods use approximations. New method by Bieri and Garfinkle using a perturbation of the Weyl tensor; this is gauge invariant. A recent paper by P. Lasky, E. Thrane, Y. Levin, J. Blackman and Y. Chen suggests a method for detecting gravitational wave memory with aligo.
Cosmology: de Sitter, FLRW and ΛCDM Observations of 1998 of Distant Supernovae Accelerating Expansion of the Universe Most popular cosmological theories: ΛCDM (with cold (i.e. non-relativistic) dark matter) Friedmann-Lemaître-Robertson-Walker (FLRW) (with a perfect fluid) de Sitter (ds) (modeling early inflation period of the Universe) Positive cosmological constant.
de Sitter Spacetime The de Sitter metric reads ds 2 = dt 2 + a 2 (t)dω 2 where Ω denotes the three-dimensional Euclidean space and a(t) is the expansion factor. This metric is conformal to the Minkowski metric diag( 1, +1, +1, +1). We introduce the conformal time η = dt/a. ds 2 = a 2 ( dη 2 + dω 2 ). (5) Thus we have the conformal behavior g ij = a 2 m ij with i, j = 0, 1, 2, 3. Here, m ij denotes the Minkowski metric with Cartesian coordinates (η, x, y, z).
de Sitter Spacetime de Sitter spacetime, a solution of the Einstein equations with a positive cosmological constant, models the early inflation period of the universe. (L. Bieri, D. Garinkle, S.-T. Yau) We find in de Sitter spacetime, that there is a factor of (1 + rh 0 ) multiplying F, the energy per unit solid angle radiated to infinity. Thus, the null memory is enhanced.
FLRW and ΛCDM FLRW: Friedmann - Lemaître - Robertson - Walker The FLRW metric reads ds 2 = dt 2 + a 2 (t) ( dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ) ) Universe started as a small perturbation from FLRW. by now: these perturbations have grown waves propagate through highly inhomogeneous medium. Consider gravitational waves in ΛCDM cosmology.
For FLRW (A. Tolish, R. Wald): For sources at the same luminosity distance, the memory effect in a spatially flat FLRW spacetime is enhanced over the Minkowski case by a factor of (1 + z).
ΛCDM ΛCDM Our inhomogeneous spacetime two zones: wave zone and cosmological zone. (L. Bieri, D. Garfinkle, N. Yunes [forthcoming preprint]) For gravitational wave memory we find that in the wave zone the memory is similar to the one with Minkowski as a background, whereas in the cosmological zone the memory is given by the memory in the wave zone multiplied by a factor including the redshift and a magnification factor due to lensing.
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