Gravitational Waves Summary of the presentation for the Proseminar Theoretical Physics Nehir Schmid 06.05.2018 Contents 1 Introduction 1 2 Theoretical Background1 2.1 Linearized Theory........................................1 2.2 Transverse Traceless Gauge...................................2 2.3 Quadrupole Radiation......................................3 3 Strain Magnitudes 3 4 Method of Measurement4 5 The Example of GW1509146 5.1 Compact Binary Systems....................................6 5.2 Estimates on the involved mass.................................7 5.3 Compactness of the System...................................8 5.4 Conclusion............................................9 1. INTRODUCTION University of Z¨urich, HS2018 1 Introduction In 2015 the first direct measurement of gravitational wave was taken by LIGO, the Laser Interferometer Gravitational Wave Observatory situated in Hanford (Washington) and Livingston (Louisiana). Two years later, in 2017, Rainer Weiss, Barry C. Barish and Kip S. Thorne were awarded the Nobel Prize in Physics for their decisive contributions to the LIGO Observatory and the detection of gravitational waves. This short summary contains some theoretical concepts from which physicists deduced that there should be gravitational waves and it would be worth searching for them. It will briefly explain the method of detection and finally discuss the sort of information, one can gain from such a measurement. 2 Theoretical Background1 In general relativity (GR) the presence of mass, or in general energy and momentum of matter, is coupled to a curvature of spacetime, and the force of gravity corresponds to an inertial motion within that curved spacetime. Gravitational waves are a local disturbance in the geometry of space-time. This means spatial lengths and time durations change as a wave passes through. The waves propagate at the speed of light through vacuum. Their source is accelerated mass. How can we show that they exist? Einstein's field equations (1) describe the relation between the mass and energy distribution in space and the curvature of space-time. 1 8πG R − g R = T (1) µν 2 µν c4 µν The energy-momentum tensor Tµν on the right hand side contains all the information about the distribution and the flux of mass and energy. The left hand side depends on gµν the metric of curved space-time. (The Ricci tensor Rµν also depends on depends on gµν .) Einstein predicted gravitational waves in 1916, as their local disturbances in the curvature of space-time, which travel through vacuum at the speed of light, are a solution of (1). The disturbances are represented in the metric gµν , which contains the information about the curvature. Finding a solution of (1) is one of the main problems in GR. The field equations are 16 coupled differential equations, which are difficult to solve. Exact solutions are in fact only possible for highly symmetric problems, for which the degrees of freedom are reduced e.g. for a non-spinning black hole. Normally approximations are used to simplify the problem. The technique used in the derivation of gravitational waves is linearized theory. 2.1 Linearized Theory The idea is to express the metric gµν of curved space-time as the metric ηµν of flat space-time plus some perturbation hµν : gµν = ηµν + hµν : (2) All the information about the disturbances caused by a gravitational wave is now contained in hµν . Then we assume that the perturbation is small. This approximation is okay for an observer on earth, because there are no such heavy objects nearby that would curve space-time strong enough that its effects could compete with the curvature caused by a typical cosmological source of gravitational waves that we could hope to detect. Since these typical waves only cause weak curvature on earth. the approximation is legitimate. st The next step is to expand Einstein's field equations in hµν and only keep 1 order terms. This leads to a simplified equation that is linear in hµν 16πG h¯ + η @ρ@σh¯ − @ρ@ − @ρ@ h¯ = − T (3) µν µν ρσ µρ µ νρ c4 µν where 1 h¯ = h − η h : (4) µν µν 2 µν 1For further information: Michele Maggiore, Gravitational Waves [1] 1 2. THEORETICAL BACKGROUND University of Z¨urich, HS2018 By using some of the gauge freedoms by moving to Hilbert gauge ν ¯ @ hµν = 0 : (5) Einstein's field equations become a simple wave equation 16πG h¯ = − T : (6) µν c4 µν The comparison to the wave equation for electromagnetic (EM) waves (7) shows that both equations are of exactly the same form. The EM waves are described in the value Aα and it's source is the electric current α density j , while the gravitational waves are in the perturbation hµν and the source term is proportional to the energy-momentum tensor Tµν . 4π Aα = jα (7) c From the form of the wave equation one can directly see that there is a solution for waves propagating at the speed of light in vacuum, where Tµν = 0. Continuing with the analogy to EM waves, we can solve the wave equation using Green's functions.2 The Green's function that not only solves the equation but also satisfies the causality condition is the retarded Green's function. 1 G(~x − ~x0) = − δ(~x0 − ~x00) (8) 4πj~x − ~x0j ret This leads to a solution (10) of Einstein's field equation in linearized theory. With the retarded Green's function, (10) depends on the integral over the energy-momentum tensor at retarded time, which means it depends at the mass and energy distribution and flux at a time tret as much before the time of measurement t, as information needs to travel from a source at ~x0 to the point ~x. 16πG Z h¯ (x) = − d4x0G(x − x0)T (x0) (9) µν c4 µν 4G Z 1 j~x − ~x0j h¯ (t; ~x) = d3x0 T t − ; x0 (10) µν c4 j~x − ~x0j µν c 2.2 Transverse Traceless Gauge Normally, the solution of the wave equation contains not only the information about the radiation itself, but also about the relative position of an observer to the source. To get rid of this additional information, we move to transverse-traceless (TT) gauge, where we have gotten rid of all the gauge freedoms. In TT-gauge, the perturbation hµν of the metric of curved space-time is traceless and perpendicular to the direction of travel of the wave. For a cartesian coordinate system in which the wave propagates in z-direction, and the x- and y-axes span the plane perpendicular to the wave vector, the perturbation is given by 0h h 01 + × h z i hTT (t; z) = h −h 0 · cos ! t − : (11) ij @ × + A c 0 0 0 ab The h0ν and hµ0 are not shown, as those entries become 0 in TT-gauge in linearized theory. As the perturbation tensor is perpendicular to the xy-plane, all the changes in distances caused by the wave lie within this plane. For a chosen reference frame there are two possible polarisations: plus- and cross-polarisation. In Figure1 you can see the effects of a passing wave in each of those polarisations. Observe that as much as the lengths stretch in one direction, as much they squeeze in the direction perpendicular to it. This could be expected for a traceless perturbation tensor. 2The concept of Green's functions is explained here: http://www.damtp.cam.ac.uk/user/dbs26/1BMethods/GreensODE.pdf (2015) 2 3. STRAIN MAGNITUDES University of Z¨urich, HS2018 Figure 1: The effects of a passing gravitational wave in plus- or cross polarization on a ring of test masses. Source: The First Detection of Gravitational Waves [2] 2.3 Quadrupole Radiation To determine the form of the waves an observer could see for a specific source, it is necessary to solve the integral in eq. (10) which is not trivial. One possibility is to expand the energy-momentum tensor in mass multipoles (eq. (12)) and drop higher order terms. With each higher order comes an additional factor of ∼ v=c. v is the typical velocity in the source. Therefore this approximation is valid for non- relativistic sources like typical cosmological sources are. The lowest order term is the quadrupole term. The constant monopole does not contribute to the radiation and the dipole term in the expansion of the energy-momentum tensor is proportional to the derivative of the total momentum with respect to time, which of course is zero. ! r ~x0 · ~n^ r x0ini 1 T t − + ; ~x0 ' T t − ; ~x0 + @ T + x0ix0jninj@2T + ::: (12) kl c c kl c c 0 kl 2c2 0 kl In this approximation the perturbation of the metric of curved space-time is given by 1 2G [hTT (t; ~x)] = Λ (^n)M¨ kl(t − r=c) ; (13) ij quad r c4 ij;kl where M kl is the second mass moment, the normal moment of inertia, of the source. This can be calculated in any reference frame in Hilbert gauge, and is then projected to TT-gauge by the projector Λij;kl wich depends on the direction of propagation n^ of the wave. With this formula we can calculate the gravita- tional waves created by any source of which we know the mass distribution and how that changes over time. TT Once having found hij (t; ~x), we can determine further properties of the waves. For example how much power is radiated. G ... ... 1 ... G ... ... dEGW P = M M − (M )2 = Q Q ≈ (14) quad 5c5 ij ij 3 kk 5c5 ij ij dt 3 Strain Magnitudes How big a perturbation h do we expect to deal with? h gets smaller with the distance to the source. Typical cosmological sources of gravitational waves cause extremely small perturbations on earth.