Introduction to gravitational waves (GWs) Gravitational waves. Other important qualitative properties of GWs and detectors.

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1 Introduction to gravitational waves (GWs) What are gravitational waves? Gravitational waves Nathan K. Johnson-McDaniel TPI, FSU Jena Wintersemester 2011 Intuitively, ripples on spacetime that carry energy and (angular) momentum away from an isolated source; the gravitational analogue of electromagnetic radiation, so sourced by accelerated masses. (We ll see the more rigorous version later.) But, unlike E&M radiation, extremely weak. While one can generate GWs by literally waving one s hands, that radiation would be completely undetectable. This is why we don t feel like we are swimming in molasses from gravitational radiation reaction when we move around. GWs of any appreciable magnitude can only be generated by astronomical objects, primarily compact objects like neutron stars and black holes (and white dwarfs, though these are not nearly so compact). 1 / 30 2 / 30 Other important qualitative properties of GWs and detectors. Despite extremely close formal analogies with E&M radiation (which we ll see starting today), GWs are often more like sound (acoustic radiation) than electromagnetic radiation. In particular, since they are generated by bulk motion of the source, they generally have wavelengths of about the size of the source or larger, and thus can t be used to form an image. They are phase-coherent, unlike most E&M radiation, so we can directly observe the field (as opposed to some sort of energy), which only falls off as 1/r, instead of 1/r 2. Thus, if we double our detector s sensitivity, we increase its reach by nearly an order of magnitude! (2 3 = 8) Gravitational wave detectors are naturally all-sky monitors, unlike most astronomical observatories, which have to work to increase their field of view. See Sec. 6 in Flanagan and Hughes for more detailed discussion. 3 / 30 Why an entire course on GWs? These days, mostly because of experiment and astrophysics. There are multimillion dollar international collaborations devoted to detecting gravitational waves (e.g., LIGO/Virgo and LISA)....and ever since observations of the Hulse-Taylor pulsar indirectly demonstrated the existence of gravitational waves, they have been a staple of descriptions of the evolution of compact binaries and even, lately, galactic dynamics (through the kicks that can be generated by merging black hole binaries). While the direct detection of gravitational waves will provide all sorts of interesting tests of fundamental gravitational physics (by probing strong-field gravity, as well as just detecting the waves themselves), gravitational waves are also a powerful tool for astrophysics, from studying compact objects, to such bread-and-butter astrophysics as binary formation and evolution and galactic mergers. 4 / 30

2 Why? (cont.) In particular, since GWs tell us what is going on with the bulk motions of mass, they would let us probe the SN mechanism (E&M observations only observe the surface, and even neutrinos take a little while to diffuse out). They ll also generally help us learn more about cold matter at supernuclear densities, from observations of neutron stars. Additionally, since GWs interact so weakly, like neutrinos, we can observe them from sources (e.g., some SNe and white dwarf binaries in our own galaxy) that are completely obscured by dust and thus unobservable electromagnetically. Of course, since they are weak, the signals are often hidden in the noise, so one needs highly accurate templates to detect the signals, and in particular to extract the properties of the source from the signal. Building such templates has driven much theoretical work over the past few decades. Why? (cont.) This course will present the basics of GWs, elementary tools for computations of the waves emitted by a source, and then discuss the detection of these waves (and the sources expected for current and planned detectors), and how to infer properties of the source from them. And while some sources require exquisitely accurate templates generated by exacting analytical and numerical work, the tools we develop will be perfectly adequate to describe other, equally important (if perhaps not quite so glamorous) sources. 5 / 30 6 / 30 GW detectors Since GWs are so weak, one requires a huge detector to observe them. (And lots of tough experimental work to reduce the noise.) The LIGO detector has 4 km arms, and the maximal changes in armlength that can be expected from a GW are m, well below the nuclear scale (10 15 m)! The proposed space-based GW detector (formerly known as LISA) has an armlength of 10 6 km....and pulsar timing has armlengths of 100 pc m. GW detector status LIGO/Virgo currently being upgraded to Advanced sensitivity expected to be operational in (And may be joined by other ground-based detectors in Japan and Australia.) The redesigned LISA should be launched by ET, DECIGO, BBO, etc. are much further in the future. Pulsar timing is taking data NOW, and only improving in sensitivity (and baseline). (There ll be a dramatic improvement in sensitivity when the Square Kilometer Array comes online in 2020.) LIGO Hanford LISA 7 / 30 8 / 30

3 The GW spectrum and various detectors. Chapter I: GW basics, Sec. 1.1: Linearized gravity GWs We follow Sec. 4.4 in Wald to a great extent here. We start, of course, from the Einstein field equations (EFEs): R µν 1 2 Rg µν = 8πG c 4 T µν. Question: What are the units on each side of the equation? length 2 [Figure from T. Creighton s website] 9 / 30 But we don t tackle them directly, instead choosing to linearize about that most symmetric of all exact solutions to the EFEs, viz., Minkowski space, so we write g µν = η µν + h µν, and ignore anything quadratic in the metric perturbation. (While we will not attempt any rigorous error bounds, this linearization is likely to be an extremely good approximation for the situations of interest, where h µν 1; cf. the spectrum shown previously.) 10 / 30 An aside about notation and conventions Some of the notation and conventions we use should be familiar from Carroll, and some will not be. We ll go over some basics now, and recall others as we go. Indices: We use concrete indices throughout, with Greek for spacetime indices (running from 0 to 3) and Latin for spatial indices (running from 1 to 3). The Einstein summation convention is always in force. Sign convention: Spacelike, i.e., a mostly plus Minkowski metric of diag( 1, +1, +1, +1). Derivatives: µ denotes the partial derivative and µ denotes the covariant derivative. Units: We ll show G and c explicitly at first, but will probably switch to mostly using G = c = 1 later. For electromagnetism, we ll use either Gaussian units (e.g., this lecture), or Heaviside-Lorentz (e.g., the homework). See the Appendix in Jackson for an extensive discussion of different units in E&M. 11 / 30 Final convention remarks, and what now? Riemann tensor: We will be following Wald in, e.g., our definition of the Riemann tensor. This is the same convention used by Creighton and Anderson, but different from that used by, e.g., Carroll, Hartle, and Maggiore. Caveat lector! Returning to linearized gravity, we want an equation for h µν, so we just compute the curvature associated with this metric perturbation. To start with, we note that the inverse metric is given by g µν = η µν h µν + O(h 2 ). (Here we raise the indices on everything except g µν using the Minkowski metric.) To see this, note that g µν g νλ = (η µν h µν )(η νλ + h νλ ) = δ µ λ + h µ λ h µ λ + O(h 2 ). 12 / 30

4 Computing Christoffel symbols What s the definition of the Christoffel symbol? Computing curvature What s the definition of the Riemann tensor? µ x ν =: µ x ν + Γ ν µλx λ, where µ denotes the (metric-compatible) covariant derivative. What do the brackets do? Antisymmetrization, so 2 [µ ν] x λ =: R µνλ ρ x ρ. We use the version with all lowered indices, for which we have 2 [µ ν] := µ ν ν µ. Γ λµν = (µ g ν)λ λ g µν /2 = (µ h ν)λ λ h µν /2. We now write the Riemann tensor in terms of the Christoffel symbols, giving (We now leave off the remainders, except where we want to emphasize a point.) What do the parentheses mean? Symmetrization, so (µ g ν)λ := ( µ g νλ + ν g µλ )/2. R µνλρ = ν Γ ρµλ µ Γ ρνλ + O(Γ 2 ) Here µν := µ ν. = ( µρ h λν µλ h ρν νρ h λµ + νλ h ρµ )/2 = µ[ρ h λ]ν ν[ρ h λ]µ. (You ll fill in the [straightforward] details as an exercise [though in a slightly different situation].) 13 / / 30 Ricci tensor and scalar linearized EFEs The obvious question :) Simplifying the linearized EFEs R µν := g λρ R µλνρ = η λρ R µλνρ + O(h 2 ) λ (µh ν)λ 1 2 ( µνh + h µν ) η µν ( λρ h λρ h) = 8πG c 4 T µν and, defining h := h λ λ and := λ λ, = λ (µh ν)λ ( µν h λ λ + λ λh µν )/2 R := g µν R µν = η µν R µν + O(h 2 ) = µν h µν h. While this is much, much simpler than the full EFEs in terms of the metric, it s still something of a mess. We ll gradually put it in a highly tractable form, using some standard tricks, starting with defining the trace-reversed metric perturbation h µν := h µν 1 2 η µνh. The linearized EFEs are thus λ (µh ν)λ 1 2 ( µνh + h µν ) η µν ( λρ h λρ h) = 8πG c 4 T µν, where T µν is the flat space stress-energy tensor. Since we linearized around flat space, which is a vacuum solution of the EFEs, the flat space stress-energy tensor should be treated as a first-order perturbation. 15 / 30 We have h := h λ λ = h 2h = h, hence the name. The linearized EFEs are thus λ (µ h ν)λ 1 2 h µν η µν λρ h λρ = 8πG c 4 T µν. 16 / 30

5 Simplifying the linearized EFEs (cont.) Gauge transformations in linearized gravity λ (µ h ν)λ 1 2 h µν η µν λρ h λρ = 8πG c 4 T µν. If we just had the d Alembertian term on the l.h.s., then we d just have an inhomogeneous wave equation, like in Loren[t]z gauge E&M and life would be easy. And, in fact, we can use the appropriate (and obvious) generalization of the Lorenz gauge, viz., to obtain the same result here. µ h µν = 0 (Recall that the E&M Lorenz gauge is µa µ = 0, where A µ is the vector [4-]potential.) We want to be able to set µ h µν = 0, but why can we always do this? Well, the full EFEs have the very large gauge freedom of all diffeomorphisms (i.e., all smooth coordinate transformations). In the linearized theory, where one wants to make linear gauge transformations of the same order as the perturbation, this corresponds to coordinate transformations of the form x µ x µ := x µ + ξ µ, where ξ µ is taken to be the same order as h µν, so we neglect all O(ξ 2 ) and O(ξh) terms. 17 / / 30 Gauge transformations in linearized gravity (cont.) Now, the metric transforms as a tensor, so we have g µν x λ x µ x ρ x ν g λρ. Here x λ x µ = δλ µ µ ξ λ + O(ξ 2 ) (where the minus sign and remainder come from inverting the expression for x µ / x ν ). We thus have g µν = η µν + h µν η µν + h µν 2 (µ ξ ν) + {second-order terms}, giving h µν h µν 2 (µ ξ ν) to first order. (And we ll hence tacitly suppress all the remainders.) See Appendix C in Wald for a fancy derivation involving the Lie derivative. Gauge transformations in linearized gravity (cont.) We thus have and so h µν h µν 2 (µ ξ ν) + η µν λ ξ λ, µ h µν µ h µν µ µ ξ ν µ ν ξ µ + ν µ ξ µ = µ h µν ξ ν, using the commutativity of partial derivatives. (Everything we re considering is at least C 2.) We can now just solve to obtain the desired gauge. ξ ν = µ hµν, 19 / / 30

6 The linearized EFEs and E&M in the Lorenz gauge Chap. I, Sec. 1.2: GW polarizations The Lorenz gauge linearized EFEs are thus h µν = 16πG c 4 T µν. For comparison, electrodynamics in the Lorenz gauge ( µ A µ = 0) is given by A µ = 4π c jµ, so the equations are really the same, except for the difference in the number of indices (though this difference is quite substantial when it comes to physical effects). So if h µν is the gravitational analogue of A µ, what is the gravitational analogue of F µν? It is R µνλρ, the Riemann tensor, which is gauge invariant to linear order. Since we have electromagnetic waves, we expect there to also be gravitational waves, by analogy. However, even though Einstein predicted gravitational radiation from his field equations soon after deriving them, this prediction has been the subject of considerable debate Einstein himself changed his mind about this! (See Kennefick s book for an excellent account of the history.) However, it is now a settled matter, and we will see the arguments that make it so (in addition to some of the pitfalls with coordinates that bedeviled early workers). 21 / / 30 A plane gravitational wave We start by considering the vacuum EFEs, h µν = 0 and making the plane wave ansatz h µν = Re e µν exp(ik λ x λ ), where e µν 0 is the polarization tensor and k λ is the wave vector. We tacitly appeal to linearity and Fourier theory to show that we do not lose any generality in making this ansatz. We also will tacitly drop the real part in subsequent expressions, again appealing to linearity in only taking it in final expressions. The linearized EFEs then say that k λ k λ = 0, so k λ is tangent to a null geodesic (i.e., gravitational waves travel at the speed of light ). The Lorenz gauge says that k µ e µν = k ν e µ µ/2. (Recall that the Lorenz gauge is applied to h µν, not h µν directly.) 23 / 30 Further gauge freedom Now, it appears as if there is considerable freedom in choosing e µν, since the Lorenz gauge only imposes 4 conditions, while e µν has How many components? 10 = components, in general, so we expect 6 = 10 4 independent components after imposing the Lorenz gauge. But here we have to realize that the Lorenz gauge is actually a family of gauges, and we still have residual gauge freedom x µ x µ + ζ µ, ζ µ = 0. Thus, all but 6 4 = 2 of the.components of the polarization tensor correspond to gauge modes; and plane waves with those choices merely give flat space in a wavy coordinate system. (You ll see this in the homework.) 24 / 30

7 Further gauge fixing: The TT gauge The standard choice to fix this additional freedom is the transverse-traceless (or TT) gauge, where one takes h = 0, h 0j = 0. Here we have 4 conditions, and expect to be able to satisfy these using ζ µ, since h µν satisfies the wave equation. (Note that one can apply the gauge conditions to either h µν or h µν, since one takes the trace to vanish.) The TT gauge (cont.) To see the details, recall that h µν h µν 2 (µ ζ ν) under the transformation, so the first condition (h = 0) implies that µ ζ µ = h/2, while the second (h 0j = 0) implies that 0 ζ j + j ζ 0 = h 0j. As you ll show in the homework, one can rewrite these in the form 0 ζ 0 = h 00 /2 + c 1, 0 ζ j = h 0j j ζ 0 (c 1 is an arbitrary function of x j ) which can clearly be solved very easily. (You ll also do this for the plane wave in the homework.) 25 / / 30 The TT gauge (cont.) If we take the GW to propagate in the z-direction, then we have k µ. = (k, 0, 0, k). Since e µ µ = 0 in the TT gauge, the Lorenz gauge condition now implies that k µ e µν = 0, which gives, using e 0j = 0, e 0µ = e 3µ = 0. Using tracefreeness a bit more, we obtain e µν = 0 e 11 e e 12 e The GW polarizations If we erect an orthonormal frame {û, ˆv, ˆn}, where ˆn is the direction of propagation, then we can write the two polarization tensors as e + µν := û µ û ν ˆv µˆv ν e µν := 2û (µˆv ν). As their symbols suggest, these are known as the plus and cross polarizations. We can then write an arbitrary plane GW propagating in the ˆn-direction as h µν = h + e + µν + h e µν. Note that we get e 00 = 0 even though we didn t specify it in our TT gauge conditions, though some authors do. See, e.g., Sec. 4.4b in Wald for the argument that shows that one can make a further gauge adjustment to set h 00 = 0, in general, in the TT gauge, if no sources are present. 27 / / 30

8 The GW polarizations (cont.) Chap. I, Sec. 2: Physical effects of GWs It s useful to see how the components transform under rotations of the frame about the direction of propagation. If we rotate the frame by an angle ψ, then the components in the new frame are given by h + h + cos 2ψ h sin 2ψ h h + sin 2ψ + h cos 2ψ. Note that h + h and h h + for ψ = π/4, while h + h + and h h for ψ = π/2 (gravity is spin-2). (The polarization angle ψ will be important when we consider the parameters measured by a GW detector.) Since h 0µ = 0 in the TT gauge, the linearized Riemann tensor has the simple form R µ00ν = ( µν h (µ h ν ) h µν )/2 = ḧ µν /2, where overdots denote time derivatives. We thus see that the two polarizations we have claimed are physical indeed are, since they generate nonzero curvature. And what are the physical effects? They re a tidal acceleration, as we ll see shortly. 29 / / 30