Post-Newtonian Approximations and Applications

Home , ADM formalism, Dynamics (mechanics), Einstein field equations, Newtonian potential, World line

Monash University

MTH3000 Research Project

Coming out of the woodwork: Post-Newtonian approximations and applications

Author: Supervisor: Justin Forlano Dr. Todd Oliynyk

March 25, 2015 Contents

1 Introduction 2

2 The post-Newtonian Approximation 5 2.1 The Relaxed Einstein Field Equations ...... 5 2.2 Solution Method ...... 7 2.3 Zones of Integration ...... 13 2.4 Multi-pole Expansions ...... 15 2.5 The ﬁrst post-Newtonian potentials ...... 17 2.6 Alternate Integration Methods ...... 24

3 Equations of Motion and the Precession of Mercury 28 3.1 Deriving equations of motion ...... 28 3.2 Application to precession of Mercury ...... 33

4 Gravitational Waves and the Hulse-Taylor Binary 38 4.1 Transverse-traceless potentials and polarisations ...... 38 4.2 Particular gravitational wave ﬁelds ...... 42 4.3 Eﬀect of gravitational waves on space-time ...... 46 4.4 Quadrupole formula ...... 48 4.5 Application to Hulse-Taylor binary ...... 52 4.6 Beyond the Quadrupole formula ...... 56

5 Concluding Remarks 58

A Appendix 63 A.1 Solving the Wave Equation ...... 63 A.2 Angular STF Tensors and Spherical Averages ...... 64 A.3 Evaluation of a 1PN surface integral ...... 65 A.4 Details of Quadrupole formula derivation ...... 66

1 Chapter 1

Introduction

Einstein’s General theory of relativity [1] was a bold departure from the widely successful Newtonian theory. Unlike the Newtonian theory written in terms of fields, gravitation is a geometric phenomena, with space and time forming a space-time manifold that is deformed by the presence of matter and energy. The deformation of this differentiable manifold is characterised by a symmetric metric, and freely falling (not acted on by external forces) particles will move along geodesics of this manifold as determined by the metric. Although this notion of curved spaces is an elegant picture, when we seek to understand it at a mathematical level we naturally need some way of describing this manifold through a coordinate system. Crucially, gravity is associated with actual curved space-time, that is to say we should not be able to make a global coordinate transformation so that in another description the space is no longer curved. This is why the foundations of general relativity are built on a covariant formalism, where their form is invariant under coordinate transformations. Tensors are therefore unavoidable in the mathematics since tensor equations have this invariance property and when one wishes to impose a certain coordinate system, it is relatively straightforward to do so. This is in stark contrast to the Newtonian theory, where it is difficult to describe systems in anything other than Cartesian coordinates. Unfortunately the price we pay for such coordinate freedom is that the equations dealt with are far more challenging to solve exactly. The Einstein field equations which govern how the space-time is curved, take the form of 6 independent, non-linear, coupled partial differential equations. Apart from the most idealised geometries such as spherical symmetry where exact solutions exist (the Schwarzschild metric [2] for instance), we must instead resort to approximations. There are many different approaches ranging from the post-Newtonian approximation (PN), perturbation approaches which assume one mass is much larger than the other masses and the field of numerical relativity which seeks to obtain solutions to the field equations for complicated systems computationally. Here we consider the post-Newtonian method which proceeds iteratively generating ever higher order approximations. As the name suggests, the lowest order result is the Newtonian gravitational potentials. The next iteration obtains the first post-Newtonian order which is labelled as 1PN and represents a first correction past Newtonian gravity. After this one can then obtain higher post-Newtonian corrections such as 2PN, 3PN and so on. In this scheme there are two main assumptions: firstly that the gravitational fields are weak which is to say that they are dominated by the Newtonian order potentials and secondly that the velocities of the system are small enough to allow the weak gravity assumption to hold. This is known collectively as a weak field, slow motion approximation. The weak field approximation can be made by considering the metric as a sum of the flat- space, unperturbed Minkowski metric plus an additional term which describes deviations from this. By slow motion, we mean that the dimensionless parameter := vT /c 1 where vT is some characteristic velocity of the system and c is the speed of light in vacuum. Here

2 Chapter 1

can be thought of as the “slowness” of the system, a larger implies a more vigorously moving system. This allows us to then consider the metric to be expanded in powers of while the limit as → 0 is to recover the Newtonian equations. An example is the case of a binary system where the post-Newtonian approximation seeks to determine the equations of motion (they give accelerations in terms of positions and velocities) of the bodies to a high order. The relative acceleration of the bodies can be schematically expanded as Gm a = −n + 2A + 3A + 4A + 5A + 6A + 7A + ··· , r2 1PN 1.5PN 2PN 2.5PN 3PN 3.5PN (1.0.1) where m := m1 +m2, r := |x1 −x2| and n := (x1 −x2)/r. We can see that the Newtonian level is obtained when → 0 and the higher order terms represent corrections past the Newtonian order given by approximations to General relativity. One can also include spin effects in this expansion, for instance as is done by Will [3] in the context of the direct integration of the relaxed field equation (DIRE) approach to post-Newtonian theory. This is allows the orbiting bodies to have some intrinsic angular momentum, much like how the Earth rotates about its own axis as it orbits the sun. The spin effects are first noticeable at the 1PN order where the spin can couple to the orbital angular momentum. The equations of motion to 1PN order for N- orbiting bodies was given by Eisntein, Infeld and Hoffman in 1938 [4]. At the other end of the scale are the highest completed calculations to 3.5PN order which have been tackled by Pati and Will [5], Futamese and Itoh [6,7], Blanchet [8] and 4PN Hamiltonians have been found [9]. The search for higher orders is complicated because the number of terms appearing in each PN term of (1.0.1) increases dramatically. The number of terms in the metric potentials at 1PN order is 10, at 2PN it is around 100 terms, and the 3PN is of the order of 103 ∼ 104 terms. Going to 4PN requires handling tens of thousands of terms, which is a hugely challenging task. A prediction of general relativity is that freely orbiting bodies are not stable forever. They in fact emit energy through gravitational radiation which gradually causes the bodies to spiral inwards towards closer orbits before plunging together in a coalescence phase. As they inspiral, the velocities increase so will in fact not be uniform forever, it will increase in this case. This implies that the post-Newtonian corrections can in fact become larger than the Newtonian terms which causes a breakdown of the Newtonian limit as the system moves to relativistic orbits. The effect of emitted gravitational waves on the system itself which is known as radiation-reaction is first seen at the 2.5PN order which is at O(5) in (1.0.1). The most likely sources for the detection of gravitational radiation are compact binary systems such as orbiting black holes or neutron stars because of the closeness of their orbits and large masses. The compact binary case is especially useful as tidal effects can be neglected for much of the motion right up until the close inspiral. The existence of gravitational waves is perhaps the greatest as yet undetected prediction of general relativity and is a large reason why the PN formulation exists at all. Just as matter curves space-time, it is possible that for accelerating, non-spherically symmetric systems, such as a compact binary or the violent collision between two black holes, pro- cesses such as these can form propagating waves in the space-time. This prediction was largely theoretical until a gradual shift in the technological capability around the 1980’s opened up the possibility of detecting these waves. The experimental difficulty is that the amplitude of a gravitational wave is extremely small, even for a binary black hole system. The effects felt in a detector are on the order of one part in 1022. Modern gravitational wave detectors such as LIGO and VIRGO have to not only be built with sensitivities in this range but they must also contend with numerous sources of noise which obscures the true signal. The noise ranges from the small: thermal noise and quantum effects, to the large scale, such as seismic noise. The goal of the post-Newtonian approximations is to provide wave templates which are theoretically derived waveforms written in terms

3 Chapter 1 of some observable parameters such as masses of bodies, orbital frequency and distance to source. These waveforms can then be used to sift the true signal from the noisy data observed. These waveforms are computed in the far-field of the source and are expressed in terms of the orders as 4G n o hij = Qij + Qij + Qij + Qij + Qij + Qij + Qij + Qij + ··· . c2r 0.5PN 1PN 1.5PN 2PN 2.5PN 3PN 3.5PN (1.0.2) The lowest order term, Qij is in fact the quadrupole which leads to the quadrupole formula. It is this reason why the qaudrupole formula is sometimes considered as a Newtonian term even though it appears nowhere in Newtonian gravity. The inspiral of compact binaries can be quite relativistic with velocities approaching 0.5c. At this point in the system evolution, the gravitational wave amplitude is around its maximum so it can potentially be registered by a detector on Earth. However the quadrupole formula is likely to give a poor prediction for this system because of a necessity of accurate time evolution which is ij only available starting at the 3PN order, or Q3PN in (1.0.2). This precise time evolution is required by the detectors such as LIGO and VIRGO. In fact, the higher order one can achieve past 3PN, the more accurately the true signal can be extracted from the noise. This is the major motivation for proceeding to ever higher order expressions. The plan of this paper is as follows. In chapter two we introduce the Landau-Lifshitz harmonic formulation of the Einstein field equations [10] which is a much better way to study the post-Newtonian expansion and also puts gravitational waves at ones fingertips. We then proceed by describing in detail how the PN method works in practice as an matching scheme. We then use these general results to consider the 1PN potentials for an N-body point particle system. We then with a look at many other techniques for employing a post-Newtonian expansion. We also look at the 1PN equations of motion and how they can be derived in two ways: a Lagrangian and a surface integral approach. Chapter 3 we then begin to discuss the formalism and generation of gravitational waves. Specifically we see that we can impose a transverse-traceless gauge on the waveforms which drastically simplifies the calculations. We then apply the results of the linearised theory of gravity to deduce the existence of plane gravitational waves in vacuum and waves emitted from a binary system. This then transitions into the slightly more physical discussion of how gravitational waves actually affect space-time, where we find it is due to stresses and strains in a system. We then reach the pinnacle of the lowest order gravitational wave theory which is the quadrupole formula for radiation emission. The quadrupole formula gives the rate that energy is emitted from a system. From this, we apply it to the binary system PSR1931+16, the famous Hulse-Taylor binary, which gives excellent agreement with experimental observations. We then have a brief look at expression that go beyond the quadrupole formula to higher orders. The conventions used herein include an event in space-time being labelled by the coordinates xα = (x0, xa) = (ct, x1, x2, x3), where Greek indices run through all space- time values (i.e. through 0, 1, 2, 3, 4), and the corresponding Latin indices run only through spatial components (i.e. 1, 2, 3) and c is the (constant) speed of light. Another constant which will appear prominently is G, the gravitational constant. We use ηαβ := αβ ∂ diag(−1, 1, 1, 1) = η the Minkowski metric of flat space-time, g := det(gαβ), ∂α := ∂xα αβ αν ν and g is the contravariant form of the metric such that gαµg = δµ. Finally we use a multi-index notation for repeated products, where for example xQq := xq1 xq2 xq3 . . . xqn , with the order of Q to be clear from the context in which it appears. Finally we note that in Appendix A.2, we define our use of symmetric, trace-free tensors (STF) which are written with Ω and are used throughout.

4 Chapter 2

The post-Newtonian Approximation

In this chapter, we discuss the theoretical basis for the post-Newtonian expansions and see how it is applied to find iterative solutions for the metric potentials. In Section 2.1, we present an extremely brief introduction to the governing equation of General relativity, that is the Einstein field equations. However for the purposes of post-Newtonian approximations, they will be transformed into an equation reminiscent of the wave equation. These preliminaries are the foundational basis for the PN approximations and for the study of gravitational waves. In Section 2.2, we discuss the a solution method this ‘wave equation’ noting that the inherent non-linearities significantly complicate the matter. With an integration approach sighted, Section 2.3 discusses the partitioning of the problem into so called near and wave zones. In these zones, we may expand the required integrals into sums of multi-poles which is the basis for Section 2.4. In Section 2.5, we begin to obtain the form of these metric potentials. At a zeroth order expansion, we find Newtonian theory indicating the internal consistency of this approach and of the field equations in general. We then discuss the 1PN order equations and apply them to the case of an N-body, point particle system. Finally, Section 2.6 concludes this chapter with a brief look at the many different approaches to integrating the ‘wave-like’ equations that can be found in the literature.

2.1 The Relaxed Einstein Field Equations

All physical theories contain at least one dynamical variable which captures the state of the system; for example Newton’s Laws of motion contain the position vector, r(t), while Quantum mechanics has the state vector |Ψ(t)i. For the General Theory of Relativity, we have the metric tensor gαβ, which encapsulates the geometric structure of space-time. The equations which govern the form of the metric are Einstein’s Field Equations (EFE)1

8πG Gαβ = T αβ, (2.1.1) c4

αβ αβ 1 αβ αβ αβ where G := R − 2 g R is the Einstein tensor, R is the Ricci tensor, R := gαβR is the Ricci scalar, and T αβ is the energy-momentum tensor which contains all the information related to the matter distribution. These form a system of 10 non-linear second-order partial diﬀerential equations for the components gαβ of the metric. By virtue of the Bianchi

1In general, the EFE contain an extra term linear in the metric with proportionality constant Λ. This is known as the cosmological constant. For post-Newtonian theory, we ignore this term due its small magnitude which is negligible for systems such as the solar-system and binary stars.

5 Chapter 2 2.1. THE RELAXED EINSTEIN FIELD EQUATIONS identities, we also have the conservation equations

αβ αβ DβG = 0, ⇒ DβT = 0, (2.1.2) where the operator Dβ represents covariant diﬀerentiation with respect to the index β. These represent the conservation of energy and momentum. The Bianchi identies in fact further reduce the number of independent equations in (2.1.1) down to just six. Since the space-time manifold is parametrisable by four coordinates xµ, we see that we have two extra free conditions which allows us to impose gauge constraints. Unfortunately (2.1.1) in this form is not the optimal formulation for studying the post- Newtonian theory and gravitational waves. In light of this, we introduce the potentials √ hαβ := ηαβ − −ggαβ, (2.1.3) along with imposing what is known as the harmonic or de Donder gauge conditions

αβ ∂βh = 0. (2.1.4) √ We can also write (2.1.3) as hαβ = ηαβ − gαβ where gαβ := −ggαβ is known as the gothic inverse metric and is useful for intermediary calculations. The definition given in (2.1.3) has the particularly nice interpretation in that the potentials represent the perturbations of the space-time relative to the flat Minkowskian background. It is also worth mentioning that from the definition in (2.1.3), knowledge of the potentials hαβ will completely determine the metric g . The process for this involves finding hαβ, rearranging √ αβ √ for −ggαβ and noting that det(gαβ) = ( −g)4/g = g and hence inverting gαβ to find the metric gαβ. Inserting (2.1.3) into (2.1.1) and after utilising many times the gauge condition of (2.1.4), we eventually arrive at an exact expression for the potentials which is known as the relaxed EFE, 16πG hαβ = − Λαβ, (2.1.5) c4 µν where = η ∂µν is the d’Alembertian wave operator. The right hand side of (2.1.5), contains the effective energy-momentum pseudo-tensor which is

αβ αβ αβ αβ Λ := (−g)(T + tLL + tH ), (2.1.6) where we make the further deﬁnitions for the Landau-Lifshitz pseudo-tensor 2

c4 1 (−g)tαβ := g gνρ∂ hαλ∂ hβµ + g gαβ∂ hλν∂ hρµ − 2g gλ(α∂ hβ)ν∂ hρµ LL 16πG λµ ν ρ 2 λµ ρ ν µν ρ λ 1 + (2gαλgβµ − gαβgλµ)(2g g − g g )∂ hντ ∂ hρσ (2.1.7) 8 νρ στ ρσ ντ λ µ and the harmonic pseudo-tensor

c4 (−g)tαβ := ∂ hαν∂ hβµ − hµν∂ hαβ . (2.1.8) H 16πG µ ν µν

We can see that these are not tensors because they involve only regular derivatives, ∂µ and not covariant derivatives. As a consequence of the harmonic gauge condition, we ﬁnd that the eﬀective energy momentum tensor is conserved, i.e

αβ ∂βΛ = 0. (2.1.9)

2Brackets connecting indices such as g(αβ gµν) denotes a symmetrization over those indices. In general (αβ...ω) 1 for a rank-N contravariant tensor, T := N! (Sum over permutations on indices α, β, . . . , ω).

6 Chapter 2 2.2. SOLUTION METHOD

This conservation can be split into two pieces, namely

αβ h αβ αβ i h αβi ∂βΛ = ∂β (−g)(T + tLL) + ∂β (−g)tH = 0. (2.1.10)

αβ A very short computation using the gauge condition of (2.1.4), reveals that ∂β(−g)tH = 0. αβ This says that (−g)tH is separately conserved independent of the total conservation of Λαβ. Therefore we have a conservation equation of both the matter sources in T αβ and αβ field sources tLL which is h αβ αβ i ∂β (−g)(T + tLL) = 0. (2.1.11) This is an extremely important equation which is deeply related to total conservation of mass-energy for both sources of matter and the gravitational field itself. The wave equation of (2.1.5) is the main starting point for the post-Newtonian theory. It is worth mentioning that (2.1.5) is non-linear in h. For instance, if hαβ is a given solution and ζαβ is some smooth tensor field, then the sum hαβ + ζαβ is not a solution αβ even if ζ = 0. This is due to the quadratic terms hidden within the effective energy- momentum pseudo-tensor. Apart from the general covariance of this theory, the non- linearity of it is the first clear departure from Newtonian gravitational physics which is itself linear. It is hard not to draw parallels between the formulation of GR presented and the equations governing electromagnetic (EM) phenomena due to Maxwell. In EM, the fundamental entity is the four-potential A which is sourced by the four-current density j and the relation is given (in cgs units) by 4π Aα = jα. (2.1.12) c α To arrive at this wave equation, one has to impose the Lorentz gauge condition ∂αA = 0 αβ α which is very similar to the de Donder gauge condition ∂βh = 0. Indeed A in (2.1.12) can be written as a retarded integral solution (see Section 2.2 for more on this) much like the gravitational potentials. On the face of it, the key equations governing the two theories may look similar but that is where it ends. The difference is, as discussed, the non-linearity of the field equations and this represents the grand departure from Newtonian mechanics.

2.2 Solution Method

In order to solve (2.1.5), we outline some extra hypothesis we require, as is discussed by Blanchet [8]. The first condition is that the matter distribution is entirely contained within some world tube T (S) = {xα|r < S, −∞ < t < ∞}. As we will see, we assume the matter is deep within the near zone of the source. This condition imposes that T αβ has spatially compact support (i.e. vanishes outside this world tube). Furthermore, we also require h αβ αβ i that ∂β (−g)(tLL + tH ) = 0 when r > S, which physically amounts to there being no matter sourcing gravitational fields outside the compact support. Secondly, we assume αβ ∞ 3 that the matter distribution within the source is smooth (T ∈ C (R )), and thus free of shocks. Thirdly, the source shall be post-Newtonian in nature as we will discuss now. The post-Newtonian approximation involves imposing a ‘slow motion’ condition on the system such that its characteristic velocities are much less than the speed of light c. As we will see, we need to formulate a process in which we can expand our governing equation (in this case (2.1.5)) in an asymptotic expansion. For this we require some small parameter to expand about and for which successive approximations are accurate to higher orders in this small parameter. A natural choice is to define v := T , (2.2.1) c

7 Chapter 2 2.2. SOLUTION METHOD

as our small parameter, where vT is the characteristic velocity of the system which is much less than c, under the ‘slow motion’ condition. The solutions we seek should thus be parametrized in terms of and when → 0, the Newtonian features of the system are to be found. Assuming the bodies are perfect ﬂuids, then they are characterized by the dynamical variables which are the density ρ(x, t), pressure P (x, t), velocity v(x, t) and the Newtonian potential Φ(x, t). Under the governing equations of these variables, namely the Poisson equation, the continuity equation and the Euler equation, we have the following scalings in terms of ,

2 ρ(t, x) = ρ(t, x), (2.2.2) 4 P(t, x) = P (t, x), (2.2.3) i i v(t, x) = v (t, x), (2.2.4) 2 Φ(t, x) = Φ(t, x). (2.2.5)

These say that our solutions are in fact invariant under such transformations as given by the (2.2.2)-(2.2.5). Following Futamese [7]), we also introduce a scaled time

τ := t, (2.2.6) which represents a Newtonian dynamical time. For example, the period of a system with = 0.01 is 10 times that of a system with = 0.1 since t = τ/. Since is the parameter that deﬁnes the order of the expansions in our family of solutions, then it also plays a helpful visual aid for spotting the terms of certain orders, as we shall see in Section 2.5. Under the scaling τ = t, the scaling laws of (2.2.5) now suggest that

ρ(ct) = 2ρˆ(cτ), (2.2.7) ja(ct) = 3ˆja(cτ), (2.2.8) T ab(ct) = 4Tâb(cτ), (2.2.9) where ja and T ab are related to T 0a and T ab respectively and will be expanded upon in Section 2.5. We now begin to differentiate between quantities written in terms of the dynamical time τ and the ordinary time t by the presence and absence of a ‘hat’ (e.g.ρ ˆ) respectively. In appendix A.1, we found the retarded solution to the in-homogeneous wave equation. By comparison between (2.1.5) and (A.1.1) and use of (A.1.6), we propose a formal solution to the relaxed field equations given by

Z αβ 0 0 αβ 4G Λ (ct − |x − x |, x ) 3 0 αβ h (ct, x) = 4 0 d x + hH (ct, x). (2.2.10) c L(ct,x) |x − x |

αβ The homogeneous solution hH is typically suppressed in post-Newtonian approximations. There are a few arguments available which attempt to justify this. The ﬁrst is the well known ‘no-incoming radiation’ condition. In (A.1.7) of Appendix 1, we mention that the homogeneous solution is given by the Kirchoﬀ formula which in this context is written as I αβ 1 ∂ αβ 0 ∂ αβ 0 hH (ct, x) = (rhH (ct , x)) + 0 (rhH (ct , x)) dΩ. (2.2.11) 4π ∂L(ct,x) ∂r ∂(ct )

Imposing the condition

∂ ∂ lim (rhαβ(ct0, x)) + (rhαβ(ct0, x)) = 0, (2.2.12) r→∞ ∂r H ∂(ct0) H

8 Chapter 2 2.2. SOLUTION METHOD where this is at past null infinity where t + r/c =constant, means that as we expand the surface of the light cone L, we encompass all space but the integrand will vanish causing the homogeneous solution to vanish. The condition of (2.2.12) is a statement that the system is unperturbed and cut off from contributions from the ‘outside.’ Another approach (given in pg. 23 of Futamese [7]) is to consider random initial data for the field that is of 1 PN order. It is found that effects due to the presence of the homogeneous solution do not arise until the 2.5 PN order. While it has not been demonstrated, it is assumed that if one specifies initial data to a sufficiently high order, then the homogeneous solution will be irrelevant to the dynamics of the system for orders close to that of the initial data. For these two reasons we can thus neglect the homogeneous solution. We would now like to consider rewriting (2.2.10) to include our post-Newtonian parameter . We could simply replace a t with a τ/ however there is an alternative which makes it easier to calculate with. If we consider the inhomogeneous wave equation, which is our main equation to solve, then making the transformation τ = t gives 1 1 − ∂2 hˆ(cτ, x) + 4hˆ(cτ, x) = [−4πf(cτ, x)], (2.2.13) cτ 2 2 where f is some arbitrary smooth source. We now consider a spatial transformation y = x3, and we will write functions of the coordinates (cτ, y) with a tilde for example, f˜ and functions of the coordinates (cτ, x) with a hat as usual. The relation between these is

k˜(cτ, y) = kˆ(cτ, y/) = kˆ(cτ, x), (2.2.14) where k is some arbitrary function. In the coordinates (cτ, y), (2.2.13) becomes

2 ˜ ˜ 2 ˜ − ∂cτ h + 4yh = 1/ (−4πf), (2.2.15) which admits a retarded integral solution

Z ˜ 0 0 ˜ 1 f(cτ − |y − y |, y ) 3 0 h(cτ, y) = 2 0 d y . L(cτ,y) |y − y |

The integration domain here is L(cτ, y) = {0 ≤ |y − y0| ≤ cτ} which is a ball centred at y with radius cτ. In order to convert back into the coordinates (cτ, x), we write

Z ˜ 0 0 Z ˜ 0 0 3 0 1 f(cτ − |y − y |, y ) 3 0 1 f(cτ − |y/ − y /|, y /) 3 d y 2 0 d y = 2 0 3 L(cτ,y) |y − y | 0≤|y/−y0/|≤cτ |y/ − y /| and notice that h˜(cτ, y) = hˆ(cτ, y/) along with using y0/ = x0 to obtain

Z ˆ 0 0 ˆ f(cτ − |y/ − x |, x ) 3 0 h(cτ, y/) = 0 d x . 0≤|y/−x0|≤cτ/ |y/ − x | The ﬁnal step is to recall our original deﬁnition y/ = x, so we arrive at

Z ˆ 0 0 ˆ f(cτ − |x − x |, x ) 3 0 h(cτ, x) = 0 d x . (2.2.16) 0≤|x−x0|≤cτ/ |x − x | Using this, we have a formal solution for (2.1.5) given by

Z ˆ αβ 0 0 ˆαβ 4G Λ (cτ − |x − x |, x ) 3 0 h (cτ, x) = 4 0 d x . (2.2.17) c L(cτ/,x) |x − x |

3The transformation y = x is implied to mean yi = xi where each spatial coordinate is transformed. For brevity, we omit these indices with understanding that x is to imply {x1, x2, x3}.

9 Chapter 2 2.2. SOLUTION METHOD where we explicitly have the integration region L(cτ/, y) = {0 ≤ |x − x0| ≤ cτ/} which is a ball centred at x with radius cτ/. The relaxed ﬁeld equations are so called because this solution does not require knowledge of the motion of the source. However there is a glaring issue with this: Λˆ αβ is itself dependent on hˆαβ, so this is no solution at all. In the post-Newtonian regime of non-relativistic motion and weak ﬁelds (||hˆαβ|| 1) we can instead proceed by an iterative method for determining the potentials by writing

∞ ˆαβ X nˆαβ ˆαβ ˆαβ 2ˆαβ 3 h = h(n)() = h(0)() + h(1)() + h(2)() + O( ). (2.2.18) n=0

This is a post-Newtonian expansion since our expansion parameter is = vT /c, which is assumed to be small. Each of the expansion terms are also functions of since we obtain these terms in another series expansion manner in powers of which arises from scaling of the energy-momentum tensor Tˆαβ. However we can simplify this by noting that if we substitute each of these separately into the reduced ﬁeld equations (2.1.5) and take note of how the energy- momentum tensor scales with as in (2.2.7)-(2.2.9), then we see that the only way we can hope to have solutions that scale properly (as in either side of the reduced EFE match) is if

ˆ00 ˆ00 ˆ0a ˆ0a ˆ0a âb âb âb âb h(0) = h(1) = 0, h(0) = h(1) = h(2) = 0, h(0) = h(1) = h(2) = h(3) = 0. (2.2.19)

There is one more argument we can make which will remove further terms from the expansions in (2.2.18). In the low order post-Newtonian expansions, the eﬀect of the emitted gravitational radiation (more on this in Chapter 4) on the ﬁelds themselves can be neglected. This back-radiation reaction begins to appear only at the 2.5PN order, which is much greater than the simple 1PN order at maximum we will be examining to come. Therefore in our system we have a time-reversal symmetry. If we consider the space time squared interval

2 α β 2 a a b ds =g ˆαβdx dx =g ˆ00d(cτ) + 2ˆg0ad(cτ)dx +g ˆabdx dx (2.2.20) and note that schematically, the metric epxands like

00 2 gˆ00 = −1 + (terms of hˆ ) = −1 + O( ), (2.2.21) 0a 3 gˆ0a = (terms of hˆ ) = O( ), (2.2.22) ˆ00 2 gˆ0a = δab + (terms of h ) = δab + O( ), (2.2.23) then when make the transformation τ → −τ, which forces velocities to transform as v → −v, we ﬁnd the interval reads as

2 2 a a b ds−τ =g ˆ00d(cτ) − 2ˆg0ad(cτ)dx +g ˆabdx dx . (2.2.24)

For the interval to be invariant, me must have the same expansion. The time reversal can also be achieved by transforming → − as suggested by −τ = (−t) = (−)t. We can then consider the metric components as an expansion in powers of . The invariance 2 2 implies that dsτ = ds−τ , so thatg ˆ00 andg ˆab must be even functions of andg ˆ0a must be odd. Thereforeg ˆ00 andg ˆab contain only even powers of andg ˆ0a has only odd powers of

10 Chapter 2 2.2. SOLUTION METHOD

. In summary, our expansions for the potentials of (2.2.18) are now

∞ ˆ00 X 2nˆ00 2ˆ00 4ˆ00 6 h = h(n)() = h(2) + h(4) + O( ), (2.2.25) n=1 ∞ ˆ0a X 2n+1ˆ0a 3ˆ0a 5ˆ0a 7 h = h(n)() = h(3) + h(5) + O( ), (2.2.26) n=1 ∞ âb X 2nâb 4âb 6âb 8 h = h(n)() = h(4) + h(6) + O( ), (2.2.27) n=2 where we have attempted to space the expansions out to make it clear the comparative sizes of these terms. As we have alluded to, we also treat the energy-momentum tensor as an expansion in , that is ∞ ˆαβ X n ˆαβ T = T(n)() (2.2.28) n=2 and in a similar manner the components are

∞ ˆ00 X 2n ˆ00 2 ˆ00 4 ˆ00 6 T = T(n)() = T(2) + T(4) + O( ), (2.2.29) n=1 ∞ ˆ0a X 2n+1 ˆ0a 3 ˆ00 5 ˆ00 7 T = T(n)() = T(3) + T(5) + O( ), (2.2.30) n=1 ∞ âb X 2n âb 4 âb 6 âb 8 T = T(n)() = T(4) + T(6) + O( ). (2.2.31) n=2 Now we are ready to see how these expansions fit together with the reduced field equations and the gauge condition of (2.1.4). To obtain analogues of the reduced field equations, we simply substitute (2.2.18) and (2.2.28) into their respective sides and make use of the distributivity of the d’Alembertian. We find

∞ ( ∞ ) X 16πG X h i hαβ = n hˆαβ = − n(−gˆ)Tˆαβ + (−gˆ) tˆαβ (hˆαβ) + tˆαβ(hˆαβ) . (n) c4 (n) LL H n=2 n=2 (2.2.32) We have emphasised here that the metric determinant, the Landau-Lifshitz and Harmonic pseudo-tensors are functions of the potentials and are to be expanded to the correct order in those potentials. The goal here is to equate like powers of either side and thus obtain equations which give the potentials to progressively higher orders. At the 1PN level, we need to determine the expansions for the components of the potentials up to the terms O(4) for hˆ00, O(4) for hˆ0a and O(4) for hˆab. This requires ˆ00 ˆ0a ˆ0a ˆ00 ˆ0a ˆ0a {h(2), h(1), h(2)} which are then used to ﬁnd {h(4), h(3), h(4)}. This involves expanding αβ ˆαβ ˆαβ gˆ ,tLL and tH to the required order. We begin with expansions for the metric and its determinant to this required order which are 1 gˆαβ = ηαβ + hˆαβ − hηˆ αβ, (2.2.33) 2 (−gˆ) = 1 − h,ˆ (2.2.34) q 1 −gˆ = 1 − h,ˆ (2.2.35) (1) 2 ˆµν µα βν ˆ ˆ where indices are lowered with the Minkowski metric, i.e. h = η η hαβ and h := µν ηµνhˆ .

11 Chapter 2 2.2. SOLUTION METHOD

ˆαβ ˆαβ For tLL and tH expansions, we ﬁrst make note of the leading order scaling of the potentials which are:

hˆ00 = O(2), hˆ0a = O(3), hˆab = O(4). (2.2.36)

We note also that spatial derivatives of these potentials will not alter the scalings but time derivatives will contribute an extra factor of . We will show how the expansions are ˆ00 ˆ00 found for tLL and tH and provide the other components. At this order, all we require of the metric gαβ are the scalings given by (2.2.21)-(2.2.23). By inspection of (2.1.7) and 2 ˆ00 ˆ00 ˆ00 (2.1.8), we can see that at O( ) (which is for h(2)), tLL = 0 = tH . This is purely due to the quadratic nature of the potentials in these expressions. The minimum order we ρσ ντ can obtain for a quadratic term, that is a general expression of the form ∂λh ∂µh , is 4 00 00 of O( ) corresponding to a term like ∂ah ∂bh . Indeed, we can therefore simply guess ˆ00 4 00 c 00 that the expression for (−gˆ)tLL accurate to O( ) is C∂ch ∂ h where C is a constant. An explicit calculation using (2.1.7), reveals that at the required order, 16πG 7 (−gˆ)tˆ00 = − ∂ hˆ00∂chˆ00 + O(6). (2.2.37) c4 LL 8 c We can then substitute (2.2.25) into this and neglecting all terms of order 6 and higher, we find 16πG 7 (−gˆ )tˆ00 = − ∂ hˆ00 ∂chˆ00 4 + O(6). (2.2.38) c4 1 LL 8 c (2) (2) For the Harmonic pseudo-tensor, we have 16πG (−gˆ )tˆ00 = O(6) (2.2.39) c4 1 H and it can therefore be ignored. The one subscript on the determinant of the metric term is to indicate that this is to be expanded accurate to the first order potentials which are ˆ00 ˆ0a âb αβ ˆ00 {h(2), h(1), h(2)}. That is, (−gˆ0) = −det(η ) = 1 and (−gˆ1) = 1 + h(2). So upon looking at the h00 component of (2.2.32), we have 16πG 16πG 7 2 hˆ00 +4 hˆ00 = 2 − (−gˆ )Tˆ00 +4 − (−gˆ )Tˆ00 + ∂ hˆ00 ∂chˆ00 +O(6). (2) (4) c4 0 (2) c4 1 (4) 8 c (2) (2) (2.2.40) Equating powers of will gives us our first and second order equations respectively, 16πG hˆ00 = − (−gˆ )Tˆ00 , (2.2.41) (2) c4 0 (2) 16πG 7 hˆ00 = − (−gˆ )Tˆ00 + ∂ hˆ00 ∂chˆ00 . (2.2.42) (4) c4 1 (4) 8 c (2) (2) For the other components of the potential we find expansions accurate to their leading ˆ0a âb ˆ0a âb orders. For the first order potentials the pairs {tLL, tLL} and {tH , tH } both vanish. At the next order, we very carefully keep track of orders of and substituting in (2.2.26) and (2.2.27), we eventually find 16πG 3 (−gˆ )tˆ0a = ∂ahˆ00 ∂ hˆ00 + (∂ahˆ0c − ∂chˆ0a )∂ hˆ00 5 + O(7), c4 1 LL 4 (2) 0 (2) (3) (3) c (2) 16πG 1 1 (−gˆ )tâb = ∂ahˆ00 ∂bhˆ00 − δab∂ hˆ00 ∂chˆ00 4 + O(6), c4 1 LL 4 (2) (2) 8 c (2) (2) 16πG (−gˆ )tˆ0a = O(7), c4 1 H 16πG (−gˆ )tâb = O(6). c4 1 H

12 Chapter 2 2.3. ZONES OF INTEGRATION

The relevant versions of (2.2.32) for these components are

16πG 3 hˆ0a + 5 hˆ0a + O(7) = 3 − (−gˆ )Tˆ0a + (3) (5) c4 1 (3) 16πG 3 5 − (−gˆ )Tˆ0a + ∂ahˆ00 ∂ hˆ00 + (∂ahˆ0c − ∂chˆ0a )∂ hˆ00 (2.2.43) c4 2 (5) 4 (2) 0 (2) (3) (3) c (2) and 16πG 1 1 4 hâb + 6 hâb = 4 − (−gˆ )Tâb + ∂ahˆ00 ∂bhˆ00 − δab∂ hˆ00 ∂chˆ00 + O(6). (4) (6) c4 1 (4) 4 (2) (2) 8 c (2) (2) (2.2.44) ˆ0a âb For the 1PN order we only require h(3) and h(4) and we can easily read off these equations as 16πG hˆ0a = − (−gˆ )Tˆ0a , (2.2.45) (3) c4 1 (3) 16πG 1 1 hâb = − (−gˆ )Tâb + ∂ahˆ00 ∂bhˆ00 − δab∂ hˆ00 ∂chˆ00 . (2.2.46) (4) c4 1 (4) 4 (2) (2) 8 c (2) (2) Equations (2.2.41), (2.2.42), (2.2.45) and (2.2.46) will form the basis for Section 2.5. As it stands, the components in (2.2.18) will not solve the reduced field equations alone. We must find a way to include the gauge condition which we recall is

ˆαβ ∂βh = 0.

Fortunately this is simple for if we take the partial derivative of both sides of (2.2.18), then ∞ ˆαβ X n ˆαβ 0 = ∂βh = ∂βh(n). (2.2.47) n=0 Therefore we ﬁnd ˆαβ ∂βh(n) = 0, (2.2.48) which says that each iteration must also satisfy this gauge condition.

2.3 Zones of Integration

In the DIRE approach, one partitions the past light cone domain L(cτ/, x) of integration of (2.2.17) into two disjoint sets, a near and a wave zone. The boundary of these regions is set to be a two-sphere of radius R. This radius represents a characteristic scale for the system which is to be understood as the length of one characteristic gravitational wavelength. The entire source is to be contained deep within the near zone which is to say that it is within the world tube T (S) and hence within the larger world tube T (R) := {xα; |x0| =: r0 < R, −∞ < τ < ∞}. Until the post-Newtonian approximation begins to break down; which could occur for instance in the in-spiral phase of two gravitating objects, we assume that S R. In this region, retardation effects are assumed to be small and in fact this may be taken instead as the definition of the interface between the near and wave zones. We define the near zone of the field point (cτ, x), namely N(cτ, x), as the intersection of 0 0 0 the spatial region L(cτ, x) := {0 ≤ |x − x | ≤ cτ/} with the spatial region r := |x | < R. That is, the intersection of the world tube T (R) containing the source and the projection of the past light cone at (cτ, x) at the time slice cτ = 0. The wave zone, W(cτ, x), is defined as the relative complement of N(cτ, x) in L(cτ, x), i.e. W(cτ, x) := L(cτ, x) \N(cτ, x). In simpler terms, the wave zone is the spatial region remaining after we remove the near

13 Chapter 2 2.3. ZONES OF INTEGRATION

(a) Wave zone ﬁeld point (b) Near zone ﬁeld point

Figure 2.1: Past, solid light cones of the field point (cτ, x) depicting the regions of interest: the matter distribution which is contained within the world tube T (S); the world tube T (R) whose projection of the intersection with the light cone L(cτ, x) onto the surface τ = 0, yields the near zone N(cτ, x). The rest of the projection of the light cone after the near zone intersection is removed is the wave zone W(cτ, x). zone region. We refer the reader to Figure 2.1 for a space-time diagram depiction of these regions. By definition, the near and wave zone sets are disjoint so that L(cτ, x) = N(cτ, x) ∩ W(cτ, x). Our solutions to (2.1.5) can then be written as Z ˆ αβ 0 0 ˆαβ 4G Λ (cτ − |x − x |, x ) 3 0 h (cτ, x) = 4 0 d x (2.3.1) c L(cτ,x) |x − x | "Z ˆ αβ 0 0 Z ˆ αβ 0 0 # 4G Λ (cτ − |x − x |, x ) 3 0 Λ (cτ − |x − x |, x ) 3 0 = 4 0 d x + 0 d x c N(cτ,x) |x − x | W(cτ,x) |x − x | (2.3.2) = hˆαβ(cτ, x) + hˆαβ (cτ, x). (2.3.3) N W where we have defined 4G Z Λˆ αβ(cτ − |x − x0|, x0) hˆαβ := d3x0, (2.3.4) N 4 0 c N(cτ,x) |x − x | 4G Z Λˆ αβ(cτ − |x − x0|, x0) hˆαβ := d3x0. (2.3.5) W 4 0 c W(cτ,x) |x − x | So for a chosen field point we have two contributions to the potential hˆαβ; one from the near zone and one from the wave zone. Each of these will individually depend on R but the sum should not depend on the arbitrary choice of R. This is because it is expected that the wave zone contribution will contain terms dependent on R, but that all these terms will be exactly cancelled by the near zone contribution and the remaining terms will be finite. This result was proved by Pati and Will [11]. In summary, we have two choices of the field point position; it can be in the near zone or the wave zone as is depicted in Figure 2.1. For the field point x ∈ N(cτ, x), the potential reads as hˆαβ(x ∈ N ) = hˆαβ(x ∈ N ) + hˆαβ (x ∈ N ). (2.3.6) N W

14 Chapter 2 2.4. MULTI-POLE EXPANSIONS

We can refer to each term on the right hand side of (2.3.6) as the near zone contribution to a near zone field point, which we label schematically as N → N , and the wave zone contribution to a near zone field point W → N . Of course, these are still dependent on the choice of the field point (cτ, x) and on ; we use these purely to aid conceptual understanding. The expressions for these contributions are explicitly given by (2.3.4) and (2.3.5) respectively. Similarly for x ∈ W(cτ, x), the potentials split into

hˆαβ(x ∈ W ) = hˆαβ(x ∈ W ) + hˆαβ (x ∈ W ). (2.3.7) N W and we label these as the near zone contribution to a wave zone ﬁeld point (N → W) and the wave zone contribution to a wave zone ﬁeld point (W → W). The explicit expressions are given by (2.3.4) and (2.3.5) respectively for each contribution.

2.4 Multi-pole Expansions

In this section we examine specific cases for the position of the field point and of these, only one contribution each. These will be the near zone contribution to a wave zone field point, N → W, and the near zone contribution to a near zone field point, N → N . For our purposes, the N → W and W → W contributions will only briefly be considered in Section 2.5. Let us first consider the field point to be in the wave zone, that is x ∈ W(cτ, x). We know from (2.3.7), that we have two contributions for the potential. We will examine only the near zone contribution here. We define

Λˆ αβ(cτ − |x − x0|, y) =: f((x − x0)), |x − x0| where the spatial dependence of Λ has been replaced with an arbitrary vector y. If we fix τ and y, we can define a function f which has argument only of (x − x0). On top of this, we may also hold x fixed since this is just an arbitrary point lying in the wave zone which we have chosen. Doing this we can then consider a function g with argument only of the source points x0. This chain of logic is summarised by

Λˆ αβ(cτ − |x − x0|, y) =: f((x − x0)) =: g(x0). (2.4.1) |x − x0|

Now since x0 lies in the near zone, we can treat it as a ‘small’ vector and write a Taylor expansion about x0 = 0 (we ignore the question of whether g is inﬁnitely diﬀerentiable). Then ∂g 1 ∂2g g(x0) = g(0) + x0a + 2 x0ax0b + ··· , (2.4.2) ∂x0a 2 ∂x0a∂x0b with the derivatives evaluated at x0 = 0. Since f is dependent on x0 through (x − x0), then ∂g ∂f ∂f = = − , (2.4.3) ∂x0a ∂x0a ∂xa where primes indicate the coordinates of x0 and unprimed quantities that of x. Critically, since the derivatives are with respect to xa rather than x0a, then we can set x0 = 0 before taking the derivatives. We then write f(x−x0) = f(x), |x| := r and hence (2.4.2) becomes

∞ ∞ ! X (−)q X (−)q Λˆ αβ(cτ − r, y) g(x0) = x0Qq ∂ (f(x)) = x0Qq ∂ . (2.4.4) q! Qq q! Qq r q=0 q=0

15 Chapter 2 2.4. MULTI-POLE EXPANSIONS

Note the multi-index notation, where a sum over a repeated multi-index is to be understood as a sum over each index contained in the multi-index. For example, if q = 2, then the 2 0a 0b αβ 0 term becomes ( /2)x x ∂ab[Λˆ (cτ − r, y)/r]. With y = x , then 2.3.4 is now 4G ∞ (−)q 1 ˆαβ X αβ, Qq h = ∂Q Mˆ , (2.4.5) N c4 q! q r q=0 where the multi-pole moments are Z Mˆ αβ, Qq := Λˆ αβ(cτ − r, x0)x0Qq d3x0. (2.4.6) N For gravitational wave detection, we will be interested in when the field point is in the far away wave zone where terms of O(r−2) and higher are dominated by those of order O(r−1). Using the product rule, we can split the derivative in (2.4.5) into two parts, one acting on r−1 and the other able to be pulled into the integrand and differentiating Λαβ. From (A.2.5) (at this point we recommend reading appendix A.2 for background of the −1 −2 coming notation), ∂Qq r = O(r ) and this term can be ignored. For the second term, we realize that Λˆ αβ is dependent on the coordinates of the field point xa through the retarded coordinate u = cτ − r. Therefore by the product rule, ∂Λˆ αβ ∂Λˆ αβ ∂Λˆ αβ = ∂ Λˆ αβ = − ∂ r = − Ω , ∂xa a ∂u a ∂u a ∂2Λˆ αβ ∂ Λˆ αβ = 2 Ω Ω + O(r−1), ab ∂u2 a b ∂qΛˆ αβ ∂ Λˆ αβ = (−)q Ω + O(r−1), (2.4.7) Qq ∂uq Qq where (A.2.3) has been used and where (2.4.7) follows by induction on q, the number of indices contained in the multi index Q. We then substitute (2.4.7) into (2.4.5) and note that the summation can be split with the derivatives terms of (2.4.5) in one sum and the O(r−1) terms in the other. However, the factor of 1/r outside the summation then makes this second sum become O(r−2), which we ignore. Finally, for a far away wave zone field point, the near zone contribution to the potential is ∞ q Z 4G X ΩQq ∂ hˆαβ = 2q Λˆ αβ(u, x0)x0Qq d3x0. (2.4.8) N c4r q! ∂u q=0 N

This is a multi-pole expansion. Recalling the definition of Λˆ αβ from (2.1.6), we see that we can split this integral into two pieces: one composed of the energy-momentum tensor Tˆαβ and another of the Landau-Lifshitz and harmonic pseudo-tensors. In this way, we can loosely consider the Tˆαβ term as a matter multi-pole expansion which gives rise to some of the potential hˆ. The other expression describes how the potentials affect themselves. This is a consequence of the non-linearity present in General Relativity and is the inherent source of much of its difficulties. If the field point x ∈ N(cτ, x), then since the source is also contained in the near zone then |x − x0| can be treated as a small quantity and we may write a Taylor expansion about |x − x0| = 0, 1 Λˆ αβ(cτ − |x − x0|) = Λˆ αβ(cτ) − |x − x0| ∂ Λˆ αβ + 2|x − x0|2∂2 Λˆ αβ − · · · , (2.4.9) cτ 2 cτ where the derivatives are evaluated at cτ. This can be inserted into (2.3.4) to yield

∞ q 4G X (−)q ∂ Z hˆαβ = Λˆ αβ(cτ, x0)|x − x0|q−1d3x0. (2.4.10) N c4 q! ∂(cτ) q=0 N

16 Chapter 2 2.5. THE FIRST POST-NEWTONIAN POTENTIALS

The domain of integration is the same as in (2.4.5). On the whole, the key equations for this section are the pair, (2.4.5, 2.4.6), describing a near zone expansion for a wave zone ﬁeld point, (2.4.8) which is a far-away wave zone expansion where we only seek terms of O(r−1) and (2.4.10), a near zone expansion for a near zone ﬁeld point. These will prove useful in the coming section.

2.5 The ﬁrst post-Newtonian potentials

The Newtonian order

As the title suggests, in this section we will integrate the first post-Newtonian equations for the potentials in the case of a system of N point-particles. This firstly requires the solution to the ‘Newtonian’ order which are given by the equations 16πG hˆ00 = − (−gˆ )Tˆ00 , (2.5.1) (2) c4 0 (2) ˆ0a 3 h(3) = O( ), (2.5.2) âb 4 h(4) = O( ). (2.5.3) At this order, the metric we use for the right hand side is the Minkwoski metric since ˆαβ αβ ˆ00 we are using h(0) = 0 here. Therefore (−gˆ1) = −det(η ) = 1 and T(2) is now a function ˆ00 of the Minkowski metric which we can write as T(2)[η]. As described in Section 2.2, we ˆαβ ∞ 3 require that T(0) ∈ C (R ) so that we may utilize the key equations of Section 2.4, namely (2.4.5) and (2.4.10). We also assume that the matter distribution is bounded and is entirely contained within the near zone so that we have the nice simplification hˆαβ = 0 W and hence hˆαβ = hˆαβ + hˆαβ = hˆαβ. N W N We now define the field variables Φˆ, Aâ and the matter variablesρ, ˆ ˆja such that 4 4 hˆ00 := Φˆ, hˆ0a := Aâ, (2.5.4) (2) c2 (3) c3 ˆ00 2 2 ˆ0a 3â T(2) := c ρ,ˆ T(3) := c j , (2.5.5) and with these definitions, we have the wave equation

ˆ 2 Φ = −4πG ρ.ˆ (2.5.6) ˆαβ The gauge condition ∂βh = 0, is now a consequence of the conservation equations αβ ∂βTˆ = 0. From these we can ﬁnd two continuity equations which are

a a ∂τ ρˆ + ∂aˆj = 0, ∂τ Φˆ + ∂aAˆ = 0. (2.5.7)

The wave equation of (2.5.6) admits a retarded integral solution

Z ρˆ(cτ − |x − x0|, x0) Φ(ˆ cτ, x) = 2G d3x0. (2.5.8) |x − x0| The task now is to evaluate this expression depending on the position of the field point. We begin by considering the field point in the near zone; x ∈ N(cτ, x)). Here we can use (2.4.10) and consider the first few terms. For the potential Φ these are

Z ρˆ(cτ, x0) G3 ∂ Z G4 ∂2 Z Φˆ = G2 d3x0 − ρˆ(cτ, x0)d3x0 + ρˆ(cτ, x0)|x − x0|d3x0 + ··· |x − x0| c ∂τ 2c2 ∂τ 2 (2.5.9)

17 Chapter 2 2.5. THE FIRST POST-NEWTONIAN POTENTIALS

This is good so far but we can do slightly better. If we integrate both sides of the continuity equation forρ ˆ as in the left of (2.5.7) and using Gauss’ theorem for the spatial derivative term we find Z Z I 3 a 3 a ∂τ ρdˆ x = − ∂aˆj d x = − ˆj dSa = 0, because there is no flux of matter across the surface bounding the matter (we assumed the matter was bounded). We can then move the time derivative outside the integral because we know thatρ ˆ is a smooth function over the matter distribution. This is a common trick with continuity equations. So if we make the following definitions Z ρˆ(cτ, x0) Uˆ := G d3x0, (2.5.10) |x − x0| Z Xˆ := G ρˆ(cτ, x0)|x − x0|d3x0, (2.5.11) then (2.5.9) reads " # 2 ∂2Xˆ Φˆ = 2 Uˆ + + O(3) . (2.5.12) 2c2 ∂τ 2 It is a rather comforting verification that a post-Newtonian method does indeed re-derive the Newtonian terms, for instance (2.5.10) is the Newtonian potential which satisfies the Poisson equation 4Uˆ = −4πGρˆ. The next order contribution (2.5.11) satisfies 4Xˆ = 2Uˆ and is called a super-potential since it is sourced by a potential. The near zone metric is now 42 24 ∂2Xˆ hˆ00 = Uˆ + + O(5), x ∈ N (cτ, x). (2.5.13) (2) c2 c4 ∂τ 2 At this order, the metric becomes 2 gˆ = − 1 + Φ , (2.5.14) 00 c2

gˆ0a = 0, (2.5.15) 2 gˆ = δ 1 − Φ , (2.5.16) ab ab c2 which is exactly that found if one considers the Newtonian limit in General relativity from the field equations. To completely determine the potentials to first order, we now consider the case where the field point is in the wave zone; x ∈ W(cτ, x). For this we make use of (2.4.5). We can therefore expand the scalar field Φˆ as

1 Z 1 Z 2 1 Z Φˆ = G2 ρdˆ 3x0 − ∂ ρxˆ 0ad3x0 + ∂ ρxˆ 0ax0bd3x0 r a r 2 ab r 3 1 Z − ∂ ρxˆ 0ax0bx0cd3x0 + ··· . (2.5.17) 6 abc r To simplify this notation we introduce a general mass multi-pole moment defined by Z Iˆq1q2...qn := ρxˆ 0Qq d3x0, (2.5.18) where Qq is a multi-index. It can be shown by taking derivatives with respect to u and using surface integral arguments that Iâ = 0 in the center of mass frame. Therefore the scalar field becomes ! ! GIˆ G Iâb G Iâbc Φˆ = 2 + 3 ∂ − 4 ∂ + .... (2.5.19) r 2 ab r 6 abc r

18 Chapter 2 2.5. THE FIRST POST-NEWTONIAN POTENTIALS

It is interesting to note that again the leading order term is the ordinary Newtonian potential since Iˆ = M, where M is the total mass-energy of the system. The higher order moments are functions of the retarded time τr := τ − r/c, so the post-Newtonian corrections contain all the retardation effects. The wave-zone potential is now ! ! 4G2 M Iâb 2 Iâbc hˆ00 = + ∂ − ∂ + O(3) , x ∈ W (cτ, x). (2.5.20) (2) c2 r 2 ab r 6 abc r

The 1PN potentials

It is now time to consider the 1PN equations for the potentials. In Section 2.2, we found that these were governed by the wave equations 16πG 7 hˆ00 = − (−gˆ )Tˆ00 + ∂ hˆ00 ∂chˆ00 , (2.5.21) (4) c4 1 (4) 8 c (2) (2) 16πG hˆ0a = − (−gˆ )Tˆ0a , (2.5.22) (3) c4 1 (3) 16πG 1 1 hâb = − (−gˆ )Tâb + ∂ahˆ00 ∂bhˆ00 − δab∂ hˆ00 ∂chˆ00 . (2.5.23) (4) c4 1 (4) 4 (2) (2) 8 c (2) (2) ˆ00 2 ˆ ˆ00 ˆ0a In these equations we recall that (−gˆ1) = 1 + h(2) = 1 + (4/c )Φ and that T(4), T(3) âb and T(4) are functions of the ‘Newtonian’ metric given by (2.5.14)-(2.5.16). Substituting ˆ00 2 ˆ h(2) = (4/c )Φ we have 16πG 14 hˆ00 = − (−gˆ )Tˆ00 + ∂ Φˆ∂cΦˆ, (2.5.24) (4) c4 1 (4) c4 c 16πG hˆ0a = − (−gˆ )Tˆ0a , (2.5.25) (3) c4 1 (3) 16πG 4 1 hâb = − (−gˆ )Tâb − ∂aΦˆ∂bΦˆ − δab∂ Φˆ∂cΦˆ . (2.5.26) (4) c4 1 (4) c4 8 c

a ab ab To continue further, we define new functions V,ˆ Vˆ , Wˆ and Wˆ := δabWˆ by 4 4 8 hˆ00 = Vˆ − Wˆ + Φˆ 2, (2.5.27) (4) c2 c4 c4 4 hˆ0a = Vˆ a, (2.5.28) (3) c3 4 hâb = Wˆ ab. (2.5.29) (4) c4 The goal with these definitions is to reduce (2.5.24)-(2.5.26) into wave equations for which some will have purely matter sources and others both matter and field sources. Inserting (2.5.29) into (2.5.26) yields a wave equation for the tensor potential 1 Wˆ ab = −4πG(−gˆ )Tâb − ∂aΦˆ∂bΦˆ + δab∂ Φˆ∂cΦˆ. (2.5.30) 1 (4) 2 c We can take the trace of this equation to find the wave equation for W , 1 Wˆ = −4πGδ (−gˆ )Tâb + ∂ Φˆ∂cΦˆ + O(2), (2.5.31) ab 1 (4) 2 c

ab where we have used δabδ = 3. We can also trivially insert (2.5.28) into (2.5.25) to obtain the wave equation satisﬁed by the vector potential Vˆ a, that is 4πG Vˆ a = − (−gˆ )Tˆ0a . (2.5.32) c 1 (3)

19 Chapter 2 2.5. THE FIRST POST-NEWTONIAN POTENTIALS

ˆ 2 In anticipation of inserting (2.5.27) into (2.5.24), we would have a Φ term which we will ˆ 2 ˆ ˆ ˆ c ˆ ˆ 2 now evaluate. Noting that Φ = 2ΦΦ + 2∂cΦ∂ Φ − 2(∂0Φ) and using the first order ˆ 4πG ˆ00 expression for Φ which is Φ = − c2 T(2) then 8πG Φˆ 2 = − ΦˆTˆ00 + 2∂ Φˆ∂cΦˆ + O(6). (2.5.33) c2 (2) c 2 6 Note that we have neglected the term (∂0Φ)ˆ which is of order O( ). Inserting (2.5.27) into (2.5.24) and making use of the wave equations for Wˆ and Φˆ 2 we find 4πG 16πG Vˆ = − (−gˆ )δ Tˆαβ + ΦˆTˆ00 + O(2), (2.5.34) c2 1 αβ (4) c4 (2) which completes our set of wave equations. Integrating each of these to an appropriate order will then yield the second order expansion. We could write down for each of these wave equations a retarded integral solution, however to evaluate these we would need to split the integration region into near and wave zones since there are now field sources Φˆ which range over all space. While we will not be integrating explicitly the general forms for these new potentials, it is worth examining just the leading order term in an expansion for W ab for a wave zone field point. This term will lead us to the beloved quadrupole formula in chapter 4. To begin we define from (2.5.30) ˆ ab ab W = −4πGχˆ , (2.5.35) 1 1 χâb := (−gˆ )Tâb + ∂aΦˆ∂bΦˆ − δab∂ Φˆ∂cΦˆ , (2.5.36) 1 (4) 4πG 2 c and we will utilise the identity [12] 1 1 χâb = ∂ (ˆχ00xaxb) + ∂ (ˆχacxb +χ ˆbcxa − ∂ χˆcdxaxb), (2.5.37) 2 00 2 c d whereχ ˆ00 = c22ρˆ + O(4). The solution to (2.5.35) is a retarded integral and we choose the field point x ∈ W(cτ, x). The source term for (2.5.35) contains field terms Φˆ which range over all space, so the wave zone contribution to Wˆ ab must be taken into account. So Wˆ ab = Wˆ ab + Wˆ ab ; however the the leading term in the wave zone contribution is of N W order 0.5PN relative to the quadrupole term of Wˆ ab so we will ignore Wˆ ab anyway. For N W higher order terms, one should not ignore this contribution and we discuss this in Section 4.6. The expansion of Wˆ ab with a wave zone field point, using (2.4.5), is N Z Z ab 1 ab 3 0 1 ab 0c 3 0 Wˆ = G χˆ d x − ∂c χˆ x d x + ··· . (2.5.38) r N r N For the leading term, we use (2.5.37) and the Divergence theorem, to find Z 4 2 Z I ab 3 0 ∂ a b 3 0 1 ac b bc a cd a b χˆ d x = 2 ρˆ x x d x + (ˆχ x +χ ˆ x − ∂dχˆ x x ) dSc, (2.5.39) N 2 ∂τr N 2 ∂N where the region of integration for the surface integral is the boundary of the near and wave zones which is set at r0 := |x0| = R with R the radius of the two-sphere which forms this boundary (recall Section 2.3). The first term is the quadrupole term

4 ∂2Iˆab 2 , (2.5.40) 2 ∂τr where the derivative is with respect to τr := τ −r/c. The utility of converting to a surface ab integral is that at the boundary, T(4) vanishes since we assumed the matter was contained

20 Chapter 2 2.5. THE FIRST POST-NEWTONIAN POTENTIALS deep within the near zone in a two-sphere of radius S R. We then only need to worry about the ﬁeld terms Φ at the boundary. The surface integral term is a little more interesting as we have not come across this yet, although we do have the machinery required to handle it. Our ﬁeld point is in the wave zone so we must use the expression for Φˆ which is also in the wave zone. We found this is in (2.5.17) and to the order we require, it is

GM Φˆ = 2 + O(4). (2.5.41) r Thereforeχ âb becomes GM χâb = 4 ΩaΩb − δab/2 + O(6), (2.5.42) 4πr4 where we have used Eqs. (A.2.2) and (A.2.5). The surface integrals can now be evaluated 2 a a using dSc = R ΩcdΩ and x = RΩ on the surface. We will only evaluate the one cd a b containing ∂dχˆ x x and this is done in Appendix A.3. The core idea of this more complicated version can be easily transferred across to the other two integrals and integrals for higher order terms. These integrals are either zero because they contain products of an odd number of the angular vectors Ωa or they are only dependent on the arbitrary radius R. In the latter case, these terms will be exactly cancelled by corresponding terms from ab the wave zone contribution WW (recall the discussion in Section 2.3). After all of this we find, 4G4 Iâb(2) hâb = + O(6), (2.5.43) (4) c4 2r which is our quadrupole term.

Near-zone 1PN potentials for an N-body system

We can specialise to a particular situation where the source of the field is generated by a system of N-point particles each having their own mass mA where A is used to label a particle, at positions zA. We assume that this system obeys the ‘slow motion’ approximation and that its entire evolution is contained deep within the near zone. For instance, a model of the solar system. Based on previous discussions, we could immediately take issue with this use of point particles. Using this approximation, we can expect to have divergences appearing typically due to the infinite self field of each particle. These are a result of evaluating the Newtonian potentials at the position of the particle exactly where the potentials diverge. We know that any energy in space-time will generate its own gravitational field (from 2.1.1) which will then contribute to the inter-body field. Another possibly clearer issue, is that we have assumed throughout that Tˆαβ ∈ C∞ and yet we seek to use Dirac delta distributions. We must therefore be quite careful of how the equations are handled. To avert these infinities, we must use the methods of regularization where we give meaning to these offending terms which typically involves assuming they should vanish. There are many prescriptions for these regularizations and common approaches are using modified forms of a Hadamard partie finie as in the BDI approach or a dimensional regularization used in effective field theories which are taken from quantum field theories and are quite powerful (see the work of Goldberg and Rothstein [13] for this in action). While these regularizations may seem unsatisfactory, the end results agree very well with experiments and observations. In no way does this validate their use mathematically, but it does give one confidence that it is not too far-fetched.

21 Chapter 2 2.5. THE FIRST POST-NEWTONIAN POTENTIALS

For an N-body system of point particles we will need an expression for the energy- momentum tensor. Although we will not prove it here we state that it is given by √ ˆαβ X α β −gˆ1 (−gˆ1)T(4) = {α, β} mAvÂvÂ q δ(x − zA), (2.5.44) µ ν 2 A −gˆµν 1vÂvÂ/c

d(ct,xA) where vA = dτ = (c/, vA) is the four-velocity of particle A and

 2  : α = 0, β = 0 {α, β} = 3 : α = 0, β = b  4 : α = a, β = b.

For the metric terms that appear we make use of the expansions given in (2.2.33) and (2.2.35) and the potentials from earlier,

42 hˆ00 = Uˆ + O(4), hˆ0a = O(3), hˆab = O(4). (2.5.45) (2) c2 (1) (2) P We also have that the density isρ ˆ = B mBδ(x−zB) and therefore we can easily evaluate (2.5.10) to obtain the ‘Newtonian’ potential for N-point masses

X GmB Uˆ = . (2.5.46) |x − zB| B The relevant metric terms appearing in the energy-momentum tensor can be expanded as

q 2 2 −gˆ vˆµ vˆν /c2 = 1 − vˆ2 − Uˆ + O(4), µν 1 A A 2c2 A c2 2 p−gˆ = 1 + Uˆ + O(4) 1 c2 and inserted into (2.5.44) to obtain

X 2 32 (−gˆ )Tˆαβ = {α, β} m vˆαvˆβ 1 + vˆ2 + Uˆ + O(4) δ(x − z ), (2.5.47) 1 (4) A A A 2c2 A c2 A A A where UÂ = Uˆ(τ, zA) is evaluated at the point of particle A. However there is a serious issue here: the potential is infinite when we reach the summation index where we put zB = zA in (2.5.46). As a distribution (because of the presence of the Dirac delta), (2.5.47) is undefined. This is a first look at the need for regularization methods in post-Newtonian theory. In this case, the method of Blanchet, Damour and Iyer (BDI) [14] proposes the regularization δ(x − z) ≡ 0, (2.5.48) |x − z| which removes the offending term in the summation and hence we instead insert

X GmB bUˆcA = , (2.5.49) |zA − zB| B6=A

a ab in place of UˆA in (2.5.44). Our wave equations for the potentials V,ˆ Vˆ , Wˆ and Wˆ are

22 Chapter 2 2.5. THE FIRST POST-NEWTONIAN POTENTIALS now

X 3 2 2 Vˆ = −4πG2 m 1 + vˆ2 − bUˆc δ(x − z ) + O(6), (2.5.50) A 2 c2 A c2 A A A ˆ a 3 X a 5 V = −4πG vˆAδ(x − zA) + O( ), (2.5.51) A X 1 Wˆ = −4πG4 m vˆ2 δ(x − z ) + ∂ Φˆ∂cΦˆ + O(6), (2.5.52) A A A 2 c A X 1 Wˆ ab = −4πG4 m vˆa vˆb δ(x − z ) − ∂aΦˆ∂bΦˆ − δab∂ Φˆ∂cΦˆ + O(6). (2.5.53) A A A A 2 c A

Here (2.5.34)-(2.5.53) are the starting points for determining the 1PN potentials of hˆαβ. We will not labour through all of the details here but we will give an overview of how one could proceed from this point and what the results are. As usual we can write the wave equations above in terms of a retarded integral solution over the past light cone of the ﬁeld point and then split this into a near zone contribution and a wave zone contribution. Next we choose where we place the ﬁeld point x, either in the near or wave zone and then proceed to evaluate the integrals to determine the potentials. To begin we consider x ∈ N(cτ, x). For V , we note that the source for its wave equation, (2.5.50), is entirely contained in the near zone; it is a matter source. ˆ ˆ ˆ Therefore its wave zone contribution vanishes (VW = 0) and hence V = VN . Keeping track of orders of we can essentially copy down the method for the evaluation of Φˆ in the near zone from the previous section. We obtain

4 4 ∂2Xˆ Vˆ = 2Uˆ + ψˆ + + O(5), (2.5.54) c2 2c2 ∂τ 2 where

3 2 GmA vˆ − bUˆcA X GmA ˆ X 2 A X Uˆ := , ψ := , Xˆ := GmA|x − zA|. (2.5.55) |x − zA| |x − zA| A A A

The wave equation for Vˆ a also contains only a matter term which implies that the wave zone contribution vanishes so Vˆ a = 3Uˆ a + O(5), (2.5.56) with a X GmAvˆ Uˆ a = A . (2.5.57) |x − zA| A The computations for Wˆ ab are more difficult and involve realising that we can split it into two parts schematically as Wˆ ab = Wˆ ab[M] + Wˆ ab[F ], (2.5.58) where Wˆ ab[M] contains only matter contributions and Wˆ ab[F ] contains the field contributions. We can then divide (2.5.53) into two pieces; one wave equation for each contribution. The matter contribution can be evaluated simply by the same argument as before that ˆ ab W [M]W = 0. However the field contribution will not vainsh in the wave zone since it is sourced by the potentials Φˆ which range over all space. For the wave zone contribution we would have to use (2.5.19) rather than (2.5.12) but it turns out that this part only contributes at the 3PN order which is well beyond the 1PN order that we are considering

23 Chapter 2 2.6. ALTERNATE INTEGRATION METHODS and so can be ignored. The final result is (noting that once we find Wˆ ab we can take the trace and find Wˆ )

Wˆ ab = 4Pˆab + O(5), (2.5.59) Wˆ = 4Pˆ + O(5), (2.5.60) where we have the rather long deﬁnitions

a b 2 2 2 âb X GmAvÂvÂ 1 X G mA a b X X G mAmB A b P := + 2 nAnA − 2 nABnAB |x − zA| 4 |x − zA| S|zA − zA| A A A B>A 2 X X G mAmB 1 + 2 (na − na )(nb − nb ) − δab(n − n ).(n + n ) , S2 A AB B AB 2 A AB B AB A B>A (2.5.61) and

2 2 2 ˆ X GmAvˆA 1 X G mA P := + 2 |x − zA| 4 |x − zA| A A 1 X X 2 1 1 1 − G mAmB + − . 2 |x − zA||zA − zB| |x − zB||zA − zB| |x − zB||x − zB| A B>A (2.5.62)

The simplifying notation we have used here is

x − zA zA − zB nA := , nAB := ,S := |x − zA| + |x − zB| + |zA − zB|. (2.5.63) |x − zA| |zA − zB| Finally, for a near zone field point, the potentials are given by ! 42 44 1 ∂2Xˆ hˆ00 = Uˆ + ψˆ + − Pˆ + 2Uˆ 2 + O(5), (2.5.64) (4) c2 c4 2 ∂τ 2 43 hˆ0a = Uˆ a + O(5), (2.5.65) (3) c3 44 hâb = Pâb + O(5). (2.5.66) (4) c4 We will not be considering the potentials for a wave zone field point since we intend to use these results to derive the equations of motion of the system. Therefore we should only need to use the results for a near zone field point.

2.6 Alternate Integration Methods

There are also more subtle issues with (2.2.17) which are the cause of mathematical divergences. This arises because the potentials cannot and in general will not have any compact support since it is assumed that they exist over all space-time which includes points very far from the source. In the context of gravitational waves, detectors on Earth can be assumed to be in this far region and therefore the limiting extent of the potentials must be known for data analysis. There is then a chance that this retarded integral may diverge or be ill-deﬁned. This typically occurs when Dirac-delta functions are used to model point particles as we have seen with the dramatic result that equations begin to diverge. This is clearly not consistent with observation and reality. To keep them in check, they are

24 Chapter 2 2.6. ALTERNATE INTEGRATION METHODS

‘regularized’ so that only finite quantities are obtained and the equations can be solved. These divergences have caused much difficulty in this field so many different approaches have been formulated to tackle them. In this section we will briefly discuss these. The method of Blanchet, Damour and Iyer (BDI) developed over many papers [14–17] computes directly the equations of motion for a system in harmonic coordinates using a post-Newtonian iteration, in many ways equivalent to the approach defined in Section 2.2. Their method begins by solving the reduced EFE at the first order in vacuum, exterior to the compact source, by a multipole expansion. They then show that the solution here can be written in terms of only two types of moments; a mass and a current moment which are (in their notation [8]) IL(u := t − r/c) and JL(u) where L is a multi-index. Due to a gauge transformation there are in fact four other moments labelled WL,XL,YL and ZL which have no physical meaning but omitting them leads to different and most importantly, incorrect results. The key is that there is no bijective map from the set of moments {IL,JL,WL,XL,YL,ZL} to {IL,JL, 0 ..., 0}, so these four new moments must be included. However there is map to a set of different moments {ML,SL, 0 ..., 0} which are in general non-linear combinations of the previous moments. The potentials are then iterated to higher orders indexing with n. It is important to note that the moments IL and so on are not given in terms of the source itself yet, they arise simply due to the presence of the wave operator. The problem now is solving a wave equation with a known source as we found before, however the moments are singular at r = 0. This is the first case for a need to regularize these divergences. The approach is to rewrite the source tensor as

αβ B αβ Π(n) → re Π(n), (2.6.1)

αβ 4 αβ αβ where re := r/r0 is dimensionless, B ∈ C and Π := (16πG/c )(−g)(tLL + tH ) . The retarded integral solution to the wave equations is then

αβ −1 h B αβ i H (B) = ret re χ(n) , (2.6.2)

−1 where ret represents the retarded integral solution. The idea is that the real part of B αβ can be taken large enough so that the order of the singularities of the moments in Λ(n) are removed. The trick is then to consider H(B) as an analytic continuation with respect to B into the complex plane, then write the solution, (2.6.2), as a Laurent series and then pick out the finite-part coefficient as B → 0. The BDI approach then shows that in a neighbourhood of B = 0, a near zone expansion can be performed in terms of the moments IL to ZL. The core of this approach is to match these expansions of the exterior field to the inner field of the source. This matching involves expressing the multipole exterior solution in terms of a near zone expansion. This is then equated to a multipole expansion of the near zone field. If M(h) is a multipole expansion of hαβ which is the exterior solution and h¯ is a post-Newtonian expansion of h, then the ‘Matching equation’ imposes M(h) = M(h). (2.6.3) The difference between h and M(h) is that h is the field valid everywhere in space-time while M(h) is a multipole expansion of h valid only outside the source and is singular at r = 0. This method gives

∞ h i 4G X (−1)q 1 M(hαβ) = FP −1 rBM(Παβ) + ∂ Hαβ, Qq , (2.6.4) B=0ret e c4 q! Qq r q=0 where Z Hαβ = FP rBΛαβ(u, x0)x0 d3x0. (2.6.5) Q B=0 e Qq

25 Chapter 2 2.6. ALTERNATE INTEGRATION METHODS

These equations require a little explaining. The operation FPB=0 represents taking the finite part as B = 0 as mentioned previously. The first term on the right hand side of (2.6.5) is the mulitpolar solution to the vacuum field equations outside the source while the second term is reminiscent of (2.4.5) and contains the actual matter source contribution where Λαβ is defined as in (2.1.6) which we see contains T αβ. These terms represent a near and a wave zone contribution except the integrals range over all space. A second method known as DIRE (direct integration of relaxed Einstein equations) used by Will, Wiseman and Pati [5,11,18] also splits the integration region into a near and far zone. The methods in Section 2.3 follow this particular partitioning of the integration region. It then evaluates them separately and exactly. As an example of the similarity (and one can prove, equivalence) between the DIRE and BDI approach is the DIRE version of (2.6.4) and (2.6.5) which are

h i ∞ q αβ −1 αβ 4G X (−1) 1 αβ, Qq M(h ) = ret M(Π ) + ∂Qq K , (2.6.6) W c4 q! r q=0 Z αβ αβ 0 0 3 0 KQ = Λ (u, x )xQq d x . (2.6.7) N where N is the near zone defined by |x| < R and the subscript W implies the retarded integral is truncated so that it extends only into the wave zone |x| > R. We can see that there is no finite part operation because the the retarded integral part no longer contains r = 0 in its its integration region where it diverges. This is exactly equivalent to (2.3.3) where instead the post-Newtonian parameter is = 1/c which is a slight abuse of notation since should be dimensionless while c, the speed of light, has dimensions. We will also point out again, that in our analysis we have largely neglected the wave zone contributions to the potentials, that is the W → N and W → W parts. The reason for this is two- fold: firstly the wave zone contribution is negligible up to the 3 PN terms which require accuracy to O(8) and so do not show at the 1 PN order of equations (2.5.64)-(2.5.66) and for the simpler reason that they are much more difficult to compute than the near zone contributions and so are in some sense beyond the content contained here. The ‘strong field point particle limit’ pioneered by Itoh and Futamese [19] is a highly useful formulation that does not require any regularizations. They define an explicit small parameter exactly as we have in (2.2.1), and from it imply how the physical quantities of the system should scale with this . Their idea is that the internal fields of a body, such as the members of an inspiralling compact binary, can be kept fixed as they are shrunk to approximate point particles. By scaling the mass of the body (M) in line with how its radius (R) shrinks, then the internal field Φ ∼ M/R can remain constant. With this, Futamase and Itoh partition into a near and a far zone however the near zone contribution is split into two pieces: a body zone B and other area N/B. The body zone is written in terms of so called ‘body zone coordinates.’ In the near zone coordinates (τ, x), the size of the body zone shrinks as while the star shrinks as 2 as → 0. In the body zone coordinates, the star instead remains fixed in size while the body zone boundary expands to infinity like −1 as → 0. In this way, the star never shrinks towards a Dirac delta and the integrals are naturally convergent. It then uses a surface integral approach to determine the equations of motion and has been calculated to 3PN order [7]. Other approaches include using the ADM formalism of general relativity to compute Hamiltonians and is of use by Damour, Jaranowski and Schafer [9,20]. With these Hamil- tonians all the equations of motion can be found by Hamilton’s equations along with other relevant quantities such as angular momentum. Their most recent achievement is determining a 4PN order correct Hamiltonian for a binary point particle system [9]. The 4PN order is known to have been notoriously difficult because the post-Newtonian expansions no longer converges. This occurs because when we consider the following near zone

26 Chapter 2 2.6. ALTERNATE INTEGRATION METHODS expansion (in the notation of Futamese) Z Z Z f(τ − r, . . . )dr = f(τ)dr − f(τ)dr + O(2), (2.6.8) we ignore retardation. However since the radius r scales as −1, then the product r will only be small as we take → 0 if the integrals themselves decay sufficiently fast. At the 4PN order the integrals do not due to the presence of tail terms and we find a blow up at this order. By controlling these terms, they were able to overcome this difficulty and achieve an impressive 4PN calculation. It is certainly an achievement unto itself that despite the extreme difficulties inherent in General relativity, so many diverse methods have been created to handle it. The complexity in the post-Newtonian regime leads to very long end results for the potentials, equations of motion and other important quantities and it is imperative that they be cross-checked to identify any errors. This is one major reason why these approaches have surfaced and it is remarkable that they are all entirely consistent in their end results. It is simply a matter then of picking the approach that suits one best for the computation at hand.

27 Chapter 3

Equations of Motion and the Precession of Mercury

One of the main reasons we want to determine the potentials and hence the metric is to explore how matter will move in the curved space-time. To this end, we seek to derive equations of motion for systems of particles based on the 1PN accurate potentials obtained in the previous chapter. These equations of motion are differential equations that describe how acceleration of a body is dependent on its position and its velocity. The integration of these then gives the world-line of the particle. In Section 3.1, we use a Lagrangian approach to determine the 1PN accurate equations of motion for an N point particle system. We also discuss the surface integral method, which is an elegant alternative method to achieve the same end results. Reducing to a two-body system, we find in Section 3.2 that the orbits actually rotate in time and we derive the famous anomalous perihelion precession formula which describes the angular amount that the orbit rotates. The application of this formula to the orbit of the planet Mercury was historically the first great prediction of General relativity.

3.1 Deriving equations of motion

Ever since the laws of Newtonian gravity were written, their major application was to the orbits of celestial bodies. The two-body problem consisting of two orbiting point masses was solved completely by Newton, who in 1687, verified analytically the elliptical orbits that Kepler had proposed 78 years previously. The more general N-body system (N ≥ 3) is much more difficult to analyse and exhibits chaotic behaviours. While it is relatively simple to write down the equations of motion for an N-body Newtonian system, it is much more difficult in General relativity. In fact, no exact two-body General relativistic expression exists yet. This is why we must use approximation methods such as the post- Newtonian formalism in order to find equations of motion accurate to a high order. Just as there are many formulations of classical dynamics, such as those due to Newton, Lagrange, Hamilton, Hamilton and Jacobi etc, there are many ways we can determine the equations of motion for a post-Newtonian system. We will explore two methods: the Lagrangian and surface integral approaches. A Hamiltonian formulation would follow simply from the results of the Lagrangian calculations. We begin with the Lagrangian approach which requires an expansion of the components of the metric accurate to O(4). A general expansion for the metric is

1 1 1 1 gˆ = η + hˆ − hηˆ + hˆ hˆµ − hˆhˆ + hˆ2 − hˆ hˆµν δ . (3.1.1) αβ αβ αβ 2 αβ αµ β 2 αβ 8 4 µν αβ

28 Chapter 3 3.1. DERIVING EQUATIONS OF MOTION

The components are using our expansions for the 1PN potentials, 1 3 1 gˆ = −1 + hˆ − (hˆ )2 + δabhˆ + O(5), (3.1.2) 00 2 00(4) 8 00(4) 2 ab(4) ˆ 5 gˆ0a = h0a(3) + O( ), (3.1.3) 1 1 1 1 gˆ = δ + δ hˆ + hˆ + hˆ − hˆ δcd − hˆ2 δ + O(5), (3.1.4) ab ab 2 ab 00(4) ab 2 00(4) 2 cd(4) 8 00(4) ab ˆ ˆµν ˆµν where we have written, for example, h00(4) = η0µη0νh(4) = h(4). The equations of motion are local to the scales of the system so we use the near zone 1PN potentials which are (2.5.64)-(2.5.66). Inserting these expressions into (3.1.2)-(3.1.4), we ﬁnd

22 24 1 ¨ gˆ = −1 + Uˆ + ψˆ + Xˆ − Uˆ 2 + O(5), (3.1.5) 00 c2 c4 2 43 gˆ = − Uˆ a + O(5), (3.1.6) 0a c3 22 24 1 ¨ gˆ = δ + δ Uˆ + 2Pˆ + ψˆ + Xˆ − Pˆ + Uˆ 2 δ + O(5). (3.1.7) ab ab c2 ab c4 ab 2 ab To proceed with Lagrangian dynamics, we need a Lagrangian. Since free particles move on geodesics of the space-time, we consider an action that encodes motion along geodesics. We take the action Z p µ ν S = −c −gµνv v dt, (3.1.8) and transform it to the dynamical time τ = t which yields c Z Sˆ = − p−gˆ vˆµvˆνdτ, (3.1.9) µν

µ d(ct,x) wherev ˆ = dτ = (c/, v). We then have the Lagrangian we were looking for as c Lˆ = − p−gˆ vˆµvˆν. (3.1.10) µν As a small aside, we will show that the Newtonian potentials of (2.5.14)-(2.5.16) do indeed give the Newtonian equations of motion with this Lagrangian. Substituting we ﬁnd s c c2 22 22 Lˆ = − − −1 + Uˆ − vˆ2 1 + Uˆ 2 c2 c2 r c2 22 2 = − 1 − Uˆ − vˆ2 + +O(4) 2 c2 c2 c2 1 = − + Uˆ + vˆ2 + O(2) 2 2 and substituting this into the Euler-Lagrange equations,

d ∂Lˆ ∂Lˆ = , (3.1.11) dτ ∂vˆa ∂xa yields the equations of motion aˆa = ∂aUˆ + O(2). (3.1.12) Taking the spatial derivatives of (2.5.10), we have the more familiar form

X GmB aˆ = − n . (3.1.13) A z2 AB B6=A AB

29 Chapter 3 3.1. DERIVING EQUATIONS OF MOTION

For the 1PN equations, we make use of the components of the metric we calculated earlier in (3.1.2)-(3.1.4) and we find that to the order we require, O(4), the Lagrangian is cp Lˆ = − −gˆ vˆ0vˆ0 − 2ˆg vˆ0vâ − gˆ vâvˆb 00 0a ab s 2 24 1 ¨ = − 1 − vˆ2 + 2Uˆ − ψˆ + Xˆ − Uˆ 2 − 4Uˆ vâ + Uˆvˆ2 + O(6) c2 c4 2 a c2 1 2 1 1 ¨ 1 3 = − + vˆ2 + Uˆ + vˆ4 + ψˆ + Xˆ − Uˆ 2 − 4Uˆ vâ + Uˆvˆ2 + O(6). (3.1.14) 2 2 c2 8 2 2 a 2

Inserting this into the Euler-Lagrange equations and turning the handle we obtain the equations of motion of a test particle moving on a geodesic of this space-time, which are

2 ˙ aˆa = ∂aUˆ + (ˆv2 − 4Uˆ)∂aUˆ − 4ˆvavˆb∂ Uˆ − 3ˆvaUˆ + ∂aψˆ c2 b 1 ¨ ˙ + ∂aXˆ + 4Uˆ a − 4(∂aUˆ b − ∂bUˆ a)ˆv + O(4). (3.1.15) 2 b

Herea â are the components of the acceleration which is defined as vˆ˙a =:a â. We have also made use of the Newtonian acceleration here once which says thata â = ∂aUˆ + O(2). In the Newtonian limit → 0, the second (and higher) term vanishes and we are left with the Newtonian result. The expression contained in the 2 bracket is the 1PN order correction to the equations of motion. There are many differences between this expression and the Newtonian one. Firstly, the acceleration of the body is dependent on its own velocity not just its position alone. In ˙ fact it is more difficult than this because of the presence of the ψˆ term and Uˆ terms which are proportional to the velocities of all the gravitating bodies. A more striking departure from Newtonian gravity is that the mass of the test particle itself influences its motion due to the ψˆ term. Therefore, we no longer have the interpretation of the test particle simply moving through a background field produced by our source. It moves through a complicated non-linear combination of a ‘background’ field and this fields interaction with the field of the test mass. This is another prime example of the non-linearity of General relativity. An explicit version of the equations of motion can be obtained by substituting into (3.1.15) the definitions of (2.5.55) and (2.5.57) and evaluating the derivatives. This leads to a long expression for the acceleration of any one of the particles but we will not reproduce it here. While its derivation was an early triumph in General relativity, there is little physical meaning we can directly extract from it in comparison to simpler equations. Instead, we note that these equations drastically simplify in the case of a binary system. If the coordinate system is shifted to the centre of mass which can be set as the origin of the coordinate system, then we can instead write down one acceleration vector which describes the relative acceleration between body 1 and body 2, that is aˆ := aˆ1 − aˆ2. This is given by

Gm 2 Gm 3 Gm Gm aˆ = − n+ − (1 + 3η)ˆv2 − η(n.vˆ)2 − 2(2 + η) n + 2(2 − η) (n.vˆ)vˆ . r c2 r2 2 r r2 (3.1.16) Here n := ˆr/r where unfortunately here we mean rˆ = er as in a unit vector in the 2 radial direction, m := m1 + m2, η := (m1m2)/(m1 + m2) and the expression n.v can be interpreted as the radial velocity. Splitting the components of the acceleration vector into radial and angular pieces using plane polar coordinates will then give the explicit diﬀerential equations that must be solved to fully determine the motion of the relative

30 Chapter 3 3.1. DERIVING EQUATIONS OF MOTION system and hence that of each particle (since the transformation from two-body to relative system is ‘invertible’). The most well known post-Newtonian effect which can be derived from (3.1.16) is the precession of the perihelion of the orbits. Another approach to obtain equations of motion is the surface integral method. The motivation for this is obtaining an extension to a useful Newtonian case. Consider an extended body in a Newtonian gravitational field given by the potential Φ. The force it experiences is given by the volume integral Z i ∂Φ F = ρ i dV, (3.1.17) V ∂x where the volume covers the whole object. This form is difficult to work with in practice since it requires detailed knowledge of the interior of the source due to the presence of the density term. Motivated by the Poisson equation and Gauss’ law, we can postulate a certain stress-tensor 1 δij Kij := ∂iΦ∂jΦ − ∂kΦ∂ Φ , (3.1.18) 4πG 2 k which has the sought after property that

ij i j i j i k 4πG∂jK = (∂j∂ Φ)∂ Φ + ∂ Φ∂ ∂jΦ − ∂ (∂kΦ)∂ Φ i j = ∂ Φ ∂j∂ Φ. (3.1.19)

The utility of this identity is that when we invoke the Poisson equation, we find 1 ρ∂iΦ = ∂ ∂jΦ ∂iΦ = ∂ Kij, (3.1.20) 4πG j j which we can substitute into (3.1.17). This is then ready for a healthy dose of the Diver- gence theorem to obtain I i ij F = K dSj. (3.1.21) ∂V What we have done here is to reduce the volume integral to a surface integral which is essentially summing the stresses on surface elements of the body. We have entirely effaced the problem of the complicated interior. The reader with a background in classical electromagnetism may realise that the form of our tensor Kij in (3.1.18) is very closely related to the Maxwell stress tensor except written in terms of the gravitational potentials (hence the derivatives) rather than the fields. Now that we have motivated the use of surface integrals, we will discuss its use in the context of General relativity. The goal is to determine equations of motion for a system, and in the Newtonian case we know that acceleration is proportional to the time rate of change of the momentum. We wish to do something similar here making use of the Landau-Lifshitz formulation as discussed in Section 2.1. We define Z α 1 0α 0α PA := (−g)(T + tLL)dV, (3.1.22) c VA as a momentum four-vector of body A and where VA is a sphere centred on zA, the position of the particle and does not intersect the spheres of other bodies, such as a VB. We will give this much more meaning in Section 4.4 but for now we just work with it as is. We would like to calculate the time derivative of this four-vector however there is an important subtlety here: body A is in motion and so the spherical volume also moves in space so that it is always enclosing the particle. Taking a time derivative must take this moving surface into account. In this case we take the surface to be moving at the same velocity

31 Chapter 3 3.1. DERIVING EQUATIONS OF MOTION

as that of the particle, vA. By the calculus of moving surfaces, the time rate of change of a ﬁeld F moving with veloctity v is given by d Z Z ∂F I F dV = dV + F v.dS. (3.1.23) dt V V ∂t ∂V Applying this to (3.1.22), we ﬁnd

α Z I dPA 1 ∂ 0α 0α 1 0α 0α c 0 = 0 (−g)(T + tLL)dV + 2 (−g)(T + tLL)vAdSc dx c VA ∂x c ∂VA Z I 1 ∂ 0α 0α 1 0α 0α c = − a (−g)(T + tLL)dV + 2 (−g)(T + tLL)vAdSc c VA ∂x c ∂VA I 1 cα 0α c = 2 (−g)(tLL − tLLvA/c)dSc, (3.1.24) c ∂VA

00 00 0a 0a where we have used the conservation identity ∂0[T +tLL] = −∂a[T +tLL] from (2.1.11), the divergence theorem and set T αβ = 0 on the surface. Introducing a dipole like 3-vector Z a 1 00 00 a a DA := 2 (−g)(T + tLL)(x − zA)dV (3.1.25) c VA and taking a derivative with respect to x0 along with converting all volume integrals to surface integrals, we ﬁnd the identity

a a a ˙ a PA = MAvA + QA + DA. (3.1.26)

The extra terms are I 1 0 a 1 0b 00 b a a MA := PA,QA := (−g)(tLL − tLLvA/c)(x − zA)dSb, (3.1.27) c c ∂VA where MA can be interpreted as a mass like quantity and QA does not have any physical significance. As it stands, (3.1.26) is just an identity relating surface integrals however it is written suggestively to imply some deeper physical meaning such as the presence of the first two terms, which appear to be a linear momentum definition. From (3.1.26) it is straightforward to take a further time derivative to obtain a law of motion for each body ˙ ˙ ˙ ¨ MAaA = P A − MAvA − QA − DA. (3.1.28)

The surface integral approach therefore reduces the problem of finding the equations of motion to the evaluation of surface integrals. While we will not proceed with explicit calculations here we will mention how these surface integrals can be evaluated in practice. The first step is evaluating the relevant components of the Landau-Lifshitz pseudo-tensor accurate to the required order. The surface integrals are then evaluated over the surface of a sphere and so these integrals become spherical averages in the sense of (A.2.6), and are evaluated accurate to an order required. The computations are arguably far longer than the Lagrangian approach detailed above for the 1PN order however the surface integral approach works particularly well in the work of Futamese [7], who uses it to calculate in the strong field point particle limit as we discussed earlier. At these low orders, it is simply a matter of choice which approach one chooses.

32 Chapter 3 3.2. APPLICATION TO PRECESSION OF MERCURY

3.2 Application to precession of Mercury

The orbit of Mercury about the Sun is observed to rotate and this effect is famously known as the precession of the perihelion of Mercury. The amount of rotation is close to 575 arc-seconds per century of which 531 arc-seconds per century [21] can be entirely explained by the effects of the other planets perturbing the Mercury-Sun system. In 1859, the French astrophysicist Urbain Le Verrier accurately calculated the effects due to all the other planets and found that there was an anomalous amount of 43” per century [22] that could not be explained using Newtonian theory. He proposed that there should exist another planet called ’Vulcan’ close to Mercury, which would be the cause of this shift. Unfortunately, no such planet was ever found which implied that Newtonian gravity may not be the final word on the motions of the heavens. There were of course many other attempts to explain this shift mostly by adding extra terms to the to the gravitational force, however these either did not work or incorrectly explained other known phenomena and were abandoned. The resolution came in 1916, when Einstein published a paper [23] based on his General theory of relativity that naturally and accurately explained this shift. He predicted that amount of this precession was

6πG(M + M ) δφ = Mercury , (3.2.1) c2a(1 − e2) where a is the semi-major axis length of the orbit and e is its eccentricity. Applying this to the known values, gives 42.8 arc-seconds per century which exactly accounts for the anomaly. This was the first of many great predictions of General relativity which gave great confidence to its correctness as a physical theory. In this section, we seek to derive (3.2.1) using the 1PN equations of motion or in actual fact, the 1PN accurate relative Lagrangian. The approach we take here is to use Hamilton-Jacobi theory, which is another formulation of classical dynamics much like the Lagrangian or Hamiltonian mechanics. The fundamental quantity is now a function in configuration-space S(qi, t) which determines the path of the system and is essentially the action evaluated along the true path. The evolution of S in the Newtonian time τ = t is determined by the Hamilton-Jacobi equation which is ! ∂Sˆ ∂Sˆ τ = −Hˆ qi, , , (3.2.2) ∂t ∂qi where our N-generalised coordinates in configuration-space are labelled {qi},p î = ∂S/∂qˆ i and Hˆ is the Hamiltonian which is itself related to the Lagrangian by X Hˆ (qi, pî, τ/) = pîq˙i − Lˆ(qi, qˆ˙i, t), i wherep î = ∂L/∂ˆ qˆ˙i. The dot represents differentiation with respect to τ here. For reference the Hamiltonian formalism satisfies its own equations which determine the motion and they are

∂Hˆ pˆ˙i = − , (3.2.3) ∂qi ∂Hˆ q˙i = , (3.2.4) ∂pˆi ∂Lˆ ∂Hˆ − = . (3.2.5) ∂τ ∂τ

33 Chapter 3 3.2. APPLICATION TO PRECESSION OF MERCURY

In the special case where the Hamiltonian is conserved, ∂H/∂τˆ = 0 which occurs when Hˆ does not explicitly depend on τ, then we can take another time derivative of (3.2.2) to obtain

∂2Sˆ = 0. ∂τ 2

Solving this we find Sˆ(qi, τ) = Wˆ (qi) − Eτ where E is a constant and Wˆ is an arbitrary function only of the coordinates qi. Now (3.2.2) reduces to ! ∂Sˆ E = Hˆ qi, , (3.2.6) ∂qi where the constant E appears to bear a relationship to the total energy of the system. Another special case can occur when one of the generalised coordinates qj where 1 ≤ j ≤ N is cyclic. This occurs when ∂H/∂qˆ j = 0. From Hamilton’s equations, pˆ˙j = −∂H/∂qˆ j = 0 and hencep ˆj = J where J is a constant. We also have the definition thatp ˆj = ∂S/∂qˆ j, so then Sˆ = Jqj + Wˆ (qi6=j), where Wˆ (qi6=j) is an arbitrary function of the remaining coordinates. In the simple case where the configuration space contains only two generalised coordinates, say q1 and q2, and where the Hamiltonian is conserved and q2 is cyclic, then we can write Sˆ(q1, q2, τ) = −Eτ + Jq2 + Wˆ (q1). (3.2.7) From these considerations, we can see that whenever we have a cyclic coordinate, the true action Sˆ is separable in that coordinate. To determine the function Wˆ we need the conjugate momenta p1 sincep ˆ1 = dWˆ /dq1 and hence we could then formally write Z Wˆ (q1) = pˆ1dq1. (3.2.8)

To complete this discussion, we write the Euler-Lagrange equations as ! d ∂Lˆ ∂Lˆ pˆ˙i = = , (3.2.9) dτ ∂q˙i ∂qi using our new notation here for a set of arbitrary generalised coordinates {qi}. With the preliminaries out of the way, we begin to formulate the actual problem at hand following an example given by Landau and Lifshitz [24]. We consider the case of two orbiting bodies of mass m1 and m2 respectively. Based on (3.1.14), we can write the Lagrangian for a two-body system in terms of the relative position. For this we set the centre of mass to be at the origin of the coordinate system so that our the Lagrangian is written in terms of the distance from this origin. A Lagrangian that gives the 1PN accurate relative acceleration which was (3.1.16), is

1 Gmµ 2 1 1 Gmµ Gηmµ G2m2µ Lˆ = µvˆ2 + + (1 − 3η)µvˆ4 + (3 + η) vˆ2 + (n.vˆ)2 − , 2 r c2 8 2 r 2r 2r2 (3.2.10) where we recall that m := (m1 + m2) is the total mass, µ := m1m2/m is the reduced mass, η := µ/m is a dimensionless reduced mass and n = rˆ is a unit vector in the radial direction. Our generalised coordinates here are the plane polar {r, φ} (which are our qi) and their derivatives {r,˙ φ˙} (which are ourq ˙i). The conjugate momenta arep ˆr = ∂L/∂ˆ r˙ ˙ 2 2 2 ˙2 andp ˆφ = ∂L/∂ˆ φ. Since the velocities are expanded asv ˆ =r ˙ + r φ , we notice that the Lagrangian is explicitly independent of the coordinate φ. From the Euler-Lagrange ˙ equations of (3.2.9), pˆφ = 0. This gives rise to conservation of angular momentum in the

34 Chapter 3 3.2. APPLICATION TO PRECESSION OF MERCURY system. The conjugate momenta can be combined into a momentum vector, where we can write pˆ = ∂L/∂ˆ vˆ noting that we mean we take each componentp â and then combine them as a vector. The momentum vector we find is 2 1 Gmµ Gmηµ pˆ = µvˆ + (1 − 3η)µvˆ2 + (3 + η) vˆ + 2 (n.vˆ)n + O(4). (3.2.11) c2 2 r c2r Noting that pˆ = µvˆ + O(2) then we define 1 Gmµ αˆ = (1 − 3η)µvˆ2 + (3 + η) 2 r 1 Gmµ = (1 − 3η)ˆp2/µ + (3 + η) + O(2), 2 r and Gmηµ βˆ = (n.vˆ) r Gmηµ = (n.pˆ) + O(2), r which allows us to write pˆ = µvˆ + (2α/cˆ 2)vˆ + (2β/cˆ 2)n. The squared magnitude of the momentum vector is then 22αµˆ 22βµˆ pˆ2 = µ2vˆ2 + vˆ2 + (n.vˆ) + O(4). (3.2.12) c2 c2 Rearranging this we find

pˆ2 22αµˆ pˆ2 22βˆ vˆ2 = − − (n.pˆ) + O(4). (3.2.13) µ2 c2 µ3 c2µ2 While these computations may seem trivial they are necessary since we wish to find an action which can be written as in (3.2.7) which will involve finding a Hamiltonian and writing it in terms of the conjugate momentump î, rather than the derivativesq ˙i allowing us to find Wˆ given by (3.2.8). In our situation, we need to write a Hamiltonian in terms ofp ˆ which involves removing all velocityv ˆ terms. The Hamiltonian is given by Hˆ = pˆ.vˆ − Lˆ, and can be written in terms of the momentum using (3.2.13), and we have pˆ2 Gmµ 2 Gm (3 + η) G G2m2µ Hˆ = − − (1−3η)ˆp4 −2 pˆ2 +2 (n.pˆ)2 +2 , (3.2.14) 2µ r 8c2µ3 2c2r µ 2c2r 2c2r2 where we neglect terms of order 4 and higher. We can instantly make a few observations. The first is that the Hamiltonian is explicitly time independent and is thus conserved. Since Hˆ = E, then at the 1PN order we have conservation of energy so dE/dτ = 0 and hence E is constant. Secondly, the angular coordinate φ is cyclic so we also have conservation of angular momentum as we have seen and we can then writep ˆφ = J where J is the constant angular momentum. We therefore have a conserved Hamiltonian with two generalised coordinates of which one is cyclic. We considered this exact scenario earlier and we found that the true action Sˆ from (3.2.7) becomes Z Sˆ(r, φ, τ) = −Eτ + Jφ + pˆrdr. (3.2.15)

At the Newtonian order, one ﬁnds that φ˙ = J/mr2 along withp ˆ2 = µvˆ2 = µr˙2 + µr2φ˙2. We then have that

2 2 2 2 pˆ =p ˆr + J /r . (3.2.16)

35 Chapter 3 3.2. APPLICATION TO PRECESSION OF MERCURY

Also at this order,

pˆ2 Gmµ E = − , 2µ r which implies that

Gmµ pˆ2 = 2µ E + . (3.2.17) r

Substituting (3.2.16) and (3.2.17) into (3.2.14) yields

1 Gµ J 2 (1 − 3η) Gmµ2 GmE G2m2µ E = 1 − 2 pˆ2+ −2 E + −2 (3+η)−2 (5/2+η). 2µ c2r r 2µr2 2c2µ r c2r c2r2 (3.2.18) Solving forp ˆr we obtain 2 2 Ar + Br − C pr = , (3.2.19) r(r − rg) where (1 − 3η)E2 A := 2µE + 2 , (3.2.20) c2 4GmµE B := 2Gmµ2 + 2 (2 − η), (3.2.21) c2 G2m2µ2 C := J 2 − 2 (6 − η), (3.2.22) c2

2 2 and rg := Gµ/c . For the integral

Z Ar2 + Br − C 1/2 Wˆ (r) = dr, (3.2.23) r(r − rg)

2 we make a transformation r(r − rg) = R . The radius rg is recognisable as being close to the Schwarzschild radius of a massive body with mass µ. For orbiting bodies we can assume that in the relative frame the distance of the body from the origin is much greater than rg, that is when rg r, then R ≈ r − rg/2 or r ≈ R + rg/2. This condition is valid for essentially all cases except for very close orbits near a black hole. Expanding terms in 2 powers of rg/R and noting that rg = O( ), then the integral of (3.2.23) becomes

Z (r A + B) (D − r B/2)1/2 Wˆ ≈ A + g − g dR. (3.2.24) R R2

The shift in the constant term in the R−2 term evaluates to 6G2m2µ2 D − r B/2 = J 2 − 2 , g c2 where we neglect higher order terms. We therefore ﬁnd that

Z B0 1 6G2m2m2 1/2 Wˆ (r) = A + − J 2 − 2 1 2 dr, (3.2.25) r r2 c2

2 2 2 0 2 2 6G m1m2 where A and B are constants. The term k := c2 is responsible for the non-zero precession of the perihelion.

36 Chapter 3 3.2. APPLICATION TO PRECESSION OF MERCURY

Utilising the fact that Sˆ acts as a generator of canonical transformations [25], then we can think of the constant J along with the constant Kˆ = ∂S/∂Jˆ as a pair of canonical coordinates. The trajectory of the system is then given by ∂S/∂Mˆ = C0 which becomes

∂Wˆ φ + = C , (3.2.26) ∂J 0 where C0 is a constant. After one full revolution this becomes

∂∆Wˆ ∆φ = − . (3.2.27) ∂J Putting this to one side for the moment, we return to (3.2.25) and split the term causing the precession apart from the other terms as r v Z B0 J 2 u k2 Wˆ = A + − u1 + dr r r2 t B0 J2 2 A + r − r2 r r Z B0 J 2 k2 Z dr ≈ A + − 2 dr + q r r 2 2 B0 J2 r A + r − r2 k2 ∂Wˆ 0 = Wˆ 0 − , 2J ∂J where in the second line we made use of the Binomial expansion since k2 = O(2), and q ˆ 0 R B0 J2 ˆ ˆ 0 k2 ∂∆Wˆ 0 have defined W (r) := A + r − r2 dr. Over a revolution, ∆W = ∆W − 2J ∂J . When there is no precession, the ∆φ = 2π, so then −∂∆Wˆ 0/∂J = ∆φ = 2π. Substituting these results into (3.2.27), we find " # ∂Wˆ 0 k2 ∂ 1 ∂Wˆ 0 ∆φ = − + ∂J 2 ∂J J ∂J k2 ∂ −2π = 2π + 2 ∂J J πk2 = 2π + . J 2 We find that under a full rotation there is a little extra amount that the orbit will rotate which is given by πk2/J 2. At the Newtonian order, we can make use of the Keplerian relation J 2 = Gm3η2a(1−e2) where a is the semi-major axis length and e is the eccentricity of the orbit. Any higher order changes to this formula would appear at the 2 terms but since k2 = O(2), these terms are relegated to O(4) and are neglected. Therefore the shift δφ in the perihelion is 6πGm δφ = 2 . (3.2.28) c2a(1 − e2) This formula is in agreement with that obtained by Damour and Deruelle [26] by integrating exactly the 1PN equations of motion. Applying this to the Mercury-Sun system yields a prediction of 43” per century as in (3.2.1).

37 Chapter 4

Gravitational Waves and the Hulse-Taylor Binary

After finding equations of motion, the next leap required of post-Newtonian approximations is to study the generation and propagation of gravitational waves. In Section 4.1, we discuss the mathematical representation of the potentials that encapsulates the presence of gravitational waves in the far-away wave zone. This involves converting into the so-called transverse, traceless gauge and rewriting our potentials there. From these, the perturbations to the metric in the far-away wave zone due to gravitational waves becomes clear: they are the result of the quadrupole mass moment. To illustrate these results, Section 4.2 works through some example waveforms. In Section 4.3, we briefly examine the effect of this metric perturbation on space-time by using the concept of geodesic deviation. This analysis reveals a key detection method for gravitational waves. In order to study the effects of these waves on the local orbital motion of the source system, we need to know the rate at which energy is lost due to emission. This leads us to the celebrated quadrupole formula of Einstein in Section 4.4 along with a brief discussion about the tricky concept of energy in General relativity. We then consider a non-relativistic two body system in bound elliptic orbits and calculate its rate of energy loss due to gravitational waves in Section 4.5. We then use these formulae to analyse the binary pulsar PSR1913+16 whose observed orbital motion agrees brilliantly with the equations derived previously. We conclude with Section 4.6 where we discuss the higher PN expansions of the quadrupole formula and there use as templates for data analysis at next generation gravitational wave detectors such as LIGO, VIRGO and GEO600.

4.1 Transverse-traceless potentials and polarisations

The transverse-traceless gauge

The goal of gravitational wave theory is to determine the potentials in a far-away wave zone (e.g. a detector on Earth). In this regime, the O(r−1) terms dominate over all terms of order r−2 and higher. The potentials can be written as hˆαβ = hˆαβ + hˆαβ , where the N W ﬁeld point is x ∈ W(ct, x). The near zone contribution can be expanded using (2.4.8), while much more diﬃcult analyses is required for the wave zone contribution. The general

38 Chapter 4 4.1. TRANSVERSE-TRACELESS POTENTIALS AND POLARISATIONS behaviour in the far-away wave zone can be written as 4GM G hˆ00 = 2 + 4 Cˆ(τ , Ω), (4.1.1) c2r c4r r G hˆ0a = 4 Dˆ a(τ , Ω), (4.1.2) c4r r G hâb = 4 Aâb(τ , Ω), (4.1.3) c4r r −2 a where τr = t − r/c and O(r ) terms have been omitted. We then seek to expand Dˆ and Aâb into irreducible components; i.e. transverse and longitudinal components, ˆ a ˆ a ˆ a D = DΩ + DT, (4.1.4) ˆ a ΩaDT = 0, (4.1.5) 1 1 Aâb = δabAˆ + Ω Ω − δab Bˆ + ΩaAˆb + ΩbAâ + Aâb , (4.1.6) 3 a b 3 T T TT â ΩaAT = 0, (4.1.7) âb ΩaATT = 0, (4.1.8) âb δabATT = 0, (4.1.9) where (4.1.5), (4.1.7) and (4.1.8) are longitudinal-free conditions for the transverse components and (4.1.9) is a trace free condition, hence the name ‘transverse trace-less’ tensor or TT tensor. Substituting these into the far zone potentials and imposing the harmonic ˆαβ −2 gauge conditions, ∂βh = 0, and omitting O(r ) terms, allows us to remove some re- dundant quantities. The potentials then become 4GM G 1 hˆ00 = 2 + 4 (Aˆ + 2Bˆ), (4.1.10) c2r c4r 3 G 1 hˆ0a = 4 (Aˆ + 2Bˆ)Ωa + Aâ , (4.1.11) c4r 3 T G 1 1 hâb = 4 δabAˆ + ΩaΩb − δab Bˆ + ΩaAˆb + ΩbAâ + Aâb . (4.1.12) c4r 3 3 T T TT Interestingly, four more terms are just a gauge, and can be removed. To implement this gauge, we introduce a small perturbation to the Minkowski metric

gˆαβ = ηαβ + δgˆαβ, (4.1.13) with a gauge produced by a four-vector ζˆα(cτ, x) such that the perturbation transforms according to OLD NEW OLD ˆ ˆ δgˆαβ → δgˆαβ = δgˆαβ − ∂αζβ − ∂βζα. (4.1.14) Expanding the metric to linear terms in the potentials because quadratic terms will be O(r−2), which we ignore, we ﬁnd 1 η + δgˆ =g ˆ = η + hˆ − hηˆ + O(hˆ2) (4.1.15) αβ αβ αβ αβ αβ 2 αβ and thus 1 hˆ = δgˆ − δgηˆ + O(r−2), (4.1.16) αβ αβ 2 αβ ˆ αβˆ ˆ since it can be shown that h = −δgˆ where η hαβ := h and similarly for δgˆ. Gauge transforming (4.1.16) according to (4.1.14), we ﬁnd that the potential transforms according to ˆαβ ˆαβ ˆαβ α ˆβ β ˆα ˆµ αβ hOLD → hNEW = hOLD − ∂ ζ − ∂ ζ + (∂µζ )η . (4.1.17)

39 Chapter 4 4.1. TRANSVERSE-TRACELESS POTENTIALS AND POLARISATIONS

This transform implies that the harmonic gauge conditions become ˆαβ ˆαβ ˆαβ ˆα ∂βhOLD → ∂βhNEW = ∂βhOLD − ζ (4.1.18) ˆα = −ζ . (4.1.19) ˆαβ In order to preserve the conditions ∂βh = 0 under a such a gauge transfomration, we α are free to choose ζ such that ζ = 0. This is a homogeneous wave equation which is satisfied in the far-away wave zone if we suppose that the components satisfy G ζˆ0 = 3 αˆ(τ , Ω) + O(r−2), (4.1.20) c3r r G ζâ = 3 βâ(τ , Ω) + O(r−2). (4.1.21) c3r r â ˆ a â â We can then decompose β into βΩ + βT with ΩaβT = 0, and substitute (4.1.20) and (4.1.21) along with (4.1.10), (4.1.11) and (4.1.12) into the gauge transformation (4.1.17). We find that it produces the following changes ˙ Aˆ → Aˆ + 3αˆ˙ − β,ˆ ˙ Bˆ → Bˆ + 2β,ˆ â â ˆ˙b AT → AT + βT, âb âb ATT → ATT, where a dot indicates differentiation with respect to τr. Therefore we are free to choose ˆα ˆ ˆ â âb the components of ζ such that A, B and AT vanish leaving only ATT standing after we exhaust all our coordinate freedom. Hence the potentials take the form in the far-away wave zone of 4GM hˆ00 = 2 + O(r−2), (4.1.22) c2r hˆ0a = O(r−2), (4.1.23) G hâb = 4 Aâb + O(r−2), (4.1.24) c4r TT âb âb where ΩaATT = 0 and δabATT = 0. These are the key equations for this subsection. Since we have used up all our coordinate freedom, the terms left standing must be physically important. In Section 4.3, we give a brief explanation of the physical effect these potentials have on space-time. While so far we have kept Aâb somewhat arbitrary, we will now match it to the prop- erties of the source itself. We noted its appearance in (2.5.43) at second order. At the 1PN order we have ab ab(2) Aˆ = 2Iˆ (τr) (4.1.25) and therefore the spatial components of the potential take the form 2G hâb = 4 Iâb(2)(τ ) + O(r−2), (4.1.26) c4r r where τr = τ − r/c is the retarded time and Z Iâb := ρxˆ axbd3x. (4.1.27)

ˆ00 2 2 Hereρ ˆ is the Newtonian order term as T(2) = c ρˆ. Therefore the waves are generated by the matter quadrupole moment of the source.

40 Chapter 4 4.1. TRANSVERSE-TRACELESS POTENTIALS AND POLARISATIONS

When we ﬁnd a suitable potential we can then take the TT-part to obtain the waveﬁeld. A method to do this is to introduce the TT projector which acts on a symmetric tensor to obtain its TT part, as below

ab ab cd ATT = (TT)cdA . The TT projector works by extracting the transverse part and then removing the trace and can be written as 1 (TT)ab := P aP b − P abP , (4.1.28) cd c d 2 cd a a a where Pb := δb − Ω Ωb is called the transverse projector. For example consider a decom- a a a position of an arbitrary contra-variant vector A = AΩ + AT. Acting with the projection operator we ﬁnd

a b a a b b Pb A = (δb − Ω Ωb)(AΩ + AT) a a a b a b = Ω A − Ω A + δb AT − Ω ΩbAT a = AT, b where we have used the fact that ΩbAT := 0, hence the name transverse projector. Wave polarisation If we have a symmetric, TT tensor, then we can decompose it into an angular basis,

ab ab A+ = fab(θ, φ)ATT = fab(θ, φ)A , (4.1.29) ab ab A× = gab(θ, φ)ATT = gab(θ, φ)A , (4.1.30) since f and g turn out to be already transverse, trace-free operators. The functions f and g are defined as 1 f (θ, φ) = (θ θ − φ φ ), ab 2 a b a b 1 g (θ, φ) = (θ φ + φ θ ), ab 2 a b a b where 1 θˆ := [cos θ cos φ, cos θ sin φ, − sin θ], (4.1.31) φˆ := [− sin φ, cos φ, 0], (4.1.32) and the projection operator satisfies the identity Pab = θaθb +φaφb. It can be easily shown ab ab ab using the definitions that fabf = 1/2, gabg = 1/2 and fabg = 0. Therefore we have found a basis of tensors to expand the potentials with, which are

e+ := θˆ ⊗ θˆ − φˆ ⊗ φ,ˆ (4.1.33)

e× := θˆ ⊗ φˆ + φˆ ⊗ θ,ˆ (4.1.34)

ab ab ab ab where we have, in terms of components, e+ := 2f and e× := 2g . The wave field tensor ab in the TT gauge hTT can then be written as a linear combination of these basis tensors as −2 hTT = h+e+ + h×e× + O(r ). (4.1.35) Now (4.1.29) and (4.1.30) have a geometrical meaning as projections of the full wave field tensor onto its polarisation basis. It is common to see a gravitational wave-field decomposed into ‘plus’ and ‘cross’ terms, h+ and h× since they can be related to how a detector measures a gravitational wave.

1Note that the hats used here are to indicate unit vectors not functions of the dynamical time τ as before.

41 Chapter 4 4.2. PARTICULAR GRAVITATIONAL WAVE FIELDS

4.2 Particular gravitational wave ﬁelds

Gravitational waves in vacuum

Firstly, we consider the waves emitted from a far-away, non-stationary source. In this region, the ﬂuctuations are assumed to be weak but are not time independent since the source may have undergone some strong motions to emit these waves and we only receive it much later as a result of retarded times. We consider only the ﬁrst order approximation to (2.1.5) in vacuum (T αβ = 0) where

αβ h = 0. (4.2.1) We can postulate a solution to this wave equation in the form of a plane wave, that is

ρ αβ αβ ikρx hTT = A e (4.2.2) where Aαβ are a set of 10 independent (complex) constants (since Aαβ is symmetric) and kµ is the wave vector. Upon finding a suitable solution we should take the real part of this since this is what has physical significance. If we consider the divergence of this solution we find ρ αβ αβ ikρx αβ ∂µhTT = ikµA e = ikµA , (4.2.3) µν and from the definition of the d’Alembertian := η ∂µ∂ν we find that for (4.2.2) to be a non trivial solution to (4.2.1), we require that

µ kµk = 0. (4.2.4) This implies that the four-vector k is a null vector and hence the speed of propagation of ρ αβ αβ ikρx this wave is c. The gauge condition ∂βh is now ∂βA e = 0 and hence αβ A kβ = 0. (4.2.5) This is a constraint on the coefficients of the tensor Aαβ. We still have two more conditions αβ we can impose which are the fact that hTT is by definition transverse and trace-free. These two conditions imply ab αβ ΩbA = 0, δαβA = 0. (4.2.6) We now make a specialisation for a wave travelling in the Cartesian z-direction and we choose k = (ω/c, 0, 0, k). This implies that θ = φ = 0 and hence Ω = (0, 0, 1). From the transverse condition of (4.2.6), we find that Aα3 = 0 and using this and (4.2.5) we have Aα0 = 0. From the vanishing of the trace and the previous property we have A11 = −A22. The only other remaining terms are A12 = A21 where we use the fact that hαβ is a symmetric tensor (this comes from the symmetry of the metric tensor and (2.1.3) which defines hαβ). Therefore the potential hαβ becomes, in matrix form, 0 0 0 0 xx xy αβ 0 A A 0 i(kz−ωt) h =   e . (4.2.7) 0 Axy −Axx 0 0 0 0 0 We have replaced the indices (1, 2, 3) with (x, y, z). The inner 2 × 2 matrix of A terms indicates the presence of two polarisation states for the wave which we will see soon are the observables of the wave. Now that we have found the wave-field (subject to taking the real part), we can deter- √ mine the metric. From (2.1.3), we can determine −ggαβ using (4.2.7). Next we can use √ αβ √ 4 2 the fact that det( −gg ) = ( −g) / det(gαβ) = g /g = g and we find that g = −1 + O(r−2). (4.2.8)

42 Chapter 4 4.2. PARTICULAR GRAVITATIONAL WAVE FIELDS

αβ We can then determine g and invert to ﬁnd the metric tensor gαβ which is, in matrix form −1 0 0 0  0 1 + pAxx pAxy 0 gαβ =   , (4.2.9)  0 pAxy 1 − pAxx 0 0 0 0 1

i(kz−ωt) 2 α β where we have written p(z, t) = e . The line element deﬁned as ds = gαβdx dx can now be calculated using this metric and is

ds2 = −c2dt2 + (dx2 + dy2 + dz2) + pAxx(dx2 − dy2) + 2pAxydxdy. (4.2.10)

2 α β We recognise the first two terms as the Minkowski line-element dsM := ηαβdx dx of flat space. This verifies that our space can be thought of as a background Minkowski spacetime imbued with weak deformations due to the presence of matter and gravitational waves. The final two terms are the perturbations due to the presence of propagating gravitational waves. Noting that p is actually a function of time, we see that these additions make the line-element time dependent. We can write this in a more enlightening form by decomposing the wave part into its two polarisation states, h+ and h×. For the choice of propagation direction, we find that θ = (1, 0, 0) and φ = (0, 1, 0) and hence

xx i(kz−ωt) xy i(kz−ωt) h+ = A e , h× = A e . (4.2.11)

Alternatively we can ﬁnd the polarisation states of the wave itself by ﬁnding the basis e+ and e× as in (4.1.35). In this case it is relatively simple since we can write

0 0 0 0 0 0 0 0 αβ 0 1 0 0 xx i(kz−ωt) 0 0 1 0 xy i(kz−ωt) h =   A e +   A e . (4.2.12) 0 0 −1 0 0 1 0 0 0 0 0 0 0 0 0 0

From (4.1.35) we have by inspection that the independent polarisations for this wave are xx i(kz−ωt) xy i(kz−ωt) h+ = A e and h× = A e . These are reminiscent of the two independent polarisations of electromagnetic waves which are the linear horizontal and vertical states. However, the interpretation is diﬀerent for gravitational waves and we discuss this in Section 4.3. The gravitational wave pieces of the line-element in (4.2.10) now become

2 2 2 dsgw := h+(dx − dy ) + 2h×dxdy (4.2.13) and we can see that each piece corresponds to the two polarisation states of the wave.

Gravitational waves from an elliptical binary

Our next example will be to examine the wave form for a binary point mass system in an elliptic orbit. The goal here will be to calculate the quadrupole moment and to examine the polarisation states h+ and h×. We may use the solutions to the Keplerian two-body problem since (4.1.26) is accurate to lowest order. These solutions show that the bodies orbit in ellipses about their centre of mass with eccentricity e and the radial and angular components are given by r p Gm (1 + e cos θ)2 r = , θ˙ = := k(1 + e cos θ)2, (4.2.14) 1 + e cos θ a3 (1 − e2)3/2

q Gm 2 −3/2 2 where we deﬁne k = a3 (1 − e ) and p = a(1 − e ) and a is the semi major axis length of the ellipse. We also note that {r, θ} are functions of time however this relation is

43 Chapter 4 4.2. PARTICULAR GRAVITATIONAL WAVE FIELDS complicated and we will not need them for the ensuing computations. In calculating the quadrupole moments, we will require the following derivatives

d 1 e sin θθ˙ = = ke sin θ, (4.2.15) dt 1 + e cos θ (1 + e cos θ)2 d 1 2e sin θθ˙ 2ke sin θ = = , (4.2.16) dt (1 + e cos θ)2 (1 + e cos θ)3 1 + e cos θ where we have made use of the time derivative of the angular coordinate in θ from (4.2.14). In the centre of mass frame, we only need to consider relative orbits because our two body system has been replaced by a single body with the reduced mass µ = m1m2/(m1 + m2) orbiting about the centre of mass which we are free to set at the origin of our coordinate system. The coordinates of this body are now

x1(t) := x = r(t) cos θ(t), x2(t) := y = r(t) sin θ(t), x3(t) := z = 0 (4.2.17) and so the density becomes ρ = µδ(x − x0) = µδ(x1 − x)δ(x2 − y)δ(x3) which makes the quadrupole moment a simple calculation, that is

Iab = µxaxb. (4.2.18)

Straight away we can make use of the fact that the motion occurs in a plane so that z = 0 and hence Ia3 = 0 for all a = 1, 2, 3. The remaining moments are

cos2 θ I11 = µr2 cos2 θ = µp2 , (1 + e cos θ)2 cos θ sin θ I12 = I21 = µr2 cos θ sin θ = µp2 , (1 + e cos θ)2 sin2 θ I22 = µr2 sin2 θ = µp2 . (1 + e cos θ)2

Taking two time derivatives which involves using θ˙ from (4.2.14) extensively along with trigonometric identities, we eventually arrive at

I¨11 = −2µp2k2 cos 2θ + e cos3 θ := −2µp2k2Q11(θ), (4.2.19) I¨12 = I¨21 = −2µp2k2 sin 2θ + e sin θ(1 + cos2 θ) := −2µp2k2Q12(θ), (4.2.20) I¨22 = 2µp2k2 cos 2θ + e cos θ(1 + cos2 θ) + e2 := 2µp2k2Q22(θ). (4.2.21)

The components of the spatial potentials in the far-away wave zone are now

−Q11 −Q12 0 4G2mµ hab = −Q12 Q22 0 . (4.2.22) c4Ra(1 − e2)   0 0 0 where R is the distance from the centre of mass of the source to the observer point (in (4.1.26) this was r) and m := m1 + m2. These potentials considerably simplify if we consider a circular orbit where e = 0. In this case, the semi major axis length reduces to the radius of the circle a = r and θ(t) = ω(t − R/c) where ω2 = Gm/r3 is the orbital angular frequency and is also Kepler’s third law. We must also use the retarded time u = t − R/c. The potential now reduces to

cos(2ωu) sin(2ωu) 0 4G2 mµ hab(u) = − sin(2ωu) − cos(2ωu) 0 . (4.2.23) c4R r   0 0 0

44 Chapter 4 4.2. PARTICULAR GRAVITATIONAL WAVE FIELDS

From this we can see that the gravitational waves are emitted at a frequency twice that of the orbital frequency. Now that we have the wave form we can decompose it into its two polarisation states using (4.1.29)-(4.1.32) (we replace θ in those definitions by ϑ to avoid confusion with the angle of the orbits θ = θ(t)). The results of these calculations are 2G2 mµ h = − (1 + cos2 ϑ)[cos(2φ) cos(2ωu) + sin(2φ) sin(2ωu)] + c4R r 2G2 mµ = − (1 + cos2 ϑ) cos[2(ωu − φ)], (4.2.24) c4R r 4G2 mµ h = − cos ϑ[− sin(2φ) cos(2ωu) + cos(2φ) sin(2ωu)] × c4R r 4G2 mµ = − cos ϑ sin[2(ωu − φ)] (4.2.25) c4R r and we can see that in general, gravitational radiation is not isotropic, the amplitudes change as we change our inclination ϑ. There are two special cases that we can consider, an observer perpendicular to the plane of the orbit and an observer in the plane of the orbit. In the former case, ϑ = φ = 0 so 4G2 mµ h = − cos(2ωu), (4.2.26) + c4R r 4G2 mµ h = − sin(2ωu). (4.2.27) × c4R r In this instance we receive both polarisation states with equal amplitude except that they act out of phase with each other (one is a cosine, the other a sine). This can be thought of as circularly polarised gravitational radiation. For the latter case of detection in the plane of the orbit, ϑ = π/2, and hence 2G2 mµ h = − cos(2ωu), (4.2.28) + c4R r h× = 0. (4.2.29) This situation is much more striking; the cross polarisation vanishes completely. This arises because the emitted waves must be transverse to the motions of the source so the system would not be able to emit waves with a cross polarisation in this case. The observer in the plane simply sees the bodies moving back and forth so they only receive one polarisation. We have also set φ = 0 since this simply amounts to rotating the plane of the orbit and orienting oneself along the x1 axis as in Figure 4.1. The important angle here is the inclination ϑ. From the (4.2.24) and (4.2.25), we can also deduce a simple fact of these waves in this elementary case: you cannot escape their effects. No matter what angles with which you are oriented with respect to the binary system, the h+ polarisation will never vanish. Therefore the emission of gravitational waves is omnidirectional. We can also rewrite these amplitudes in terms of the frequency of the gravitational 2 3 waves fgw = ω/π, which is twice the orbital frequency using the Keplerian relation ω r = Gm. We find 2GM h = − c (GM πf )2/3 (1 + cos2 ϑ) cos(2ωu), (4.2.30) + c4R c gw 4GM h = − c (GM πf )2/3 cos ϑ sin(2ωu), (4.2.31) × c4R c gw 2/5 3/5 where Mc := m µ is known as the chirp mass. In this form we can see that the only way to maximise the amplitude of the wave is to increase the masses of the system which increases Mc or the frequency of the gravitational waves which is in turn related to the radius of the orbit.

45 Chapter 4 4.3. EFFECT OF GRAVITATIONAL WAVES ON SPACE-TIME

Figure 4.1: Geometry of a binary system orbiting in the x1 − x2 plane. The observer (ﬁeld point) is at a distance R from the centre of mass of the system and is much greater than the distance between the bodies r. The angles {ϑ, φ} are the angles appearing in the polarisation states and are dependent on the position of the ﬁeld point relative to the system. This diagram is largely the same for an eccentric orbit

4.3 Eﬀect of gravitational waves on space-time

Here we seek to study the effects of the potentials derived in (4.1.22)-(4.1.24) on the proper distances of space-time. To do this we require the equation of geodesic deviation, which loosely speaking, describes the variation of the components of some vector ξα which connects two infinitesimally close geodesics due to curvature of the space. In a Euclidean space, our geodesics are straight lines which in this case will run parallel forever when extended, so there is zero geodesic deviation. The equation of geodesic deviation at a point P on one of the geodesics is α α µ ν β ∇U ∇U ξ = RµνβU U ξ , (4.3.1) where ξα is a connecting vector which begins at some point P on one geodesic and reaches over to the other, U is a tangent vector to the geodesic, ∇U is the covariant derivative α along the direction of U and Rµνβ is the Riemann tensor we met in Section 2.1. In general, (4.3.1) can be very complicated since we have two covariant derivative operations and a Riemann tensor showing up. Suppose we have two freely falling particles with the connecting vector ξα between them. The condition of freely falling implies they travel along geodesics of the space- time. We choose a local inertial coordinate system such that at P, the origin of the vector ξ, the space is locally Minkowski space. The vector U in this context is now the four-momentum of the particles. Neglecting quadratic metric terms, the coordinate distances become proper distances and since the particle separation is assumed to be small, the components of ξα represent proper distances. The variation in the components of ξα is governed (4.3.1). The first derivative gives differentiation with respect to the proper time, which we label as t. The second covariant derivative gives rise to Chirstoffel symbols but since the space is locally Minkowskian, then they all vanish at P. Hence α d2ξα 0 a ∇U ∇U ξ = dt2 . If we further assume the particles are moving slowly so that |U | |U |, α µ ν β 2 α β 2 α β then RµνβU U ξ ≈ c R00βξ = −c R0β0ξ and the proper time reduces to time t. Therefore the variation in the proper distance between the particles is d2ξa = −c2Ra ξb. (4.3.2) dt2 0b0

46 Chapter 4 4.3. EFFECT OF GRAVITATIONAL WAVES ON SPACE-TIME

Expanding the Riemann tensor into its corrdinate representation in terms of Christoffel 1 symbols and using gαβ = ηαβ − hαβ + 2 hhηαβ, the only surviving term which is of order r−1 is −1 G Aäb . 2c2 c4r TT Therefore (4.3.2) reduces to d2ξa G = Aäb ξb, (4.3.3) dt2 2c4r TT where we can see that it is the TT part of the potentials that alter proper distances in the far-away wave zone, which is due to the change in distance between infinitesimally close geodesics. We can also examine the effect of the plus and cross polarisations have on the proper distance between points. We orient our axis so that the wave is propagating in the z- direction and take a slice in time and a planar slice in the z-axis. This corresponds to setting dt = 0 and dz = 0. Our line element now reads as

2 2 2 2 2 ds = dx + dy + h+(dx − dy ) + 2h×dxdy. (4.3.4)

Suppose we have two freely-falling test masses in the x − y plane. The distance between the masses is given by

Z Z 1 α β 1/2 0 dx dx L = ds = gαβ dλ, (4.3.5) 0 dλ dλ where we have parametrised the curve connecting the points with parameter λ. For our metric of (4.3.4), we have

Z 1 0 p 2 2 2 2 L = x˙ +y ˙ + h+(x ˙ − y˙ ) + 2h×x˙y˙ dλ, (4.3.6) 0 with a dot indicating diﬀerentiation with respect to λ. If the particles are oriented along a line, then in ﬂat space we can parametrise the straight line connecting the two particles with x√(λ) = aλ and y(λ) = bλ. The unperturbed distance between these particles is L = a2 + b2 andx ˙ = a,y ˙ = b. The distance between them in the presence of the propagating wave is now

Z 1 0 p 2 2 2 2 L = a + b + h+(a − b ) + 2abh× dλ 0 r Z 1 a2 − b2 ab = L 1 + h+ 2 + 2 2 h× dλ (4.3.7) 0 L L a2 − b2 ab ≈ L + h + h , (4.3.8) 2L + L × where we have used the binomial approximation since |h+| 1 and |h×| 1. This can be written in terms of the relative change in the displacement ∆L/L where ∆L := L0 − L and we ﬁnd ∆L a2 − b2 ab ≈ h + h . (4.3.9) L 2L2 + L2 × Recalling that the polarisations are sinusoidally time varying, then this proper distance is in fact time dependent and causes it to both contract and expand as the wave passes. We emphasise that the particles are still at rest; it is space itself that is changing as the wave passes. In the case of the masses aligned along the x-axis, then b = 0, a = L and ∆L/L = h+/2; the same is true for alignment along the y-axis albeit with a minus sign.

47 Chapter 4 4.4. QUADRUPOLE FORMULA

For alignment at either +45◦ to the positive x-axis or at −45◦, the relative displacement is ∆L/L = ±h×/2. These calculations show that the effects of the gravitational wave can be directly related to its polarisation. The effects of these polarisations are striking. The time dependence of the polarisations, which are proportional to cos(kz − ωt), implies that L0 becomes larger and smaller than the unperturbed length L periodically in time with a frequency 2πω. This expansion and contraction is the characteristic defining of a gravitational wave as it interacts with matter. The polarisation h+ induces changes only along the vertical and horizontal direc- tions but the effects of compression and expansion occur out of phase due to the minus sign, that is when the horizontal is compressed the vertical expands. The same is true for the h× polarisation except its effects are rotated π/4 relative to those of the h+. This is the origin of the ‘plus’ and ‘cross’ names for the polarisation states. These independent modes are demonstrated on a ring of test particles in Figure 4.2. Each state periodically deforms the circle into an ellipse. A linear combination of polarisations would of course induce a linear combination of these deformations. More exotic arrangements can be examined by using (4.3.6) and parametrising the paths between the particles. If we arrange many particles along a circle of radius r we have the parametrisations x(λ) = r cos(λ) and y(λ) = r sin(λ). Inserting this into (4.3.6) and expanding the square root to quadratic order (the integral over the linear terms 2 2 vanishes) we find the relative change in the radius is ∆r/r ∝ (h+ + h×). For detection purposes then, it is far better to arrange the masses co-linearly than in more complicated arrangements such as along a circle. Indeed since ∆L ∝ L we should also displace our masses across great distances to amplify the effects.

Figure 4.2: The eﬀect of a passing gravitational wave into the page on test particles at rest lying in a circle. The two modes of oscillation shown are the eﬀects of a purely plus polarised wave and a purely crossed polarised wave. The shape is deformed in such a way that it is compressed in one direction and expanded in another orthogonal to the compression direction.

4.4 Quadrupole formula

Now that we know the wave-ﬁeld we expect in the far-away wave zone, we can ask about the rate at which energy is dissipated through gravitational radiation. The results will lead us to the well-known quadrupole formula for gravitational radiation emission and from this we will be able to predict the eﬀect of such emissions on the orbital motion of the system itself. Finally we apply these results to the case of the Hulse-Taylor binary which yields excellent agreement with observations.

48 Chapter 4 4.4. QUADRUPOLE FORMULA

We mentioned long ago that (2.1.11) had a connection to the conservation of mass- energy in space-time. It is now time to explore this further. The conservation equations we had there where h αβ αβ i ∂β (−g)(T + tLL) = 0. (4.4.1) In Special Relativity (SR), the metric is the flat-space Minkowksi metric which implies that αβ (4.4.1) reduces down to the SR conservation equivalent of ∂βT = 0. In that context, it then makes perfect sense to define a momentum four-vector for the matter in some volume V as Z α 1 α0 3 PSR[V ] := T d x, (4.4.2) c V 0 along with a definition of energy as E[V ] := cPSR[V ]. These definitions can be motivated to be consistent with what we expect in SR if we define T 00 = c2ρ. Then Z E[V ] = c2 ρ d3x = Mc2, V

R 3 with M := V ρ d x, which is of course mass-energy equivalence. We seek a generalisation of (4.4.2) that is valid in General relativity (GR). In GR, the conservation of the energy- momentum tensor is now a covariant derivative since the metric is in general different from flat-Minkowski space. This conservation alone is not useful because it only takes into account matter contributions and neglects energy that is stored in the gravitational field itself. The combination we are looking for that contains the field is (4.4.1). We define in analogy to the SR case, the four-momentum in a volume V with boundary ∂V and with T αβ = 0 on ∂V , as Z α 1 α0 0a 3 P [V ] := (−g)(T + tLL)d x. (4.4.3) c V This is similar to our definition in (3.1.22) except we here we consider a time-independent volume (equivalent to choosing a hyper-surface where x0 =constant) so we will not need to make use of results from the calculus of moving surfaces. An ‘energy’ like component of this four-vector is given by the zero index term and we define E[V ] := cP 0[V ], so then Z 00 00 3 E[V ] = (−g)(T + tLL)d x. (4.4.4) V

00 In the limit of no gravitational field, tLL = 0 and this reduces to the SR case. As it stands, (4.4.4) has no physical meaning because a notion of local energy densities in GR does not exist. In general, this is because there is no way of splitting the metric gαβ into a “background” and a “dynamical” part, for which we would expect changes in energy to be found in the dynamical part, without introducing coordinate specific quantities which is entirely counter to the covariant formulation of GR (as is discussed by Wald [27]). More specific to the formulation in Section 2.1, if we have all components of T αβ vanishing at some point, then it will vanish in all frames because it is a valid tensor. However the same αβ cannot be said for tLL since it is not a tensor so in some frames it vanishes and in others αβ it does not. Even further we can make tLL vanish in flat-space Cartesian coordinates (i.e. using gαβ = ηαβ), but it will not vanish in, say, flat polar coordinates. It therefore does not make sense to talk of local energy densities here. We will see soon however that it in fact does make physical sense to define a total mass-energy if we consider all space. Consider taking a derivative of (4.4.3) with respect to the time-like coordinate x0. Since the integration is over the spatial coordinates, the derivative may come inside the

49 Chapter 4 4.4. QUADRUPOLE FORMULA integral (we assume that we can indeed do this) and we can make use of (4.4.1) and the Divergence theorem to obtain α I dP αc αc [V ] = − (−g)(T + tLL)dSc. (4.4.5) dt ∂V

However by assumption, T αβ = 0 on this boundary, so this reduces to α I dP αc [V ] = − (−g)tLLdSc. (4.4.6) dt ∂V Looking at the zero component of this we have I dE 0c [V ] = −c (−g)tLLdSc. (4.4.7) dt ∂V Our conserved quantity has led us to see that the rate of change of P 0[V ] is directly related 0c to flux of the gravitational field represented by tLL. Under the assumption of asymptotic 0c flatness which is that (−g)tLL (and its derivatives) will vanish in the limit as r → ∞ at least as fast as 1/r2, 2 then we can extend the volume V to ‘all-space’ which leads us to the total quantities Z 00 3 E = (−g)tLLd x, (4.4.8) All space I dE 0c = −c (−g)tLLdSc. (4.4.9) dt All space The advantages of these quantities is that under a special gauge transformation, they are both invariant [27] which means that they have qualities of the concepts we require them to be. For now, we define (4.4.8) as the total mass-energy in the space-time, and later we will motivate why this is the case in terms of other definitions. Since (4.4.9) represents a flux of gravitational energy, then the rate at which energy is emitted by gravitational waves is the negative of this, so we write I ˙ 0c Egw = c (−g)tLLdSc. (4.4.10) All space With the potentials written in the far-away wave zone as 4GM h00 = , h0a = 0, hab = hab , (4.4.11) c2r TT the specific Landau-Lifshitz pseudo-tensor becomes

c2 (−g)t0c = h˙ TTh˙ ab Ωc. (4.4.12) LL 32πG ab TT

Substituting this into (4.4.10) and evaluating in spherical coordinates so that dSc = 2 r ΩcdΩ, we have 3 I c 2 TT ab E˙ gw = lim r h˙ h˙ dΩ. (4.4.13) 32πG r→∞ ab TT

2 0c −2 2 Roughly speaking, we require (−g)tLL = O(r ) as r → ∞ because dSc = r ΩcdΩ so for the integral 0c 0c to converge, we require that the combination (−g)tLLdSc converges uniformly. Since tLL is quadratic in the potentials, then this requires that hαβ = O(r−1) in this limit. This is ﬁne since we saw that in the far-ﬁeld this is the leading order scaling of the potentials in (4.1.1)-(4.1.3).

50 Chapter 4 4.4. QUADRUPOLE FORMULA

00 00 In this calculation we have made use of the fact that h is time-independent so ∂0h = 0 and that we can swap between spatial derivatives and retarded time derivatives with ∂b = ∂b(t − r/c)∂u = −Ωb∂u/c. We can write this in terms of the wave polarisations by using (4.1.35) and it becomes 3 I c 2 2 2 E˙ gw = lim r h˙ + h˙ dΩ. (4.4.14) 16πG r→∞ + × The validity of our definition in (4.4.8) as the total mass-energy is consistent with the result derived using the more rigorous Bondi-Sachs mass loss equation [28], where we find E˙ gw = E˙ gw(Bondi-Sachs), where E˙ gw is given in (4.4.14) and E˙ gw(Bondi-Sachs) is the Bondi-Sachs mass loss equation in the case of gravitational waves. [28]. To lowest order in the potentials, we saw that in the wave zone, 2G hab = Iäb, (4.4.15) c4r where this term appears in (2.5.43) and where Z Iab = ρxaxbd3x, (4.4.16) is the quadrupole mass moment. Taking the TT part of hab in (4.4.15) and contracting over itself leads to (A.4.1) which is derived in appendix 3,

4G2 1 h˙ TTh˙ ab = δ δ − δ δ − δ Ω Ω − δ Ω Ω ab TT c8r2 ac bd 2 ab cd ac b d bd a c 1 1 1 + δ Ω Ω + δ Ω Ω + Ω Ω Ω Ω Iab(3)Icd(3). (4.4.17) 2 ab c d 2 cd a b 2 a b c b We can then insert this into (4.4.13) to arrive at the quadrupole formula for the rate of energy loss due to gravitational wave emission G 1 E˙ = Iab(3)I − I(3)2 , (4.4.18) gw 5c5 ab(3) 3

ab where I := δabI . The details are contained in Theorem A.2 of Appendix A.4. We could further write (4.4.18) in terms of STF tensors, such as I(3) however its given form will prove more useful for calculations we shall do in the next section. The first term to contribute to gravitational radiation is therefore the quadrupole moment. The monopole moment vanishes since this is related to the total mass of the system, which is held constant. Similarly the dipole moment vanishes in the centre of mass frame as is seen in Section 2.5 which is related to conservation of linear momentum. At higher orders, we could rightly expect the presence of current multipoles in a quadrupole radiation formula to appear. Indeed this is the case as we discuss in Section 4.6. The far-field potentials of (4.4.15) can be derived by an alternative means as is discussed by Wald [27] by considering the retarded integral solution and Fourier transforming in time while also making a far- field approximation. One can then use this result to derive quadrupole formula exactly the same as in (4.4.18). If we also formulated this in terms of the dynamical time τ, we would find 2c2 ˙ ˙ (−g)tˆ0c = hˆTThâb Ωc, (4.4.19) LL 32πG ab TT âb 4 2G äb and using h(4) = c4r I , the quadrupole formula would become

˙ G 1 Eˆ = 10 Iˆab(3)Iˆ − Iˆ(3)2 . (4.4.20) gw 5c5 ab(3) 3

51 Chapter 4 4.5. APPLICATION TO HULSE-TAYLOR BINARY

4.5 Application to Hulse-Taylor binary

The quadrupole formula ‘derived’ in the previous section gives to lowest order, the rate at which energy is emitted via gravitational waves from an orbiting system. We would very much like to test this formula and hence the prediction of gravitational waves in general and therefore require a suitable system. A very useful system to consider is an inspiraling compact binary. These systems are compact in the sense that tidal forces due to the finite size of the bodies are negligible in comparison to the effects of radiation damping on the orbits. Indeed a quick calculation [8] reveals that finite size effects should only appear in the equations of motion at 5PN order with an error equivalent to less than one part in 16,000 rotations. Typical compact objects include neutron stars, pulsars and black holes. Another important question that arises is that the objects we consider have very strong internal gravitational fields. Since general relativity is a non-linear theory we could expect that there should be some contribution due to these internal fields to the external fields and if there is, then the post-Newtonian approximation of weak fields should not be valid. The resolution of these complications is rather simple: in GR, the internal dynamics of the gravitating objects are irrelevant for the external gravitational fields as long as they are well separated to ignore tidal interactions. This is the strong equivalence principle (SEP) in action (some alternative theories violate SEP, such as scalar-tensor gravity). The only two relevant parameters for the inter-body fields are the total mass m and any spin due to angular momentum. This has been termed the ”effacement” principle. Despite the post-Newtonian approximation begin valid for slow-moving, weak gravity systems it has also been seen to be ”unreasonably effective” [29] at describing the fast-motions and strong fields for inspiraling neutron stars and black holes. Perhaps the greatest verification of this principle of neglecting the internal dynamics is the remarkable agreement between observations and the theoretical predictions based off the quadrupole formula. The hallmark example is the PSR 1913 + 16 pulsar discovered by Hulse and Taylor in 1974 [30]. It was discovered by radio pulse emissions from the active radio pulsar and there was very likely an inert neutron star companion in orbit with a period close to 8 hours. From subsequent observations, the period has seen to decrease at a rate of 76.5 microseconds per year with decreases in the separation as well. This observed period decrease is in remarkable agreement with the GR prediction and is thus seen as an indirect observation of the existence of gravitational waves. By consideration of the ‘effacement’ principle we approximate the neutron stars as point particles of mass m1 and m2 and neglect individual angular momentum. Therefore the density becomes ρ = m1δ(x − z1) + m2δ(x − z2) and thus the quadrupole moment reduces to Z ab a b 3 a b a b I = ρx x d x = m1z1 z1 + m2z2 z2. (4.5.1) If we convert to the Newtonian centre of mass frame (of course in special relativity, center of mass is relative) using z := z1 − z2 as the separation vector, then to lowest order, z1 = (m2/m)z and z2 = (m1/m)z where m := m1 + m2. Then m m2 m m2 Iab = 1 2 zazb + 2 1 zazb m2 m2 m m = 1 2 mzazb m2 = ηmzazb, (4.5.2)

m1m2 where we deﬁne the dimensionless reduced mass η := m2 . Inserting this quadrupole mass moment into the quadrupole formula (A.4.3), yields (details given in appendix 3) 8 G (Gm)2(mη)2 E˙ = (12v2 − 11z ˙2), (4.5.3) gw 15 c5 z4

52 Chapter 4 4.5. APPLICATION TO HULSE-TAYLOR BINARY

a where n = z/z is a unit vector, andz ˙ = n va is the radial velocity. Immediately we can specialize to a quasi-circular orbit that is Newtonian so thatz ˙ = 0 and v2 = Gm/z so that

32 η2c5 E˙ = (v/c)10. gw 5 G We can see that as the energy radiated increases as the velocity of the objects becomes larger. Therefore the emission of gravitational waves is stongest in the inpsiral phase of 2 the binary. The Newtonian orbital energy is E = −Gm1m2/(2z) = −Gm η/(2z) and if we suppose that energy is conserved so that the rate of decrease of the orbital energy is exactly matched by the rate at which energy is transported due to gravitational waves, then E˙ = −E˙ gw. This leads to the prediction that the orbital separation decreases as gravitational radiation is emitted, where the rate of decrease is given by 64η G3m3 z˙ = − . (4.5.4) 5 c5z3 Equivalently the orbital velocity v increases along with the angular velocity ω ∝ z−3/2. We can in fact easily solve (4.5.4) to determine the separation as a function of time. If we integrate over an intial separation R0 to an arbitrary one R while considering the start of the measurement as t = 0, then Z R 3 3 Z t 3 64η G m 0 z dz = − 5 dt (4.5.5) R0 5 c 0 and the integration yields

256η G3m3 1/4 t 1/4 R(t) = R(0) 1 − 5 4 = R(0) 1 − , (4.5.6) 5 c R (0) tc where 5c5R4(0) t = , (4.5.7) c 256ηG3m3 is an in spiral time defined such that R(tc) = 0. This time is purely representative of the idealized case of point masses. Real inspiral times are likely to be less than this since finite bodies will begin to exert non-negligible tidal forces on each other when they are sufficiently close and then the quadrupole formula would be no longer valid. Nonetheless the prediction of a time to coalesce is certainly a post-Newtonian effect. We plot in Figure 4.3(a) the solution given by (4.5.6) and in Figure 4.3(b) we plot a parametric solution. Both plots clearly indicate the presence of gradual orbital decay before a sudden inspiral phase. Newtonian theory would predict that stable orbits are infinitely stable when isolated while we see here that this is not the case any more. However we expect for ‘Newtonian’ systems that this time should be extremely large and indeed for the Sun-Earth system, this time is on the order of 1023 years or approximately 1013 times the age of the universe. Clearly the emission of gravitational waves in this system is completely negligible for everyday applications, such as satellite positioning. The solution to the Keplerian two-body problem shows that the orbits are planar ellipses with an eccentricity e and obey a(1 − e2) r = , (4.5.8) 1 + e cos θ where a is the length of the semi-major axis. Further the rate of change of the angular coordinate is given by r Gm (1 + e cos θ)2 θ˙ = , (4.5.9) a3 (1 − e2)3/2

53 Chapter 4 4.5. APPLICATION TO HULSE-TAYLOR BINARY

(a) Radial plot (b) Parametric plot

Figure 4.3: Plots of the orbital separation in the relative frame for a quasi-circular orbit. The orbits slowly decay before a inspiral phase at times close to tc. which is due to the conservation of angular momentum of the system. We also deﬁne m := m1 + m2 as the sum of the individual masses. Taking the derivative of (4.5.8) with respect to time and using (4.5.9) yields

Gme2 sin2 θ r˙2 = . (4.5.10) a(1 − e2) In the case of a quasi-Keplerian orbit where the rate of decrease in the separation between the bodies is small in comparison to the period of the orbits, we can use the Keplerian expressions in the quadrupole formula. The total velocity of the centre of mass is then given in polar coordinates as v2 =r ˙2 + r2θ˙2 andz ˙ =r ˙. Substituting we ﬁnd

8 G (Gm)2(mη)2 E˙ = (r ˙2 + 12r2θ˙2) (4.5.11) gw 15 c5 z4 and substituting in (4.5.8)-(4.5.10), gives

8 G (Gm)3(mη)2 E˙ = (1 + e cos θ)4 e2 sin2 θ + 12(1 + e cos θ)2 . (4.5.12) gw 15 c5 a5(1 − e2)5 As it stands, (4.5.12) is an intractable quantity for physical measurement, so we average it over a period of the orbit. The Keplerian period is

2πa3/2 T = , (4.5.13) G1/2m1/2 which can be obtained by integrating the reciprocal of (4.5.9) from zero to 2π. Hence the average over the period is

1 Z T hE˙ gwi = E˙ gwdt T 0 1 Z 2π E˙ = gw dθ T 0 θ˙ 1/2 1/2 r 3 3 2 Z 2π G m a 2 3 8 G (Gm) (mη) 2 2 2 2 = (1 − e ) 2 (1 + e cos θ) e sin θ + 12(1 + e cos θ) dθ. 3/2 5 5 2 5 2πa Gm 15 c a (1 − e ) 0

54 Chapter 4 4.5. APPLICATION TO HULSE-TAYLOR BINARY

The integral can be evaluated with the result

Z 2π 73 37 (1 + e cos θ)2 e2 sin2 θ + 12(1 + e cos θ)2 dθ = 24π 1 + e2 + e4 0 24 96 and hence the averaged energy loss rate for the binary system is

32 G4 m5η2 hE˙ i = f(e), (4.5.14) gw 5 c5 a5 where 1 73 37 f(e) = 1 + e2 + e4 , (4.5.15) (1 − e2)7/2 24 96 is known as the enhancement factor since f(e) ≥ 1 for elliptic orbits with the minimum for a circular orbit. Due to the Keplerian motion we have a relationship between the energy and the semi-major axis which is E = −(Gηm2)/(2a), therefore the period is proportional to the energy by T = (constant) × (−E)−3/2. Therefore we have the relation

T˙ 3 E˙ = − . T 2 E The rate at which the energy decreases is equal to the rate at which energy is removed due to gravitational waves so E˙ = −E˙ gw and hence over one period, the period itself decreases at a rate predicted by 192π G5/3 2π 5/3 T˙ = − m5/3η . (4.5.16) 5 c5 T For the Hulse-Taylor binary, the masses of each object are known from other means and the eccentricty and period are also known [31], they are

e = 0.6171,

mpulsar = 1.44M ,

mcompanion = 1.39M , T = 27907 s, along with the constants and the mass of the sun

G = 6.674 × 10−11 m3kgs−2, c = 2.9979 × 108 ms−1, 30 M = 1.989 × 10 kg.

Plugging these directly into (4.5.16), we ﬁnd a rate of decrease of

T˙ = −2.40 × 10−12 s/s, which is in excellent agreement with the observed value [32] of

T˙ = −(2.427 ± 0.026) × 10−12 s/s.

This success of general relativity is seen as indirect evidence for the existence of gravitational waves. To achieve higher orders of precision requires formulae that go beyond the 1PN order of the quadrupole formula and are discussed in the next section.

55 Chapter 4 4.6. BEYOND THE QUADRUPOLE FORMULA

4.6 Beyond the Quadrupole formula

Even though the quadrupole formula yields excellent agreement in the case of the Hulse- Taylor binary, more accurate equations are required for use as templates in acquiring the real gravitational wave signals from noisy data. Initially, higher order PN expansions of the quadrupole formula were only of academic interest. However with the need for high order waveform templates for the analysis of gravitational wave data at detectors such as LIGO, these expressions have become a key priority. At higher orders, the energy flux takes on different terms related to the increased complexity of the iterations. For instance, Blanchet [8] has derived an equivalent quadrupole formula accurate to the much higher 3.5PN order. The energy flux is split into three parts

E˙ gw = Einst + Etail + Etail-tail, (4.6.1) where Einst is created only by the multipole moments and has the form

G 1 1 1 1 16 E = I(3) I(3) + I(4) I(4) + J (3) J (3) inst c5 5 hiji 5 hiji c2 189 hijki hijki 45 hiji hiji 1 1 1 + I(5) I(5) + J (4) J (4) c4 9072 hijkmi hijkmi 84 hijki hijki 1 1 4 1 + I(6) I(6) + J (5) J (5) + O , (4.6.2) c6 594000 hijklmi hijklmi 14175 hijkli hijkli c8 where repeated indices are assumed to be summed over; they are placed as subscripts so to make it easier to read. We can make a few observations for this portion of the energy flux. The first term is the lowest order quadrupole formula of (4.4.18), with the subsequent terms grouped into higher PN orders. The mass and current moments I and J are written in terms of STF tensors. We expect the presence of these higher order moments simply because they will not in general vanish as we do not have any conservation equations left to make use of. We also note that each successive PN order requires higher order moments and further differentiations. The tail terms Etail and Etail-tail are due to the wave zone contribution of the potentials and involve integrals of logarithmic terms. Applying to circular orbits allows the tail integrals to be evaluated and (4.6.2) to be simplified. The result which is the culmination of a real tour de force is

32c5 1247 35 E˙ = η2x5 1 + − − η x + 4πx3/2 gw 5G 336 12 44711 9271 65 8191 583 + − + η + η2 x2 + − − η πx5/2 9072 504 18 672 24 6643739519 16 1712 856 134543 41 94403 775 + + π2 − γ − ln(16x) + − + π2 η − η2 − η3 x3 69854400 3 105 E 105 776 48 3024 324 16285 214745 193385 1 + − + η + η2 πx7/2 + O , (4.6.3) 504 1728 3024 c8 where the ﬁrst term is a disguised version of (4.5.14) with e = 0 and γE is the Euler- Mascheroni constant. The post-Newtonian expansion parameter here is instead the dimensionless quantity Gmω 2/3 x := , (4.6.4) c3 which is much more useful than say v/c since it is related to the orbital frequency ω which is directly measurable and is not coordinate speciﬁc (gauge invariant) unlike v/c. Counting of PN orders is simple for example after factorising the common x5 term in

56 Chapter 4 4.6. BEYOND THE QUADRUPOLE FORMULA

(4.6.3), the bracketed entries have PN orders directly equal to the power of x appearing. 1 It is interesting to note that there is no 2 PN contribution to the radiated energy. Even though the waveform does contain a term at this order it vanishes due to destructive interference with the ‘Newtonian’ waveform term. The other remaining half integer terms in x (e.g. 3/2, 5/2 and 7/2) are known as wave-propagation corrections since they appear as a result of interactions between the integer order waveforms and tail terms. Remarkably the averaged energy over an obit can be computed for eccentric binaries and are higher order analogues of (4.5.14). It can be written as

32c5 hE˙ i = η2x5 P + P x + P x3/2 + P x2 + P x5/2 + P x3 , (4.6.5) gw 5G 0 1 3/2 2 5/2 3 where each Pi is a function of the dimensionless mass η and so called time eccentricity √terms et which to lowest order are (actual calculation requires an extra order higher) 2 1 − e where e is the ordinary orbital eccentricity. The lowest order term P0 = f(et), where f is the enhancement factor from (4.5.15). Therefore these terms can be thought of as enhancement factors to the Keplerian orbit.

57 Chapter 5

Concluding Remarks

We will first proceed with a brief summary of the content contained within. We introduced the reduced field equations of GR and we then found that they were brilliantly suited for the implementation of a post-Newtonian approximation scheme. This involved defining as some small parameter to quantify the slowness of the system and we made use of this to write expansions about this parameter. We then obtained the first and second order equations from these considerations and showed that the first order yielded the Newtonian limit and the second order corresponded to the first post-Newtonian contribution. We then applied these equations to the case of a system of point particles and found the metric in this case and the equations of motion. As a verification of these results, we obtained a well known formula for the perihelion precession of a binary system. In the second half we were concerned about the prediction of gravitational wave phenomena; ripples in space-time. By utilising gauge freedom we could place these the leading order potentials in such a way that the wave was transverse and trace-free. We then applied these results to the case of waves in vacuum and to the elliptical binary system and examined the polarisation states. We then found that it is in fact these polarisation states that correspond to physical observables and it is the change in distances between geodesics that is observed of a passing gravitational wave. With all of the previous results, we could discuss the tricky concept of energy in GR and finally arrived at the quadrupole formula for gravitational wave emission. The validity of this formula was then tested on the binary pulsar PSR1913+16. The prediction was astoundingly accurate even to the lowest order which is the quadrupole formula and it is indirect evidence for the existence of gravitational radiation. The concepts covered within would form a great introduction into applications of GR such as to the post-Newtonian approximations and to the theory of gravitational waves and would form a firm basis for more advanced study. Perhaps the most important test of GR will be the detection of gravitational waves. Since Joseph Weber created the first gravitational wave detector in 1959 [33] and showed that it may be possible to detect them, detectors have sprung up all over the world. The most recent of these are the highly sensitive interferometers such as LIGO in the US and VIRGO in Italy which are due to be improved into the advanced LIGO and VIRGO detectors which will be able to detect wave amplitudes of ∼ 10−22 [34]. While no positive detections have been made yet, detection is becoming more and more likely as peak sensitivities increase. The most important application of the post-Newtonian expansions is, as we have mentioned previously, obtaining high accuracy wave-templates for data analysis. While have said that high accuracy such as to the 3.5PN order is essential for reducing systematic errors between the observed data and the theoretical templates, it is important to see the magnitude of these corrections. In Figure 5.1 we show estimates for the contributions each PN order makes for the accumulated gravitational wave cycles measured by a detector like

58 Chapter 5

LIGO or VIRGO. We can see that for high mass binaries, the first PN order contribution is about 10% of the Newtonian order which would be found using the quadrupole formula (although the tail contributions go some way towards lowering the instantaneous contributions). Although they may look small, the higher contributions such as 3PN are required due to the extreme levels of sensitivity needed in the measurements. To declare a positive detection would require all sources of noise to be accounted for, of which there are many, and these higher order contributions help to rule out false signals. Obtaining higher order corrections say to 4PN is an eventual goal for the post-Newtonian approximation aimed at obtaining more accurate waveforms. The main difficulty here is the number of terms one deals with goes through the roof, so to speak, and it is an extreme challenge to carry out such calculations. It is possible that there may be an as yet introduced scheme which is able to obtain higher order contributions more easily. Regardless of whether one can obtain these or not, it is in the hands of the detectors to find direct evidence of gravitational waves and verify these theories.

Figure 5.1: Representative magnitude estimates for the corrections gained over many gravitational wave cycles with each PN contribution. These are given for three diﬀerent binary systems that would be detectable in the frequnency band of LIGO and VIRGO (Reproduced from [35]).

59 Bibliography

[1] A. Einstein, “ Die Grundlage der allgemeinen Relativittstheorie,” Annalen der Physik, vol. 354, pp. 769–822, 1916.

[2] K. Schwarzschild, “Uber¨ das Gravitationsfeld eines Massenpunktes nach der Ein- steinschen Theorie,” Sitzungsber.Preuss.Akad.Wiss.Berlin (Math.Phys.), pp. 189– 196, 1916. For an English translation see ”On the gravitational ﬁeld of a mass point according to Einstein’s theory” , available at: http://arxiv.org/abs/physics/9905030.

[3] C. M. Will, “Post-Newtonian gravitational radiation and equations of motion via direct integration of the relaxed Einstein equations. III. Radiation reaction for binary systems with spinning bodies,” Phys. Rev. D, vol. 71, p. 084027, Apr. 2005.

[4] I. L. Einstein, A. and B. Hoﬀmann, “ The Gravitational Equations and the Problem of Motion,” Ann. Math., vol. 39, no. 1, pp. 65–100, 1938.

[5] M. E. Pati and C. M. Will, “Post-newtonian gravitational radiation and equations of motion via direct integration of the relaxed einstein equations. ii. two-body equations of motion to second post-newtonian order, and radiation reaction to 3.5 post- newtonian order,” Phys. Rev. D, vol. 65, p. 104008, Apr 2002.

[6] H. Asada and T. Futamase, “Post Newtonian approximation: Its Foundation and applications,” Prog.Theor.Phys.Suppl., vol. 128, pp. 123–181, 1997.

[7] T. Futamase and Y. Itoh, “The post-newtonian approximation for relativistic compact binaries,” Living Reviews in Relativity, vol. 10, no. 2, 2007.

[8] L. Blanchet, “Gravitational radiation from post-newtonian sources and inspiralling compact binaries,” Living Reviews in Relativity, vol. 17, no. 2, 2014.

[9] T. Damour, P. Jaranowski, and G. Sch¨afer,“Nonlocal-in-time action for the fourth post-newtonian conservative dynamics of two-body systems,” Phys. Rev. D, vol. 89, p. 064058, Mar 2014.

[10] L. Landau and E. Lifshitz, The Classical Theory of Fields. Pergamon Press, London, third ed., 1971. pp 304-6.

[11] M. E. Pati and C. M. Will, “Post-newtonian gravitational radiation and equations of motion via direct integration of the relaxed einstein equations: Foundations,” Phys. Rev. D, vol. 62, p. 124015, Nov 2000. pp. 37-39.

[12] Ref. 36, p. 6.

[13] W. D. Goldberger and I. Z. Rothstein, “An Eﬀective ﬁeld theory of gravity for extended objects,” Phys.Rev., vol. D73, p. 104029, 2006.

60 Chapter 5 BIBLIOGRAPHY

[14] L. Blanchet and T. Damour, “Radiative gravitational ﬁelds in general relativity. I - General structure of the ﬁeld outside the source,” Royal Society of London Philosoph- ical Transactions Series A, vol. 320, pp. 379–430, Dec. 1986.

[15] L. Blanchet and T. Damour, “Tail-transported temporal correlations in the dynamics of a gravitating system,” Phys. Rev. D, vol. 37, pp. 1410–1435, Mar 1988.

[16] L. Blanchet and T. Damour, “Post-newtonian generation of gravitational waves,” Ann. Inst. Henri Poincare A, vol. 50, pp. 377–408, 1989.

[17] T. Damour and B. R. Iyer, “Post-newtonian generation of gravitational waves. ii. the spin moments,” Ann. Inst. Henri Poincare A, vol. 54, pp. 115–164, 1991.

[18] C. M. Will and A. G. Wiseman, “Gravitational radiation from compact binary systems: Gravitational waveforms and energy loss to second post-newtonian order,” Phys. Rev. D, vol. 54, pp. 4813–4848, Oct 1996.

[19] T. Futamase and B. F. Schutz, “Newtonian and post-newtonian approximations are asymptotic to general relativity,” Phys. Rev. D, vol. 28, pp. 2363–2372, Nov 1983.

[20] P. Jaranowski and G. Sch¨afer,“Third post-newtonian higher order adm hamilton dynamics for two-body point-mass systems,” Phys. Rev. D, vol. 57, pp. 7274–7291, Jun 1998.

[21] G. M. Clemence, “The relativity eﬀect in planetary motions,” Rev. Mod. Phys., vol. 19, pp. 361–364, Oct 1947.

[22] U. J. L. Verrier, “Theorie du mouvement de mercure,” Annales de l’Observatoir lmpirial de Paris., vol. 76, 1859.

[23] A. Einstein, “The foundation of the general theory of relativity,” Reprint of the 1916 paper in The Principle of Relativity, New York, Dover Publications Inc., pp 109-164, 1923.

[24] Ref. 10, p. 287-290.

[25] D. Tong, “Classical dynamics.” University Lecture, http://www.damtp.cam.ac.uk/user/tong/dynamics/clas.pdf, 2012. see p. 123.

[26] D. N. Damour, T., “General relativistic celestial mechanics of binary systems. i. the post-newtonian motion,” Ann. Inst. H. Poincar Phys. Thor, vol. 43, no. 1, pp. 107– 132, 1985. p. 117.

[27] R. M. Wald, General Relativity. The University of Chicago Press, Chicago, 1984. see pp. 84-88 and section 11.2, pp. 285-295.

[28] H. Bondi, M. G. J. van der Burg, and A. W. K. Metzner, “Gravitational Waves in General Relativity. VII. Waves from Axi-Symmetric Isolated Systems,” Royal Society of London Proceedings Series A, vol. 269, pp. 21–52, Aug. 1962.

[29] C. M. Will, “Inaugural Article: On the unreasonable eﬀectiveness of the post- Newtonian approximation in gravitational physics,” Proceedings of the National Academy of Science, vol. 108, pp. 5938–5945, Apr. 2011.

[30] R. A. Hulse and J. H. Taylor, “Discovery of a pulsar in a binary system.,” The Astrophysical Journal Letters, vol. 195, pp. L51–L53, 1975.

61 Chapter BIBLIOGRAPHY

[31] J. M. Weisberg and J. H. Taylor, “The Relativistic Binary Pulsar B1913+16: Thirty Years of Observations and Analysis,” in Binary Radio Pulsars (F. A. Rasio and I. H. Stairs, eds.), vol. 328 of Astronomical Society of the Paciﬁc Conference Series, p. 25, July 2005.

[32] J. M. Weisberg, D. J. Nice, and J. H. Taylor, “Timing Measurements of the Rela- tivistic Binary Pulsar PSR B1913+16,” The Astrophysical Journal Letters, vol. 722, pp. 1030–1034, Oct. 2010.

[33] J. Weber, “Detection and generation of gravitational waves,” Phys. Rev., vol. 117, pp. 306–313, Jan 1960.

[34] G. M. Harry and the LIGO Scientiﬁc Collaboration, “Advanced ligo: the next generation of gravitational wave detectors,” Classical and Quantum Gravity, vol. 27, no. 8, p. 084006, 2010.

[35] Ref. 8, p. 121.

[36] E. Poisson, “Post-newtonian theory for the common reader.” University Lecture, http://www.physics.uoguelph.ca/poisson/research/postN.pdf, 2007.

62 Appendix A

Appendix

A.1 Solving the Wave Equation

As we have seen, ﬁnding solutions to the wave equation is a central mathematical problem in post-Newtonian theory. Therefore we collect here a method of integrating the inhomogeneous wave equation ψ = −4πµ(ct, x), (A.1.1) by means of a Green’s function. Here φ(ct, x) is our scalar potential which is generated by the source term µ(x) and x = (ct, x). A Green’s function for the d’Alembertian should satisfy 0 0 0 G(x, x ) = −4πδ(ct − ct )δ(x − x ), (A.1.2) with G = 0 if x is in the past light cone generated at the source point x0. The Green’s function we are looking for is

δ(c(t − t0) − |x − x0|) G(x, x0) = . (A.1.3) |x − x0|

So our potential is, by the deﬁnition of a Green’s function,

Z ∞ Z t ψ(x) = G(x, x0)µ(x0)d(ct0)d3x0, (A.1.4) −∞ 0 where the spatial integrations run over all space. Substituting into this (A.1.3) we obtain

Z ∞ Z t 0 0 δ(c(t − t ) − |x − x |) 0 0 0 3 0 ψ(ct, x) = 0 µ(ct , x )d(ct )d x . (A.1.5) −∞ 0 |x − x | We may now proceed with the temporal integration yielding

( 0 0 R µ(ct−|x−x |,x ) d3x0 : |x − x0| ≤ ct ψ(ct, x) = |x−x0| 0 : otherwise and hence Z 0 0 µ(ct − |x − x |, x ) 3 0 ψ(ct, x) = 0 d x , (A.1.6) L(ct,x) |x − x | where the integration is performed over the domain L(ct, x) := {0 ≤ |x − x0| ≤ ct} which is a ball centred at x with radius ct. This is the retarded integral solution to the wave equation of (A.1.1), so called because the solution at the ﬁeld point depends on the values 0 of the source µ at the retarded time tr := t − |x − x |/c. This is a simple consequence of the ﬁnite speed of propagation of information, c. There also exists an advanced Green’s

63 Chapter A A.2. ANGULAR STF TENSORS AND SPHERICAL AVERAGES function which solves the wave equation with integration region being the future light-cone, however this is taken as non-physical due to violations of causality. There is also a homogeneous solution that is to be added (A.1.6). We label it as ψH and it is the unique solution to ψH = 0, where appropriate initial data is given by, say at time t = 0, ψH (0, x) and ∂(ct)ψH (0, x) which are smooth functions. The solution to this is given by Kirchoﬀ’s equation Z 1 ∂ 0 ∂ 0 ψH (ct, x) = (ρψH (ct , x)) + 0 (ρψH (ct , x)) dΩ, (A.1.7) 4π ∂L(ct,x) ∂ρ ∂(ct ) t=0, ρ=r where ρ := |x−x0| and dΩ := sin(θ)dθdφ in spherical coordinates. The region of integration is now the surface of the past light cone of the ﬁeld point (ct, x). The full solution to the wave equation of (A.1.1) is then

Z 0 0 µ(ct − |x − x |, x ) 3 0 ψ = ψH + 0 d x . (A.1.8) L(ct,x) |x − x |

A.2 Angular STF Tensors and Spherical Averages

Due to the presence of a retarded time in our integrals, it will sometimes be necessary to take spatial derivatives of the intergrands. Since the time component is dependent on source positions, spatial derivatives must take this into account. Typical derivatives include ∂ar where r := |x| and spherical coordinates are implied. In order to proceed with those calculations we deﬁne the angular vector x Ω := = (cos φ sin θ, sin φ sin θ, cos θ), ⇒ rΩa = xa, (A.2.1) r which satisﬁes the following identities

a b δabΩ Ω = 1, (A.2.2)

∂ar = Ωa, (A.2.3) 1 ∂ Ω = (δ − Ω Ω ), (A.2.4) a b r ab a b −1 −2 ∂ar = −r Ωa, (A.2.5)

b where Ωa := δabΩ . We will verify each of these. Beginning with (A.2.2), we have

a b 1 2 2 2 3 2 δabΩ Ω = (Ω ) + (Ω ) + (Ω ) = 1, using the deﬁnition of the components given in (A.2.1). In order to derive (A.2.3), we consider taking the spatial derivative of (A.2.1), with the left hand side as

b b b ∂a(rΩ ) = (∂ar)Ω + r∂aΩ

= (∂ar)Ωb

b b and the right hand side trivially as ∂ax = δa, thus putting it together we have

b b (∂ar)Ω = δa.

b Multiplying both sides by Ωb and making use of δaΩb = Ωa and (A.2.2), we arrive at (A.2.3). Use of the chain rule and of (A.2.3) will yeild (A.2.5). For the third identity we

64 Chapter A A.3. EVALUATION OF A 1PN SURFACE INTEGRAL use the deﬁnition (A.2.1), that is x ∂ Ω = ∂ b a b a r −1 −1 = ∂a (xb) r + xb∂ar −1 −1 −1 = δabr − r xbΩar 1 = (δ − Ω Ω ). r ab a b We next introduce the concept of symmetric trace-free tensors which are written, for example, as Ωhabci and correspond to products of the angular tensors Ωa in such way that they are symmetric and their trace vanishes. For example, 1 Ωhabi := ΩaΩb − δab, 3 1 Ωhabci := ΩaΩbΩc − (δabΩc + δacΩb + δbcΩa). 5

habci habci habci They are trace free since δabΩ = δacΩ = δbcΩ = 0. Finally, we introduce the notion of a spherical average of a function ψ, which is deﬁned as 1 Z hhψii := ψ(θ, φ) dΩ, (A.2.6) 4π where dΩ = sin θ dθ dφ is an element of solid angle. We present here a few identities related to this deﬁnition that are simply imported into the derivation of the Quadrupole approximation. Using the fact that the angular average of an STF tensor is zero, allows one to immediately write

hhΩaii = 0, (A.2.7) 1 hhΩaΩbii = δab, (A.2.8) 3 hhΩaΩbΩcii = 0, (A.2.9) 1 hhΩaΩbΩcΩdii = (δabδcd + δacδbd + δadδbc). (A.2.10) 15 A.3 Evaluation of a 1PN surface integral

In this section our goal is to evaluate the integral I cd a b ∂dχˆ x x dSc. (A.3.1) ∂N

cd We first need to calculate ∂dχˆ where we recall from (2.5.42), we have GM χâb = 4 ΩaΩb − δab/2 + O(6). (A.3.2) 4πr4 To evaluate the derivative we require (A.2.2)-(A.2.5). We first find

c d −4 c d −4 c d −4 ∂d(Ω Ω r ) = ∂d(Ω )Ω r + Ω Ω ∂dr −5 c c d −5 d c = r (δd − ΩdΩ )Ω − 4r ΩdΩ Ω = −4r−5Ωc.

65 Chapter A A.4. DETAILS OF QUADRUPOLE FORMULA DERIVATION

Therefore GM ∂ χˆcd = 4 (−4Ωc + 2Ω δcd) d 4πr5 d 2GM = −4 Ωc. 4πr5 2 a a We now use dSc = R ΩcdΩ, x = rΩ and that at the surface r = R to obtain I I 2 cd a b 4 1 R c a b 2 ∂dχˆ x x dSc = −2 GM 5 Ω Ω Ω R ΩcdΩ ∂N 4π ∂N R 2GM 1 I = −4 ΩaΩbdΩ R 4π ∂N 2GM = −4 hhΩaΩbii R 2GM = −4 δab. 3R The general ideas in this evaluation can be readily applied to other surface integrals of a similar type.

A.4 Details of Quadrupole formula derivation

In this appendix, we present a few useful identities related to the derivation of the quadrupole formula that are simply inserted into the relevant sections of the text. ab 2G ¨ab Lemma A.4.1. For the lowest order far-away wave zone contribution h = c4r I we have the identity

4G2 1 h˙ TTh˙ ab = δ δ − δ δ − δ Ω Ω − δ Ω Ω ab TT c8r2 ac bd 2 ab cd ac b d bd a c 1 1 1 + δ Ω Ω + δ Ω Ω + Ω Ω Ω Ω Iab(3)Icd(3), (A.4.1) 2 ab c d 2 cd a b 2 a b c b where (3) stands for the third derivative with respect to the retarded time u = t − r/c. Proof. We recall the definition of the TT and projection operators: ab ab cd ATT = (TT)cdA , 1 (TT)ab := P aP b − P abP , cd c d 2 cd a a a Pb := δb − Ω Ωb, c a c a ab and the identities Pa Pb = Pb and Pa = 2 and as a consequence of these P Pab = 2 along ab a with P Pcb = Pb . The latter two are obtained by simply raising and lowering indices of the first two using the Minkowski metric ηab. We first require the following identity, ˙ TT ˙ ab ab ef ˙ ˙ cd hab hTT = (TT)cd(TT)ab hef h 1 1 = P aP b − P abP P eP f − P ef P h˙ h˙ cd c d 2 cd a b 2 ab ef 1 1 1 = P aP bP eP f − P abP P eP f − P ef P P aP b + P abP P ef P h˙ h˙ cd c d a b 2 cd a b 2 ab c 4 cd ab ef 1 1 1 = P eP f − P P ef − P P ef + P P ef η η h˙ abh˙ cd c d 2 cd 2 cd 2 cd ea fb 1 = P P − P P h˙ abh˙ cd, ac bd 2 cd ab

66 Chapter A A.4. DETAILS OF QUADRUPOLE FORMULA DERIVATION where we have made use of the above identities many times. Now inserting the deﬁnition of the projection operator Pab := δab − ΩaΩb into each term and expanding while also ab 2G ¨ab substituting h = c4r I , we arrive at the required result. Theorem A.4.2 (Quadrupole formula). Inserting (4.4.17) into 3 I c 2 TT ab E˙ gw = lim r h˙ h˙ dΩ, (A.4.2) 32πG r→∞ ab TT yields the quadrupole formula G 1 E˙ = Iab(3)I − I(3)2 , (A.4.3) gw 5c5 ab(3) 3 ab ab R a b 3 where I := δabI and I = ρx x d x. Proof. The calculations here are not the most instructive but they do make use of a lot of the machinery we have taken care to develop with a key reason being the derivation of this result. The angular integrals that will appear have already been given in (A.2.8) and (A.2.10) and we simply insert those results. To begin we substitute (4.4.17) into (4.4.13) and start the process G 1 E˙ = δ δ − δ δ − δ hhΩbΩdii − δ hhΩaΩcii gw 2c5 ac bd 2 ab cd ac bd 1 1 1 + δ hhΩcΩdii + δ hhΩaΩbii + hhΩaΩbΩcΩdii Iab(3)Icd(3) 2 ab 2 cd 2 G 11 2 1 = δ δ − δ δ + δ δ Iab(3)Icd(3) 10c5 6 ac bd 3 ab cd 6 ad bc G 11 2 1 = Iab(3)I − I(3)2 + Iab(3)I 10c5 6 ab(3) 3 6 ab(3) G 1 = Iab(3)I − I(3)2 . 5c5 ab(3) 3

Lemma A.4.3. For a two body system, with Iab = mηzazb in the centre of mass frame, the rate of energy loss due to gravitational waves emitted, in the far-away wave zone is 8 G (Gm)2(mη)2 E˙ = (12v2 − 11z ˙2). (A.4.4) gw 15 c5 z4 Proof. We first note that we are working in the post-Newtonian limit where the orbital velocities of the bodies are much smaller than the speed of light. Since the quadrupole formula is of 1PN order, we can use the Newtonian equations of motion. If we had a formula for E˙ gw which was accurate to say 2PN order, we would have to use the 1PN equations of motion. This follows from the iterative solution method discussed in Section 2.2. In light of this, the Newtonian acceleration is given by aa =z ä = −(Gm/z2)na where a n = z/z is a unit vector, andz ˙ = n va. To begin we have to find an expression for the third derivative of the mass moment Iab. These derivatives are Iab = mηzazb, Iab(1) = mη[z ˙azb +z ˙bza], Iab(2) = mη[¨zazb + 2z ˙az˙a +z ¨bza], ...... Iab(3) = mη[ z azb + 3z ˙az¨b + 3z ˙bzä + z bza] −3Gm ...... = mη (z ˙anb +z ˙bna) + z azb + z bza . z2

67 Chapter A A.4. DETAILS OF QUADRUPOLE FORMULA DERIVATION

For the third derivative terms we use ... ∂ za z a =a ˙ a = −Gm ∂u z3 z˙a 3zz ˙ a = −Gm − z3 z4 and hence

...... z˙azb 3zz ˙ azb z˙bza 3zz ˙ bza z azb + z bza = −Gm − + − z3 z4 z3 z4 Gm = − z˙anb +z ˙bna − 6zn ˙ anb . z2 Therefore 2Gm2η h i Iab(3) = −2(vanb + vbna) − 3zn ˙ anb . (A.4.5) z2 ab(3) We now calculate the product I Iab(3) using (A.4.5), i.e.

4(Gm)2(mη)2 h i Iab(3)I = −2(vanb + vbna) − 3zn ˙ anb [−2(v n + v n ) − 3zn ˙ n ] ab(3) z4 a b b a a b 4(Gm)2(mη)2 h i = 9z ˙2nanbn n − 12zn ˙ n (vanb + navb) + 4(vanb + navb)(v n + n v ) z4 a b a b a b a b 4(Gm)2(mη)2 = 9z ˙2 − 24z ˙2 + 8z ˙2 + 8v2 z4 4(Gm)2(mη)2 = 8v2 − 7z ˙2 , z4 a where we repeatedly made use of n.n = 1 (unit vector) andz ˙ = n va. The trace term is (3) ab(3) deﬁned as I := δabI so 2Gm2η h i I(3) = −2(δ vanb + δ vbna) − 3zδ ˙ nanb z2 ab ab ab 2Gm2η = [3z ˙ − 4z ˙] z2 −2Gm2η = z˙ z2 and hence I(3)2 = 4(Gm)2(mη)2z˙2/z4. So putting it all together we ﬁnd

G 4(Gm)2(mη)2 4(Gm)2(mη)2 E˙ = 8v2 − 7z ˙2 − z˙2 gw 5c5 z4 3z4 8 G (Gm)2(mη)2 = (12v2 − 11z ˙2), 15 c5 z4 after some simpliﬁcation.