key: cord-0528127-5tm9nmsw
authors: Undheim, Vegard
title: First post-Newtonian correction to gravitational waves produced by compact binaries: How to compute relativistic corrections to gravitational waves using Feynman diagrams
date: 2021-10-29
journal: nan
DOI: nan
sha: 678fcfbfedb15abb5df43ac57877190cebee092b
doc_id: 528127
cord_uid: 5tm9nmsw

The purpose of this thesis is to calculate the relativistic correction to the gravitational waves produced by compact binaries in the inspiral phase. The correction is up to the next to leading order, the so-called first post-Newtonian order (1PN), which are correctional terms proportional to $(v/c)^2$ compared to leading order, Newtonian, terms. These corrections are well known in the literature, even going beyond the first order corrections, so why is it computed again here? In later years, an alternative approach for computing these terms using effective field theory has emerged. This thesis investigates this approach by replicating it, and attempts to make this approach more accessible to those not familiar with effective field theories. It has been claimed that this approach greatly simplifies the complicated calculations of gravitational waveforms, and even provides the required intuition for 'physical understanding'. By this master student that was found not to be entirely correct. The calculations were made easier for those with a rich background in quantum field theory, but for those who are not well acquainted with quantum field theory this was not the case. It was, however, found to be a worthwhile method as a means for deepening one's understanding of gravity, and might provide a shorter route for some alternative theories of gravity to testable predictions.

TT transverse-traceless. 21, 25, 29-31, 57, 59, 60 Glossary black hole A region of space-time curved to the point that no matter or radiation can escape.

Usually taken to be a gravitationally collapsed star. viii, ix, 1, 26

quantum field theory The theory of fields endowed with quantum properties that can be used to describe forces and matter. ii, viii, 3, 15, 68

Einstein's field equations The equations of motion resulting from the Einstein-Hilbert action which dictates the dynamics of space-time. Coupled to a matter source it reads R µν − 1 2 Rg µν = 8πG c 4 T µν . ix, 1-3, 15, 20

Einstein-Hilbert action Einstein-Hilbert action is the action which when extremized generates the Einstein's field equations, i.e. the action which governs general relativity 16, 20, 35, 37, 38 field theorist Physicists using fields on a static background space-time to model physical effects like forces and particles. In this thesis especially those who use fields to model gravity. 3, 4, 6, 15, 20, 29, 35, 67 GW150914 The gravitational wave event which occurred 14/09/2015. The first GW event by LIGO [1] . 1, 12 quasi-stable circular orbit Approximating the inspiral as circular orbits with gradually falling radii. The change in radius is negligible unless viewed over several periods. 8, 13, 64, 67

relativist Physicists using a geometrical interpretation of gravity, following in the footsteps of Einstein. 3, 4, 6, 15, 26, 67, 68 Schwarzschild radius The Schwarzschild radius R S is the radius associated with the event horizon of a non-rotating, static black hole. R S = 2GM c 2 . 36

On the 14 th of September 2015 the world was shocked, ever so slightly. So slightly in fact that the only reason we know about it is thanks to the effort of the Laser Interferometer Gravitational-Wave Observatory (LIGO), who measured this faint strain in their detectors. After careful testing and retesting, LIGO published their results on the 11 th of February 2016 [1] . They concluded that the event, called GW150914, was a gravitational wave (GW) produced by the merger of two black holes, and was the first directly detected gravitational wave event in human history.

With the announcement of the historic detection of GW150914 came promises of a new era of astronomy, now equipped with a brand new type of data to constrain astronomical theories. Popular science lectures and books were given and written, and at the height of this hype I started my bachelor's degree in physics. Fascinated by these mysterious waves I wanted to learn more about them, and when the time came to pick a topic for my master's thesis I requested to work on gravitational waves.

My supervisor and I decided to work on relativistic corrections to the binary inspiral, using field theoretical methods. To date, all confirmed GW events are thought to be produced by compact binaries. A compact object is a black hole (BH) or neutron star (NS), and a compact binary is a system consisting of two compact objects. When compact objects revolve around each other they produce so called gravitational waves which dissipate orbital energy from the system. As a result the compact objects fall toward each other, and in the end collide and merge together.

The problem with compact binaries is that they are too heavy and fall too close to each other, making the gravitational interaction too strong to be adequately described by Newton's law of gravity. Although the two body problem has a general solution in Newtonian mechanics, there is no known equivalent solution for the two body problem in general relativity, only the Inspiral Merger Ringdown one body problem. To combat this issue, researchers have followed one of two approaches. 1 1. Solve the full, non-linear, Einstein's field equations numerically for the binary system in question. 2. Use an approximate, analytical, solution and perturbatively expand it to account for relativistic corrections.

This thesis will focus on analytical approximations. With numerical simulations one obtains a picture of the dynamics at an arguably very high accuracy, but due to the complexity of Einstein's field equations this is computationally costly, i.e. takes a lot of time and computing power. Furthermore, analytical expressions provide information about important quantities and intuition about the most important physical effects at play, that one simply does not gain from computer simulations. In order to expand an analytical solution relativistically one first needs an approximate solution to expand. For this it is useful to divide the evolution of the compact binary into three phases, see Figure 1 .1. The first phase is called the inspiral phase. Here the compact objects orbit each other at a distance, gradually falling closer together due to the emission of gravitational waves. Once the bodies are so close that a collision is imminent (typically when they 'touch' or form a common event horizon) the system becomes highly non-linear, and enters the so-called merger phase. After the two objects have merged into one, the system enters the ringdown phase, in which the system can be described as a one body problem, but with remnant asymmetries from the merger. Typically, the merged object's asymmetries oscillate around the Kerr solution and gradually dampen down, hence the name ringdown. This is a useful division of the binary evolution as the different phases lend themselves to different approximations. The first phase, the inspiral, can be approximated as Keplerian orbits since the leading order term in the equations of motion is the Newtonian law of gravitation. The last phase can be approximated as a Schwarzschild or Kerr solution with perturbations. The merger phase is sandwiched between these two widely different approximations and is dominated by non-linear effects. Thus the merger phase has no good analytical approximation and must be simulated numerically.

In this thesis I will work with the analytical approximation of the inspiral phase.

As we will see in Chapter 3 the frequency of the gravitational waves produced by compact binaries are directly dependent on the frequency at which the source oscillates, which is found to be inverse proportional to the total mass of the binary raised to the 5/8ths power: ω source ∝ M −5/8 (see equation (2.18) ). Therefore, the waveform of GWs measured here on Earth provides information about the dynamics of the binary which produced it, and can be compared with the predicted dynamics according to general relativity. This is why GW observation is a precise tool for constraining theories of gravity.

To motivate these computations, Chapter 2 starts off by computing the waveform, using results from following chapters. Then in Chapter 3 an alternative path to gravity is presented, that of a gauge field theory on a static space-time background. It is demonstrated to recover the main results of standard linearized gravity, which is the Einstein's field equations expanded to linear order in metric perturbations over flat space-time. In Chapter 4 and 5 the main results needed to compute the waveform in Chapter 2 are derived, using the effective field theory (EFT) based on the material presented in Chapter 3. Then the thesis ends with Chapter 6, which is concluding remarks on the effective field theory approach to gravitational waves. This is a form of top down approach, starting with the final result (the waveform) and working back to the fundamental assumptions behind it. This structure has been chosen because of the large amount of laboursome calculations leading to the gravitational waveform, and it will hopefully provide the overview needed to understand the motivation for each calculation as it appears.

In 2006, Goldberger and Rothstein [3] wrote a paper showing how the gravitational waveform could systematically be calculated to any post-Newtonian (PN) order using EFT formalism. Post-Newtonian expansion is ordering results like energy, the equation of motion, radiated energy flux, velocity, etc. as the Newtonian result plus relativistic corrections, usually expanded in factors of v/c. E.g.

Here E Newt · E i v c i would be the i 2 PN term of the energy. This scaling as half the v/c power is chosen to represent the PN order such that the leading order correction is 1PN.

In this thesis, working with fields on a non-dynamic, flat, space-time will be referred to as field theory, or the approach of field theorists, like Goldberger. This is supposed to be contrary to traditional geometrical theories of gravity, in the spirit of Einstein, which will be referred to as the approach of relativists. By any normal definition however, general relativity and its interpretation by relativists, is a field theory. But they work with dynamical spacetimes, making it conceptually and mathematically quite differently formulated. Therefore, these constructed labels of field theorists and relativists will be employed in this thesis to emphasize the difference in approach.

Formulating the computations in the language of field theorists, Goldberger and Rothstein unlocked all tools, tricks, and language usually reserved for quantum field theory (QFT). Since then, this approach has been argued by field theorists to be easier and faster than the traditional relativist approach. One of these field theorists, R. Porto, has even claimed [4] "[...] that adopting an EFT framework, when possible, greatly simplifies the computations and provides the required intuition for 'physical understanding'."

My supervisor, a self-proclaimed relativist, got curious, and wondered just how easy the effective field theory approach would make the computation. Therefore, he asked me if I would try to go through these computations, to test if they made the computation manageable even for master's students. My comments on Porto's claim are given in the discussion of Chapter 6.

With verifying or refuting Porto's claim as the ultimate goal of this thesis, it is mostly written as a relativist's guide to a field theorists' approach to gravitational waves. It should also be useful for those with a field theoretical background who wish to understand how Feynman diagrams can be used in classical gravity, and gravitational wave physics. Four-vectors are written with Greek letter indices, and spatial vectors with Latin letter indices. The Einstein summation convention applies.

Four-vector

Notably, the action is defined as

with L and L being the Lagrangian, and Lagrangian density, respectively. These tensor index notations are also used.

Colons will also appear in indices, but these have no mathematical meaning. Colons are simply used to separate pairs of indices that have distinct roles. E.g. could T µν x λ ≡ S µν:λ . Lastly, the Fourier transform, and inverse Fourier transform are defined by 2

In this chapter the gravitational waveform will be computed, both in the time domain (2.19) and in the frequency domain (2.27). The computation follows standard methods, like presented in Arun et al. [5] .

As inferred by the name, gravitational waves are waves, which is to say they are solutions of the wave equation.

Here the d'Alembert operator, also called the d'Alembertian, has been defined, which is the operator of the wave equation.

A simple solution to equation (2.1) is h µν = µν e −ik σ x σ , with k µ k µ = −k 2 0 + k 2 = 0, and where µν is some x µ -independent tensor structure. The exponential is a plane wave solution, according to Euler's formula (C.1).

Gravitational waves are rank two tensors, which means they have two indices and therefore 4 × 4 = 16 components. It is also symmetric in these two indices: h µν = h νµ , which means that only 10 of these components are independent. The reason gravitational waves are rank two tensors follows in the relativists' approach because h µν is a perturbation of the metric g µν = η µν + h µν , where η µν is the flat space-time metric. In the field theorists' approach it is because gravity is the effect of a massless spin two field. µν is the polarization tensor of GWs, and since it is a massless field it only has two independent polarizations. Gravitational waves are transverse, and thus ij k j = 0, i.e. the amplitude direction given by the polarization is orthogonal to the direction of propagation k.

To solve the wave equation, the wave four-vector had to be null-like. This implies further that the wave itself must travel at the speed of light, v = ∂ω ∂|k| = ∂ck 0 ∂|k| = c. This is also a consequence of h µν being a massless field.

Because of the linearity of the wave operator, any sum of such exponential (or trigonometric) terms will also be a solution of the wave equation. The most general solution is thus, a sum over all null-like wave-vectors k µ , and an expression which also leaves h µν as a real function. 1

The expression above being real follows from the observation h † µν (x α ) = h µν (x α ), which can only hold for real numbers. Here ω k = |k| = k 0 , which is to make the wave null-like, also known as 'on shell'. For a derivation of this solution, see Appendix A.

The coefficients a µν (k) are used to select particular solutions based on some initial condition, and are left to be determined.

The frequency of the wave turns out to be integer multiples of the frequency at which the source binary orbits, which will be demonstrated in Chapter 5. Thus, it can be approximated as

where Φ(t) is the phase of the source binary. The waveform describes what kind of wave it is. a n (t) can be found, but the most important factor for detection of gravitational waves is contained in Φ(t). The reason for this is that gravitational wave detectors receives faint signals with amplitudes close to the amplitude of noise. However, the frequency of GWs is different from the major noise factors, and can thus be extracted using Fourier analysis. Therefore, in the rest of this chapter, and much of the literature, the word waveform will be used interchangeably about the phase, as it encodes information about the frequency spectrum. The orbital energy for circular, Newtonian motion is related to the frequency as E = − 1 2 µv 2 = − 1 2 µ(GM ω) 2/3 , using v = ωr and Kepler's third law,

to eliminate r in favour of ω. 2 As usual, M = m 1 + m 2 is the total mass of the binary,

is the reduced mass, G is Newton's gravitational constant, r = |r 2 − r 1 | is the spatial separation of the binary, and v = |ṙ| is the relative velocity. More details on Newtonian orbital mechanics and its associated masses and quantities can be found in Appendix B.

The approximation of circular motion here might seem over idealized, but it turns out that the effect of gravitational wave emission on elliptical orbits is to circularize them.By the time the binary's frequency enters the detector range, near the time of coalescence, the orbits have become very circular, making circular orbits a sensible approximation.

Noting that the energy was easier to handle with v rather than ω, as it has integer powers instead of fractional powers, one may use v = (GM ω) 1/3 as a proxy variable for the frequency. Note that as a Newtonian approximation this variable coincides with the relative velocity parameter, but this is no longer the case after relativistic corrections are accounted for.

Then the phase of the orbit can be expressed as

Sadly v = v(t), which at this point is still an unknown function of time. However it is known that v must evolve with time according to how the orbital energy evolves with time.

The differential equation governing the dynamics of the orbital phase is

with E the energy associated with conserved orbital motion, and F the total energy flux out of the system by means of GWs. This is nothing but energy conservation for a gravitationally bound system. 3 Both E and F can be analytically expanded in a relativistic parameter, like (v/c). This requires a separation in scale, where on the short timescale the motion is conservative and has energy E, while on the long timescale the system loses energy to gravitational radiation at a rate F, leading to an inspiral. This requires the inspiral to happen slowly compared to the orbital motion, so that at any one moment the motion can still adequately be described by Newtonian motion. Thus, it only works for relatively small values of F, such that the objects do not fall down too rapidly. 4 Luckily, to leading order the flux term is suppressed by a factor of c −5 compared to the leading order term of the energy. Thus, the approximation of so called quasi-stable circular orbits and post-Newtonian formalism holds surprisingly well, even when compared to numerical simulations of the full Einstein equations (see Borhanian et al. [8] ).

As will be demonstrated in Chapter 4 and 5, the orbital energy (4.63) and energy flux (5.36) can be expanded in terms of (v/c) as

(2.8)

In the expansion (2.8) there is defined a Newtonian energy flux F Newt = 32η 2 5Gc 5 where η ≡ µ/M is the symmetric mass ratio (see Appendix B, and especially equation (B.5), for more details). This is strange since there are no GWs in Newtonian theory, so what is this flux? It is merely a convention to call leading order terms Newtonian, and this is why it is referred to as 'Newtonian'.

Since v is just a proxy for the frequency the expression (2.7)-(2.8) would be different expressed in terms of the actual centre-of-mass frame relative velocity. This point will be revisited in Chapter 4 and 5.

Up to (v/c) 2 order corrections define the first post-Newtonian order, or 1PN for short, and is the leading order correction. This has started the convention of calling terms ∼ (v/c) 2i for iPN order corrections, e.g. the leading order, Newtonian, term is 0PN order. This has a somewhat awkward effect, since not all terms are even powers of v/c, already the next order correction is ∼ (v/c) 3 , and is thus of 1.5PN order. Higher order terms of both the energy and flux, and the final result of this chapter: The waveform, can be found in papers like Arun et al. [5] .

Using equation (2.6) the time evolution dt can be expressed in terms of v as

Substituting (2.9) for dt in (2.5) results in the final expression for which the waveform can be derived (using (2.7)-(2.8))

Solving (2.9) will provide v as a function of time. We proceed however by computing Φ as a function of v directly rather than of time, as ultimately to be compared with experiments it is the waveform in the frequency domain (which will be called Ψ) which is needed. As already mentioned, this is because the signal is filtered in the frequency domain, and therefore the highest resolution is in the frequency spectrum.

In order to compute Φ(t) it is convenient to first compute Φ(v) (equation (2.14)), then v(t) (equation (2.17)), and lastly Φ(t) = Φ(v(t)) (equation (2.19) ).

Computing the waveform as a function of frequency

To evaluate this integral it would be advantageous to write the last fraction in an easier form. Utilising that v/c is small the last fraction can be Taylor expanded around v/c = 0 up to 1PN.

Performing the Taylor expansion results in

This result inserted in (2.11) yields the easily integratable 1PN expression

Integrating to obtain Φ(t) = Φ(v(t)) one must choose a reference point in time, usually referred to as t 0 . For binary inspirals this reference point is canonically chosen to be the moment of coalescence t c (see Maggiore [9] chapter 4), which for the duration of the inspiral is in the future. Therefore, the integration variables should go from v(t) to v c = v(t c ), but a multiplication of −1 to both sides can flip this order. Performing the integral finally provides

Collecting all constant terms into one phase constant Φ 0 , writing out E Newt and F Newt from (2.7) and (2.8) respectively, results in the final result for the waveform as a function of v

(2.14)

The phase is dimensionless, as one should expect. 5 To obtain the waveform as a direct function of time the frequency parameter v must be given as a function of time.

Computing the frequency as a function of time

The frequency parameter v as a function of time can be obtained from the differential equation (2.9), in an equivalent fashion to how (2.14) was derived.

Notice that t c − t was chosen for the left hand side (LHS) such that the expression on the right hand side (RHS) becomes strictly positive. This is desired because both sides must be raised to the negative one 4 th power, in order to produce a quadratic equation of v 2 . Taking the square root of the resulting solution for v 2 , and Taylor expanding it to the 1PN order yields the expression for v(t).

Following the aforementioned procedure, and using τ = t c − t, the frequency can be determined to be v(τ ) = c 2 5GM That is to say, both the signal and the model in (b) has been filtered by frequency, making the plots somewhat jagged. Plot (a) has not been filtered. In figure (a) the amplitude diverges at t = 0.425s, while in the in (b) it does not. This is because the inspiral model breaks down here, and the merger phase takes over.

Computing the waveform as a function of time Substituting (2.17) for v in (2.14) yields

All that remains now to obtain the waveform is to find the amplitude of the different harmonics, and multiply them by cos(nΦ(τ )). By using expression (5.9) from Chapter 5 the amplitude of the h 1PN can be found to be

(2.20)

Here z is the cosmological redshift, see Maggiore [9] for a derivation. This function is plotted in Figure 2 .1a using numbers from table 1 of Abbott et al. [1] for R, z, m 1 , and m 2 . The functions v(τ ) and Φ(τ ) are given by equation (2.17) and (2.19) respectively. The 0PN amplitude is obtained by using v 0PN (τ ) and neglecting the correctional terms inside the curly bracket that contain a factor of c −1 .

In Figure 2 .1a it is clear that 1PN corrections does not affect the amplitude much directly, but it has significant effect on the time evolution of the phase, and hence the frequency spectrum. It is however noticeable that the phase of Figure 2 .1a does not match up with Figure 2 .1b. Either higher order corrections are required, or the model breaks down for such low values of τ .

To obtain the high sensitivities in GW detections the signal is Fourier transformed, in order to show which frequencies dominate the signal. This frequency spectrum can be compared to theoretical predictions to determine factors like the total mass at 0PN, symmetric mass ratio at 1PN, and more parameters at higher orders, e.g. spin at 1.5PN [5] and finite size effects like tidal deformation at 5PN [10] .

In order to compare data with theoretical predictions these predictions must also be expressed in the frequency domain. Therefore, the desired waveform is Ψ(f ), which is the phase of the Fourier transformed waveform.

To compute the Fourier transformedB(f ) of some function B(t) the stationary phase approximation (SPA) can be used, and it is commonly utilized to compute the Fourier transform of (2.3). Standardized in GW physics by Cutler and Flanagan [11] it approximates for B(t) = A(t) cos(Φ(t)),

This is exactly the type of expression which describes GWs (2.3), and the conditions do indeed apply to the inspiral phase.

The leading order amplitude scales as v 2 ∼ τ −1/4 (2.17) (also, see equation (3.67) in the next chapter for why the amplitude scales as v 2 ), while dΦ/dt = ω(τ ) ∼ τ −3/8 (see (2.18) ).

Thus, for large τ , which is the time remaining till coalescence, dln(a n (t)) /dt ∼ 1 4 τ −1 ω(τ ) ∼ τ −3/8 .

As for the last prerequisite it can be shown to hold for quasi-stable circular orbits. Taking the time derivative of Kepler's third law (2.4) results in

For quasi-stable circular orbits the inspiral must be slow compared to the orbital motion, and thus the radial velocity (ṙ) must be small compared to the tangential velocity (ωr), since for perfectly circular motion their fraction is identically zero. From Kepler's law this implies also thatω/ω 2 1 →Φ Φ 2 , which is exactly the condition required to use the SPA. This in hand also provides an estimate for the validity of this approximation, as ω is a known function of time (2.18)

This expression is indeed small for most values of t < t c .

Since the most important part of the waveform for comparisons to experimental data is the frequency spectrum, the last computation of this chapter will be of the Fourier transformed phase Ψ(f ).

From equation (2.21) the phase of the Fourier transformed waveform can be approximated as

Φ(v) being given by equation (2.14) , and t(v) by (2.16), Ψ(v) can easily be computed.

Lastly the phase can be expressed in terms of the physical frequency by using v = (GM ω) 1/3 = (2πGM f ) 1/3 .

This is indeed equivalent to the expression found by Arun et al. [5] (equation (6.22), page 21) up to 1PN, with some difference in notation.

In order to compute this waveform all that is needed is the PN expansion of the orbital energy, and the GW energy flux, both associated with stable, energy conservative, motion. In [5] these were provided with references to other papers.

In a sense, (2.27) is the final result of this thesis, computation wise, but it now remains to justify the expressions used for the post-Newtonian expansion of the orbital energy (2.7), and energy flux expansion (2.8), which will be derived in Chapter 4 and 5 respectively.

In this chapter the fundamental theory by which the orbital energy and energy flux will be calculated is derived. How can Einstein's general relativity be described as a classical field theory, and then recast into the language of EFT.

The derivations presented in this chapter largely follows those presented in Feynman [12] , with supplements from Maggiore [9] and Porto [4] .

The modern theory of gravity is partially split between two traditions. On the one hand there is the geometrical tradition following Einstein's approach by interpreting gravity as the effect of a curved space-time, which is curved according to the Einstein's field equations. The followers of this tradition may be called relativists. On the other hand there is the tradition of using the formalism of Lorentz invariant fields on a static, Minkowskian, space-time, inspired by its monumental success for electrodynamics and quantum theory. The followers of this tradition may be called field theorists.

Though these traditions are not entirely separated, the two different interpretations lend themselves to different natural extensions of general relativity. Thus the two traditions tend to separate relativists and field theorists by which theories they work on.

In this thesis the 1PN phase of GW produced by compact binaries are computed using the formalism of field theory. Familiarity with basic quantum field theory (QFT) is expected, but the derivations are otherwise supposed to be elementary.

One of the most famous field theorists, R. P. Feynman had a "gravity phase" from 1954 to the late 1960s (Di Mauro et al. [13] ). After having worked on the foundations of quantum electrodynamics, Feynman sought to uncover the quantum nature of gravity pursuing a sim-ilar method. He reckoned that gravity could, similarly to electromagnetism, be perturbatively expanded with respect to its coupling constant, and then quantized by quantizing the frequencies.

Quantizing gravity turns out to be a little more complicated than that, but Feynman's approach to classical gravity as a massless, spin 2, gauge field has made a lasting impression on gravity physics, especially in the context of GWs. This approach can be studied in the lecture notes from his lecture series of the 60's [12] .

To linearized order of the field (h µν ) in the resulting EoM, the Einstein-Hilbert action of general relativity is equivalent to the massless Fierz-Pauli action from field theory [14] .

When Feynman set out to study gravity, he took the mindset of a field theorist who until recently was unaware of gravitation, and just now have been presented with data suggesting that all masses attract other masses according to an inverse square law, proportional to the product of their masses,

Feynman envisioned this as the mindset of aliens on Venus who had just now acquired the technology to pierce through the atmosphere and measure the movement of the planets, but were still our equals in particle physics.

Their first impulse would probably be to guess that this is an unknown effect of some known field. After finding no field that could replicate the solar system observation, their next guess would be that there exists a new kind of field which mediates this mysterious force. Calling this hypothetical field the graviton field, and its associated quantum particle the graviton, the Venusians would next try to uncover its structure.

To construct the Lagrangian for this new force of nature they would determine that it has to be of even spin, and thus an even tensor rank, for the resulting static force to be attractive for equal charges, where the charge for the graviton field would be mass. For the force to go as an inverse square the field must also be massless.

Lastly, it must couple to all matter equally, but it must do so in a relativistic way. The natural suggestion is to somehow couple the field with the four-velocity of the source, like how the electric charge which the electric field couples to is promoted to the charge density four current j µ = γ −1 ρu µ , and couples to the vector potential A µ = (φ/c , A i ). See Gourgoulhon [15] or other textbooks on relativistic field theory.

However, to let the graviton field couple to all fields a natural candidate is the energymomentum tensor T µν , induced by field invariance under space-time translations. Incidentally, for a point particle it is constructed by the four-velocity of the source: T µν = γ −1 p µ u ν = γ −1 ρu µ u ν . Now for a scalar field it can be contracted to form a scalar, the trace, which is proportional to the mass density. Alternatively, a field of higher tensor rank can couple to the indices, also coupling the field to the mass density in the static frame T µν = T 00 δ µ0 δ ν0 .

The spin zero / scalar field is a candidate for the graviton, but fails to couple to the electromagnetic energy-momentum tensor, as the electromagnetic energy-momentum tensor is traceless. It also fails to predict the perihelion procession of Mercury correctly [15] .

Thus, the Venusians would probably try a massless spin 2 field next. Since massless fields only have one (spin s = 0) or two degrees of freedom (helicity = ±s) the symmetric spin two field should be easiest to work with, as it will have 10 − 2 = 8 redundant degrees of freedom. The antisymmetric field by comparison only have 6 − 2 = 4 redundant degrees of freedom.

Thus demanding the Lagrangian to be composed of a massless, symmetric, rank two tensor field there are only four unique terms, containing only second / two derivatives, after considering partial integrations:

Two where the index of the tensor and the index of the derivative differ.

1

Why are these terms of quadratic order in h µν ? Because it is action terms of quadratic order in a field which yields EoMs of linear order of that field. Note that term number 2. and 3. are the same after two successive partial integrations h µν,ρ h µρ,ν = −h µν h µρ,ν ρ = h ,ν µν h µρ ,ρ . Some texts use term 2. (like Maggiore [9] ), but here term 3. will be employed (like in Feynman [12] ). Thus, the free part of the Lagrangian 1 must be of the form 1 Terms ∼ h 1 and h 0 only contribute constants to the equation of motion, and can thus be removed by field shifts. Terms proportional to h 2 , but with no derivatives, determine the mass of the field ∼ m 2 hh, and must therefore be zero for massless fields. Lastly, the Lagrangian must be a scalar in order to be Lorentz invariant. There are no contractions of only 1 derivative and two h's that can produce a scalar. Therefore, to leading order in h n , the Lagrangian must consist of terms proportional to h 2 with two derivatives. See e.g. Schwichtenberg [16] , page 573-575, for a more detailed discussion.

can be used to fix the coefficients of equation (3.2c), and thus also the Lagrangian.

Thus a Lagrangian of a symmetric, massless, rank 2 tensor field which couples to a divergenceless rank 2 tensor field (e.g. L int = − λ 2 h µν T µν ), consisting of only second derivatives, in a flat space-time, must to second power of h take the form of the massless Fierz-Pauli Lagrangian [14] 

Here the overall factor has been set to a 1 = −1/2.

The EoM for L FP + 1 2 λh µν T µν follows directly from the Euler-Lagrange equation (3.2a) as

This is equivalent with the equation of motion found in the linear approximation of general relativity, for an appropriate choice of λ (see e.g. equation (1.17) of Maggiore [9] , or equation (9.16) of Grøn and Hervik [18] )

Varying the Fierz-Pauli action directly should also provide the equations of motion

This automatically holds because of (3.2c) (Ξ µν = 0). But (3.7) can also be solved using the condition (3.3b), Ξ µν ,ν = 0. Letting δh µν = −ξ µ,ν − ξ ν,µ it is easy to show that the action stays invariant under this type of transformation, using partial integration.

Thus the following transformation of the field leaves both the EoM and the gauge condition invariant.

Again, this is equivalent to the gauge condition found in linear theory when linearizing metric invariance under change of coordinates (see equation (9.9) of Grøn and Hervik [18] ).

Also introducing the commonly used bar operator, which symmetrize tensors and changes the sign of their trace,S µν ≡

the gauge condition for the barred h-field is obtained by transforming (3.9) as (3.10), resulting withh

Thus the divergence of this barred field transforms as

As ξ µ can be any vector without changing the EoM, it can be chosen such that ξ µ =h ,ν µν , shifting the field such thath ,ν µν = 0, which is to impose the Lorenz gauge 2 .

Lorenz gauge condition :

The exact expression for ξ µ can be obtained by method of Green's functions, but this is unnecessary to compute. Simply keeping in mind thath ,ν µν = 0 shall suffice to simplify the Lagrangian (3.5). Notice that the following terms must be zero, using again (3.10), but in reverse.h ,ν

Adding and subtracting 0 from the Lagrangian should change nothing, and thus the following expression can be used as a gauge fixing term (gf)

with the subscript (2) to signify that this is the action to quadratic order in h. The EoM of L (2) + L int is the familiar

from linearized theory (see equation (1.24) of Maggiore [9] or equation (9.22) of Grøn and Hervik [18] ).

Comparing with the EoM of linearized GR it is tempting to conclude that λ ≡ 4κ = 32πG/c 4 , but then it is also common to make the Einstein-Hilbert action dimensionless by scaling it with a factor of (16πG/c 4 ) −1 . Comparing (3.16) with the Einstein-Hilbert action expanded to second order 3 the Lagrangian (3.16) carries an additional factor of 2, and is dimensionful. Field theorists usually fix the dimensionality of the action by rescaling their fields to become dimensionful. Doing this

h adsorbs the dimensionful prefactor. To compare (3.17) with the linearized Einstein's field equations, it must first be rescaled back to a dimensionless field according to (3.18) , and then the coupling constant is revealed to be

where κ = 8πG c 4 is the constant which appears in Einstein's field equations. Some call this coupling constant M −1 Pl rather than λ ( [4] , [3] , [19] ). However it does not have dimesion of mass, nor is the Planck constant anywhere in the expression, so why do they do this? These articles use natural units = c = 1, and M −1 The action must be dimesionless in QFT, since in the path integral approach it is exponated. Every field Lagrangian has a kinetic term ∼ ∂φ∂φ, which scale as

Thus for couplings L int ∼ λh(∂h) 2 to have the same dimension as the kinetic term;

Calling the coupling constant M −1 Pl might have the unfortunate consequense of making it look like a quantum theory, but make no mistake, this is all classical field theory. Therefore, it is simply labelled λ in this thesis.

According to the equation of motion (3.17) the field will in a vacuum (T µν = 0) behave as a relativistic waveh

which admits solutions of the form (2.2). See Appendix A for a derivation of this solution.

Up until now the gauge has only been used to make sure the entire EoM (3.17) remains divergence free, just like the source term T ,ν µν = 0. In doing so it was determined that the field might only be shifted according to (3.9) . Furthermore, the divergence of the barred h-field could be eliminated only imposing further that ξ µ = 0.

Keeping ξ µ = 0 still leaves

with four degrees of freedom, as it is a function of the four independent parameters ξ µ , which satisfy ξ µ = 0. To make the graviton field divergence free imposes four additional conditions on h µν , by the four equations h ,σ µσ = 0. This leaves h µν with 10 − 4 = 6 degrees of freedom. Subtracting further the four gauge freedoms reduces h µν to only two effective degrees of freedom, as any massless spin two field should have. These four gauge freedoms ξ µ can be used to impose four additional conditions onh µν . ξ 0 can be used to set the traceh = 0, and sinceh µν is just h µν with the reversed sign trace, in this gauge h µν =h µν .

The three remaining freedoms, ξ i , can be used to set h 0i = 0 as well. Since h ,ν

A constant contribution to a GW are uninteresting and for all intents and purposes it can be considered to be zero, making all h 0µ = 0.

This specific gauge is referred to as the transverse-traceless (TT) gauge, and is defined by TT gauge condition:

Note that this gauge can only be imposed in a vacuum, since the vacuum conditionh µν = 0 ⇒ ξ µν = 0 was used.

as the scalar solution to the wave equation

which is (A.6) from Appendix A. The TT gauge condition then implies

For a wave travelling in the z-direction k µ = k, 0, 0, k , making µ3 = 0. With (3.22) this is enough to determine the polarization down to the two essential degrees of freedom

The general solution of the equation of motion (3.17) with sources can be obtained by method of Green's functions.

The Green's function of a linear differential operator LO x is defined as the function that satisfies

where LO x only acts on x.

The differential equation in question

admits solutions of the form

This solution can easily be demonstrated to recover the original differential equation by using the definition of the Green's function,

which was the original differential equation.

Green's function of the d'Alembert operator

. In order to find the corresponding Green's function ∆(x − x ) the easiest way is to go through Fourier space.

Matching the last equality of both lines indicates that the Fourier transform of ∆(x − x ) must be 4

Obtaining the Green's function in real space is now just a matter of transforming (3.27) .

One last remark about Green's functions is in the context of four dimensional space-time the solution in terms of Green's functions can be understood as counting up contributions from source terms, all over space, and across all time

Thus ∆(x − x ) weighs the importance of source contributions at different points in space and time. For physical solutions only contributions of source configurations from the past contribute to h(t). This is imposed by demanding t ≥ t .

Back to deriving the real space Green's function. Performing the transform, and defining

where the spatial integral was performed in spherical coordinates. Relabelling |k| = k and |r| = r, the expression can be worked further

Extending to the complex plane, this integral can be evaluated using Cauchy's residue theorem, from complex analysis. Shifting k 0 → k 0 +iε is equivalent to imposing the retardation condition:

Why this is the case should become apparent soon. According to the residue theorem

where a k is a pole of f (z) enclosed by the integral, and

(3.28)

Utilizing this theorem and integrating over the upper complex plane, the Green's function becomes

It is now apparent that this is the retarded solution. Had k 0 rather been shifted by −iε, the Dirac delta function would have been δ(ct + r). The delta function picks out contributions on the light cone of x α , the retarded Green's function picks out on the past light cone, while the advanced Green's function picks out on the future light cone.

Beyond singling out contributions from the light cone, it is apparent that the importance of each contribution to the field is weighted by how far away it is from the point in question, according to the inverse power of the spatial distance.

Back to the EoM (3.17), using the retarded Green's functions it admits solution of the form

Assuming the GW is measured far away from the source compared to the size of the source system, then the approximation |x − x | ≈ |x| ≡ R holds. This simplifies the integral in equation (3.30) to only be dependent on the energy distribution of the source, and not where it is measured.

Furthermore, waves measured in vacuum can be set into the TT gauge, which implies that all physical information of the source can be captured by its spatial indicesh ij .

The volume V must cover the entirety of the spatial extension of the source, in order to be equivalent to the infinite integral of equation (3.30) .

To simplify the expression of equation (3.31) further it is useful to note some properties of the energy-momentum tensor.

First: it is divergenceless.

Second: use of the following integral will be made.

In the last line the divergence theorem has been used. If the integration boundary is taken to encapsulate the entire source, T ik ∂V = 0, equation (3.33) becomes equal to 0. It then follows

(3.34)

In the last line the symmetry T {ij} = 1 2 (T ij + T ji ) was written out explicitly. So far not much has been accomplished, but notice how this procedure can be repeated to eliminate all dependence of the spatial components of T µν .

Using that T 00 = ρc 2 is the mass-energy density, the integral in equation (3.31) takes a simple form of the second time derivative of the so-called quadrupole moment

Finally, the source of linearized GWs is provided as

Utilizing this result the GWs generated from any source, with a non-vanishing energymomentum tensor, can be calculated at distances sufficiently far away from the source.

Note that this does not mean that the energy-momentum tensor of a source is converted into GWs. This formula is simply a result of the interaction term of the graviton action to source energy-momentum tensors, e.g. from the energy-momentum tensor of stars or black holes. However, in section 3.5 it will be demonstrated that these GWs do carry energy, and since this energy must come from somewhere it will be assumed to come from the source. This amendment of the theory is the topic of section 3.7.

Equipped with a field Lagrangian it is now desirable to check that it in fact reproduces the Newtonian law of universal gravity for non-relativistic sources.

In special relativity the action of point particles are the geodesics 5 of Minkowski space-time. The geodesic of some non-trivial space-time, with metric g µν = η µν + λh µν is

Why compute the geodesic for a perturbed Minkowski space? In the relativists' approach GWs is such a perturbation of the metric (at least for vacuum solutions), thus this is the action of point particles to linearized order in h µν , according to relativists. It is included here to motivate the interaction term L int = λ 2 h µν T µν , with the point particle energy momentum-

Otherwise, it just shows that the free point particle Lagrangian (the kinetic part) is just

The constant term −mc 2 does not contribute to the EoM, and can therefore be neglected. 5 Extrema, e.g. shortest, path between two points in space-time. In Euclidian space it is the straigh line connecting the two points. 6 The factor of γ −1 a is a result of rewriting the proper time integral into a generic time integral dt dτa = γa, common for all particles. Writing out the four-velocities should result in an overall factor of γ 1 a . For more info on the pp energy-momentum tensor see e.g. Gourgoulhon [15] .

Thus, the total Lagrangian of two point particles interacting only via the graviton field, up to the 0PN order, is

The EoM for the h field is known from (3.17) . Since the bar operator is its own inverse operator when used on symmetric tensors,S µν = S µν , equation (3.17) may also be written as

Substituting (3.42b) for h µν in the interaction term results in the following action for the point particles.

The next trick is to first approximate the space-time distance to mostly be in time, for slow moving, not too far separated point particles. Then the middle Dirac delta is approximately δ x 0 − y 0 . After using that approximation to eliminate the y 0 integral, the spatial integrals are next, which due to the remaining Dirac deltas just makes all x → x a (t), y → x b (t), and r = |x 1 − x 2 |. Lastly, to leading order in powers of c the only contributing factor ofẋ µ is the time componentẋ 0 a = γ a c 2 ≈ c 2 . The result is

which is exactly the action of Newtonian theory for two point particles with mass. Notice in equation (3.43) , the potential can be understood as the energy-momentum tensor T µν a (x) connected to the energy momentum tensor T αβ b (y) by some sort of propagator, defined by the contents of the curly brackets. This can be expressed graphically, as in Figure 4 .1, and will be further developed in Chapter 4.

Once the action is known, the energy can simply be derived by finding the corresponding Hamiltonian

where p i is the canonical momentum, and q i are generalized coordinates. This is nothing but a Legendre transformation of the Lagrangian forq i → p i . Unsurprisingly, for the Newtonian action it reads

In relative coordinates it reads (see Appendix B for derivations and tricks)

which is exactly the first order term of the energy expansion (2.7).

Equipped with the Lagrangian for linearized theory (3.16) it is straight forward to utilize Noether's theorem to obtain the energy-momentum tensor of the graviton field.

For coordinate transformations

Noether's theorem is a central result of modern physics, and for a detailed derivation consult any field theory book, e.g. Kachelrieß [17] , Maggiore [9] or Goldstein et al. [2] . For pure translations A β α = δ β α , F i,α = 0, and thus

The terms in this equation is known, the first from (3.2b), 7 and the second from (3.16) .

For a wave-packet centred around a reduced wavelength λ ≡ λ/2π, the total 4-momentum flux is obtained by integrating over a volume V ∼ L 3 where L λ, such that t µν is zero on the boundary ∂V. Then the effective energy-momentum tensor is 8

Here the x is to be understood as spatial averaging. Also, for radiation detectable far away from the source the EoM can be considered to be for a vacuum, and thush µν = 0. Setting the vacuum as the zero solution of the h-field the Lagrangian averages to zero L (2) = 0, and is thus dropped from (3.51). Additionally, in vacuum the Lorenz gauge can be promoted to the TT gauge, further imposing that h = h 0µ = h ,j ij = 0 (3.22), results in the final expression for the energymomentum (pseudo-) 9 tensor

(3.52a) (3.52b) 7 To match this expression to (3.16) set a1 = −2a4 = −1/2 and a2 = a3 = 0, which is to impose the Lorenz gauge. This gauge can only be imposed after evaluating the ∂L ∂h µν,β term. 8 The Noehter current need not be physical in itself, it is the volume integral of j ν a , where j ν a goes sufficently fast to zero on the boundary, which is conserved, and thus physical. In field theory one defines the effective energy-momentum tensor, which is the average value over the volum integral, i.e. the spatial average, as the physical energy-momentum tensor. All terms lost under spatial averaging would not contribute to physical effects anyway. This is why (3.51) is averaged over space.

An interesting lesson here is that the energy of GWs can not be isolated to one space-time point, which can be understood by them being non-localisable. 9 Note that this is the energy-momentum tensor in the TT gauge, which is to say it is gauge dependent! This might sound strange, but as Maggiore [9] points out this is also the case for electromagnetism.

In the geometric approach the gauge dependence of tµν is a coordinate dependence, since the gauge transformation is a coordinate transform in the geometrical picture. Thus tµν is not a real tensor, but rather a pseudotensor. Also in the geometrical picture, tµν is averaged over several wavelengths to extract the GW contribution to the background metric, making it equivalent to the energy-momentum tensor obtained by field theorists.

The total radiated energy flux F can be obtained from P 0 = E/c and the divergencelessness of the energy-momentum tensor, which follows from Noether's theorem t µ0 ,µ = 0,

where the last equality used the divergence theorem, expressed in spherical coordinates. For the next step notice that t r0 = ∂ r h TT mn ∂ 0 h mn TT , and recall that h's sufficiently far from the source can be written as h TT mn = 1 r f mn (t ret ) with t ret = t − r/c and f mn some function. Then

Thus in the TT gauge, and sufficiently far from the source, t r0 = t 00 + O r −3 , and

Note thatḣ = dh dt does not include any additional factors of c. It is apparent that GWs carry energy out of a volume, hence the negative sign convention for F. The term of order r −3 can be neglected for large values of r, while in the remaining term the r dependence cancels, making it non-vanishing.

Before computing the angular integral it is important to note that the radiation is not isotropic, but rather a function of direction n. Following the outline of Maggiore [9] this can be accounted for by introducing the so-called Lambda tensor

(3.56)

The Lambda tensor is a projection operator which is defined in such a way that it projects a wave already in the Lorenz gauge (3.13) into the TT gauge (3.22) .

which makes sure the wave is transverse with respect to the direction of propagation n.

The Lambda tensor has the property

making it a projection operator. Thus, the contraction of two waves in the TT gauge becomes

i.e. the Lambda tensors can be used to contract tensors not in the TT to get the contraction of the TT gauged tensors, which is exactly what is needed for equation (3.55 ). The last equality utilized the property Λ ij:kl = Λ kl:ij of the Lambda tensor. When integrating over the Lambda tensor the following integral appears

Here n is taken to be a unit vector |n| = 1. To illustrate the solution take = 1 → dΩ /(4π)n i n j = S ij . The trace of S i i = 1, because n i n i = |n| = 1, reducing the integral to the scalar integral dΩ = 4π. The tensor structure of S ij should be that of the Kronecker delta, as it should be symmetric in its indices, and have a non-vanishing trace. Thus,

Generalizing this argument it follows that the integral (3.60) should be the sum of all possible Kronecker delta combinations, to maintain index symmetry. This is indicated by the dots on the RHS of (3.60). The symbol !! is here meant to imply the product of every second number: (2 + 1)!! = 1 · 3 · · · (2 − 1) · (2 + 1), and 2 !! = 2 · 4 · · · (2 − 2) · 2 . This fixes the correct trace value, just like the factor of 1/3 for the = 1 case. Note that all integrals over an odd number of unit vector components are 0 because the integral is also odd then.

Thus, the integral of the Lambda tensor becomes

Utilizing this result for (3.55), and the fact that Q ij = Q ji , yields the flux in terms of the mass quadrupole with Q ≡ Q i i . This is an important dynamical property of GWs, that they can carry energy, momentum, and angular momentum out of a system. For a compact binary this influences the orbital energy, extracting energy from it over time, make them spiral in towards each other. The fact that there is no analytical exact solution to the relativistic binary problem is often attributed to this effect. The orbit is affected by GW radiation, which is determined by the orbit, making the problem complicated.

As an illustrative example this section will calculate the exact form of GW produced by binary systems in circular orbits, according to linearized theory (3.16) .

Those already familiar with GWs may skip this section, as this system will be covered in greater detail in the following chapters, and is here mearly used as an example to better illustrate the result (3.38) and (3.62).

Taking the centre of mass as the origin, the positions of the stars are r 1 = r 1 cos(ωt), r 1 sin(ωt), 0 , (3.63a) r 2 = − r 2 cos(ωt), r 2 sin(ωt), 0 . This determines the mass density as

The quadrupole moment of the binary system is thus For convenience rω = (r 1 + r 2 )ω has been replaced by v, which is the sum of the velocities of the stars. This is equivalent to the frequency parameter v 3 = GM ω introduced in Chapter 2.

It should be noted that the wave frequency ω GW = 2ω = 2ω s , with ω s as the source frequency, is the dominant frequency for circular orbits, and that the amplitude is proportional to the frequency v 2 = (GM ω) 2/3 . This makes the GW highly dependent on the frequency of the source binary, both for the amplitude, and frequency spectrum.

The energy flux produced by such a system is now easily computable by using equation (3.62) together with (3.66).

This is exactly the first order term of the flux expansion (2.8). Furthermore, circular orbits is a solution of the Newtonian, point particle, action (3.44). If the source interact with the graviton field, and only the graviton field, this implies that the orbital energy of the system is E Newt = − 1 2 µv 2 (3.47). With (3.68) and (3.47), the Newtonian order (0PN) version of the phase (2.27) can be derived, assuming compact binaries can effectively be treated as point masses.

The flux is obviously not zero, which means that energy is dissipated out of the system. However, according to the Lagrangian (3.16) the energy of the source T µν is conserved. Therefore, this energy radiation is an inconsistency of the theory.

This hits deeply into the problem of gravity. As a field theory it couples to energy, while as a field it itself stores energy, and should therefore couple to itself. Amending this feature is the topic of the next section.

In Section 3.5 it was found that gravitons / GWs can carry energy, and can even be given an effective energy-momentum (pseudo-)tensor of its own. But since the graviton field couples to energy-momentum tensors, should it not couple to itself?

In Section 3.6 it was even found that a binary system of point particles radiated energy in the form of GWs, but where does this energy come from? One natural assumption would be that the energy is extracted from the orbital energy associated with the system, leading to an inspiral. But this would mean that the energy-momentum tensor of the binary T ,ν µν = 0, as it changes over time. Furthermore, T µν ,ν = 0 was used to determine the graviton action in the first place, making the conservation of energy of the source, and the energy carrying capability of the graviton field a theoretical inconsistency!

The solution is to change the energy conservation criteria (3.3b) to

which is to make the energy of the source plus the energy stored in gravitons (i.e. in GWs) conserved, together.

To implement this in a new and improved Lagrangian for the graviton field the first naive idea would be to couple the graviton directly to the graviton energy-momentum tensor L int → λ 2 h µν (T µν + t µν ). This will however lead to problems for the EoM, as

Since it is the EoM which dictates the physics of any given action, one would rather demand that

This requires the addition of a more general cubic term to the Lagrangian λ 2 L (3) such that

Writing out all possible terms cubic in h, containing only two derivatives (necessary to obtain t µν ), which also produces t µν when varied (3.72), together with new energy conservation condition (3.69) fixes the 18 coefficients uniquely (up to the same overall factor of L (2) ). This is obviously a lot of work, but the result can be quoted from Feynman [12] (equation (6.1.13)). 10

(3.73)

Using again Noether's theorem on this new Lagrangian L (2) +L (3) , the new effective energymomentum tensor picks up a term cubic in h:

Again it can be argued that the cubic term here is not conserved, and the process of this section can be started all over again. In fact, it turns out that

with the EoMh

This makes the equation of motion nonlinear, and in general both the action and the EoM contain infinitely high powers of h.

By now the Venusians would probably be disappointed, but for terrestrial physicists this should be expected, as GR is also a nonlinear theory. When expanded in powers of metric perturbations, also the Einstein-Hilbert action contain infinitely high powers of h, which coincide with the action found thus far in this thesis. 11 An important observation is that all these corrections scale with increasing powers of λ, which is small. Thereby the more powers of h it contains, the less it contributes to the final result. Thus, the theory can be perturbatively expanded in powers of λ.

In this chapter the 1PN energy of compact binaries in circular motion is computed.

The derivation follows closely those presented in Porto [4] , and Goldberger [19] , [3] .

In the last chapter examples focused on point particles. Compact objects like BHs and NSs are however not point particles. But for sufficiently far separated binaries, the separation distance r will be much greater than the 'size' of the compact object, which can be approximated as the Schwarzschild radius ∼ R S r. Then the system can be described by an effective action, treating the compact object as a point mass at the scale ∼ r.

On the other end of the scale the system is producing GWs, carrying energy out of the system. But when calculating the energy flux in Section 3.5, the source was assumed to effectively be a point endowed with quadrupole structure (which will be expanded to a general multipole structure in Chapter 5), since the GW was measured far away. That is, the flux is measured at a scale L λ r. 1 Thus, in this chapter the effective action at the scale r, with R S r L, will be derived, which approximates the immediate 2 orbital dynamics. To compute the 1PN orbital energy the procedure of Section 3.4 will be expanded. In Section 3.4 it was found that the graviton potential could be expressed graphically like Figure 4 .1. In Figure 4 .1 the two sources are depicted as solid lines, like in a space-time diagram, while the Green's function is depicted as a squiggly line, connecting the two point particles. Graphical representations of this kind are called Feynman diagrams, as they were introduced by Feynman [21] to illustrate expansion terms of QED. Note that the Green's function has been given 1 The wavelength can be shown to be greater than the size of the binary by noticing two things. First: ωGW ∼ 2ωs, as approximated in (3.67). The reduced wavelength λ = λ/2π is related to the angular frequency as λ = c/ω. Second: Kepler's third law (2.4) relate ω and r: λGW = c 2 √ GM r 3/2 = r 2R S r r. 2 Immediate because over time the energy loss through GW emssion can not be ignored. However over short periodes of time, the orbital energy can be approximated to be conserved. the subscript 'inst', to remind that the Newtonian action was recovered by approximating δ 4 (x µ − y µ ) ≈ δ(t − t ), making the interaction instantaneous.

For those familiar with Feynman diagrams, note that the solid lines are not propagating, even though they are depicted the same way as propagating fermions in QFT. This can be confusing, but it is the convention introduced by Goldberger and Rothstein [3] , and has by now become the standard.

Based on the findings of Section 3.7, the Einstein-Hilbert action 3 S EH can be used to find the expression of the propagator for the h µν field, by solving equation (3.76). Expanding the Einstein-Hilbert action in powers of h yields the contribution for the different levels of self interaction, which all scale with additional powers of the coupling constant λ.

Note that the terms in (4.1) are symbolic representations for the powers of h µν , and that all terms contain only two derivatives. Why does the first order correction lead to a three graviton vertex? Looking at the leading order correction in the EoM (3.76)

where again ∼ implies symbolic relation of powers of h µν , rather than the exact relation. The leading order term of this three-point propagator will be derived in Section 4.2.7. It can however be seen from this simple relation, where the h's of the RHS was substituted for the linear order solution, that at this order h is produced by two sources, which in principle can be at different space-time points. Therefore, an interaction h µν T µν (x) in the action will to next order connect three energy-momentum tensors, using two Green's functions. This can neatly be visualized by a three graviton vertex.

In Section 3.4 the action of point particles (3.39) was found to also be expandable in powers of h, similarly to the Einstein-Hilbert action.

Here again the choice of having two gravitons coupling to the world line for the ∼ h µνẋ µẋν 2 term is natural as the integral can be split in two integrals multiplied together

, which couples this source effectively to two different gravitons. In Section 3.4 this was argued to be a natural choice for the point particle action, as it is the expansion of the action of GR. It is also a natural choice for a classical field theory with a coupling to a symmetric rank 2 tensor field, and is equivalent to equation (11.40) of Gourgoulhon [15] .

Because of the separation of scale, ∼ r R S and ∼ L r, it will be useful to split the graviton field into a short-range, potential field (H µν ) and a long range, radiation field (H µν )

From here on out the potential field will be drawn using dashed lines, while the radiation field will continue to be drawn using squiggly lines. The frequency of GWs are proportional to the frequency of the source binary ω GW 2ω = 2ω s , and the relative velocity of a binary in circular motion is v = ωr. Using the relation between null wave frequency and wavelength ω = c|k| = cλ −1 the wavelength of GWs scale as

for binaries moving at non-relativistic speeds.

Because H is null-like, it follows that k σ k σ = 0, and

i.e. H must be on shell.

H on the other hand is a potential field, and can not be on shell, since it shall reproduce the gravitational potential in the static limit. In Section 3.4 this was achieved by approximating δ 4 (x − y) ≈ δ(t − t ). This 'instantaneous' propagator is not the Green's function of the d'Alembertian operator

but it is the Green's function of the Laplace operator ∇ 2 ≡ ∂ i ∂ i . This integral scales proportional to |k|

Assuming k 0 is small compared to k i , the instantaneous propagator may be obtained as a leading order term of an expansion in k 0 /|k| 

Not only is this a highly constructed expansion to obtain a desired leading order term, it will be demonstrated that this expansion actually scales as an expansion in (v/c) 2 , and will be drawn graphically by ⊗ on the propagator to show which order in the (k 0 /|k|) 2 expansion it represents. This also implies that k 0 /k i ∼ v ⇒ k 0 ∼ vk i ∼ v/r, and in conclusion

Now Feynman diagrams may be constructed by putting together terms as presented in (4.1), (4.3), and (4.9), but the following three rules must be upheld to make sense as an expansion term in the point particle action. Rule 2-3 follow by definition of H µν and H µν . Rule 1 however seems more mysterious, but it is a consequence of the solid lines not propagating, and thus is just a requirement that the Feynman diagram is connected, as all Feynman diagrams must. Unconnected diagrams are simply separate diagrams, and represents multiple terms in the expansion at once. 

In order to acquire relativistic, post-Newtonian, corrections to the Newtonian Lagrangian (3.44) it is a straightforward matter to just add additional terms of the form found in expansion (4.1), (4.3), and (4.9) to the action. But (4.1) and (4.3) are expansions in λ and H rather than in (v/c) 2 , so how can it be determined which Feynman diagrams contribute at which PN order?

Using the scaling of k µ for the potential field (4.10) and Kepler's law (2.4) , it turns out that it is possible to assign a power of v, r and m to each Feynman diagram, and thus select the appropriate diagrams and terms relevant to each PN order, all without doing the full calculation! This is why the diagrams are introduced in the first place, as they are tools to make it easier to order expansion terms.

The scaling of coordinate (integration variables) follows from the relations of k µ .

Notice that Dirac delta functions carries inverse dimension and scaling of its argument (and the exponential function is of course dimensionless) 4 . For each graviton H µν in a diagram, it scales as

Note that to leading order in interaction terms H µν couples only to temporal components µ = ν = 0, sinceẋ 0 = γc ∼ c, andẋ i = γv i ∼ v, relegating spatial indices to higher PN orders compared to the temporal ones. Since (4.1) and (4.3) expands in powers of λ, it would be useful to associate a scaling with 4 Since e x = 1 + x + x 2 2! + . . . , and all the terms in a sum must have the same dimension.

the coupling constant. This can be achieved using Kepler's third law (2.4) , and v = ωr.

The orbital angular momentum scale L ∼ rmv has been introduced as a convenient scaling, as will be demonstrated shortly.

It is now a straightforward exercise to assign PN orders to different diagrams:

(4.14)

Thus the Newtonian diagram, Figure 4 .1, scales as (4.14) squared, making L the scaling of the 0PN order. This is why introducing L = rmv as a scaling is convenient, as it easily makes the leading order scaling apparent. Now, all 1PN diagrams should scale as Lv 2

The next interaction graph is similarly found to scale

And so it goes on...

(4.16)

The last type of diagram that needs to be assigned a scaling is the multi graviton vertex propagators. Note the inclusion of a three-dimensional Dirac delta, to make sure momentum (k i for graviton i) is conserved. This is because the gravitons only transfers momentum from one source to another (Newton's third law), and therefore the sum of momentum in and out of this vertex should be zero. Otherwise, the graviton field would spontaneously generate momentum and energy. The two spatial derivatives comes from the definition of t µν (2) , which consist of two derivatives ∂ j ∼ x −1 j ∼ k j , and two factors of H, which is the reason the propagator exists in the first place. Notice that ∂ j ∼ r −1 , while ∂ 0 ∼ v r , relegating temporal derivatives to higher PN orders.

At the next order, the only change is an additional factor of λ, and H. This follows from the expansion of the graviton energy-momentum tensor (3.76) .

As already mentioned, spatial indices in the interaction term between the source and graviton leads to additional powers of velocity (H 0iẋ 0ẋi = H 0i γcγv i ), and thus belong to higher PN orders. These will graphically be represented by a v n next to the interaction vertex, with n describing the power of v correction the diagram represents.

All thinkable diagrams belonging to the 1PN correction scale as Lv 2 , and are depicted in 

In this first diagram nothing has changed from the Newtonian diagram (see Figure 4 .1), except a coupling between the velocity of the two sources 5

Integrals over Dirac deltas should be straightforward, leaving x i 1 P 0i:0j x j 2 as the only new and interesting part. Note that P µν:αβ is symmetric in µ ↔ ν, α ↔ β, and µν ↔ αβ, and therefore all permutations of the indices belong to this same diagram. The last symmetry, µν ↔ αβ, is just a relabelling of m 1 m 2 ↔ m 2 m 1 , and is superfluous. The other symmetries also result in the same potential, but should be summed over, producing a factor of 2 · 2 = 4.

Recalling the definition of the projector P µν:αβ , (3.42c) 6 , this is

Thus the potential of diagram 4.3a is

(4.21)

According to the sign, this is seemingly a repulsive force, but it is dependent on the dot product of the two velocity vectors, which for any Keplerian orbit will be negative (see Figure 3 .1). Of course, it also scales by an additional factor of v 2 /c 2 compared to the Newtonian term, as a 1PN term should.

But for particles moving in the same direction it is a repulsive force, making it comparable to a magnetic type force. For oppositely charged particles, which should attract in the static approximation, will also repel each other magnetically when moving in the same direction, quite analogously. The analogy goes even further, as magnetic forces can also be interpreted as relativistic corrections to the electric force [22] .

This diagram mostly follows suit of Section 4.2.2, but with a few additional details. These are • The relativistic expansion of the free point particle kinetic energy is added here, as it also scales ∼ v 2 a compared to 0PN. • The relativistic expansion of the Lorentz factor of the energy-momentum tensor is expanded, as it also scales ∼ v 2 a compared to 0PN. • And of course, the P 00:ij and P ij:00 couplings belong to this diagram.

The first term is just the velocity expansion of the free point particle action Next notice that there is also a Lorentz factor in the energy-momentum tensor, that until now has been approximated to 1. x a (t) ) .

(4.23)

Other than the factor of 1 2 v 2 a c 2 this is exactly the same as the 0PN potential. Lastly, the velocity dependent coupling. Using again (3.42c)

To the first PN order, only one particle at the time may be expanded this way, therefore the v 0 -v 2 vertices in the diagram. But really both the v 0 -v 2 diagram and the v 2 -v 0 diagram belong to the 1PN potential, therefore these will both be summed up here, under the same diagram.

Summing up all these contributions to the 4.3b diagram yields Here (4.25a) 7 is the result of (4.22), (4.25c) of (4.23), and (4.25b) of (4.24). Substituting H 00 (x 0 , x a ) for the usual term (3.42b) yields the potential

This potential is attractive, and proportional to the Newtonian kinetic energy. Thus, this diagram can be thought of as the gravitational attraction from the kinetic energy of one of the particles on the other, showing that in GR gravity is an attraction of energies, and not just masses. Using Einstein's mass-energy equivalence, the 'Newtonian kinetic mass' is E = 1 2 mv 2 = m kin c 2 , and inserted into Newton's law of gravity produces V (b) /3. This analogy can not explain the missing factor of 3, because the analogy only account for the (4.23) part of the potential.

Notice that terms of the type (4.23) can in principle be expanded to infinite orders, thus there is no reason to stop expanding vertices past the v 2 order.

After having computed diagram a and b, a natural idea of a v 0 -v 1 coupling emerges as a 0.5PN contribution.

The situation is analogous to Section 4.2.2, only here the connection is not symmetric. Thus, the potential is also here the Newtonian potential multiplied by some velocity factor. which follows from P 00:i0 = 1 2 (2η 00 η 0i − η 00 η i0 ) = P 00:0i = 0. A lesson to be taken from this is that P µν:αβ only couples sources which both have an even or both an odd number of temporal indices. E.g. 00:00, 00:ij, and 0i:0j. Terms like 0i:00, or 0i:ij will turn out to be zero. Therefore, this kind of interaction expansion can only produce even powers of v/c.

The last 'H'-shaped diagram is not expanded in its vortexes, but rather the propagator of H is expanded to second order in v, according to expansion (4.9) . In that expansion, the propagator was only expanded in Fourier space, so it remains to determine the value of the first order correction of the propagator in real space.

(4.29)

From the gradient relations (4.10) it is evident that (k 0 /|k|) 2 ∼ v r / 1 r 2 = v 2 and thus belong to the 1PN correction, but the scaling of k 0 ∼ v r was not truly argued for. So this section will demonstrate this scaling by computing the propagator in real space.

To evaluate the integral in (4.29) some tricks are in order. Writing out the exponent as e ikµ(x µ −x µ ) = e −ik 0 (ct−ct ) e ik·(x−y) , a useful way to proceed is to rewrite the integral as k 2 0 e −ik 0 (ct−ct ) = − ∂ 2 ∂ct∂ct e −ik 0 (ct−ct ) . In the following note that the time derivative is only operating on the k 0 integral.

To proceed the derivative of Dirac's delta function must be determined. To that end, partial integration is the key.

the last equality only holding for x 0 / ∈ ±∞. The result generalizes for any number of derivatives as

Back to equation (4.31b), the situation is analogous, thus partial integrations can also be utilized here to rewrite the derivative of the Dirac delta function in the same manner.

(4.34d)

This final integral might not look like that much of an improvement, but these are the kinds of integrals often encountered in QFT, and there exists a known solution [4] .

Here Γ(z) is the gamma-function, and d is the spatial dimension. In the integral of equation (4.34d), d = 3 and α = 2. Recalling that Γ(1) = 1, Γ 1 2 = √ π, and zΓ(z) = Γ(z + 1), it follows that Γ(2) = 1 · Γ(1) = 1 = Γ(α). In the same fashion − 1

π. Then, the final expression of the potential follows as

Again, this is obviously of first post-Newtonian order, as it has an additional factor of v 2 c 2 compared to the Newtonian potential. From the procedure it is hopefully clear how additional powers of (k 0 /k) 2 leads to additional time-derivatives, additional powers of c −2 , and additional powers of v 2 . Thereby the scaling k 0 ∼ v r should be justified. Like diagram 4.3a, this potential is also dependent on v 1 ·v 2 , but has the opposite sign. Thus, it is attractive for particles moving in the same direction, and repulsive for particles moving in opposite directions.

Some insight might be gained from the projector in line (4.36b)

where n = r/r. Contracting P ij (n) with a vector x has the effect of projecting x onto the orthogonal plane of n

Ergo, P ij (n)x j is orthogonal to n.

With this in mind, the potential only couples velocities orthogonal to r, and is equivalent to

where v i was exchanged for v according to (B.4), and p ≡ µv. In the last line, ω = r × v/r 2 was used, and the term inside the parenthesis is the kinetic energy associated with rotational motion of a particle with effective mass µ.

A last observation is that the Lambda tensor, introduced in Section 3.5 (3.56), can be defined using this projection operator (Maggiore [9] )

The last two diagrams have a higher order of H's and are thus non-linear in λ. Therefore, these diagrams should be proportional to G 2 . Diagram 4.3d represent a second order in H coupling between the graviton field and the point particle, while the last diagram represents the higher order propagator, where H couples with itself. It was argued in equation (4.15) and (4.17) that these diagrams belong to the 1PN correction, but then only coupling to the 00 component of T µν . Thus, the same expansions preformed for diagram (a)-(c) will need to be implemented to these non-linear diagrams, when computing higher order PN corrections.

Notice that particle a interacts with the graviton field twice, and possibly at different times, while particle b only interacts once, but to a higher order, thus connecting it to the other particle through two graviton propagators, as depicted in the diagram. The sum adds the mirrored diagram as well.

There is nothing surprising in this diagram, and the integrals can be carried out without any fuzz, eliminating two of the three time integrals, making both graviton exchanges instant-aneous, and simultaneous.

One way to interpret this result is as a gravitational coupling between the total mass of the system, and half the Newtonian potential energy itself

This does not however make much sense as being separated the distance r, but it is an interesting analogy.

Note that in this section spacial indices are suppressed, as subscripts are used to enumerate vector variables. The last diagram makes use of the three graviton propagator, and is the first one not to use the linear propagator.

(4.43)

Here the propagator in the second line needs a lot of work. It can be read off from Goldberger and Rothstein [3] , equation (37)-(39). But it will also be derived in the following. Much of the structure is already given in (4.2), which was found inserting leading order results 8 for h in the expanded EoM.

To make sure momentum and energy is not spontaneously generated by the propagator, one should demand the momentum vectors k i sum to zero. This is achieved by multiplying the integral by (2π) 4 δ 4 ( i k i ).

(4.45)

The factor of (k 0 1 ) 2 + k 2 1 = k 2 1 1 + (k 0 1 ) 2 /k 2 1 , which was found to induce factors of v 2 past leading order in Section 4.2.5. Therefore, this factor will be approximated as k 2 1 . Furthermore, when one of the k i integrals are preformed; the Dirac delta function will eliminate that vector by k 1 + k 2 + k 3 = 0. For example preforming the k 1 integral yields

It is the product of two propagators, connecting two different points to the same third point, just like in the diagram. This result could also have been argued to result from Wick's theorem, like Porto [4] does, but then it would be all pairwise combinations of points, including ∆ inst (x 2 −x 3 ). It was discarder here because the k 1 propagator was eliminated by the derivative, which again followed from k 1 being the main transform, i.e. the lone graviton going in the final interaction term of the action. In [4] this 'missing' contribution was handled using dimensional regularization, and turns out to be zero, as demanded by our result.

The potential should now follow straightforwardly as

(4.47) Surprisingly, this diagram has the same potential as diagram 4.3d, times negative 2. Thus, the joined effect of these two last diagrams is a positive, and thus repulsive, potential proportional to the total mass and the Newtonian potential.

It is surprising that the effect of non-linear terms is to weaken the static force, but this is the case.

Summing up all the potentials V (a) -V (e) ((4.21), (4.26), (4.36d), (4.41), and (4.47)), and remembering to add the kinetic energy expansion (4.25a), the final 1PN Lagrangian will be the result This is the Einstein-Infeld-Hoffmann Lagrangian from 1938 [23] , derived in an entirely different manner, warranting some confidence in the result.

In order to determine the orbital energy of the binary system it is useful to reduce this Lagrangian to its equivalent one body problem.

As it is already just spatially dependent on the relative displacement of the two bodies r, the last thing needed is just to express their velocities by the relative velocity.

Using the centre of mass frame the position of each body can be expressed through the relative displacement r as 9

(4.50)

The velocity of particle number i is defined as the time derivative of its position v i ≡ṙ i . Defining the time derivative of the relative displacement as the relative velocity one finds 

(4.52b)

Equipped with the effective Lagrangian up to first post-Newtonian order (4.52b), all that remains is to determine the equation of motion and associated energy.

The corresponding equation of motion can be obtained by finding the extremum of the action. Using polar coordinates it is obvious that θ is a cyclic coordinate, as it does not appear in the Lagrangian.

(4.53c)

Approximating ≈ µr 2 ω + O 1 c 2 it is clear that is the angular momentum of Newtonian theory, with a 1PN correction.

To get the relations of other order of the LHS, is simply a matter of raising the equation to the desired power, and then Taylor expanding away terms that are not of first post-Newtonian order. But the relations as written here are those that usually come up, many having already been used in this thesis at the 0PN approximation.

One illustrative example to get a sense of the scale of the 1PN correction is to use (4.58a) to compute the correction of the lunar orbital distance. The Moon does not follow a circular orbit, but rather has an eccentricity of 0.02 < e < 0.08, so it will not be an exact approximation. It can however give an idea of the scale of the effect.

The Moon has a (sidereal) period of T = 27.32 days, and mass of m = 1.23 · 10 −2 M C , where the Earth mass is M C = 5.97 · 10 24 kg [24] .

3.85 · 10 8 1 − 1.17 · 10 −11 m 3.85 · 10 8 m − 4.47 · 10 −3 m (4.60b)

Thus, even though the approximation is crude, this shows that the correction is in order of millimetres for the Earth-Moon system. It is perhaps not surprising considering that the Moon is not exactly moving at relativistic speeds.

However, due to reflective mirrors left by the Apollo missions, the Earth-Moon distance is measured at a millimetre precision [25] .

The best models for the Earth-Moon system operates at this precision, and thus needs to account for 1PN corrections like this one [26] .

To obtain the orbital energy it will suffice to derive the corresponding Hamiltonian of the 1PN Lagrangian (4.52b) by Legendre transformation (3.45) . Utilizing the results of equations (4.53b) and (4.54c) the Hamiltonian is found to be

(4.61d)

The Hamiltonian is expressed in terms of the relative velocity v 2 =ṙ 2 + r 2 ω 2 instead of the canonical momentum because the end goal is simply to obtain the 1PN energy in terms of the frequency. Note that this expression is valid for all type of motion, not just circular.

Imposing circular motion again, the relations between v, r, ω from (4.57)-(4.59) may be used to express the energy in terms of one of these variables. The most commonly used variable is the frequency ω, as it is most directly related to the observable: the GW frequency. However here the velocity will be used for more convenient calculations.

This is not the energy expansion (2.7) presented in Chapter 2, so what is going on?

Recalling that in Chapter 2 v * was only used as a proxy variable for the orbital frequency, and was defined as v * ≡ (GM ω) 1/3 . In the Newtonian theory v = v * , but at 1PN the relative velocity and orbital frequency are related according to (4.59a), hence v = v * . Therefore, (4.62c) is the orbital energy in terms of the actual relative velocity.

Using equation (4.59a) to transform v → ω the energy in terms of frequency is obtained to be

This is the energy presented in (2.7), where (GM ω) 1/3 was named v, somewhat confusingly from the point of view of this chapter.

Beware that in the literature, energy and flux can, and are, presented in terms of GM r , (GM ω) 1/3 , or v. But they are all the frequency energy/flux, relabelled using the 0PN approximation of the relations (4.57) -(4.59). This is of course since they are ultimately used to compute waveforms, which are computed from differential equations of the frequency. And in the end, the frequency is the directly observable parameter.

To get a sense of the scale of this energy correction, lets use this on the Earth-Moon system. Which is of course comparably tiny. Using the mass-energy equivalence, the correction is comparable to ∼ 3kg, of an ∼ 4 · 10 8 metric tonnes 0PN energy.

In order to fully describe the 1PN dynamics of the compact binary the energy dissipation by generated GWs need to be accounted for. In this section this total radiated power is to be calculated.

Derivations presented here closely follows those presented in Maggiore [9] , Porto [4] , and Ross [27] .

In Section 4.1 it was argued that the binary system could be separated into three different length scales ∼ L, ∼ r, and ∼ R S , related by L r R S . In Chapter 4 the ∼ R S scale was 'integrated out', leaving BHs and NSs only with a point mass structure at the scale of the orbit ∼ r. Similarly, in Section 3.5 the total energy flux of a system was found evaluating the graviton field at a scale L λ GW , leaving the source effectively as a point source, endowed with a quadrupole structure.

By the requirement of evaluating at a scale L λ, it is also automatically realized to be evaluated at a scale much larger than the binary system that created it, λ GW r. 1

In Section 3.5 the solution of the graviton field was found to be (3.30), which reads

With V ∼ L 3 , such that T ij evaluated at ∂V is zero.

On the other hand, in the far region H = 0, which admits solution of the general form

Here the multi index notation has been introduced, which is to say a capital letter index L represent a number of indices. By comparing these two expressions, which should be equivalent for R r, F L can be demonstrated to be [28] 

Here t ret = t − R/c. The subscript STF stands for symmetric trace free. This is the only part that is not eliminated by the Lambda tensor (3.56) . Recall that in the flux, the Lambda tensor eliminated the trace of the quadrupole moment.

As a remainder !! = · ( − 2) · ( − 4) · · · 2 or 1, depending on whether is even or odd respectively.

Before proceeding, it will be useful to investigate STF tensors.

The STF part of a tensor is the irreducible representation of the tensor under rotations. Therefore, in GW physics it represents the physical degrees of freedom, where the other terms can be gauged away.

As an example, a rank two tensor can be decomposed into three parts

The first term in the last equality is the trace part, the second the anti-symmetric part, and finally the third term is the STF part. Now, noticing Λ ij kl δ kl = 0, and since the Lambda tensor is symmetric in k ↔ l Λ ij kl ε kl m = 0. This is why only the STF part of F L ij contribute to the final flux.

The STF part of a rank n tensor can be obtained by [29] T i 1 ···in

The construction of this expression is not self-evident, but calculating it for the quadrupole and octupole moments will be instructive. The operator x rounds x down to the closest integer, and is called the floor function. E.g. 3/2 = 1.

First notice that the p = 0 term always correspond to the symmetric version of the tensor in question for p = 0:

Not surprisingly, the terms of p > 0 in (5.5) is used to subtract all possible traces, thus making the expression symmetric and trace free. E.g. the quadrupole moment becomes

which is equivalent to the expression used in the quadrupole radiation (3.62).

The octupole moment follows similarly as 

Working the expression further, following the somewhat complicated steps of Ross [27] the result isH

.

Here I is the mass multipole, while J is current multipole, defined as

(5.10)

Here T µν is the energy-momentum tensor of the source. Now, using (3.55) the total energy flux is determined as [29] 

where the bracket is understood as averaging over time.

To assign PN orders to the different terms, notice that every time derivative contributes with a factor of c −1 , ∂ 2p 0 ∼ 1/c 2p . Also since T µν ∼ mẋ µẋν , andẋ 0 ∼ c, we can expect every spatial index of the energy-momentum tensor to contribute with a factor of c −1 compared to the 00 term. Utilizing these observations it should be clear at which PN order the various terms of (5.10) and (5.11) enter.

For the leading order term, only moments with the lowest power of (c −1 ) n can contribute. From the general flux expression (5.12b) it is clear that every derivative of the multipole moments contributes with additional factors of c −1 , thus the leading order term must be of only two indices, a quadrupole. Because the leading order term in the energy-momentum tensor is the point particle contributions, and since the point particle energy-momentum tensor is proportional to the tensor product of the particle's four velocity (3.40) any spatial index of T µν contributes with an additional factor of c −1 . This excludes the current multipole (5.11) entirely, and all but the first line of the mass multipole (5.10).

Since the ct derivatives contribute with superfluous factors of c −1 , only the p = 0 term of the first line of (5.10) for L = 2 contributes to the leading order energy flux.

Using this leading order term of (5.10) and (3.40) for the T 00 term the resulting leading order 2 expression for the mass quadrupole moment is

In the last line the masses were rewritten to the reduced mass (see Appendix B for more details).

To compute the resulting flux, equation (5.12b) requires the third time derivative of this term. Applying circular motion implies n x = cos(ωt), n y = sin(ωt), and n z = 0. Thus, after consulting (C.2) for how to rewrite squared trigonometric functions, the result is

Notice that I ij is indeed symmetric and trace free. To calculate the energy flux the sum 2 To leading order γa = 1.

over squares of each component is needed.

In the last line v = ωr was used, and finally Kepler's third law (2.4) to exchange ω for v. Then the leading order term of the energy flux is For the next to leading order correction the octupole moment I ijk (t) and the current quadrupole moment J ij (t) must be added, and also there are relativistic corrections to the quadrupole formula used here for the 0PN flux term, like the other terms in (5.10) and relativistic corrections to T µν .

Using (5.10) and (3.40) the mass octupole moment reads

Inserting circular motion (n x = cos(ωt), n y = sin(ωt) and n z = 0) and then taking the 4 th time derivative, as necessitated by equation (5.12b) produces 4 4 cos(3ωt) + 3ω 4 20 cos(ωt) , (5.18a) 

Because of the symmetric property of I L , terms like I xyx are equivalent to I yxx . Thus, any term not appearing in equation (5.18) are either equivalent to one of the listed terms by symmetry, or zero (like odd numbers of z-indices). Notice also that the sum of (5.18a)-(5.18c) is zero. Similarly the sum of (5.18d)-(5.18f) is also zero, as they should be since I L is trace free.

Note that for the square sum cos(nωt) cos(mωt) = δ nm /2, thus all contributing terms will be of the form sin 2 (nωt), and cos 2 (nωt). For example 2 c 4 (1 − 4η)r 6 ω 8 (3 8 + 3 9 ) · 5 2 + (3 2 + 3) + 3 · 2 4 2 4 · 5 2 = 2 · 3 · 1367 5

⇒ F oct.

(2) = G c 3 1 3 3 · 7 2 · 3 · 1367 5

3. Relativistic corrections to the inserted motion of the source. For quasi-stable circular orbits the motion does not change, but the relation between v, ω, and r pick up some relativistic corrections (4.57)-(4.59).

Starting with the relativistic corrections to T 00 , and T k k (it will be clear in a moment why these are lumped together) recall that (3.40) simply Taylor expanding γ a = 1 − v 2 a /c 2 −1/2 around v/c = 0. The next to leading order terms in T 00 is thus proportional to (c −1 ) 0 . From the Virial theorem, or equivalently from (4.58b), the leading order (Newtonian) term of the gravitational potential scales also as v 2 , and should therefore also be included. This concludes point 1., the relativistic correction of T µν . Finally, to leading order T k k is the point particle tensor (3.40) Since the time dependent part (n i ) is equivalent to the first order term of the quadrupole formula, the third derivative of the expression correct to first order in (v 2 /c 2 ) 1 can be inferred directly from (5.14) For the p = 0 terms this only leaves the ∂ 0 T 0k term in line three of (5.10). The last line containing ∂ 2 0 T ij will not contribute as it scales as (∂ 2 0 T ij )/T 00 ∼ c −4 . To leading order also this energy-momentum tensor component is the free point particle tensor, and thus 0k-part of I ij (2) 

For circular motion v a = ±ωr a − sin(ωt), cos(ωt), 0 , and is thus orthogonal to x a : v a · x a = 0. Therefore the leading order contribution of the T 0k -part of I L is 0.

To finish point 2. only accounting for the p = 1 term remains. This term follows as p = 1 part of I ij Here the subtlety of point 3., corrections to the equations of motion, enters. At this stage equations (4.57)-(4.59) must be used to convert between r, ω, and v. One might expect at this point to use ωr = v and (4.59a) to convert the last factor of ω = v 3 /GM , but this is not the case. Recalling that v is just a proxy variable for the frequency, one should expand the flux in terms of ω, and perhaps change ω to v = (GM ω) 2/3 . the time evolution of the relative separation due to GW emission is found to be r (τ ) = GM (GM ω) 2/3 1 − 1 − This is, not surprisingly, very slowly. At this rate, it would take ∼ 2.8 · 10 16 years until the effect would be in the range of millimetres.

We have now seen how the 1PN gravitational waveform (2.19) can be obtained from the 1PN energy (4.63) and flux (5.36), assuming quasi-stable circular orbits and separation of scales.

We have also demonstrated how the 1PN energy can be determined using Feynman diagrams (Chapter 4), and how the 1PN flux can be computed using multipoles (Chapter 5), all based on an effective field theory of gravity as a gauge field (Chapter 3).

In his text [4] , Porto claimed "[...] that adopting an EFT framework, when possible, greatly simplifies the computations and provides the required intuition for 'physical understanding'."

In my own experience, the computations do not seem all that more simplified compared to the more traditional geometrical approach (see Maggiore [9] for an outline, or Blanchet [30] for more details). For someone without a deep background in EFT, like a master's student, any simplification of the calculation is outweighed by the work of familiarizing oneself with standard results and conventions from QFT.

Of course, if one does have a deep familiarity with EFTs, the field theorist approach presented in this thesis is a great way to transfer those skills to gravitational wave physics. These are after all powerful tools for handling perturbative phenomena. The use of Feynman diagrams makes the terms in the perturbation series more manageable, and can give intuition for what kind of physical effects the different terms account for. In this sense, the computations can be considered to have been 'simplified', and provided the intuition for 'physical understanding'.

It is also possible that the field theorists' approach becomes significantly simpler than the relativists' approach at higher PN orders. In order to verify this, I would need to compute higher order corrections.

Even so, for relativists, some familiarity with the field theory way of thinking of gravitational dynamics is helpful for deepening their understanding of gravity. As Feynman once said at a Cornell lecture during his gravity phase:

"Every theoretical physicist who is any good knows six or seven different theoretical representations for exactly the same physics. He knows that they are all equivalent, and that nobody is ever going to be able to decide which one is right at that level, but he keeps them in his head, hoping that they will give him different ideas for guessing." -Feynman [31] For such reasons, it is valuable to have alternative ways of thinking about gravity and gravitational waves. Some extensions of, and alternative theories for, Einstein's theory of gravity might present themselves more naturally in the language of field theory, rather than differential geometry. E.g. quantum 'loop' corrections of gravity [32] . With gravitational wave data imposing some of the strongest constraints on gravity theories, having an alternative route for translating theories of gravity to gravitational waveforms is a useful tool.

In conclusion, this effective field theory approach to computing gravitational waveforms will probably not replace the more traditional relativist approach as the standard or introductory way of deriving these results any time soon. As a method it is however worth developing, as it provides an alternative perspective on the physics of gravitational waves. It might also provide a shorter path for some alternative theories of gravity to testable predictions, and can be used by physicists with a heavier quantum field theory background to simplify the computations of such theories.

Implementing these restrictions (A.1) becomes

Performing the k 0 integral results with

Step-by-step the above calculation first, (A.4a), collapse all dependence on k 0 into a function f (k 0 ), other than the Dirac delta function and Heaviside step function. In line (A.4b) the argument of the Dirac delta was expanded, and in line (A.4c) the Dirac delta was itself expanded according to the relation

Lastly, in line (A.4d), the k 0 ≥ 0 term was singled out by Θ(k 0 ). All the steps of (A.4) can be performed for (A.3). Also requiring h(x α ) to be a real function can easily be done by demanding h † (x α ) = h(x α ), which is obtained most generally by having b(k µ ) = a † (k µ ).

Thus, the most general solution of the wave equation for a real scalar field is h(x α ) = 0, ⇒ h(x α ) = d 3 k (2π) 3 · 2ω k a(k)e −ik σ x σ + a † (k)e ik σ x σ , (A. 6) with k 0 = |k| = ω k . This is the solution presented in equation (2.2) . This is still a large class of solutions, but it is restricted to travel in the k-direction through space, with a velocity of Notice that both the group velocity v g ≡ ∂ω ∂|k| and the phase velocity v p ≡ ω |k| are both equal to c.

Equivalent one body problem and mass term manipulation B.1 Rewriting to the equivalent one body problem To solve the equation of motion of the two body problem it is useful to rewrite the equations in terms of relative quantities, like the spatial separation r, and relative velocity v ≡ṙ. This will be done for the centre of mass frame in this appendix.

Letting r i , i ∈ {1, 2} be the position of object i relative the centre of mass, which is placed in the origin, as in Figure B .1 the following identity holds r ≡ r 2 − r 1 , (B.1)

Relativistic corrections to T µν , like kinetic energy, and gravitational energy. 2. Including other terms from (5.10), like p = 1, T 0i and T k k

Observation of gravitational waves from a binary black hole merger

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Theory and Experiments, ser. Oxford Master Series in Physics

Gravitating binaries at 5PN in the post-Minkowskian approximation

Gravitational waves from merging compact binaries: How accurately can one extract the binary's parameters from the inspiral wave form?

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A roadmap for Feynman's adventures in the land of gravitation

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Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries

The Character of Physical Law

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The radial equation of motion is similarly obtained by Imposing circular motion entailsṙ =r = 0, thus the EoM simplifies toThe solution to this equation follows asNotice that to the 0PN order equation (4.56a) reduces to Kepler's third law for circular orbits (2.4). Using (4.56), r, ω, and v can be related with 1PN corrections. To compute the other relations, organize the equation into a quadratic equation, and solve for the quadratic parameter (e.g. solve for GM/r in (4.56b) to obtain GM/r as a function of v 2 ).The 1PN correct relations of ω, r, and v turns out to be

Using (5.11) and (3.40) the current quadrupole moment readsNotice that circular motion is here already assumed as v a = ω × x a is used to simplify the expression of the first equality of the second line. Again, using n x = cos(ωt), n y = sin(ωt), and n z = 0, and taking the third time derivative as instructed by formula (5.12b) results inNotice that J ij (2) is trace free and symmetric, as it should be. The flux is determined by the sum of squares of all the tensor components, which is(5.24)

Circling back to the mass quadrupole moment, all first order assumption that went into 5.2.1 must now be expanded to next to leading order. This primarily means 3 things:Which is exactly the flux (2.8) presented in Chapter 2.To get some perspective, lets consider the Earth-Moon system again. According to (5.36), the energy flux due to GWs isUsing equation (2.18) for the change in orbital frequency, and relation (4.58a) of r and ω,

This derivation can be found in most textbooks on field theory, e.g. Kachelrieß [17] , or Schwichtenberg [16] . Indices will be ignored in this appendix, as the spatio-temporal dependence of the solution is assumed to be independent of indices.Assuming the solution to be a superposition of plane waves e −ikσx σ , the most general form the wave can take isFor (A.1) to be a solution of the wave equation (3.20) the following must holdwhich is a solution as long as k 2 0 = |k| 2 . Since k 0 is identified as the temporal frequency it is required to be positive k 0 = ω/c ≥ 0 in order to be physical. Both these conditions can be imposed by δ k 2 0 − ω 2 k · Θ(k 0 ), where δ(x) is the Dirac delta function, Θ(x) is the Heaviside step function, and ω k is determined by the dispersion relation and is ω k = |k| for massless fields. 1In the last line of (B.2) equation (B.1) was used to eliminate r 2 . Similarly r 1 can be eliminated in favour of r 2 . Thus r i can be expressed asHere M is the total mass of the binary. Since the velocity of each object in the centre of masswhere again v =ṙ. Substituting r i and v i for the expressions of equations (B.3)-(B.4) the Lagrangian, and thus the EoM, becomes a function of just r and v. Thus the two body problem is reduced to solving for just the relative motion of one object, an equivalent one body problem.Explicitly the Newtonian Lagrangian becomesBottom line is that preforming the substitution to r and v reduces the problem to describing the motion of one particle with an effective mass of µ = m 1 m 2 M in a gravitational potential produced by an effective mass of M = m 1 + m 2 , which is static and located at the position of the other particle. 

Chirp massHere is a stepwise approach to deal with m 1 v n+1and m 1 (−r 1 ) n+1 + m 2 r n+1 2 type expressions.1. The product m 1 m 2 = M µ. Identify and extract all common factors of µ from the expression. 2. This will leave something like (m n 2 ± m n 1 ) /M n . a. If it is a sum, calculate M n = m n 1 + m n 2 + . . . . Thus m n 2 + m n 1 = M n − . . . Then again look out for reduced masses and remember that µ/M = η. b. If it is a difference this will usually imply that n is even. Let n = 2k and then m 2k 2 − m 2k 1 = (m k 2 − m k 1 )(m k 2 + m k 1 ). The (m k 2 + m k 1 ) term can be expanded as in step a, and hopefully this will be enough. The difference can be further expanded In equation (B.6) the different sum of powers are listed, and in (B.7) the mixed products are listed. These expressions are useful for mass term manipulations like those that appear in this thesis. Appendix C

The trigonometric identities that are used in this thesis can all be derived using Euler's formula e iθ = cos(θ) + i sin(θ).For the squared trigonometric functions one only needs to square this formula e iθ 2 = e i2θ = cos(2θ) + i sin(2θ) = (cos(θ) + i sin(θ)) 2 = cos 2 (θ) − sin 2 (θ) + i2 sin(θ) cos(θ) = 2 cos 2 (θ) − 1 + i2 sin(θ) cos(θ) = 1 − 2 sin 2 (θ) + i2 sin(θ) cos(θ),where in the last line the Pythagorean identity sin 2 (θ) + cos 2 (θ) = 1 was used to write the expression only by second powers in cosine or sine respectively. Comparing the real parts and imaginary parts of the first and third line produces the useful identities cos 2 (θ) = 1 2 (1 + cos(2θ)) , sin 2 (θ) = 1 2 (1 − cos(2θ)) , sin(θ) cos(θ) = 1 2 sin(2θ). and then once again comparing the imaginary part of the first and third line the following identities are obtained.cos 3 (θ) = 1 4 (3 cos(θ) + cos(3θ)) , sin 3 (θ) = 1 4 (3 sin(θ) + sin(3θ)) .(C.3a) (C.3b)