post-newtonian and post-minkowskian two-body potentials ... lytic post-newtonian (pn) models...

Post-Newtonian and Post-Minkowskian

Two-Body Potentials From Scattering

Amplitudes

Andrea Placidi

Dipartimento di Fisica e Geologia

Universita degli Studi di Perugia

Supervisor

Gianluca Grignani

26/09/2019

Contents

Introduction 4

1 Gravitational Waves 8

1.1 GWs in linearized gravity . . . . . . . . . . . . . . . . . . . . . . 10

1.2 GW propagation in the transverse-traceless gauge . . . . . . . . . 12

1.3 Energy and momentum flux of GWs . . . . . . . . . . . . . . . . 15

1.4 GW generation in linearized theory . . . . . . . . . . . . . . . . . 20

1.5 Multipole expansion in the low-velocity regime . . . . . . . . . . . 22

1.6 GWs from a Newtonian two-body system . . . . . . . . . . . . . . 27

1.7 Radiated energy in the quadrupole approximation . . . . . . . . . 30

1.8 Orbital evolution in a GW emitting binary system . . . . . . . . . 32

1.9 Beyond linearized theory and Newtonian dynamics . . . . . . . . 35

2 Post-Newtonian And Post-Minkowskian Methods 39

2.1 Hamiltonian PN calculations . . . . . . . . . . . . . . . . . . . . . 40

2.2 Effective field theory approach in a wordline-oriented description . 43

2.3 Introduction to the scattering amplitude approach . . . . . . . . . 46

2.4 Quantum field theory for gravitational scattering amplitudes . . . 50

2.5 Focus on classical terms . . . . . . . . . . . . . . . . . . . . . . . 54

2

CONTENTS

3 PN and PM computations in parametric frames of reference 56

3.1 Feynman rules for tree-level gravity amplitudes . . . . . . . . . . 58

3.2 Tree-level potential in a generalized reference frame . . . . . . . . 61

3.3 Tree-level potential in the COM frame with off-shell external legs 69

3.4 1PM potential in the general reference frame . . . . . . . . . . . . 72

3.5 Completing the 1PN potential . . . . . . . . . . . . . . . . . . . . 75

Conclusions 84

A A meaningful reference frame for Iwasaki’s article 86

References 95

3

Introduction

Gravitational waves (GWs) were conjectured by Albert Einstein in 1916 as wave

solutions to the linearized weak-field equations of General Relativity (GR) [1].

After a long debate on their own physical reality, officially concluded with the

Chapel Hill conference in 1957 [2], GW existence indirectly found a first experi-

mental confirmation only in 1982, through the energy loss observations of Taylor

and Weisberg on the binary pulsar system PSR B1913+16 [3], previously discov-

ered by Hulse and Taylor [4].

Since then a lot of effort was devoted to overcome the outstanding smallness

of GW amplitude, as it was required for their direct detection. This was finally

achieved on 14 September 2015 thanks to the joint activity of the two LIGO

detectors, one located in Livingstone and the other in Hanford [5]: GWs produced

by the coalescence of two black holes were detected and a brand new branch of

observational astronomy was born.

Indeed GWs encode a remarkable amount of information about their sources

so that they represent a powerful tool to extract physical parameters out of inspi-

ralling and merging astronomical objects such as black holes [6; 7] and neutron

stars [8]. In parallel, GW detection open up the possibility to test GR at an

unprecedented level, e.g. allowing us to probe the strong gravity regime as it is

provided for example by black holes on the edge of the coalescence.

Nevertheless, past and present success of GW astronomy should not be thought

4

as a simple matter of technical improvements in the GW detector network: it en-

compasses a matched-filter data analysis performed through theoretically mod-

eled waveform templates. Given the evident impossibility to find an exact solution

for the general relativistic two-body dynamics, which underlies GW emission,

these templates can only be obtained through approximate approaches whose

precision must grow along with the instrumental one of the detectors.

To be more specific, the currently employed waveform models relies on the

complementarity between numerical-relativity simulations [9; 10], and the ana-

lytic post-Newtonian (PN) models resummed within the effective-one-body pre-

scription [11; 12]. They are respectively efficient at short and large separation

between the orbit’s and the body’s scales, thus accomplishing the coverage of the

full inspiral-merger-ringdown sequence which composes a compact binary coales-

cence.

The PN approximation schemes for the two-body dynamics are hinged on the

determination of perturbative GR correction to its simple Newtonian description,

in the context of an expansion in the dimensionless parameter v/c, where v is the

orbital velocity of the compact binary and c the speed of light. Clearly a compact

binary can not but be a self-gravitating bounded system; therefore, according to

the virial theorem, its kinetic and potential energy must share the same order of

magnitude, namely

v2 =GM

r,

G being Newton’s constant, M the typical mass of the system and r the separation

between the two bodies. From this perspective the PN approach translates into

a two parameters expansion (in v and G) whose nth order, which is referred to

as nPN order, presents contributions proportional to

Glvn−l+1,

5

where l = 1, 2, ..., n+ 1.

The first-ever PN calculation dates back to 1938, with the historical paper of

Einstein, Infeld and Hoffmann on the perihelion precession [13], but a systematic

study on the PN approximation came only with the works of Chandrasekhar

and associates [14; 15]. From then on, a good number of PN approaches have

been proposed to predict the two-body motion: e.g. one can find PN calculations

within the effective field theory (EFT) language [16; 17], with surface integral

techniques [18; 19] or carried through the hamiltonian formalism, among which

the Arnowitt, Deser, and Misner scheme, originally outlined in [20], is one of the

most extensively explored up to date [21; 22; 23].

In 1971 the pioneering paper of Iwasaki [24] called another PN approach into

existence, showing the possibility to obtain a 1PN gravitational potential from

the evaluation of relativistic scattering amplitudes in a quantum field theory of

gravity coupled to matter, built up at the classical level by a coupling between the

Einstein-Hilbert action and the stress-energy tensor of a real scalar field. Later

on, Holstein and Donoghue proved that the loop expansion subtended to this

approach presents entirely classical terms at arbitrarily high order, because of a

subtle cancellation of ~ factors [25].

In recent times this particular method has been receiving a renewed interest,

due to its newly found compatibility with some powerful techniques originally

developed for Yang-Mills theories computations, such as the unitarity methods

[26]. Indeed this turned out to extremely simplify the evaluation of gravity am-

plitudes and made such an approach a good candidate for further advance in the

formulation of PN models.

Moreover, the scattering amplitude prescription is conveniently predisposed

to be carried through while keeping all orders in velocity, within the weak-field

only framework of the so called Post Minkowskian approach where one expands in

6

the sole Newton’s constant G. Lately proposed by Cheung, Rothstein and Solon

in Ref. [27], where an amplitude matching to the EFT is performed, this scheme

has already been pushed up to the computation of a 3PM two-body potential

(i.e. up to O(G3) terms) [28], and then cut loose from the EFT matching in Ref.

[29]. Taking the virial theorem into account, there are no general arguments to

predict the eventual accuracy increase in keeping all the velocity terms when we

deal with a bound system like a compact binary. However in Ref. [30] it has

been shown, by comparison with numerical relativity results, that pushing the

PM expansion to higher orders may actually lead to a valuable refinement of the

currently employed GW templates.

This thesis is mainly devoted to the exploration of the aforementioned pertur-

bative analytical approaches to two-body dynamics, with particular reference to

the scattering amplitude methods. More precisely, the contents will be presented

according to the following structure:

• In Chapter 1 a review on the basic GW physics will be presented. This will

shed light on the limits of the simple Newtonian description with respect

to the modellization of GWs which have been radiated in the coalescence

of compact binaries.

• In Chapter 2 we will outline some of the currently leading techniques ori-

ented toward finding PN and PM perturbative models for the two-body

dynamics, lingering specifically over the scattering amplitude method.

• Chapter 3 represents the actual original contribution of this thesis. It is

intended to analyze and circumscribe the ambiguities, in terms of coordinate

dependence, which may arise in the two-body potential as it is provided by

scattering amplitude calculations.

7

Chapter 1

Gravitational Waves

On 15 September 2015 the two interferometric detectors of LIGO observed inde-

pendently a signal compatible with a GW passage, tagged as GW150914. After

5 month of checks and inspections on the collected data, the LIGO Scientific

Collaboration and the Virgo Collaboration announced together to the world the

first experimental detection of gravitational radiation [5]. Moreover the analy-

sis of this signal brought to light that the received GWs had been radiated in

the inspiral-merging-ringdown sequence of two stellar black holes, thus providing

the first direct experimental proof on the very existence of such a binary sys-

tem. Later on, nine additional GW observations were performed by the LIGO

and Virgo network, such as GW151226 [6] and GW170104 [31], and many others

seem to await in the data of the still on-going third observing run.

Generally speaking, the opportunity to detect GW has paved the way to a

priceless new approach to multi-messenger astronomy, expanding the kingdom of

the astrophysical bodies we have a chance to probe and the amount of information

we could hope to gather from them. At the same time GWs observations represent

also a new testing laboratory for general relativity (GR) since, being the offspring

of exceptional events such as the coalescence between two black holes, they allow

8

us to access the previously unexplored regime of strong-gravity, where outstanding

space-time distortions come into play.

The line of research to which the present thesis belongs has found in these

discoveries a decisive propelling fuel. After all, theoretical physics interacts with

GW detection both as a supply of the necessary waveform templates and as

a consumer of its experimental data, to confirm or revise some of its general

relativistic models. Therefore seems natural to open this thesis with a first chapter

dedicated to a self-contained review on the fundamental aspects of GW theory.

For this purpose we will follow Refs. [32; 33; 34].

In general the highly non-linear character of GR prevents us from finding

full-fledged solutions to its equations of motion, unless we restrict our analysis to

conveniently symmetric systems, essentially with spherical or axial symmetry, re-

spectively corresponding to the Schwarzschild and Kerr metric. This descriptions

have no use in a space-time affected by the emission and propagation of gravita-

tional radiation, so that we are forced to look for some befitting approximation

scheme whenever we deal with GWs. In this regard, we will start by working

within the framework of linearized theory, based on an expansion of the Einstein

equations around the Minkowskian metric ηµν .

In the first sections of this chapter we will use such an approach to red-

erive GW basic properties, from their equation and propagation to the energy-

momentum flux that is carried by them. Then we will proceed with the analysis

of GW production, limiting ourselves to Newtonian physics and checking to what

extent this description can be successfully applied. That will be the starting point

for our immersion into more articulated general relativistic models, which will be

the leading subject of the following chapters.

9

1.1 GWs in linearized gravity


Let us start from the GR action S = SEH + SM , where

SEH =c3

16πG

∫d4x√−gR (1.1)

is the widely-known Einstein-Hilbert action and SM the action of matter. The

variation of the latter under a change of the metric gµν → gµν + δgµν defines the

energy-momentum tensor of matter T µν by means of the relation

T µν = − 2c√−g

δSMδgµν

. (1.2)

The principle of least action on S brings us to the Einstein equations

Rµν −1

2gµνR =

8πG

c4T µν . (1.3)

An essential feature of GR is its diffeomorphism invariance, which is often

referred to as its gauge symmetry. In other words GR is symmetric under the

group of coordinate transformation xµ → x′µ(x), for x′µ(x) differentiable and with

a differentiable inverse. The metric transforms accordingly as

gµν(x)→ g′µν(x′) =

∂xρ

∂x′µ∂xσ

∂x′νgρσ(x). (1.4)

In this context we move towards Linearized GR by considering the weak-field

approximation

gµν = ηµν + hµν , |hµν | 1, (1.5)

along with a linear expansion of Eq. (1.3) in powers of hµν . Within this scheme,

the flat metric ηµν is the one which raises and lowers indices.

10


It should be noted that by imposing the condition (1.5) we are implicitly

choosing a reference frame in which it effectively holds in a sufficient large region

of space, thus somehow breaking the aforementioned diffeomorphism invariance.

Nevertheless a residual gauge symmetry remains: let us take into consideration

a coordinate transformation

xµ → xµ + ξµ. (1.6)

From (1.4) we can get the lowest order transformation law of hµν , namely

hµν → hµν − (∂µξν + ∂νξµ). (1.7)

As long as |∂µξν | ∼ |hµν | 1, this is what becomes the gauge symmetry of GR

in linearized gravity.

To linear order in hµν the Riemann curvature tensor reduces to

Rµνρσ =1

2(∂µ∂σhνρ + ∂ν∂ρhµσ − ∂µ∂ρhνσ − ∂ν∂σhµρ). (1.8)

We are ready to linearize Einstein’s equations (1.3) and with the definitions

h ≡ ηµνhµν , hµν ≡ hµν −1

2ηµνh, (1.9)

intended to a compact notation, we arrive to the linearized gravity equations of

motion:

hµν + ηµν∂ρ∂σhρσ − ∂ρ∂νhµρ − ∂ρ∂µhνρ = −16πG

c4Tµν . (1.10)

Note that at this stage hµν and hµν possess 10 degrees of freedom due to their

symmetric nature. To partially remove the gauge redundancy (1.7) we impose

11

1.2 GW propagation in the transverse-traceless gauge

the De Donder gauge fixing condition

∂νhµν = 0. (1.11)

Such an imposition is always possible, as it is equivalent to require

ξµ = fµ, (1.12)

for a generic function fµ, while in turn (1.12) is guaranteed to admit solutions

thanks to the invertibility of the d’Alambertian operator. From the ten indepen-

dent components of hµν we are left with six. Furthermore the equations (1.10)

are simplified into

hµν = −16πG

c4Tµν , (1.13)

the wave equation which stands as the basic result for studying GW generation

within linearized theory.

1.2 GW propagation in the transverse-traceless

gauge

In order to understand the fundamental features of GW propagation we put our

lens far away from the matter source, where Tµν = 0 and Eq. (1.13) becomes

hµν = 0. (1.14)

Since = −(1/c2)∂t + ∇2 we immediately deduce that GWs travel at the speed

of light. The gauge symmetry (1.7) in terms of hµν can be rewritten in the form

hµν → hµν − ξµν , (1.15)

12


where we defined ξµν ≡ ∂µξν + ∂νξµ + ηµν∂ρξρ. Hence with a simple derivative we

find

∂νhµν → ∂νhµν −ξµν = ∂νhµν −ξµ, (1.16)

and see that a coordinate transformation xµ → xµ + ξµ does not spoil the De

Donder condition, provided that ξµν = ξµ = 0. This discloses the opportu-

nity to impose four other conditions on hµν through suitable choices for ξµ. In

particular, we choose ξ0 such that h = 0 and consequently

hµν = hµν . (1.17)

Additionally we fix the remaining three components ξi so as to set h0i = 0.

Combining this with the De Donder condition (1.11), for µ = 0, yields

∂0h00 + ∂ih0i = ∂0h00 = 0. (1.18)

The time independent part of h00 is non other than the Newtonian potential of

the GW source, therefore as far as we are concerned with GWs in vacuum Eq.

(1.18) means h00 = 0. Overall we end up with

h0µ = 0, hii = 0, ∂jhij = 0, (1.19)

the so called transverse-traceless gauge, or TT gauge. In there the metric hµν

has no more spurious degrees of freedom (only two are left now) and it is usually

denoted by hTTµν .

Let’s go back to Eq. (1.14). It admits plane wave solutions of the kind

hTTµν = eTTµν (k)eikx, (1.20)

13


where kµ = (ω/c,k), with ω/c = |k|, and eTTij is the polarization tensor in the

TT gauge. According to (1.19) the latter must satisfy

e0µ = 0, eii = 0, njeij = 0, (1.21)

n = k/|k| being the direction of propagation. A general solution to (1.14) can

be built with a straightforward superposition of plane waves:

hTTij =

∫d3k

(2π)3(eTTij (k)eikx + c.c.)

∣∣∣∣k0=|k|

. (1.22)

If we instead restrict our analysis to a single plane wave propagating along

the z axis (n = (0, 0, 1)), taking the real part of (1.20) while factoring in the

conditions (1.21) leads us to

hTTij =

h+ h× 0

h× −h+ 0

0 0 0

ij

cos(ω(t− z/c)

), (1.23)

in which we have introduced h+ and h× as the “plus” and “cross” physical po-

larization of the GW. We observe that all the non-zero components of hTTij lie on

the plane transverse to n. In terms of the invariant interval dS2 = gµνdxµdxν we

have

dS2 = −c2dt2 + dz2 +[1 + h+cos

(ω(t− z/c)

)]dx2

+[1− h+cos

(ω(t− z/c)

)]dy2 + 2h×cos

(ω(t− z/c)

)dxdy

(1.24)

It’s worth mentioning that the TT gauge we have worked in so far is not well

defined inside the GW source, as in this case hµν 6= 0. There the De Donder

gauge could still be imposed and four other degrees of freedom could be gauged

14

1.3 Energy and momentum flux of GWs

away, but we have no means of setting to zero any further component of hµν .


Here we intend to characterize GWs from the energetic point of view, computing

the expressions for their energy and momentum flux within linearized theory.

In addition to its geometrical interpretation, one can think of the latter as the

classical field theory of the field hµν living in the flat Minkowski space-time with

metric ηµν . That is the approach we are going to employ in the present section.

The first step to move in this direction is finding the action which governs the

hµν dynamics. Since we want it to reproduce the linearized Einstein equations

(1.10), linear in hµν , as its Euler-Lagrange equations, such an action has to be

quadratic in hµν . Then we recall the Einstein-Hilbert action

SEH =c3

16πG

∫d4x√−gR (1.25)

and expand it to second order in hµν . Therefore we have

R = gµνRµν =[ηµν − hµν +O(|hµν |2)

][R(1)µν +R(2)

µν +O(|hµν |3)], (1.26)

where

R(1)µν =

1

2(∂ρ∂µhνρ + ∂ρ∂νhµρ − ∂ρ∂ρhµν − ∂µ∂νh) (1.27)

and

R(2)µν =

1

2ηρσηαβ

[1

2∂µhρα∂νhσβ + ∂ρhνα(∂σhσβ − ∂βhµσ)

+ hρα(∂µ∂νhσβ + ∂σ∂βhµν − ∂β∂µhνσ − ∂β∂νhµσ)

+

(1

2∂αhρσ − ∂ρhασ

)(∂µhνβ + ∂νhµβ − ∂βhµν)

] (1.28)

15


are the linear and quadratic components in hµν of the Ricci tensor. At the same

time, by employing the general identity log(detA) = Tr(logA) we get

√−g = 1 + h+O(|hµν |2). (1.29)

After some algebra and integration by parts we finally obtain

S(2)EH = − c3

64Gπ

∫d4x (∂ρhαβ∂

ρhαβ − ∂ρh∂ρh+

+ 2∂ρhρα∂αh− 2∂ρhαβ∂

βhρα),

(1.30)

whose Lagrangian will be referred to as L(2)EH .

Now we want to build an appropriate energy-momentum tensor for the GWs.

First of all we note that L(2)EH is symmetric under global space-time translations

xµ → xµ + aµ. Noether’s theorem then provides an energy-momentum tensor as

the conserved current

θ(2)µν ≡ −

∂L(2)EH

∂(∂µhαβ)∂νhαβ + ηµνL

(2)EH . (1.31)

For the time being, we put ourselves in the gauge

∂µhµν = 0, h = 0. (1.32)

The second, third and fourth terms in the action (1.30) are quadratic in quantities

that vanish in this gauge, so it is straightforward to find

∂L(2)EH

∂(∂µhαβ)

∣∣∣∣∣∂µhµν=h=0

= − c4

32πG∂µh

αβ,

L(2)EH

∣∣∣∂µhµν=h=0

= − c4

64πG∂ρhαβ∂

ρhαβ,

(1.33)

16


so that (1.31) becomes

θ(2)µν =

c4

32πG(∂µh

αβ∂νhαβ − 2ηµν∂ρhαβ∂ρhαβ). (1.34)

At this point one could be tempted to follow the standard procedure and

associate the GW four-momentum to the conserved charge of θ(2)µν . However a

swift analysis of (1.34) reveals a major obstacle: θ(2)µν is not actually gauge in-

variant so we can’t define unambiguously the GW momentum with it. Besides,

we could have foreseen such an outcome, since Noether’s theorem only returns

local quantities, while the GR equivalence principle ensures the existence of a

locally inertial frame in which any given local quantity associated to the gravi-

tation field may be set to zero. To overcome this obstacle we should rather look

for a non-local energy-momentum tensor. For this purpose we rethink the GW

as a wave-packet with reduced wavelength peaked around a value λ and perform

a spatial average over a box centered on λ and with size L λ. We will denote

this operation by 〈...〉 and stress that in a plane wave scenario this is equivalent

to a temporal average over numerous periods. Therefore, inside 〈...〉, integration

by parts is possible within the scope of our reasoning. The energy-momentum

tensor of GWs is then defined by the non-local average of (1.34),

tµν ≡ 〈θ(2)µν 〉 =

c4

32πG〈∂µhαβ∂νhαβ〉, (1.35)

where, through integration by parts, the second term in (1.34) has been made

proportional to hαβ and then deleted with the equations of motion. As we

wished tµν is a gauge invariant quantity (at least to leading order in λ/L), and

17


this also enables us to recover the TT gauge in its expression:

tµν =c4

32πG〈∂µhTTij ∂νhTTij 〉. (1.36)

The definition of the GW four-momentum in a volume V outside the source,

pµV = (EV ,pV ), now easily follows:

pµV ≡∫V

d3x t0µ =c3

32πG

∫V

d3x 〈∂thTTij ∂µhTTij 〉. (1.37)

Moreover tµν satisfies by construction the conservation law ∂µtµν = 0, so that

0 =

∫V

d3x (∂0t0µ + ∂it

iµ) =1

c

dpµVdt

+

∫V

d3x ∂itiµ (1.38)

and with the divergence theorem

1

c

dpµVdt

= −∫∂V

dσ nitiµ, (1.39)

where ∂V is the boundary surface of V and n is the outward pointing unit

normal of ∂V . Now we want to compute the GW energy flux as it is measured by

a detector at a distance r from the source. Then we choose V to be a spherical

shell centered on the source, with the inner and outer boundaries, S ′ and S, in

the far region, and set S as the surface on which the detector lies. The energy

flux passing through S is given by:

1

c

dE

dt= −

∫S

dS t0r (1.40)

where

t0r =c3

32πG〈∂thTTij ∂rhTTij 〉 (1.41)

For large r, in analogy with electromagnetic waves, the outward propagating GW

18


can be assumed to take the general form

hTTij (t, r) =1

rfij(t− r/c) (1.42)

for an unspecified function fij of the retarded time tret = t − r/c. Therefore we

have

∂rhTTij = − 1

r2fij(t− r/c) +

1

r∂rfij(t− r/c) =

= − 1

r2fij(t− r/c)−

1

cr∂tfij(t− r/c),

(1.43)

or equally

∂rhTTij =

1

c∂th

TTij +O(1/r2). (1.44)

In other words we have just found that at large distances the relation t0r = t00

holds. By substituting this in Eq. (1.40) we arrive to

dE

dt= −c

∫S

dS t00 = − c3

32πG

∫S

dS 〈∂thTTij ∂thTTij 〉. (1.45)

The decreasing E tells us that the outward propagating GW carries away energy

flux, and we can easily extract it from Eq. (1.45):

dE

dSdt= ct00 =

c3

32πG〈∂thTTij ∂thTTij 〉. (1.46)

An analogous calculation shows that the corresponding GW momentum flux

isdpk

dSdt= t0k =

c3

32πG〈∂thTTij ∂khTTij 〉. (1.47)

19

1.4 GW generation in linearized theory


At the end of the first section of the current chapter we derived the linearized

equations of motion in the De Donder gauge,

hµν = −16πG

c4Tµν , (1.48)

where Tµν is the energy-momentum tensor of the GW matter source. In terms of

the Green’s function G(x− x′) for the operator x, defined by the identity

xG(x− x′) = δ(4)(x− x′), (1.49)

a solution to Eq. (1.48) is given by

hµν(x) = −16πG

c4

∫d4x′G(x− x′)Tµν(x′). (1.50)

Since we aim to describe GW generation, concerning the boundary conditions we

select the retarded Green’s function

G(x− x′) = − 1

4πc|x− x′|δ(4)(tret − t′), (1.51)

where

tret = t− |x− x′|c

(1.52)

is the retarded time that we already met in the preceding section. Therefore Eq.

(1.50) becomes

hµν(t,x) =4G

c4

∫d3x′

1

|x− x′|Tµν

(t− |x− x′|

c,x′). (1.53)

20


Outside the source we would like to impose the TT gauge. To this end we

introduce the transverse projector P (n) ≡ δij−ninj and then the Lambda tensor

Λij,kl(n) ≡ Pik(n)Pjl(n)− 1

2Pij(n)Pkl(n), (1.54)

which satisfies the properties

Λij,klΛkl,mn = Λij,mn, niΛij,kl = njΛij,kl = ... = 0, Λii,kl = Λij,kk = 0. (1.55)

As for its main feature, one can prove that this Lambda tensor extracts the

transverse-traceless part of any given symmetric tensor Aij:

ATTij = Λij,klAkl. (1.56)

By means of (1.54) we rewrite Eq. (1.53) as

hTTij (t,x) =4G

c4Λij,kl(x)

∫d3x′

1

|x− x′|Tkl

(t− |x− x′|

c,x′), (1.57)

where T00 and T0k can be omitted by virtue of the conservation law ∂µTµν = 0

that relates them to Tkl. Let us consider a spherical source with radius d and

denote |x| by r. We put ourselves far away from the source, i.e. r d, and

observe that in (1.57) the integral is restricted to |x′| ≤ d, because Tµν vanishes

outside the source. In this situation, schematically depicted in Fig. (1.1), one can

expand

|x− x′| = r − x′ · x +O

(d2

r

), (1.58)

and express accordingly Eq. (1.57):

hTTij (t,x) =1

r

4G

c4Λij,kl(x)

∫d3x′ Tkl

(t− r

c− x′ · x

c,x′), (1.59)

21

1.5 Multipole expansion in the low-velocity regime

Figure 1.1: Schematic for the situation behind the expansion (1.58). The circlerepresents the source.

in which all the O(1/r2) terms have been neglected.


At this point we specify the above analysis on GW generation to the case of

sources with typical velocities much smaller than c, moving from the electrodynamics-

inspired guess that this will provide a substantial simplification. The magnitude

of the typical velocity inside a source with size d is v ∼ ωsd, where ωs is the

typical frequency one could associate to its internal motion. Furthermore, as we

will show below in this chapter, the frequency ωgw = c/λ of the radiated GWs

presents a magnitude comparable to ωs, so that we have

λ ∼ c

vd. (1.60)

By imposing v c, we end up with

λ d. (1.61)

22


Such a condition states that, from the perspective of a GW observer, the motion

inside a non-relativistic source is of little importance: the source is essentially

probed as a whole. Therefore we are naturally led to perform a multipole expan-

sion while neglecting, in first approximation, all the multipole moments beyond

the lowest one since we already know they would provide corrections carrying

along more and more details on the source internal motion. With this in mind

we pull the energy-momentum tensor out of Eq. (1.59) and write its Fourier

transform:

Tkl

(t− r

c− x′ · x

c,x′)

=

∫d4k

(2π)4Tkl(ω,k)e−iω(t−r/c−(x′·x)/c)+ik·x′

. (1.62)

Being focused on a non-relativistic source, we can think Tkl(ω,k) to be peaked

around a frequency ωs with ωsd c. Mindful of the condition |x′| ≤ d, we

observe thatω

cx′ · x . ωsd

c 1 (1.63)

and expand accordingly:

e−iω(t−r/c−(x′·x)/c)+ik·x′= e−iω(t−r/c)

(1− iω

cx′ixi − 1

2

ω2

c2x′ix′jxixj + ...

), (1.64)

The substitution of this equation in (1.59) yields

Tkl

(t− r

c− x′ · x

c,x′)

=

[Tkl +

x′ixi

c∂tTkl −

1

2c2x′ix′jxixj∂2

t Tkl + ...

](t−r/c,x′)

,

(1.65)

where all the quantities inside [...] are evaluated at the point (t− r/c,x′).

23


Moreover we introduce the momenta of T ij:

Sij(t) ≡∫d3xT ij(t,x) [stress monopole]

Sij,k(t) ≡∫d3xT ij(t,x)xk [stress dipole]

Sij,kl(t) ≡∫d3xT ij(t,x)xkxl [stress quadrupole]

...

(1.66)

Note that the comma in Sij,klm... stands between two completely symmetric groups

of indices, while the exchange of two comma-separated indices is not a symmetry.

Next we insert the expansion (1.65) in Eq. (1.59), along with the stress mul-

tipole definitions (1.66), obtaining

hTTij (t,x) =1

r

4G

c4Λij,kl(x)

[Skl +

1

cxm∂tS

kl,m+

+1

2c2xmxn∂2

t Skl,mn + ...

](t−r/c)

.

(1.67)

Since every xm, xn, ... in Skl,mn... brings an O(d) factor and O(∂t) ∼ O(ωs), while

O(ωsd) ∼ O(v), we recognize in (1.67) a non-relativistic expansion in the standard

O(vn/cn) form. Truncating (1.67) to the lowest order yields

[hTTij (t,x)

]leading

=1

r

4G

c4Λij,kl(x)Skl(t− r/c). (1.68)

Now we devote ourselves to rewrite the momentum Skl in terms of quantities

with an easier physical interpretation. For this purpose we define the momenta

24


of T 00/c2, the mass density in the low-velocity weak-field limit, as

M(t) ≡ 1

c2

∫d3xT 00(t,x) [mass monopole]

M i(t) ≡ 1

c2

∫d3xT 00(t,x)xi [mass dipole]

M ij(t) ≡ 1

c2

∫d3xT 00(t,x)xixj [mass quadrupole]

...

(1.69)

Correspondingly the momenta associated to the momentum density T 0i/c are

P i(t) ≡ 1

c

∫d3xT 0i(t,x) [momentum monopole]

P i,j(t) ≡ 1

c

∫d3xT 0i(t,x)xj [momentum dipole]

P i,jk(t) ≡ 1

c

∫d3xT 0i(t,x)xjxk [momentum quadrupole]

...

. (1.70)

Let us rework Sij definition in a convenient fashion:

Sij =

∫d3xT ij =

∫d3xT ikδjk =

∫d3xT ik∂kx

j =

= −∫d3x ∂kT

ikxj.

(1.71)

The last equality holds because T µν vanishes outside the source, so that we are

free to extend the integral on a box with volume V larger than the source, on

whose boundaries T µν = 0, thus overall allowing integration by parts. At the

same time from the conservation law ∂µTµν = 0 we have

∂kTkν = −1

c∂tT

0ν . (1.72)

Inserting Eq. (1.72) in (1.71) and taking the symmetric part of the right term

25


bring us to

Sij =1

c

∫d3x ∂tT

0(kxj)(1.70)=

1

2∂t

(P i,j + P j,i

). (1.73)

With the same trick we have:

P i,j =1

c

∫d3xT 0kδik x

j =1

c

∫d3xT 0k

(∂kx

i)xj =

= −1

c

∫d3x

(∂kT

0kxixj + T 0jxi)

(1.72),(1.69)=

= ∂tMij − P j,i.

(1.74)

Altogether we obtain

Sij =1

2∂2tM

ij, (1.75)

and thus, from Eq. (1.68),

[hTTij (t,x)

]leading

=1

r

2G

c4Λij,kl(x) ∂2

tMkl∣∣t−r/c . (1.76)

In other words the leading term of the expansion (1.67) involves exclusively the

mass quadrupole. This is the reason why truncating the multipole expansion to

the lowest possible order is usually referred to as quadrupole approximation. It

is possible to demonstrate that the leading quadrupole nature of GWs we just

derived in linearized theory is actually preserved even in the full theory. This

is in agreement with the field-theoretic picture of GR, where the gravitational

interaction is mediated by a massless particle with helicties ±2, the graviton,

which can not be put in a state with total angular momentum j = 0 or j = 1, as

it would be required for monopole or dipole gravitational radiation.

26

1.6 GWs from a Newtonian two-body system


In the present section we will exploit our previous results to make a first contact

with the emission of gravitational radiation from a physically relevant source: a

binary system of massive objects. Specifically, here we will pursue the descrip-

tion of a simple circular moving system of massive point particles, under the

Newtonian regime.

The energy-momentum tensor we associate to such a source is

T µν(t,x) =∑k=A,B

mkdxµkdt

dxνkdt

δ(3)(x− xk(t)) +O

(v2A

c2

)+O

(v2B

c2

), (1.77)

where the indices A, B labels the two massive particles and xk(t) stands for the

k-particle trajectory. We stress that, by neglecting all the O(v2/c2) corrections,

we end up loosing some contributions which may seem necessary, e.g. the binding

energy between the two particles. However, under the quadrupole approximation,

Eq. (1.77) can be proved to provide the exact same result of the full theory.

The quantity we aim to determine is the mass quadrupole momentum M ij,

and it can be computed directly from its definition (1.69). Before that, we switch

to the convenient coordinatesxrel(t) = xA(t)− xB(t)

xcm(t) =mAxA(t) +mBxB(t)

m

(1.78)

expressed in terms of the total mass m ≡ m1 +m2. Therefore we have

M ij(t) =1

c2

∫d3xT 00xixj =

∑k=A,B

mkxik(t)x

jk(t) =

= mxicm(t)xjcm(t) + µxirel(t)xjrel(t),

(1.79)

27


where also the reduced mass µ ≡ m1m2/m has been introduced. Henceforth we

will consider only

M ij(t) = µxirel(t)xjrel(t) (1.80)

since we are concerned with an isolated system and the motion of the center of

mass has no relevance.

As regards the relative motion, we consider a circular orbit of radius R which

lies in the (x, y) plane:

xrel(t) =(R sin(ωst), R cos(ωst), 0

). (1.81)

From (1.80) we get

M ij(t) = µR2

sin2(ωst) sin(ωst)cos(ωst) 0

sin(ωst)cos(ωst) cos2(ωst) 0

0 0 0

ij

(1.82)

and thus

∂2tM

ij∣∣t−r/c = 2µω2

sR2

cos(2ωs(t− r/c)

)−sin

(2ωs(t− r/c)

)0

−sin(2ωs(t− r/c)

)−cos

(2ωs(t− r/c)

)0

0 0 0

ij

.

(1.83)

Taking Eq. (1.76) as a reference, it is now time to contract ∂2tM

ij|t−r/c with

the Lambda tensor Λij,kl(x). We proceed in two step. First we go back to the

case of a GW propagating along the z-axis, so that x = (0, 0, 1). Writing the

time derivatives with the usual dot notation, the contraction we want to evaluate

28


reduces to

Λij,kl

[(0, 0, 1)

]Mkl =

(M11 − M22)/2 M12 0

M21 (M11 − M22)/2 0

0 0 0

ij

. (1.84)

From it we can read directly the two polarization amplitudes

h+(t) =G

rc4

[M11(t− r/c)− M22(t− r/c)

],

h×(t) =2G

rc4M11(t− r/c).

(1.85)

Then we generalize these results for

x = (sinθ sinϕ, sinθ cosϕ, cosθ) (1.86)

by replacing M ij in (1.85) with

M ijrot = (RTMR)ij = RikRjlMkl, (1.87)

where R is the rotation matrix

R =

cosϕ sinϕ 0

−sinϕ cosϕ 0

0 0 1

1 0 0

0 cosθ sinθ

0 −sinθ cosθ

. (1.88)

Finally we find

h+(t, θ, ϕ) =2Gµω2

sR2

rc4(1 + cos2θ)cos

(2ωs(t− r/c) + 2ϕ

),

h×(t, θ, ϕ) =2Gµω2

sR2

rc4cosθ sin

(2ωs(t− r/c) + 2ϕ

).

(1.89)

29

1.7 Radiated energy in the quadrupole approximation

We observe that the frequency of the emitted GW, ωgw, and the one of the motion

inside its source, ωs, are related by

ωgw = 2ωs. (1.90)

As we have mentioned above they share the same order of magnitude.

1.7 Radiated energy in the quadrupole approx-

imation

In Eq. (1.76) we have seen that, within the quadrupole approximation, the wave-

form of the emitted GW is completely determined by the mass quadrupole mo-

mentum M ij. Moreover, the Lambda tensor makes sure that only its traceless

part is taken into account, therefore it is customary to introduce a traceless mass

quadrupole momentum

Qij ≡M ij − 1

3δijMkk =

∫d3x

T 00

c2

(xixj +

1

3r2δij

)(1.91)

and directly replace the Mkl in (1.76) with it,

[hTTij (t,x)

]leading

=1

r

2G

c4Λij,kl(x)Qkl(t− r/c) (1.92)

Let’s recall the expression for the total power radiated through GW emission

as we found it at the end of Sec. 1.3 :

P ≡ dE

dt=

r2c3

32πG

∫dΩ 〈hTTij hTTij 〉. (1.93)

Under the quadrupole approximation, one can insert Eq. (1.92) in (1.93) and get

30

1.7 Radiated energy in the quadrupole approximation

Pquad =G

8πc5

∫dΩ Λij,kl(x)〈

...Qij(t− r/c)

...Qij(t− r/c) 〉. (1.94)

The lambda tensor is the only angular dependent quantity in (1.94), so its integral

is actually ∫dΩ Λij,kl(x) =

2π

15(11δikδjl − 4δijδkl + δilδjk), (1.95)

computed with the general x given in (1.86) and the explicit form of Λij,kl outlined

in (1.54).

Consequently, we arrive to the renowned Einstein’s quadrupole formula:

Pquad =G

5c5〈

...Qij(t− r/c)

...Qij(t− r/c) 〉. (1.96)

Now we look back to the simple binary system of the previous section: our

current objective is to employ (1.96) and calculate its total radiated power. We

start by noticing that the M ij(t) given in (1.82) exhibits an unitary trace. As a

consequence the traceless momentum defined in (1.91) differs from it only for a

constant term δij/3, which is clearly lost after a single time derivative. Therefore

we can simply determine the needed...Qij

by deriving Eq. (1.83) with respect to

time:

...Qij

(t−r/c) = 4µω3sR

2

−sin

(2ωs(t− r/c)

)−cos

(2ωs(t− r/c)

)0

−cos(2ωs(t− r/c)

)sin(2ωs(t− r/c)

)0

0 0 0

ij

. (1.97)

In conclusion the total radiated power of our binary system turns out to be

Pquad =64

5

Gµ2

c5R4ω6

s〈 cos2(2ωs(t− r/c)

)〉 =

32

5

Gµ2

c5R4ω6

s . (1.98)

31

1.8 Orbital evolution in a GW emitting binary system

1.8 Orbital evolution in a GW emitting binary

system

All the results we have so far obtained for the simple binary system introduced

in Sec. 1.6 rest on the assumption that its internal motion stays indefinitely

circular, with constant radius R and frequency ωs. Nevertheless, in Eq. (1.98)

we found that such a system constantly radiates energy into space, manifestly in

contradiction with our starting assumption. Therefore we end up with the urge

to check the time dependence of R and ωs as it’s determined by Eq. (1.98) and

consequently understand the actual significance of our analysis in relation to the

predicted orbital evolution.

For a start we recall that a Newtonian bound system must satisfy the virial

theorem

Ekinetic = −1

2Epotential ⇒ 1

2µv2 =

Gµm

2R, (1.99)

namely, as long as v = ωsR (circular motion),

ω2s =

Gm

R3, (1.100)

which is the well-known Kepler’s law.

In parallel, the total energy

Esystem ≡ Ekinetic + Epotential =Gµm

2R(1.101)

is tied to the radiated power Pquad of Eq. (1.98) by the balance equation

Pquad = −dEsystem

dt. (1.102)

32


Figure 1.2: R(t) and ωs(t) as they have been computed in (1.103). The red lineroughly marks the region of quasi-circular motion.

With the support of Eq. (1.100) one can separately recast Eq. (1.102) as two

separate differential equations for ωs(t) and R(t) and find their solutions:

R(t) = R(t0)

(tcoal − ttcoal − t0

)1/4

, ωs(t) = ωs(t0)

(tcoal − ttcoal − t0

)−3/8

, (1.103)

where t0 is the initial time and tcoal is the finite value of time we associate to the

coalescence, since

limt→tcoal

R = 0, limt→tcoal

ωs = +∞. (1.104)

The time dependence (1.103) of R and ωs is plotted in Fig. (1.2). For the purpose

of this thesis the relevant time window is that characterized by a nearly smooth

slope both in R(t) and ωs(t) (the red line in the figure). In this region we can

think the contributions of the time derivatives of R and ωs to be negligible (the

typical reference conditions are R R2 and ωs ω2s ), so roughly validating the

results (1.89). We are in the so called quasi-circular motion regime. However, in

order to reach an acceptable first approximation we partially reintroduce in (1.89)

the effects of the orbital evolution by replacing ωs and R with their expressions

(1.103), together with the substitution

ωgwt = 2ωst→ Φ(t) ≡∫ t

t0

dt′ ωgw(t′) (1.105)

33


where we defined the time-dependent phase Φ(t) of the emitted GW.

Therefore, after a shift of the time origin which let us replace 2ωs(t−r/c)+2ϕ

with 2ωst, we get

h+(τ, θ) =1

r

(GMc

c2

)5/4(5

cτ

)1/4(1 + cos2θ)

2cos(Φ(τ)

),

h×(τ, θ) =1

r

(GMc

c2

)5/4(5

cτ

)1/4

cosθ sin(Φ(τ)

).

(1.106)

where τ ≡ tcoal − t is the time to coalescence and

Mc ≡(m1m2)3/5

(m1 +m2)1/5= µ3/5m2/5 (1.107)

the usually defined chirp mass. From (1.103) and (1.105) we obtain also the time

dependence of the GW phase Φ,

Φ(τ) = −2

(5GMc

c3

)−5/8

τ 5/8 + Φ0 (1.108)

in which we used dτ = −dt and the integration constant Φ0 is associated to the

vale of Φ at coalescence, namely Φ0 = Φ(τ = 0).

Approaching the coalescence this quasi-circular description eventually loses

its validity, due to the visible growth of R and ωs, however at this stage we would

be definitely outside of the inspiral phase, namely the one we are interested

in. Besides, even within the quasi-circular regime the currently achieved results

represent only a first step in the modellization of binary-radiated GWs, suitable

to support the detection experiments, since they are the ultimate offspring of the

quadrupole approximation in the context of Newtonian dynamics. In the next

section we will address the problems one encounters in trying to go beyond and

introduce possible analytical methods to accomplish that.

34

1.9 Beyond linearized theory and Newtonian dynamics

1.9 Beyond linearized theory and Newtonian dy-

namics

In the linearized theory we have discussed so far, a major assumption is manda-

tory: the back-ground space-time is always taken as flat, i.e. the sources which

produce GWs are considered to contribute negligibly to the space-time curva-

ture. This allowed an approximate description of GW production grounded on

the leading term of a multipole expansion. Moreover, as we observed in Sec. 1.5

, the retention of higher order multipoles could apparently yield non-relativistic

O(vn/cn) corrections, without any apparent need to modify the flat back-ground

metric. However the sources of gravitational radiation we are taking into account

are self-gravitating compact binaries, for which the virial theorem states

v2

c2∼ RS

R, (1.109)

RS = 2Gm/c2 being their Schwarzschild radius and m their total mass. Since the

ratio Rs/R can be used to roughly quantify the strength of the gravitational field

around the corresponding system, we conclude that the only path toward im-

proving accuracy prescribes to take into account the progressive deviations from

the Minkowskian back-ground metric. Therefore we can not proceed straightfor-

wardly in the multipole expansion while remaining in the theoretical framework

of the previous sections but instead we should turn to more accurate models

which supply us with general relativistic corrections: the post-Newtonian and

post-Minkowskian schemes.

The idea behind these two approaches is to build general relativistic two-

body models through perturbative expansions around the simple Newtonian re-

sults. Aside from the subtleties of the specific strategy one may choose for their

35


Figure 1.3: Summary of G and v dependence of every PM and PN order, takenfrom Ref. [38]. Blue and green lines outline respectively the PN and PM resultswhich have been calculated so far. The red shaded region corresponds to therecently computed 3PM order (Ref. [28])

implementation, the main difference between the two lies in how the respective

expansions are performed: the PN one is a low-velocity and weak-field expan-

sion whose orders are organized following Eq. (1.109), whereas the PM one is a

weak-field only expansion in which all orders in velocity at fixed order in G are

included. For a better understanding, the powers in G and v corresponding to

each PM and PN order are summarized in Fig. (1.3). The overlapping of the two

expansions is evident and ensures both the possibility of valuable crosschecks and

an intrinsic mutuality in their development. The recent 3PM calculation [28] (up

to the shaded red region in the figure) is emblematic in this sense, as we see that

it provides unprecedented contribution to the 5PN dynamics, beyond the static

5PN component determined within the PN framework.

Once the dynamics of the binary system is formalized (either in Hamiltonian,

Lagrangian or Routhian language) up to the desired order in one of the two

expansions, the general course of action to extract the corresponding waveform

models is the following:

36


• The total energy Esystem and the radiated power P of the system are com-

puted up to the available expansion order. As regards P we note that going

up with the sought accuracy eventually makes higher order terms in the

multipole expansion to be quite important and thus non-negligible.

• With a similar machinery to the one adopted in Sec. 1.8, the time depen-

dencies of the relevant observables, principally the GW phase Φ and the

radial separation between the two bodies R, are determined in a GR cor-

rected form as solutions of the respective differential equations, which in

turn are provided by the balance equation.

• The GW waveforms are built. Basically in their determination one should

keep as well higher order multipoles and thus obtain “side bands” with re-

spect to the leading quadrupole waveforms, which oscillate at half-integer

multiples of Φ. Nevertheless it is usual to work in the so called restricted

approximation, under which only the leading quadrupole component is con-

sidered and the relativistic corrections are exclusively left to the expressions

of the observables employed in the waveform construction.

In practice carrying out this process is far from effortless, especially when some

demanding subtleties come into play: for instance the back-reaction effects of

GWs on the dynamics of their sources, the secondary production of GWs from

the gravitational field of other GWs (due to the non-linear character of general

relativity), and the consequences of the tidal forces acting on the gravitating

bodies. Moreover in the current state of the art the PM and PN results are

combined with numerical relativity results within the effective one body scheme

([11; 12; 30]), where the resulting description of the two-body dynamics manages

to cover the entire inspiral-merger-ringdown sequence of the coalescence.

We will not add further details to this aspect of the GW waveform compu-

37


tation. Instead, in the following chapters we will dedicate ourselves to analyze

some of the most important techniques which may be employed to formalize the

PN and PM two-body dynamics.

38

Chapter 2

Post-Newtonian And

Post-Minkowskian Methods

The efforts at theorizing models well beyond linearized gravity and Newtonian

physics are entirely justified: GW experiments hunt for signals so small that they

come out inevitably buried in noise, typically several order of magnitude larger

than them. Therefore, to extract relevant information from the collected data

a far from trivial analysis is needed on them. For this purpose the most used

technique is the matched filtering [35], in which high precision GW templates,

provided by the available general relativistic descriptions of the compact binary

dynamics, serve as references to recognize the GW signals and accordingly filter

out the noise.

Proceeding in this direction, the present chapter revolves around post-Newtonian

(PN) and post-Minkowskian (PM) approaches to the two body dynamics, specif-

ically suitable to the description of the initial inspiralling phase, while the final

plunge, merge and ringdown are mainly covered by numerical relativity calcula-

tion, that one can deepen for instance in Refs. [10; 36; 37].

As for the chapter structure, the first section hosts a brief historical introduc-

39

2.1 Hamiltonian PN calculations

tion on the 1PN two-body hamiltonian and a swift review on the PN-oriented

Arnowitt, Deser, and Misner (ADM) hamiltonian formalism (following [39]) while

in the second we will cover concisely the wordline-oriented effective field theory

(EFT) description ([16; 40]).

For the reminder of this chapter, we will start the discussion of the scattering

amplitude approach, suitable for the calculation in both PN [41] and PM [28;

29] schemes. Besides, being the main subject of this thesis, it will also receive

additional focus in the next chapter.


In 1938 Einstein, Infeld and Hoffmann performed the first full-fledged PN com-

putation, by obtaining the equations of motion for a two-body, self-gravitating

system [13]. Essentially they calculated surface integrals around the field singu-

larities in the context of a point-like particle approximation. Later on Fichtenholz

extracted out of their equations the corresponding Lagrangian and Hamiltonian

[42]. Here we present his 1PN Hamiltonian, which became a fundamental bench-

mark in the literature:

H(p1,p2, r) =p2

1

2m1

+p2

2

2m2

− p41

8m31c

2− p4

2

8m32c

2− Gm1m2

r+

− Gm1m2

2rc2

[3p2

1

m21

+3p2

2

m22

− 7p1 · p2

m1m2

− (p1 · r)(p2 · r)

m1m2r2

]+

+G2m1m2(m1 +m2)

2r2c2+O(c4),

(2.1)

p1, p2 being the momenta of the two bodies and r their radial separation. In the

center of mass reference frame, where p1 = −p2 = p, one has the equivalent form

40


H(p, r) =(m1 +m2)

2m1m2

p2 − (m31 +m3

2)

8m31m

32c

2p4 − Gm1m2

r+

− Gm1m2

2rc2

[3p2

m21

+3p2

m22

+7p2

m1m2

+(p · r)2

m1m2r2

]+

+G2m1m2(m1 +m2)

2r2c2+O(c4).

(2.2)

Let us now explore the ADM method, originally developed in [20] and then

successfully employed in the derivation of 3PN [43] and 4PN [22; 23] two-body

dynamics. The hamiltonian nature of such a method imposes a space and time

(3+1) splitting on the metric gµν :

dS2 = gµνdxµdxν = −(Ncdt)2 + γij

(dxi +N icdt

)(dxj +N jcdt

), (2.3)

where the flat metric signature is taken as (−,+,+,+) and

γij ≡ gij, N ≡ 1√−g00

, N i ≡ g0i = γijNj. (2.4)

In this framework it can be demonstrated that the Hamiltonian which generates

all Einstein’s field equations assumes the form

H[γij, π

ij, N,N i,xA,pA]

=

∫d3x

(NH −N iHi

)+

+c4

16πG

∮i0dSi ∂i(γij + δijγkk),

(2.5)

in which we have introduced the ADM canonical field momentum

πij ≡√γN(Γ0

ij − γklΓ0klγij), γ ≡ det(γij), (2.6)

and the canonical matter variables xA and pA (with A = 1, 2), encompassed by

41


the matter Hamiltonian density HM and matter momentum density HMi inside

the quantities

H ≡ c4

16πG

[−√γR +

1√γ

(γikγjlπ

ijπkl − 1

2πijπ

ij

)]+ HM

Hi ≡c3

8πGγij∇kπ

jk + HMi,

(2.7)

R being the Ricci’s scalar and ∇k the covariant derivative with respect to γij; i0

denotes space-like flat infinity.

By varying the Hamiltonian (2.5) one can get the constraint equations

H = 0, Hi = 0. (2.8)

Furthermore we put ourselves in the ADM coordinate frame (or ADMTT

gauge), where

πii = 0, 3∂jγij − ∂iγjj = 0. (2.9)

The transverse and traceless components(hTTij , π

ijTT

)become the only indepen-

dent parts of the gravitational field variables.

Through (2.8) and (2.9) the fully reduced version of the Hamiltonian (2.5) is

written as

Hred

[hTTij , π

ijTT ;xA,pA

]=

c4

16πG

∮i0dSi ∂i(γij + δijγkk) =

=c4

16πG

∫d3x ∂i∂j(γij + δijγkk).

(2.10)

and yields the Hamilton’s equations

pA = −∂Hred

∂xA, xA =

∂Hred

∂pA;

πijTT = −δHred

δhTTij, hTTij =

δHred

δπijTT.

(2.11)

42

2.2 Effective field theory approach in a wordline-oriented description

Next we shift the gravitational component of Hred to the Lagrangian formal-

ism, by employing the following Legendre transform, which defines the Routh

functional:

R[hTTij , h

TTij ,xA,pA

]≡ Hred −

c3

16πG

∫d3x πTTij h

TTij (2.12)

The strategy from here is to take advantage of Eqs. (2.11) and thus determine

the functional dependency of hTTij and hTTij on xA and pA; then, the two-body

Hamiltonian is obtained as

H[xA,pA

]= R

[xA,pA, h

TTij (xA,pA), hTTij (xA,pA)

]. (2.13)

We won’t delve into the details of this calculation, limiting ourselves to observe

that eventually some point-like singularities arise in the process, and a regular-

ization technique is needed (a typical choice is the dimensional regularization

described in [44]).

2.2 Effective field theory approach in a wordline-

oriented description

In this section we will illustrate the general features of the wordline-oriented EFT

approach to the dynamics of compact binaries. The original formulation of the

EFT language came in 2006, with the work of Goldberger and Rothstein [45].

Since then, it has caught the interest of many authors [17; 46; 47] who have

tried to push it the furthest in the PN expansion, up to the determination of the

5PN static component (i.e. velocity independent) of the two-body gravitational

potential [48].

Leaving aside finite size effect, which for spin-less object are guaranteed to

43


appear only from 5PN order onwards by the effacing principle [49], the EFT

description we will cover throughout this section assumes the massive bodies in

binary systems as non-dynamical point particles whose wordlines xµa(λ) interact

with gravitons.

The action functional one associates to such a description, assuring its in-

variance to both general coordinate change and wordlines reparameterization, is

structured as follows:

S[gµν , x

µ1,2(λ)

]= SEH

[gµν]

+ SGF[gµν]

+∑a=1,2

Sapp[gµν , x

µa(λ)

]. (2.14)

The first term is the Einstein-Hilbert action

SEH[gµν]≡ 2Λ2

∫dd+1x

√−gR(gµν), (2.15)

where Λ = 1/√

16πG and the space dimension d is kept generic for dimensional

regularization purpose.

The second term is intended for the gauge fixing. In this respect a popular

choice corresponds to

SGF[gµν]≡ −Λ2

∫dd+1x

√−g ΓµΓµ, Γµ = Γµρσg

ρσ, (2.16)

and is called “De Donder gauge” like the one we employed in Chapter 1 (indeed,

one can verify that the above gauge is equivalent to the latter under linearized

theory). One may think to involve also the Faddeev-Popov ghost fields in the

gauge fixing, but they would produce only undesired quantum corrections.

The last term of Eq. (2.14) is the wordline point particle action, given by

Sapp[gµν , x

µa(λ)

]≡ −ma

∫dτa = −ma

∫dλ

√−gµν(xµa)

dxµa

dλ

dxνadλ

. (2.17)

44


Due to the general coordinate invariance, we are even free to select a conve-

nient parameterization of the metric. A standard example is the Kaluza-Klein

(KK) parameterization [50]:

gµν = e2φ/Λ

−1 Ai/Λ

Ai/Λ e−cdφ/Λ(δij + σij/Λ)− AiAj/Λ2

, cd = 2d− 1

d− 2, (2.18)

in which the gravitational degrees of freedom are expressed through the scalar φ,

the vector Ai and the symmetric tensor σij.

Adopting the functional integral language, the idea is to define a two-body

effective action Seff

[xµ1,2(λ)

]by integrating all the gravity fields out of the action

(2.14), namely

eiSeff[xµ1,2(λ)] =

∫DφDAiDσij e

iS[gµν , xµ1,2(λ)]. (2.19)

This computation is performed via Feynman diagrams, in the EFT one can build

rewriting the complete action (2.14) in the KK parameterization (see [50]) and

expanding it perturbatively in powers of 1/Λ. Specifically, the needed Feynman

rules can be read right from the expanded action in momentum space, where the

role of fundamental variables is played by the Fourier-transformed fields:

φk(t) ≡∫ddxφ(t,x)e−ikx,

Ai,k(t) ≡∫ddxAi(t,x)e−ikx,

σij,k(t) ≡∫ddxσij(t,x)e−ikx.

(2.20)

This process turns out to involve the evaluation of a great number of effective

diagrams and only advanced multi-loop techniques (on which we won’t linger)

make the computation viable.

45

2.3 Introduction to the scattering amplitude approach

Having found the effective action Seff

[xµ1,2(λ)

], the corresponding two-body

equations of motion are readily determined by the functional derivatives:

δ

δxµ1Seff

[xµ1,2(λ)

]= 0

,δ

δxµ2Seff

[xµ1,2(λ)

]= 0.

(2.21)

2.3 Introduction to the scattering amplitude ap-

proach

In this section we present the main topic of the thesis: computing the two-body

gravitational potential by means of scattering amplitudes. Here we will introduce

only the general concepts and the basic features which characterize this particular

method, leaving to the following chapter the exploration of further details.

The founding idea that backs up this method is a general relativistic gener-

alization of the systematic connection between scattering amplitudes and inter-

action potentials, as it is typically found in non-relativistic quantum mechanics

and scattering theory [51]. The starting point is the one-particle Hamiltonian for

a two-body system of massive particles,

H ≡ H0 + V , (2.22)

where

H0 = c√p2 +m2

1c2 + c

√p2 +m2

2c2 (2.23)

is the free Hamiltonian and V the potential which accounts for the various inter-

actions in the system, gravitational-like in our case. Then we define the C-valued

46


Green’s operators

G0(z) ≡(z − H0

)−1, G(z) ≡

(z − H

)−1. (2.24)

As one can deepen in Sec. 8-a of Ref. [51], these operators are analytic throughout

the complex plane apart from the spectra of the respective Hamiltonians. There-

fore, the knowledge of G0(z) and G(z) for all z ∈ C is equivalent to a complete

solution of the eigenvalue problem of H0 and H. Observe that G0(z) and G(z)

are simply related: by using the general identity

A−1 = B−1 + B−1(B − A

)A−1 (2.25)

for A−1 = G and B−1 = G0, one readily gets

G = G0 + G0V G. (2.26)

Along with these Green’s operators, we introduce also the off-shell scattering

operator

T (z) ≡ V + V G(z)V , (2.27)

whose on-shell matrix elements correspond to the non-trivial components of the

scattering S-matrix. Moreover, the insertion of Eq. (2.26) in (2.27) leads to

T (z) = V + V(G0 + G0V G

)V = V + V G0

(V + GV

)=

(2.27)= V + V G0T ,

(2.28)

the well-known Lippmann-Schwinger equation. With respect to the two-particle

scattering process from the state |p1, p2〉 to the state |p3, p4〉, the on-shell matrix

47


element of T assumes accordingly the integral form

〈p3, p4|T (z)|p1, p2〉 = 〈p3, p4|V |p1, p2〉+

+

∫d3k1

(2π)3

d3k2

(2π)3

〈p3, p4|V |k1, k2〉〈k1, k2|T (z)|p1, p2〉z − Ek1 − Ek2

.(2.29)

where

Ek1 + Ek2 = c

√k2

1 +m21c

2 + c

√k2

2 +m22c

2 (2.30)

is the total energy of the intermediate on-shell states |k1, k2〉 over which the

integral in (2.29) spans.

Eq. (2.29) commonly serves as a tool to determine scattering amplitudes from

a known interaction potential. Indeed, considering

Epi ≡ c√p2i +m2

i c2 (i = 1, 2) (2.31)

one can set z = Ep1+ Ep2

+ iε, while taking the limit ε → 0+, and turn to the

relation

limε→0+〈p3, p4|T (Ep1

+ Ep2+ iε)|p1, p2〉 = M(p1, p2, p3, p4), (2.32)

that introduces the non-relativistic scattering amplitude M . In this way Eq.

(2.29) becomes

M(p1, p2, p3, p4) = 〈p3, p4|V |p1, p2〉+

+ limε→0+

∫d3k1

(2π)3

d3k2

(2π)3

〈p3, p4|V |k1, k2〉M(k1, k2, p1, p2)

Ep1+ Ep2

− Ek1 − Ek2 + iε.

(2.33)

However our interest is directed to the opposite calculation, we want an ex-

pression for the interaction potential in terms of M . To this end, we invert the

recursive Eq. (2.33) and solve it iteratively for the matrix element of V , thus

48


finding the so called Born series :

〈p3, p4|V |p1, p2〉 = M(p1, p2, p3, p4)+

− limε→0+

∫d3k1

(2π)3

d3k2

(2π)3

M(p1, p2, k1, k2)M(k1, k2, p3, p4)

Ep1+ Ep2

− Ek1 − Ek2 + iε+ ...

(2.34)

To the lowest possible order, in the so called Born approximation, the potential

is directly provided by the tree-level scattering amplitude. The second term,

known as the Born subtraction, provides crucial corrections for higher order com-

putations. Eventually in the chase of growing precision both in the PN and

PM framework, even other terms of the Born series would happen to offer non-

negligible contributions; nevertheless for the purpose of this thesis we won’t go

any further. In any case, with the momentum transfer

qµ ≡ pµ1 − pµ2 = pµ4 − p

µ3 (2.35)

we can recast the potential as

V (p1, p2, q) ≡ 〈p3, p4|V |p1, p2〉 (2.36)

and find its expression in position space by a simple Fourier transform of the type

q → r.

Now that we have established the link between the interaction potential and

the scattering amplitudes we will proceed to outline the quantum field theory

(QFT) of gravity coupled to matter from which those amplitudes can be con-

structed and evaluated. However before leaving this section we intend to supply

our reasoning with some general observations:

• The scattering amplitude obtained from the given QFT is not exactly M

49

2.4 Quantum field theory for gravitational scattering amplitudes

but rather the relativistic amplitude M. The former is simply related to

the latter by the non-relativistic normalization, namely

M =M

4c2√p0

1 p02 p

03 p

04

. (2.37)

• The whole antiparticle sector must be eliminated by hand, as we funda-

mentally associate the point-particles with the macroscopic classical objects

that compose binary systems. Consequently we neglect from the start any

process in the t and u channels, where particle annihilation takes place.

• What has been found so far holds regardless of the eventual recourse to the

non-relativistic limit i.e. to the expansion in v2/c2. Whether to take this

limit or not is entirely up to us, so that we are able to apply the scattering

amplitude method to compute either PN or PM gravitational potential.

2.4 Quantum field theory for gravitational scat-

tering amplitudes

In the current section our goal is to find the action at the heart of the QFT that

reproduces scattering amplitudes suitable for the computation of the potential in

a two-body gravitationally interacting system. Indeed this action will be basically

formed by a pure gravity term, which describes the dynamics of the graviton (i.e.

the mediator of the gravitational interaction) and an additional coupling term

which provides the link between gravity and our interacting astrophysical objects,

represented by some kind of massive fields. In this sense our first concern is to

understand how to recast general relativity in a meaningful QFT language.

The direct approach is to make an attempt to quantize gravity, as many au-

thors have tried to accomplish in the past (see [52; 53] for some notable examples).

50


Following this line of thinking we would start from the Einstein-Hilbert action

SEH =2

κ2

∫d4x√−gR, κ =

√32πG, (2.38)

and endeavor to work with it along the lines of other field theories. However a

dimensional analysis reveals a serious issue: the coupling constant κ has negative

mass dimensions, [κ] = −1, so that the superficial degree of divergence of every

correlation function in our theory grow as we add more and more loops. Therefore,

increasing the number of loops imply the need to remove an increasing number of

divergences. To be more precise, following the dimensional regularization adopted

in [53], an appropriate suppression of all the divergences at one loop order can

be effectively obtained, by adding to SEH the counterterm

∆S =1

8π2ε

∫d4x√−g(R2

120+

7

20RµνR

µν

), ε = 4−D. (2.39)

At two loops we should add accordingly other counterterms where cubic curvature

invariants like RµνρσRρσαβR µν

αβ make their appearance, multiplied to suitable

divergent coefficients. However we have no means to reabsorb these divergences

into the renormalization of the fields or the coupling constant, therefore the finite

part of the above-mentioned coefficients can only be extracted with the help of

phenomenological indications. Going up with the loop order this situation gets

even worse, and our theory of gravity looses completely its predictive character.

We have to look for an alternative approach. A clue on how to proceed comes

from the chiral perturbation theory: as well as the QFT of gravity discussed

above, this theory is non-renormalizable and non-linear, being the low energy

limit of QCD. Nonetheless predictive and experimentally verified calculation have

been accomplished within it, in the theoretical framework of the effective field

theories (EFT) [54] that we have already met in Sec. 2.2 . Basically it involves

51


a separation between the low-energy, well-behaved degrees of freedom and the

diverging high-energy sector. For what concerns gravity, the energy scale one

typically sets as a reference for this separation is the Plank mass

MP ≡1

κ≈ 1.2× 1019GeV. (2.40)

Correspondingly, an effective action suitable for gravity would be

Seff =

∫d4x√−g(

2

κ2R + c1R

2 + c2RµνRµν + ...

), (2.41)

where we have to include all the possible invariants under general coordinates

transformations (the ellipses stand for invariants with higher powers of R,Rµν ,

and Rµνρσ). With c1 and c2 we denote the dimensionless coupling constants

related to the quadratic invariants. The cosmological constant term has been

omitted due to experimental indications on its notable smallness [55].

Indeed, we can associate any field singularity which arises in loop diagrams

with some component of the action (2.41) and hence absorb it via a simple redefi-

nition of the respective coupling constant. Most importantly, just like one should

expect from the action of an EFT, Seff is organized as an energy expansion: setting

the flat metric as our background metric we observe that

R,Rµν , Rµνρσ ∼ ∂2 ∼ p2. (2.42)

Thus we infer that, for long distance or equivalently low energy computations

such as the one we are interested in, where the energy scale is well below MP , all

the terms in (2.41) except for the Einstein-Hilbert one provide extremely little

contributions. That explains also the really poor experimental boundaries which

have been found for the c1, c2 factors in (2.41) and the complete absence of any

52


experimental indication for the coefficients of higher order terms.

In conclusion by following the EFT prescription we can describe the pure

gravity sector with a straightforward truncation of Seff to the simple Einstein-

Hilbert action without the concern for singularities, which spring up solely at

short-distance, high-energy scales we neglect.

The other ingredient we need is the gravity-matter coupling term. Since

our underlying purpose is to describe the motion of two spinless self-gravitating

objects, we will employ the point-particle action

Scoupling =

∫d4x

√−g2

∑a=1,2

(gµν∂µφa∂νφa −m2

aφ2a

), (2.43)

which minimally couples gravity to the real scalar fields φ1, φ2, whose masses

m1 and m2 are precisely those of the compact binary bodies we take into con-

sideration. Note that vertices involving a φ1 − φ2 direct coupling are absent in

(2.43), as we wish short local interactions between the two matter fields not to

come into play, coherently with our prescriptions. In principle the restriction to

the point-particle approximation may be relaxed with the systematic inclusion

of finite size correction, using higher-dimension operators in the coupling [45].

Similarly one may use a different coupling term with spin carrying massive fields,

in the attempt to develop a suitable description for systems such as a compact

binary of spinning black-holes. See Ref. [41] for a work on this subject.

Overall, the EFT action we will consider is given by

S =

∫d4x√−g[

2R

κ2+

1

2

∑a=1,2

(gµν∂µφa∂νφa −m2

aφ2a

)]. (2.44)

53

2.5 Focus on classical terms


We should bear in mind that the ultimate goal of our computation is to provide

general relativistic correction to the Newtonian potential. In other words, among

the various terms that could emerge from our amplitudes, the contributions we

seek are exclusively the classical and long range ones. Specifically, with respect

to the momentum transfer q, our interest is entirely focused on those terms which

dominate the infrared regime q2 → 0. Therefore we are justified to ignore all the

analytic terms (i.e polynomial in q) and to keep only the leading non-analytic

terms such asα1

q2,

α2√−q2

, α3ln(−q2), (2.45)

for some q-independent coefficients α1, α2, α3. This situation is obviously un-

changed after the Fourier transform q → r, since analytic terms give rise to

unwanted ultra-local contributions, that is r-dependent delta functions or their

derivatives, whereas non-analytic terms produce the long-distance corrections we

seek, proportional to some positive power of 1/r. Moreover the separation be-

tween these two classes of terms is always clear in our theory and the cumbersome

renormalization process we have mentioned in the previous section ends up affect-

ing only the negligible analytic sector (which dominates the ultraviolet regime),

thus validating further the truncation of (2.41) to the Einstein-Hilbert action.

That classical terms emerge from a quantum mechanical loop expansion is some-

what surprising. This is made possible by a subtle cancellation of ~ factors,

clearly discussed in Ref. [56].

Besides, our exclusive focus on classical terms amounts to a crucial simplifi-

cation in our computation, because allow us to slim down enormously the set of

Feynman diagrams whose evaluation is needed, by limiting ourselves to those that

provides the non-analytic contributions we desire. For example we can exclude

54


from the start diagrams where

• at least a momentum in a loop flows only through graviton propagators:

• at least a graviton connects two external legs relative to the same particle:

• at least one graviton line starts and ends on the same matter line:

Further simplifications come from generalized unitarity [26], a method originally

developed in the context of Yang–Mills theories which outlines a systematic

scheme for building loop amplitudes from simpler tree-level amplitudes. On the

same note, the double copy construction [57] may be used to establish a system-

atic connection between gravity and gauge theory scattering amplitudes, giving

the chance to work with the latter, which are more manageable, and recover with

contained effort the corresponding results for gravity.

55

Chapter 3

PN and PM computations in

parametric frames of reference

In recent times many authors devoted themselves to the computation via scat-

tering amplitudes of general relativistic corrections to the Newtonian two-body

potential. A first crucial consistency check for the results of those calculations is

often found in the comparison with the potential of the Einstein-Infeld-Hoffman

(EIH) Hamiltonian, given in (2.1). However a large part of this works such as

Refs. [29; 38; 41] exhibits potentials which do not match the EIH one (for PN

results this comparison is direct, whereas in the PM case it requires first a non-

relativistic expansion). This fact is typically justified a posteriori in terms of

equivalence up to canonical transformations: essentially one may define a coor-

dinate shift such that it preserves the Poisson brackets and maps the alternative

solution in these articles to the EIH one, thus establishing the physical equivalence

between the two. In this sense the cause of the apparent discrepancy is attributed

to the gauge dependence of the potential after the Fourier transform. For more

extensive discussions on this topic we refer to Appendix B of [41] and to Sec.

11.1 of [38]. All things considered, it would be interesting to outline a systematic

56

way to find directly the EIH potential without the need of any remapping, and

correspondingly to track down the passage in which the EIH-compatibility could

be lost.

The original calculation of this thesis, main subject of the present chapter,

goes exactly in this direction and shows some strategies, within the scattering

amplitude method, to obtain directly an EIH-consistent gravitational potential

while adopting the De Donder gauge fixing. More specifically, after an initial

section in which the prerequisite Feynman rules of our EFT will be determined,

a first segment of this chapter will be devoted to the computation of the O(G/c0)

and O(G/c2) two-body potential in a generalized reference frame where the scat-

tering momenta are expressed as functions of the free parameters α and β. In

the subsequent part we will present an alternative approach which yields the

same results while remaining confined to a generally parameterized center-of-

mass (COM) reference frame. Then, we will complete the tree-level discussion

with the determination, within the above mentioned two-parameter frame, of

a full relativistic 1PM potential such that, when non-relativistically expanded,

returns all the O(G) terms of the EIH potential. In the final section we will

conclude the determination of the EIH-consistent 1PN potential by computing

also its O(G2/c2) static component.

57

3.1 Feynman rules for tree-level gravity amplitudes

3.1 Feynman rules for tree-level gravity ampli-

tudes

In this section we derive the needed Feynman rules as they can be extracted from

the action (2.44) by considering the metric expansion

gµν = ηµν + κhµν ,

gµν = ηµν − κhµν + κ2hµρh νρ +O(κ3),

√−g = 1 +

κ

2h+

κ2

8

(h2 + 2hραh

ρα)

+O(κ3),

(3.1)

where the small fluctuation hµν is identified with the graviton field and κ =√

32πG as usual.

Let us start from the graviton propagator. The Einstein-Hilbert term in (2.44)

expanded to second order in the graviton field reduces to the well-known Pauli-

Fierz action

SPF =

∫d4x

1

2

(∂ρh∂

ρh− ∂ρhαβ∂ρhαβ − 2∂ρhρα∂αh+ 2∂ρhαβ∂

βhρα). (3.2)

Just like in other gauge theories the quadratic form in (3.2) is not invertible in

itself, and we must impose a proper gauge fixing. In this respect we mention that

a correct quantization scheme for the pure gravity sector of our effective field

theory should encompass the so called background field method: the metric is

expanded around a background metric gµν , namely

gµν = gµν + κhµν , (3.3)

so that all the dynamical fields in our theory propagate in a curved background

geometry whose general covariance is not affected by the gauge fixing conditions,

58


which are only imposed on the quantum field hµν . Moreover also the background

metric is expanded, along the lines of what is done with gµν :

gµν = ηµν + κHµν , (3.4)

where Hµν represents the “external” graviton field, i.e. the field one associates

to gravitons which appear as external states in the scattering process. However,

for the Feynman rules required in our computation we should always expand to

zeroth order in the field Hµν , therefore this scheme has no practical implications

on our work and we will proceed without focusing further on it.

As for the gauge fixing we choose once more the De Donder gauge, by adding

to (3.2) the action

SGF = −∫d4x

(∂ρhρα −

1

2∂αh

)2

. (3.5)

The Faddeev–Popov gauge fixing procedure would include also the the ghost field

action

Sghost =

∫d4x η†ρ

(∂ρ∂α −Rρα

)ηα (3.6)

but it would bring nothing more than purely quantum contributions so that we

can safely omit it in our classical calculation. After an integration by part we get

SPF + SGF =1

2

∫d4xhρσPρσαβh

αβ, (3.7)

for the properly defined

Pρσαβ ≡1

2

(ηραησβ + ηρβησα − ηρσηαβ

). (3.8)

Moving to the momentum space, the graviton propagator is found as the tensor

59


Dµνρσ(q) such that

Dµνρσ(q)(− q2Pρσαβ

)=

1

2

(δµαδ

νβ + δµβδ

να

). (3.9)

With some algebra we find

≡ −iDµνρσ(q) =iPµνρσq2 + iε

. (3.10)

Now we turn our attention to the determination of the vertices between gravi-

tons and the two massive fields. Clearly in order to accomplish that we have to

expand the gravity-matter coupling term of (2.44), according to Eqs. (3.1). This

operation leads us to

Scoupling = S(0)coupling + S

(1)coupling + S

(2)coupling +O(κ3) (3.11)

where

S(0)coupling ≡

∫d4x

1

2

∑a=1,2

(∂ρφa∂

ρφa −m2aφ

2a

), (3.12)

S(1)coupling ≡

κ

2

∫d4xhρσ

∑a=1,2

[1

2ηρσ(∂αφa∂

αφa −m2aφ

2a

)− ∂ρφa∂σφa

], (3.13)

and

S(2)coupling ≡

κ2

2

∫d4x

∑a=1,2

[1

8

(h2 − 2hρσh

ρσ)(∂αφa∂

αφa −m2aφ

2a

)+

+

(hραh ν

α −1

2hhρσ

)∂ρφa∂σφa

].

(3.14)

60

3.2 Tree-level potential in a generalized reference frame

At this stage the single-graviton vertex can be read directly from (3.13):

≡ τµν(p1, p2,m) = −iκ2

[p1µp2ν+p1νp2µ−ηµν(p1·p2−m2)

].

(3.15)

Similarly from the action term (3.14) we extract the two-graviton vertex

≡ τµνρσ(p1, p2,m) = iκ2

2

[2IµναβI

βλρσ(pα1p

λ2 + pλ1p

α2 )+

− (ηµνIαλρσ + ηρσIαλµν)pα1p

λ2 − Pµνρσ(p1 · p2 −m2)

],

(3.16)

in which we have introduced

Iρσαβ ≡1

2(ηραησβ + ηρβησα) (3.17)

These are all the Feynman rules we need for our purposes.

3.2 Tree-level potential in a generalized refer-

ence frame

In compliance with what has been presented in Sec. 2.3, the two-body potential

at order O(G/c0) and O(G/c2) is directly provided by the tree-level amplitudes

of our theory, since in it an L-loop amplitude ML−loop is of order O(GL+1). Fur-

61


thermore, being the s-channel the only one to be considered, due to our request

for long-distance interactions between the two matter scalar fields, we can restrict

ourselves to the sole tree-level amplitude

≡M tree(p1, p2, p3, p4). (3.18)

The Feynman rules we stockpiled above tell us its general form:

M tree(p1, p2, p3, p4) =i

4c2√p0

1 p02 p

03 p

04

τµν(p1, p2)iPµνρσ

q2τρσ(p3, p4), (3.19)

where the non-relativistic normalization has been included as well. Restoring c

for the sake of clarity, the contractions in (3.19) evaluates to

τµν(p1, p2)Pµνρστρσ(p3, p4) = (p1 · p3)(p2 · p4) + (p1 · p4)(p2 · p3) + (p3 · p4)m21c

2+

+ (p1 · p2)m22c

2 − (p1 · p2)(p3 · p4)− 2m21m

22c

4.

(3.20)

For the present calculation all the scattering momenta on the external legs

will be taken on-shell:

pµi =(Ei/c,pi

),

E1 = c√m1c2 + p1, E2 = c

√m1c2 + p2,

E3 = c√m2c2 + p3, E4 = c

√m2c2 + p3,

(3.21)

i.e. p21 = p2

2 = m1c2 and p2

3 = p24 = m2c

2. Moreover, we will work in the parametric

62


reference frame defined by

p1 = p + αq

p2 = p + (α− 1)q

p3 = p′ − βq

p4 = p′ − (β − 1)q

, (3.22)

in which q is the spatial component of the momentum transfer. Observe that

setting p′ = −p and β = α equals to recovering the generalized COM reference

frame p1 = −p3 = p + αq

p2 = −p4 = p + (α− 1)q

. (3.23)

Many authors have employed the COM frame for their gravity amplitude com-

putations and from (3.23), by fixing the free parameter α, we are able to match

all of their choices:

• for α = 0 we achieve the simple framep1 = −p3 = p

p2 = −p4 = p− q

(3.24)

one can find in Refs. [29; 38]

• with α = 1/2 we reach the so called symmetric COM frame

p1 = −p3 = p +

q

2

p2 = −p4 = p− q

2

(3.25)

employed for instance by Holstein and Ross in Ref. [41].

63


Furthermore from the general frame (3.22) we could recover even other meaningful

frames of reference. An example in this sense is the laboratory frame we get by

imposing p = −αq,

p1 = 0

p2 = q

p3 = p′ − βq

p4 = p′ − (β − 1)q

(3.26)

where we see as expected that one of the particles is initially at rest.

Now let us move to the graviton propagator. By following a procedure

launched in the seminal paper of Iwasaki [24] we consider the expansion

1

q2=

1

q20 − q2

= − 1

q2

1

1− q20/q

2= − 1

q2

(1 +

q20

q2+q4

0

q4+ ...

), (3.27)

which may be thought as a series of smaller and smaller corrections to an interac-

tion devoid of energy exchange between the scattering particles (in which case one

has q0 = 0). Below we will return to the effective smallness of the contribution

in (3.27), for now we stress that the essential condition q0 < |q| is ensured by the

space-like character of qµ. Regardless, in full generality we can write

q0 = (p1)0 − (p2)0 =(p1)2

0 − (p2)20

(p1)0 + (p2)0

=p2

1 − p22 + p2

1 − p22

(p1)0 + (p2)0

(3.28)

and by considering p21 = p2

2 = m21c

2 and q = p1 − p2,

q0 =(p1 + p2)q

(p1)0 + (p2)0

= c(p1 + p2)q

E1 + E2

. (3.29)

64


The same reasoning from q0 = (p4)0 − (p3)0 bring us to

q0 = c(p3 + p4)q

E3 + E4

. (3.30)

The propagator (3.27) truncated to the next-to-leading order is then given by

1

q2≈ − 1

q2

[1 +

c2

q2

(p1 + p2)q(p3 + p4)q

(E1 + E2)(E3 + E4)

], (3.31)

where we stress the fact that both (3.29) and (3.30) have been used to obtain it,

so as to reach a symmetric form with respect to the scattering particles. Now one

may ask: is q0/|q| so small that such a truncated expression can be effectively

used in our amplitude? The answer is yes, as we point out that, within the non-

relativistic limit, the terms we neglect in (3.31) result at least of order O(1/c4) in

a framework where we keep contributions only up to order O(1/c2). That aside

this expression has the only constraint of on-shell external momenta, for the rest

it is valid in every coordinate system we listed above. In the reference frame

(3.22) that we selected for our current calculation it can be rewritten in the form

1

q2≈ − 1

q2

[1 +

c2

q2

(p · q)(p′ · q)

(E1 + E2)(E3 + E4)

], (3.32)

which is exactly the one we are going to substitute in our amplitude. Despite

holding also in the COM frame (3.23) regardless the value of α, in this case the

propagator expansion becomes useless because kinematics imposes q0 = 0 and

from (3.28) one has straightforwardly

1

q2= − 1

q2. (3.33)

65


In other words the term proportional to

(p · q)2

q4,

that would be derived from (3.31) with the relations among the momenta given

in (3.23), actually vanishes under this coordinate choice. This passage is funda-

mental: as we will see below, the presence of such a term is mandatory if we

want to obtain an EIH-consistent potential. Thus its absence in the COM frame

definitively marks the unsuitableness of this reference frame for achieving this

objective, as long as all the external momenta are taken on-shell. That is exactly

the reason behind the mismatch between the EIH potential and what has been

found in Refs. [29; 38; 41].

That being said, let us proceed with our calculation. Overall, after an expan-

sion in powers of c−2 around zero in which we retain only non-analytic terms in

q, our tree-level amplitude equipped with (3.32) results in

M tree(p,p′, q) = −4πGm1m2

q2+

6πGp · qc2 q2

[m2

m1

(1− 2α) + (1− 2β)

]+

− 6πGp′ · qc2 q2

[m1

m2

(1− 2β) + (1− 2α)

]− 6πG

c2 q2

(m2

m1

p2 +m1

m2

p′2)

+

+16πGp · p′

c2 q2− 4πG (p · q)(p′ · q)

c2 q4+O

(1

c4

).

(3.34)

From here the two-body potential up to order O(G/c2) is readily obtained by

performing the Fourier transform

V (p,p′, r) =

∫d3q

(2π)3M tree(p,p′, q)e−iq·r, (3.35)

66


with the help of the relations:

•∫

d3q

(2π)3

e−iq·r

q2=

1

4πr

•∫

d3q

(2π)3

p · qq2

e−iq·r = ip · r4πr3

•∫

d3q

(2π)3

(p · q)(p′ · q)

q2e−iq·r =

1

8πr

[p · p′ − (p · r)(p′ · r)

r2

] (3.36)

Our result is

V (p,p′, r) = −Gm1m2

r− Gm1m2

2rc2

[3p2

m21

+3p′2

m21

− 7p · p′

m1m2

− (p · r)(p′ · r)

m1m2r2

]+

+i3Gp · r2r3c2

[m2

m1

(1− 2α) + (1− 2β)

]− i3Gp

′ · r2r3c2

[m1

m2

(1− 2β) + (1− 2α)

].

(3.37)

If we identify the momenta of the two self-gravitating objects with p and p′ then

the real part of our potential corresponds exactly to the O(G) part of the EIH

potential as it can be easily extracted from the Hamiltonian (2.1). In addition

we see a parametric imaginary part too. Since we obtained this potential from a

scattering amplitude the presence of imaginary terms should not be so surprising,

nevertheless we have to eliminate them if we hope for a correctly defined potential.

That is, we have to fix the parameters α and β in order to simultaneously nullify

the square brackets in the last two terms of (3.37). Actually, the two consequential

conditions on our parameters are linearly dependent, therefore we have an infinite

set of plausible values for α and β:

m2

m1

(1− 2α) + (1− 2β) = 0 =⇒

α = x

β =m2

m1

(1

2− x)

+1

2

(3.38)

for a generic real number x.

67


We observe that at least one of the two parameters is forced to depend on

the masses of our astrophysical bodies unless we set α = β = 1/2. Then one

may naively conclude that, according to our result, the reference frame (3.24)

corresponding to α = β = 0 should produce a potential with non-zero imaginary

terms, in a way that would have affected all the computations in which it has

been employed, such as the aforementioned Refs. [29; 38]. Indeed that is incorrect,

since in these works terms like (p ·q) never arise, while we have just showed them

to be necessary for the appearance of imaginary components in the potential. The

absence of those terms is strictly connected to the aforementioned impossibility

of a direct matching with the EIH potential, and results as a general feature of

the COM reference frame whenever all the external momenta are taken on-shell

(as it is done in Refs. [29; 38]). To explain this fact let us put ourselves in the

general COM frame given in (3.23). Here kinematics imposes

q0 = 0 ⇐⇒ E1 = E2 ⇐⇒ p21 = p2

2 (3.39)

as soon as we consider on-shell external momenta. In turn (3.39) yields

p · q =1− 2α

2q2, (3.40)

therefore all the (p · q) terms are either zero (for α = 1/2) or lost in irrelevant

analytic contributions.

Fundamentally the kinematic condition (3.39) one has in the COM frame is

at the same time what prevents to reproduce the EIH potential and what keeps

imaginary terms from appearing regardless the value of α. The only hope to relax

this condition without leaving behind the COM frame seems to rest in giving up

on the precept of on-shell external momenta. This represents the subject of the

following section.

68

3.3 Tree-level potential in the COM frame with off-shell external legs

Moreover in light of the result obtained in this section, the Appendix A will

be devoted to a possible clarification on Iwasaki’s calculation at the heart of Ref.

[24].

3.3 Tree-level potential in the COM frame with

off-shell external legs

Here our aim is to find a two-body potential compatible with (3.37) within the

COM frame of reference. As we have already mentioned in order to achieve this

we need to set off-shell some of the external momenta. If we were taking into

account an actual scattering process such an assumption would be clearly incon-

sistent. However in our scheme the scattering process serves as a mere conceptual

instrument to extract the corresponding interaction potential, under the hypoth-

esis that it responds to the same physics of the compact binary dynamics, the

real physical process we wish to describe. That being said, our particular choice

for the momenta is

pµ1 =

(√m2

1c2 + p2

1,p1

),

pµ2 =

(√m2

1c2 + p2

2 + ε(p21 − p2

2),p2

),

pµ3 =

(√m2

2c2 + p2

1 + ε(p21 − p2

2),−p1

),

pµ4 =

(√m2

2c2 + p2

2,−p2

).

(3.41)

Only the external legs 2 and 3 are taken off-shell and this is accomplished with

an additional term in the energy definition, proportional to the new parameter ε

which determines the extent of the deviation from the on-shell condition. Besides,

we highlight that the momenta in (3.41) go automatically on-shell, regardless the

69


value of ε, as soon as p21 = p2

2. In our calculation this feature has proved to be

crucial. Equivalently one could put off-shell in the same way the external legs 1

and 4, or more in general all the pairs of external legs that, being put off-shell,

ensure the fundamental condition

q0 = (p1)0 − (p2)0 = (p4)0 − (p3)0 6= 0. (3.42)

Note that the expanded expression (3.31) for the graviton propagator no

longer holds, since in the derivation of (3.29) and (3.30) we used p21 − p2

2 =

p24 − p2

3 = 0, whereas our current momenta choice (3.41) yields

p21 − p2

2 = p24 − p2

3 = −ε(p21 − p2

2) (3.43)

Therefore restarting from Eq. (3.28) we get

1

q≈ −1

q

[1− c2

q2

(1− ε)(1 + ε)[(p1 + p2)q

]2(E1 + E2)(E3 + E4)

]. (3.44)

In the generic COM reference frame (3.23) we put ourselves in, the tri-

momenta p1 and p2 are rewritten as

p1 = p + αq

p2 = p + (α− 1)q

. (3.45)

70


Overall the tree-level amplitude (3.34) is now given by

M tree(p, q) = −4πGm1m2

q2− 6πGp2

c2 q2

(m2

m1

+m1

m2

+8

3

)+

+6πGp · qc2 q2

[(2 +

2

3ε2

)(1− 2α) +

(m2

m1

+m1

m2

)(1 + ε− 2α)

]+

+4πG (p · q)2

c2 q4(1− ε2) +O

(1

c4

)(3.46)

Finally the Fourier transform q → r, resorting once more to the Eqs. (3.36),

yields the potential

V (p, r) = −Gm1m2

r− Gm1m2

2rc2

[3p2

m21

+3p2

m22

+7p2

m1m2

+(p · r)2

m1m2r2

]+

+ ε2 G

2rc2

[(p · r)2

r2− p2

]+ i

3Gp · r2r3c2

(2 +

m2

m1

+m1

m2

)(1− 2α)+

− iε3Gp · r2r3c2

(m2

m1

+m1

m2

)+ iε2 3Gp · r

r3c2(1− 2α).

(3.47)

Interestingly, if we set p as the momentum carried by our pair of astrophysical

bodies in the COM frame, the real part of (3.47) reduces to the O(G/c0) and

O(G/c2) components of the EIH potential in Eq. (2.2), provided that we consider

the limit ε → 0, i.e. the limit in which all the external momenta return on-

shell. We emphasize that such a limit could only be taken at this stage in the

calculation, since restoring the on-shell character of the external momenta before

the Fourier transform would lead back to the kinematic condition (3.39) discussed

in the conclusive observations of Sec. 3.2.

Regarding the imaginary part cancellation we see that in the limit ε → 0

we are left with the sole choice α = 1/2. Actually this is in perfect agreement

with (3.38) since in the COM frame of reference one has the default condition

α = β and in (3.38) the only value which can be simultaneously shared by the

71

3.4 1PM potential in the general reference frame

two parameters is exactly 1/2.


So far we have always worked under the non-relativistic limit, but as we specified

in the previous chapter the scattering amplitude method is naturally predisposed

to be performed also in the fully relativistic framework of the post-Minkowskian

scheme. Therefore in order to conclude our dissertation on the tree-level compu-

tation, here we will try to apply the strategy outlined in Sec. 3.2 to compute a

1PM two-body potential such that it is EIH-consistent under the non-relativistic

expansion.

Once again the starting point is the tree-level amplitude (3.18) which can be

expressed as

M tree(p1, p2, p3, p4) =1√

E1E2E3E4

4πG

q2

[(p1 · p3)(p2 · p4) + (p1 · p4)(p2 · p3)+

+ (p3 · p4)m21c

2 + (p1 · p2)m22c

2 − (p1 · p2)(p3 · p4)− 2m21m

22c

4].

(3.48)

Just like in Sec. 3.2 we set on-shell all the external momenta and employ the

two-parameter generalized reference frame given in (3.22). For what concerns

the graviton propagator we have somehow to rewrite it in a form suitable for the

Fourier transform q → r. Again the idea would be to employ Iwasaki’s expansion

and thus to insert in the amplitude the truncated propagator (3.32), which we

recall here:1

q2≈ − 1

q2

[1 +

c2

q2

(p · q)(p′ · q)

(E1 + E2)(E3 + E4)

]. (3.49)

However such a truncation can no longer be justified from the perspective of the

non-relativistic limit as we have done in Sec. 3.2, since the present calculation

72


is completely relativistic. Therefore in principle one should rather keep all the

terms in Iwasaki’s expansion and work with the full propagator

1

q2= − 1

q2+

+∞∑j=1

[c2

q4

(p · q)(p′ · q)

(E1 + E2)(E3 + E4)

]j. (3.50)

After all even in the context of a nPN calculation, where one retains all the terms

which contain power of 1/c2 up to 1/c2n, as we increase the order n we would

have to include more and more terms of this series, specifically up to j = n.

Nonetheless in the following we will recur to the truncated propagator (3.49),

because we are currently interested in finding a 1PM potential which matches

the O(G) components of the EIH one when non-relativistically expanded, and to

this end the propagator form (3.49) is plenty enough since it “fixes” the matching

up to order O(1/c2).

Besides, not taking the non-relativistic limit until the final result determines

that, unlike what we saw happen in the previous two sections, even by carrying

out the products in the amplitude (3.48) and using the above propagator we

actually fail to reach an expression suitable for the Fourier transform q → r,

because of the q factors which remain nestled in the square roots of the energy

explicit forms. To overcome this obstacle we consider a Laurent expansion in

|q| around zero where we neglect all the terms from O(|q|0) onward, basically

exploiting the fact that all the contributions we seek are non-analytic functions

of q = |q|q. In this way we find

M tree(p,p′, q, α, β) = M tree1 (p,p′, q) +M tree

2 (p,p′, q, α, β), (3.51)

73


where the parameter-free component can be written as

M tree1 (p,p′, q) = −4πG

q2

[2(p1 · p3)2 −m2

1m22c

4

E1E3

]|q|=0

+

− 4πGc2 (p · q)(p′ · q)

q4

[2(p1 · p3)2 −m2

1m22c

4

(E1E3)2

]|q|=0

(3.52)

while the parametric one assumes the form

M tree2 (p,p′, q, α, β) = (2α− 1)

[A(p,p′)

(p · q)

q2+B(p,p′)

(p′ · q)

q2

]+

+ (2β − 1)

[C(p,p′)

(p · q)

q2+D(p,p′)

(p′ · q)

q2

]+

+ (2α− 1)E(p,p′, q) + (2β − 1)F (p,p′, q),

(3.53)

for some different functions A,B,C,D,E, F whose expressions is not relevant for

our purpose. Starting from the latter we observe that the terms inside the square

brackets would produce imaginary terms when Fourier-transformed. Therefore,

consistently with what has been found in the previous sections, we have to sacrifice

our parameterization freedom in favor of their exclusion from the potential, by

setting α = β = 1/2. Indeed in this case the entire contribution (3.53) vanishes,

making clear why we do not need its complete expression.

Then let us shift our attention to the remaining component (3.52) of the

tree-level amplitude. The first term corresponds to the momentum space 1PM

potential which may be extracted from equation (19) of Ref. [29] or equivalently

from equations (10.9)-(10.10) of Ref. [38]. Note that in their coordinate choice

(COM frame with α = 0) the quantity that we evaluate in |q| = 0 is already

q-independent. The other term in (3.52) originates from the second piece of

the truncated propagator (3.49) and, as we will show below, it is crucial to the

matching with the EIH potential.

The Fourier transform q → r on the amplitude (3.52), with the usual relations

74

3.5 Completing the 1PN potential

(3.36), yields the potential in position space:

V1PM(p,p′, r) = −Gr

[2(p1 · p3)2 −m2

1m22c

4

E1E3

]|q|=0

+

− Gc2

2r

[p · p′ − (p · r)(p′ · r)

r2

][2(p1 · p3)2 −m2

1m22c

4

(E1E3)2

]|q|=0

.

(3.54)

To check the EIH-compatibility of our potential we have to determine its non-

relativistic expanded form up to order O(1/c2). This operation is straightforward

and results in

V1PM(p,p′, r) = −Gm1m2

r− Gm1m2

2rc2

(3p2

m21

+3p′2

m21

− 8p · p′

m1m2

)+

− Gm1m2

2rc2

[p · p′

m1m2

− (p · r)(p′ · r)

r2

]+O

(1

c4

),

(3.55)

where in the third term we have collected all the contributions which follow

from the second component of our 1PM potential (3.54), namely the additional

component that the truncated propagator (3.49) yields when q0 6= 0. From here

one may easily confirm that the matching with (3.37) and hence with the EIH

potential is achieved only when this third term is included as well. Again the

condition q0 6= 0 and Iwasaki’s expansion seem to represent the key point in the

EIH-compatibility of the two-body potential whenever the scattering amplitude

method is employed.


In Sec. 3.2 and Sec. 3.3 we showed two methods to obtain the EIH-potential up

to order O(G/c2). To complete the derivation of the 1PN two-body potential

we ought to compute the O(G2/c2) part as well. This is precisely what we aim

to do in the current section. With reference to the two-parameter coordinate

75


system (3.22), in light of the constraints we have found for the parameters in the

preceding sections, we will assume from the start the condition α = β = 1/2,

which we have just seen to prevent the appearance of imaginary terms in all

of our tree-level calculations. Moreover, since we are after a static component,

i.e. momentum-independent, we can put ourselves straight in the on-shell COM

frame (p′ = −p) without any concern for missing terms with respect to the EIH

potential. To sum up, we will employ the symmetric COM frame (3.25), where

the following convenient relations hold:

p2i ≡ p2 = p2 +

q2

4for i = 1, 2, 3, 4 (3.56)

p =p1 + p2

2= −p3 + p4

2(3.57)

p · q = 0. (3.58)

Besides, being this reference frame the one chosen by Holstein and Ross in Ref.

[41], by working in we will be able to take advantage of their results.

Following the procedure outlined in a general fashion in Sec. 2.3, the O(G2/c2)

term we seek is the offspring of a non-relativistic expansion truncated to order

O(1/c2) of the quantity

M1−loop −B,

which is the combination of the one-loop amplitudes of our theory M1−loop and

the Born subtraction term B. The latter, in the reference frame we opted for, is

76


given by

B =

∫d3k

(2π)3

M tree(p1,k,−p1,−k)M tree(k,p2,−k,−p2)

c√

p2 +m1c2 + c√

p2 +m2c2 − c√

k2 +m1c2 − c√

k2 +m2c2,

(3.59)

where the tree-level amplitudes are readily obtained from Eq. (3.34), for α = β =

1/2, by rewriting everything (q included) in terms of the external tri-momenta,

namely

M tree(p1,p2,p3,p4) = − 4πGm1m2

(p1 − p2) · (p4 − p3)

[1 +

3

4

(p2

1 + p22

m21c

2+

p23 + p2

4

m22c

2

)+

− (p1 + p2) · (p3 + p4)

m1m2c2+

(p1 − p2) · (p1 + p2) (p4 − p3) · (p4 + p3)

4m1m2c2 (p1 − p2) · (p4 − p3)

]+

+O

(1

c4

).

(3.60)

Note that in the Born subtraction term only tree-level amplitudes appear, since

we are looking for a term quadratic in the gravitational constant G and each

tree-level amplitude brings along an O(G) factor. This same argument explains

why no other term of the Born series has been considered here.

For a start let us perform explicitly the computation of B. Our primary task

is to determine its integrand up to order O(1/c2). For this purpose we need to

expand in 1/c2 also the denominator of (3.59):

1

c√

p2 +m1c2 + c√

p2 +m2c2 − c√

k2 +m1c2 − c√

k2 +m2c2=

=2m1m2

(m1 +m2) (p2 − k2)+

(m21 −m1m2 +m2

2) (p2 + k2)

2m1m2(m1 +m2)c2 (p2 − k2)+O

(1

c4

) (3.61)

Now that all the quantities in (3.59) are organized in powers of 1/c2, the associated

77


integrand is found by evaluating their product

16π2G2m21m

22

(p1 − k)2(p2 − k)2

[1 +

3

4

m21 +m2

2

m21m

22c

2(p2 + k2) +

(p1 + k)2

m1m2c2− (p2 − k2)2

4m1m2c2 (p1 − k)2

]×

×[1 +

3

4

m21 +m2

2

m21m

22c

2(p2 + k2) +

(p2 + k)2

m1m2c2− (p2 − k2)2

4m1m2c2 (p2 − k)2

]×

×[

2m1m2

(m1 +m2) (p2 − k2)+

(m21 −m1m2 +m2

2) (p2 + k2)

2m1m2(m1 +m2)c2 (p2 − k2)

]

and neglecting all the contributions more than quadratic in 1/c. This process

yields

B =8π2G2m1m2

(m1 +m2)c2

[4c2m2

1m22I1 + 4m1m2I2 + (7m2

1 −m1m2 + 7m22)I3 −m1m2I4

]+

+O

(1

c4

),

(3.62)

where

I1 ≡∫

d3k

(2π)3

1

(p1 − k)2(p2 − k)2(p2 − k2), (3.63)

I2 ≡∫

d3k

(2π)3

(p1 + k)2 + (p2 + k)2

(p1 − k)2(p2 − k)2(p2 − k2), (3.64)

I3 ≡∫

d3k

(2π)3

p2 + k2

(p1 − k)2(p2 − k)2(p2 − k2), (3.65)

I4 ≡∫

d3k

(2π)3

[(p1 − k)2 + (p2 − k)2

](p2 − k2)

(p1 − k)4(p2 − k)4. (3.66)

We have now to compute this four tridimensional integrals. The fact that

we seek only classical contributions provides a notable simplification: the first

78


integral I1 is already given in Eqs. (103) of Ref. [41] and reads

I1 = ilog(q2)

8π p q2. (3.67)

Restoring ~ through dimensional analysis reveals that this contribution is pro-

portional to ~ and thus it is purely quantum mechanical. Therefore we are free

to neglect I1 and all the components of I2 and I3 which do not present a k-

dependence in the numerator. Moreover regarding I2 we have

(p1 + k)2 + (p2 + k)2 (3.57)= 2p2 + 2k2 + 4k · p, (3.68)

so that the classic components of I2 and I3 are completely determined by the

following two integrals, again available in Eqs. (103) of Ref. [41]:

∫d3k

(2π)3

k2

(p1 − k)2(p2 − k)2(p2 − k2)= − 1

8|q|, (3.69)

∫d3k

(2π)3

k

(p1 − k)2(p2 − k)2(p2 − k2)= − p

16p2|q|. (3.70)

Thanks to them we get

I2

∣∣classical

= − 1

2|q|,

I3

∣∣classical

= − 1

8|q|.

(3.71)

For what concerns I4, we start from

(p1 − k)2 + (p2 − k)2 (3.57)= 2p2 + 2k2 − 4k · p (3.56)

= 2(p− k)2 +q2

2(3.72)

and express all the rest in k, p and q by resorting to (3.56) and (3.25). Our

79


integral becomes

I4 =

∫d3k

(2π)3

(2p2 + 2k2 − 4k · p + q2/2)(p2 + q2/4− k2)

(p + q/2− k)4(p− q/2− k)4(3.73)

Then to recast it an advantageous form we consider the shift k → k + p + q/2

along with the simple relation 2k · q = (k + q)2 − k2 − q2 so as to obtain

I4 = −1

2

∫d3k

(2π)3

[k2 − (k + q)2][k2 − q2 + (k + q)2 + 4p · k]

(k2)2[(k + q)2]2=

= −1

2

∫d3k

(2π)3

[1

[(k + q)2]2+

1

(k2)2− q2

k2[(k + q)2]2− q2

(k2)2(k + q)2+

+2

k2(k + q)2+

4p · kk2[(k + q)2]2

+4p · k

(k2)2(k + q)2

].

(3.74)

For the first five sub-integrals we can use the formula

∫d3k

(2π)3

1

(k2)n[(k + q)2]m=

Γ(

32− n

)Γ(

32−m

)Γ(m+ n− 3

2

)(4π)3/2Γ(n)Γ(m)Γ(3− n−m)

1

|q|2(n+m)−3,

(3.75)

with which we discover that only the fifth one is effectively non-zero. The last

two integrals vanish as well, because of the formula

∫d3k

(2π)3

k

(k2)n[(k + q)2]m=

Γ(

52− n

)Γ(

32−m

)Γ(m+ n− 3

2

)(4π)3/2Γ(n)Γ(m)Γ(4− n−m)

q

|q|2(n+m)−3

(3.76)

and the condition p · q = 0, always valid in our currently employed reference

frame. Therefore we end up with

I4 = −∫

d3k

(2π)3

1

k2(k + q)2

(3.75)= − 1

8|q|. (3.77)

By inserting the classical values of the just calculated integrals Ii in Eq. (3.62)

80


Figure 3.1: List of the classically relevant one-loop Feynman diagrams, groupedby topology: (a) two-graviton triangular diagrams, (b) box and crossed-box dia-grams, (c) three-graviton triangular diagrams

the Born subtraction term results

B = −7π2G2m1m2(m1 +m2)

c2 |q|. (3.78)

Let us now move to the one-loop contribution, which at this stage is the

only ingredient we lack for the 1PN potential. The one-loop Feynman diagrams

with non-zero classical contributions are listed in Fig. (3.1). The idea behind the

computation of their associated integrals is to decompose them in terms of scalar

integrals multiplied by coefficient which do not depend neither on the momentum

transfer qµ nor on the loop momentum lµ. A list of already evaluated integrals

which turn out to be quite useful for this task can be found in Appendix A of

Ref. [58]. We will not dwell upon the details of their explicit calculation and

refer once more to the results given in Ref. [41], that we present here divided per

81


topology in compliance with Fig. (3.1):

M1−loop(a) = −8G2π2m2

1m2

c2 |q|− 8G2π2m1m

22

c2 |q|= −8G2π2m1m2(m1 +m2)

c2 |q|,

M1−loop(b) =

4G2π2m1m2(m1 +m2)

c2 |q|− 4G2π2m1m2(m1 +m2)

c2 |q|= 0,

M1−loop(c) =

2G2π2m21m2

c2 |q|+

2G2π2m1m22

c2 |q|=

2G2π2m1m2(m1 +m2)

c2 |q|,

(3.79)

where c has been restored and all the signs changed to make them consistent

with our conventions. We observe that the classical components of the square-

like diagrams delete each other, so that only the triangular ones actually provide

noteworthy contributions.

By adding M1−loop(a) and M1−loop

(c) we easily find

M1−loop = −6G2π2m1m2(m1 +m2)

c2 |q|(3.80)

and hence, recovering our result (3.78) for B, we finally obtain

V[G2/c2](q) = M1−loop −B =G2π2m1m2(m1 +m2)

c2 |q|. (3.81)

In conclusion, the Fourier transform

∫d3q

(2π)3

e−iq·r

|q|=

1

2π2r2(3.82)

allows us to determine the corresponding component of the two-body potential

in position space:

V[G2/c2](r) =

∫d3q

(2π)3V[G2/c2](q)e−iq·r =

G2m1m2(m1 +m2)

2r2c2. (3.83)

A swift comparison with Eq. (2.1) shows that also this component of the

82


potential is EIH-compatible. Overall, we can claim to have rederived completely

the EIH two-body potential within the scattering amplitude method.

83

Conclusions

The scattering amplitude approach seems to be one of the most promising van-

guards in the derivation of general relativistically corrected two-body potentials

and its plausible implications for the future development of the newborn gravita-

tional wave astronomy provide a strong incentive for studying it extensively and

pushing it to the highest possible accuracy.

Having inserted ourselves in this theoretical framework, we showed that the

scattering amplitude method is capable to reproduce directly an EIH-consistent

potential, as long as one chooses reference frame and external momenta for the

subtended scattering process such that the time component q0 of the momentum

transfer does not vanish and, correspondingly, such that terms proportional to

(p · q)2

q4

can be introduced in the calculation through Iwasaki’s expansion. Since we always

worked in the De Donder gauge, our results indirectly prove that this gauge fixing

choice is not responsible for the mismatch with the EIH potential, which should be

rather traced back to the residual gauge ambiguity that is left after its imposition.

In order to fully recognize in the aforementioned aspects of the two-body

potential computation the crux of the EIH-compatibility, we implemented them

in three different ways: with the employment of a generalized two-parameter

reference frame, both from a non-relativistic and a full relativistic perspective, and

by setting off-shell the external scattering momenta while working in a generally

84

parameterized COM frame. We succeed in all of our three attempts and overall

we determined full-fledged 1PN and 1PM potentials which were confirmed in

agreement with the EIH one.

Furthermore, as a side effect of the condition q0 6= 0, in all of our calculations

we saw the natural appearance of imaginary terms in the potential. However

by limiting the generality of our choices in terms of parameterization we always

succeeded in removing them. More specifically it turned out, with reference to the

two-parameter frame (3.22), that the choice α = β = 1/2 (equivalent to α = 1/2

in the general COM frame) prevents imaginary terms from appearing in each of

the computational possibilities we have explored.

As a prospect it would be interesting to check the applicability of the outlined

strategies to higher order computation, and in particular to determine whether or

not the constraints we found for the parameters are enough to ensure the reality

of the potential at any given order in the PN and PM expansions.

85

Appendix A

A meaningful reference frame forIwasaki’s article

Ref. [24], written by Yoichi Iwasaki in 1971, lunched the scattering amplitude

method for the computation of the two-body potential and introduced some

clever ideas like the graviton propagator expansion we employed in the main

text. Despite its great importance it presents some passages that may easily

become sources of confusion. In particular Iwasaki never specifies the reference

frame he works in, notwithstanding that eventually in the calculation a reference

frame has to be selected and employed to clarify how the external legs in the scat-

tering process are related among themselves and how they exchange momentum

and energy.

This problem manifests itself starting from Iwasaki’s result for the tree-level

amplitude (Eq. (3.23) in Ref. [24]):

M treeIwa = −4πGm1m2

q2

[3

4

(p2

1 + p22

m21c

2+

p23 + p2

4

m22c

2

)+

+

((p1 + p2) · q

)((p3 + p4) · q

)4c2m1m2 q2

− (p1 + p2) · (p3 + p4)

c2m1m2

]+O

(1

c4

),

(A.1)

where we corrected an Iwasaki’s typo because of which he replaced the 3/4

86

factor with a 4/3. At this stage of the calculation one has to specify the reference

frame, in order to take into account the q-dependence of the external trimomenta

pi while evaluating the Fourier transform. Instead Iwasaki proceed by quoting

directly his result in Eq. (3.24),

VIwa = −Gm1m2

r− 3Gm1m2

2r

(p2

m21c

2+

p′2

m22c

2

)+G

(p · r)(p′ · r)

2c2r3+ 7G

(p · p′)2c2r

,

(A.2)

where we restored the notation that we employed in chapter 3. This is indeed the

correct O(G/c2) potential but its derivation is far from being clear. To understand

this passage one has to guess the reference frame which allows to obtain (A.2)

from (A.1). Actually, the most straightforward way to achieve this would rest on

the assumption

p1 = p, p3 = p′,

p2 = p, p4 = p′.(A.3)

However this is clearly paradoxical with respect to the scattering process, since

it implies that the two matter scalar fields are non-interacting. Another path to

follow is necessary and we believe that a possible solution is hinted by our result

in Sec. 3.2: starting from the general frame (3.22) we can choose α = β = 1/2

and get the reference frame

p1 = p +q

2

p2 = p− q

2

p3 = p′ − q

2

p4 = p′ +q

2

, (A.4)

where it’s easy to check that

p1 + p2

2= p,

p3 + p4

2= p′, (A.5)

87

and

p21 + p2

2 = 2p2 +O(q2), p23 + p2

4 = 2p′2 +O(q2). (A.6)

If we insert these relations in (A.1) and neglect any analytic term in q we are

able to rederive the result (A.2) through the simple Fourier transform. Moreover,

coherently with what we found in every tree-level computation that has been

performed in the main text, the choice α = β = 1/2 we made here prevents the

imaginary terms from appearing in the potential, and this is consistent with their

absence in Iwasaki’s paper. This reasoning let us conclude that the reference

frame employed by Iwasaki should be exactly (A.4).

88

References

[1] A. Einstein, “Approximative Integration of the Field Equations of Gravi-

tation,” Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.), vol. 1916,

pp. 688–696, 1916. 4

[2] P. R. Saulson, “Josh Goldberg and the physical reality of gravitational

waves,” Gen. Rel. Grav., vol. 43, pp. 3289–3299, 2011. 4

[3] J. H. Taylor and J. M. Weisberg, “A new test of general relativity: Grav-

itational radiation and the binary pulsar PS R 1913+16,” Astrophys. J.,

vol. 253, pp. 908–920, 1982. 4

[4] R. A. Hulse and J. H. Taylor, “Discovery of a pulsar in a binary system.,”

The Astrophysical Journal, vol. 195, pp. L51–L53, Jan 1975. 4

[5] B. P. Abbott et al., “Observation of Gravitational Waves from a Binary

Black Hole Merger,” Phys. Rev. Lett., vol. 116, p. 061102, Feb 2016. 4, 8

[6] B. P. Abbott et al., “GW151226: Observation of Gravitational Waves from

a 22-Solar-Mass Binary Black Hole Coalescence,” Phys. Rev. Lett., vol. 116,

p. 241103, Jun 2016. 4, 8

[7] B. P. Abbott et al., “Gw170814: A three-detector observation of gravita-

tional waves from a binary black hole coalescence,” Phys. Rev. Lett., vol. 119,

p. 141101, Oct 2017. 4

89

REFERENCES

[8] B. P. A. et al, “Multi-messenger observations of a binary neutron star

merger,” The Astrophysical Journal, vol. 848, p. L12, oct 2017. 4

[9] T. Chu, H. Fong, P. Kumar, H. P. Pfeiffer, M. Boyle, D. A. Hemberger, L. E.

Kidder, M. A. Scheel, and B. Szilagyi, “On the accuracy and precision of

numerical waveforms: effect of waveform extraction methodology,” Classical

and Quantum Gravity, vol. 33, p. 165001, jul 2016. 5

[10] S. Husa, S. Khan, M. Hannam, M. Purrer, F. Ohme, X. J. Forteza, and

A. Bohe, “Frequency-domain gravitational waves from nonprecessing black-

hole binaries. i. new numerical waveforms and anatomy of the signal,” Phys.

Rev. D, vol. 93, p. 044006, Feb 2016. 5, 39

[11] B. et al., “Improved effective-one-body model of spinning, nonprecessing bi-

nary black holes for the era of gravitational-wave astrophysics with advanced

detectors,” Phys. Rev. D, vol. 95, p. 044028, Feb 2017. 5, 37

[12] N. et al., “Time-domain effective-one-body gravitational waveforms for co-

alescing compact binaries with nonprecessing spins, tides, and self-spin ef-

fects,” Phys. Rev. D, vol. 98, p. 104052, Nov 2018. 5, 37

[13] A. Einstein, L. Infeld, and B. Hoffmann, “The gravitational equations and

the problem of motion,” Annals of Mathematics, vol. 39, no. 1, pp. 65–100,

1938. 6, 40

[14] S. Chandrasekhar, “The Post-Newtonian Equations of Hydrodynamics in

General Relativity.,” Astrophysical Journal, vol. 142, p. 1488, Nov. 1965. 6

[15] S. Chandrasekhar, “The Post-Newtonian Effects of General Relativity on the

Equilibrium of Uniformly Rotating Bodies. I. The Maclaurin Spheroids and

the Virial Theorem.,” Astrophysical Journal, vol. 142, p. 1513, Nov 1965. 6

90

REFERENCES

[16] W. D. Goldberger and I. Z. Rothstein, “Effective field theory of gravity for

extended objects,” Phys. Rev. D, vol. 73, p. 104029, May 2006. 6, 40

[17] B. Kol and M. Smolkin, “Dressing the post-newtonian two-body problem

and classical effective field theory,” Phys. Rev. D, vol. 80, p. 124044, Dec

2009. 6, 43

[18] Y. Itoh, “On the equation of motion of compact binaries in the post-

Newtonian approximation,” Classical and Quantum Gravity, vol. 21,

pp. S529–S534, Mar 2004. 6

[19] T. Futamase and Y. Itoh, “The post-newtonian approximation for relativistic

compact binaries,” Living Reviews in Relativity, vol. 10, p. 2, Mar 2007. 6

[20] R. Arnowitt, S. Deser, and C. W. Misner, “Dynamical structure and defi-

nition of energy in general relativity,” Phys. Rev., vol. 116, pp. 1322–1330,

Dec 1959. 6, 41

[21] P. Jaranowski and G. Schafer, “Third post-newtonian higher order adm

hamilton dynamics for two-body point-mass systems,” Phys. Rev. D, vol. 57,

pp. 7274–7291, Jun 1998. 6

[22] P. Jaranowski and G. Schafer, “Derivation of local-in-time fourth post-

newtonian adm hamiltonian for spinless compact binaries,” Phys. Rev. D,

vol. 92, p. 124043, Dec 2015. 6, 41

[23] T. Damour, P. Jaranowski, and G. Schafer, “Nonlocal-in-time action for the

fourth post-newtonian conservative dynamics of two-body systems,” Phys.

Rev. D, vol. 89, p. 064058, Mar 2014. 6, 41

[24] Y. Iwasaki, “Quantum theory of gravitation vs. classical theory. - fourth-

91

REFERENCES

order potential,” Prog. Theor. Phys., vol. 46, pp. 1587–1609, 1971. 6, 64, 69,

86

[25] B. R. Holstein and J. F. Donoghue, “Classical physics and quantum loops,”

Phys. Rev. Lett., vol. 93, p. 201602, Nov 2004. 6

[26] Z. Bern, L. Dixon, D. C. Dunbar, and D. A. Kosower, “One-loop n-point

gauge theory amplitudes, unitarity and collinear limits,” Nuclear Physics B,

vol. 425, pp. 217–260, Aug 1994. 6, 55

[27] C. Cheung, I. Z. Rothstein, and M. P. Solon, “From Scattering Amplitudes to

Classical Potentials in the Post-Minkowskian Expansion,” Phys. Rev. Lett.,

vol. 121, p. 251101, Dec 2018. 7

[28] Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M. P. Solon, and M. Zeng, “Scat-

tering Amplitudes and the Conservative Hamiltonian for Binary Systems at

Third Post-Minkowskian Order,” Phys. Rev. Lett., vol. 122, p. 201603, May

2019. 7, 36, 40

[29] A. Cristofoli, N. E. J. Bjerrum-Bohr, P. H. Damgaard, and P. Vanhove,

“On Post-Minkowskian Hamiltonians in General Relativity,” arXiv e-prints,

p. arXiv:1906.01579, Jun 2019. 7, 40, 56, 63, 66, 68, 74

[30] A. Antonelli, A. Buonanno, J. Steinhoff, M. van de Meent, and J. Vines,

“Energetics of two-body Hamiltonians in post-Minkowskian gravity,” Phys.

Rev. D, vol. 99, p. 104004, May 2019. 7, 37

[31] B. P. Abbott et al., “GW170104: Observation of a 50-Solar-Mass Binary

Black Hole Coalescence at Redshift 0.2,” Phys. Rev. Lett., vol. 118, p. 221101,

Jun 2017. 8

92

REFERENCES

[32] M. Maggiore and O. U. Press, Gravitational Waves: Volume 1: Theory and

Experiments. Gravitational Waves, OUP Oxford, 2008. 9

[33] J. D. E. Creighton and W. G. Anderson, Gravitational-wave physics and

astronomy: An introduction to theory, experiment and data analysis. 2011.

9

[34] A. Dirkes, “- Gravitational waves - A review on the theoretical foundations of

gravitational radiation,” International Journal of Modern Physics A, vol. 33,

p. 1830013, May 2018. 9

[35] B. J. Owen and B. S. Sathyaprakash, “Matched filtering of gravitational

waves from inspiraling compact binaries: Computational cost and template

placement,” Phys. Rev. Lett., vol. D60, p. 022002, 1999. 39

[36] J. G. Baker, J. Centrella, D.-I. Choi, M. Koppitz, and J. van Meter,

“Gravitational-Wave Extraction from an Inspiraling Configuration of Merg-

ing Black Holes,” Phys. Rev. Lett., vol. 96, p. 111102, Mar 2006. 39

[37] M. Campanelli, C. O. Lousto, P. Marronetti, and Y. Zlochower, “Accurate

Evolutions of Orbiting Black-Hole Binaries without Excision,” Phys. Rev.

Lett., vol. 96, p. 111101, Mar 2006. 39

[38] Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M. P. Solon, and M. Zeng,

“Black Hole Binary Dynamics from the Double Copy and Effective Theory,”

arXiv e-prints, p. arXiv:1908.01493, Aug 2019. 36, 56, 63, 66, 68, 74

[39] G. Schafer and P. Jaranowski, “Hamiltonian formulation of general relativ-

ity and post-newtonian dynamics of compact binaries,” Living Reviews in

Relativity, vol. 21, p. 7, Aug 2018. 40

93

REFERENCES

[40] T. Damour and G. Esposito-Farese, “Testing gravity to second post-

newtonian order: A field-theory approach,” Phys. Rev. D, vol. 53, pp. 5541–

5578, May 1996. 40

[41] B. R. Holstein and A. Ross, “Spin Effects in Long Range Gravitational Scat-

tering,” arXiv e-prints, p. arXiv:0802.0716, Feb 2008. 40, 53, 56, 63, 66, 76,

79, 81

[42] I. G. Fichtenholz, “The Lagrangian form of the equations of motion in second

approximation (Russian version),” Zh Eksp Teor Fiz, p. 233–242, 1950. 40

[43] T. Damour, P. Jaranowski, and G. Schafer, “Dimensional Regularization

of the Gravitational Interaction of Point Masses in the Adm Formalism,”

pp. 2490–2492, Sep 2008. 41

[44] G. ’t Hooft and M. J. G. Veltman, “Regularization and Renormalization of

Gauge Fields,” Nucl. Phys., vol. B44, pp. 189–213, 1972. 43

[45] W. D. Goldberger and I. Z. Rothstein, “Effective field theory of gravity for

extended objects,” Phys. Rev. D, vol. 73, p. 104029, May 2006. 43, 53

[46] J. B. Gilmore and A. Ross, “Effective field theory calculation of second post-

newtonian binary dynamics,” Phys. Rev. D, vol. 78, p. 124021, Dec 2008. 43

[47] S. Foffa and R. Sturani, “Effective field theory calculation of conservative

binary dynamics at third post-newtonian order,” Phys. Rev. D, vol. 84,

p. 044031, Aug 2011. 43

[48] S. Foffa, P. Mastrolia, R. Sturani, C. Sturm, and W. J. Torres Bobadilla,

“Static two-body potential at fifth post-newtonian order,” Phys. Rev. Lett.,

vol. 122, p. 241605, Jun 2019. 43

94

REFERENCES

[49] T. Damour, “Gravitational radiation and the motion of compact bodies.,”

in Lecture Notes in Physics, Berlin Springer Verlag, vol. 124, pp. 59–144,

1983. 44

[50] B. Kol and M. Smolkin, “Einstein’s action and the harmonic gauge in terms

of Newtonian fields,” Phys. Rev. D, vol. 85, p. 044029, Feb 2012. 45

[51] J. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Colli-

sions. Dover Books on Engineering, Dover Publications, 2012. 46, 47

[52] S. N. Gupta, “Quantization of einstein’s gravitational field: General treat-

ment,” Proceedings of the Physical Society. Section A, vol. 65, pp. 608–619,

aug 1952. 50

[53] G. ’t Hooft and M. J. G. Veltman, “One loop divergencies in the theory

of gravitation,” Ann. Inst. H. Poincare Phys. Theor., vol. A20, pp. 69–94,

1974. 50, 51

[54] J. F. Donoghue, “General relativity as an effective field theory: The leading

quantum corrections,” Phys. Rev. D, vol. 50, pp. 3874–3888, Sep 1994. 51

[55] L. Montanet et al., “Review of particle properties. Particle Data Group,”

Phys. Rev., vol. D50, pp. 1173–1823, 1994. 52

[56] D. A. Kosower, B. Maybee, and D. O’Connell, “Amplitudes, observables,

and classical scattering,” Journal of High Energy Physics, vol. 2019, p. 137,

Feb 2019. 54

[57] Z. Bern, T. Dennen, Y.-t. Huang, and M. Kiermaier, “Gravity as the square

of gauge theory,” Phys. Rev. D, vol. 82, p. 065003, Sep 2010. 55

[58] B. R. Holstein and A. Ross, “Spin Effects in Long Range Electromagnetic

Scattering,” arXiv e-prints, p. arXiv:0802.0715, Feb 2008. 81

95

post-newtonian and post-minkowskian two-body potentials ... lytic post-newtonian (pn) models...

Documents