All order gravitational waveforms from scattering amplitudes Tim Adamo1Andrea Cristofoli1Anton Ilderton2and Sonja Klisch1_2

2025-04-30 0 0 610.87KB 22 页 10玖币

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All order gravitational waveforms from scattering

amplitudes

Tim Adamo,1Andrea Cristofoli,1Anton Ilderton2and Sonja Klisch1

1School of Mathematics and Maxwell Institute for Mathematical Sciences

University of Edinburgh, EH9 3FD, UK

2Higgs Centre, School of Physics and Astronomy

University of Edinburgh, EH9 3FD, UK

E-mail: t.adamo@ed.ac.uk,acristof@exseed.ed.ac.uk,

anton.ilderton@ed.ac.uk,s.klisch@ed.ac.uk

Abstract: Waveforms are classical observables associated with any radiative physical pro-

cess. Using scattering amplitudes, these are usually computed in a weak-ﬁeld regime to

some ﬁnite order in the post-Newtonian or post-Minkowskian approximation. Here, we use

strong ﬁeld amplitudes to compute the waveform produced in scattering of massive particles

on gravitational plane waves, treated as exact nonlinear solutions of the vacuum Einstein

equations. Notably, the waveform contains an inﬁnite number of post-Minkowskian con-

tributions, as well as tail eﬀects. We also provide, and contrast with, analogous results in

electromagnetism.

arXiv:2210.04696v2 [hep-th] 22 Jun 2023

1 Introduction

The observation of gravitational waves has brought renewed importance to the study of

general relativity and its observables. Surprisingly, scattering amplitudes – one of the key

outputs of quantum ﬁeld theory – are providing a new way to study classical general rel-

ativity; for reviews see [1–3]. Starting from novel perspectives on [4–6], and a remarkable

state-of-the-art calculation for [7], the conservative Hamiltonian of the gravitational two-

body problem, a new program for providing higher-order post-Minkowskian (PM) approx-

imations to gravitational observables has emerged based on the classical limit of scattering

amplitudes. This has led to a variety of exciting new results for gravitational observables,

e.g. [8–26], which build on many of the powerful structures in scattering amplitudes such

as generalized unitarity and double copy, as well as techniques from eﬀective ﬁeld theory.

A key tool in this program has been the development of a formalism to systematise

the extraction of classical physical observables from scattering amplitudes [27]. So far, all

observables computed with this approach are valid for weak ﬁelds only: they are obtained

from amplitudes at ﬁnite PM order, so truncate at a corresponding ﬁxed order in the

coupling [28–39]. This is in sharp contrast with other approaches to gravitational dynamics

such as the self-force paradigm [40–45], where perturbation theory is implemented around

a curved background and the weak ﬁeld limit is not considered.

To address this gap, the amplitudes-based approach can be generalised to curved back-

grounds by means of strong ﬁeld scattering amplitudes and their classical limits [46]. This

provides an alternative route to the computation of classical observables, as strong ﬁeld

amplitudes encode a substantial amount of information about higher-order processes [47–

54] and ﬁnite size eﬀects [55–57] in trivial backgrounds, and can also admit remarkably

compact formulae [58–60]. A key aspect is that even ﬁrst order perturbation theory around

a curved background – which we refer to as ‘ﬁrst post-background’, or 1PB, order – encodes

inﬁnitely many orders of the PM expansion. This is analogous to the relation between the

PM and post-Newtonian (PN) expansions for bound orbits, where a ﬁxed contribution of

the former encodes inﬁnitely many orders of the latter due to the virial theorem.

Here we show for the ﬁrst time how classical observables encoding all-order results can

be extracted from scattering amplitudes. We derive expressions for the classical gravita-

tional waveform emitted by a point particle scattering on a gravitational plane wave (an

exact solution to the nonlinear Einstein equations), encoding all-order contributions in the

PM expansion when the ﬂat spacetime limit is taken, as well as tail eﬀects which usually

enter at high order in the PM approximation. We also perform analogous calculations

for charged particles scattering on electromagnetic plane waves. While our aim is not to

study the phenomenology of electrodynamics, the waveforms do not seem to appear in an

otherwise extensive literature [61–64], and it is revealing to compare and contrast with the

gravitational case [65–68].

Note that plane waves are not just good models of gravitational waves, but also describe

any spacetime in the neighbourhood of a null geodesic [69]. This directly connects our

results to the gravitational 2-body problem: in the limit where one mass is negligible,

the massless probe will experience the heavy body’s metric as a plane wave. Indeed, plane

– 1 –

wave/ultrarelativistic limits have been used to analyse gravitational self-force [70] and black

hole quasinormal modes [71].

2 Asymptotic waveforms

Let |Ψ⟩be a normalised superposition of free particle (mass m) states,

|Ψ⟩=ZdΦ(p)ϕ(p) eip·b/ℏ|p⟩,(2.1)

where dΦ(p) is the Lorentz-invariant on-shell measure, the wavepacket ϕ(p) has a well-

deﬁned classical limit (cf., [27]) and bµis the impact parameter. This state is evolved on

an electromagnetic or gravitational plane wave background. In terms of the S-matrix Son

that background, the time-evolved state is simply S |Ψ⟩.

Our interest is in the classical gravitational or electromagnetic radiation emitted by a

scalar particle as it scatters on these backgrounds, as measured by an asymptotic observer

at future null inﬁnity. The particular observable of interest is the waveform, encoded in

the expectation value of the Maxwell and Riemann tensors, ⟨Fµν (x)⟩and ⟨Rµνσρ(x)⟩. In

coordinates xµ= (t, x), approaching future null inﬁnity corresponds to taking r≡ |x|→∞

while u=t−ris held constant. Following [32], the waveform Wis deﬁned simply as the

coeﬃcient of the leading 1/r term in ⟨F⟩or ⟨R⟩. It is a function of uand the two angular

degrees of freedom encoded in the null vector ˆxµ= (1,ˆ

x). Inserting complete sets of ﬁnal

states into the expectation value, and using the mode expansion of Fµν and Rµνσρ, one easily

obtains an expression for the waveform in terms of scattering amplitudes on the background.

The leading contribution is at 1PB, meaning order e(the fundamental charge) in QED or

order κ(the gravitational coupling) in gravity, but all orders in the background ﬁelds,

and comes from interference between tree-level 2-point and 3-point amplitudes. Unlike in

vacuum, 2-point amplitudes on backgrounds are not trivial even at tree-level, encoding

e.g. memory eﬀects [46]. Deﬁning the (theory-dependent) combination

α(k) = ZdΦ(p′)⟨Ψ|S†|p′⟩⟨p′, kη|S |Ψ⟩(2.2)

we arrive at, in QED and gravity respectively,

Wµν (u, ˆx) = −ℏ1

πRe

∞

dωe−iωu k[µε−η

ν]α(k),(2.3)

Wµνσρ(u, ˆx) = −κ

πℏ1

∞

dωe−iωu k[µε−η

ν]k[σε−η

ρ]α(k),

in which kµ=ℏωˆxµfor ωa classical frequency (as will be useful later when taking the

classical limit), εη

µ≡εη

µ(k) is the photon polarisation vector and ˆ

dx:= dx/(2π). One can

check that the combination of amplitudes in α(k) reproduces the radiation emitted due to

geodesic motion, i.e. the ﬁrst contribution of self-force eﬀects [42].

– 2 –

Plane wave backgrounds. Plane waves are highly symmetric vacuum solutions of the

Einstein or Maxwell equations with two functional degrees of freedom. In gravity, they are

described by metrics of the form [72]:

ds2= 2dx+dx−−dxadxa−κ Hab(x−)xaxb(dx−)2,(2.4)

where Latin indices label the ‘transverse’ directions x⊥= (x1, x2), while the 2 ×2 matrix

Hab(x−) is symmetric, traceless and compactly supported on x−

i< x−< x−

f(ensuring the

spacetime admits an S-matrix [73]). The metric has a covariantly constant null Killing

vector n=∂+(or nµ=δ−

µ) which will recur throughout. To ease notation, we absorb

the gravitational coupling into the background, taking κHab →Hab from here on; as such,

note that expressions below containing all orders in Himplicitly contain all-order PM

contributions in κ.

Plane wave metrics have several associated geometric structures. First, there is a

zweibein Ea

i(x−) and its inverse Ei a(x−), labelled by the index i= 1,2 satisfying ¨

Ei a =

HabEb

i,˙

[iEj]a= 0 . The zweibein encodes gravitational (velocity) memory through the

diﬀerence

∆Ei

a=Ei

a(x−> x−

f)−Ei

a(x−< x−

i),(2.5)

which compares the relative transverse positions of two neighbouring geodesics. The

zweibein also deﬁnes a transverse metric γij(x−) := Ea

(iEj)aand deformation tensor

σab(x−) := ˙

aEi b, the latter encoding the expansion and shear of the null geodesic

congruence associated to (2.4). These deﬁnitions are completed by the initial condition

a(x−< x−

i) = δi

a, which yields γij (x−< x−

i) = δij and σab(x−< x−

i) = 0.

Turning to electromagnetism, plane waves can be deﬁned by the potential Aµ(x) =

−xbEb(x−)nµin lightfront coordinates (given by the ﬂat space part of (2.4)) and nµis

as above. Eb(x−) is the two-component, compactly supported electric ﬁeld. A useful

associated quantity is

a⊥(x−) := Zx−

−∞

dsE⊥(s),(2.6)

such that ea⊥is the eﬀective ‘work done’ on a charge. The electromagnetic velocity mem-

ory eﬀect is encoded in the constant ea⊥(x−> x−

f) [74]; this is the change in transverse

momentum of a particle crossing the background from the asymptotic past to the future.

To simplify the presentation of our results we make the assumption that velocity mem-

ory eﬀects induced by our backgrounds are parametrically small, and thus negligible. (We

relax this assumption in Appendix B.) This means setting ab(x−> x−

f) = 0 in electromag-

netism, and Ei

a(x−> x−

f) = δi

ain gravity. The main simpliﬁcation is that the tree-level

2-point amplitudes reduce to ⟨p′|S |Ψ⟩ → eiθ(p′)ϕ(p′), for a theory dependent phase θwhich

can be absorbed by redeﬁning u1.

1In general the phase will however encode position memory eﬀects on the scattered scalar [75].

– 3 –

3 Electromagnetism

We now construct the classical limit of the electromagnetic waveform Wµν (u, ˆx) from (2.3).

Given our assumption of no memory, the only ingredient required is the 3-point amplitude

for a charged scalar, on an electromagnetic plane wave background, to emit a photon.

Let the incoming (outgoing) scalar have momentum pµ(p′

µ), and the emitted photon have

momentum kµand helicity η. The amplitude is calculated by evaluating the cubic part of

the action on the appropriate scattering states in a plane wave, see e.g. [64]. The result is

⟨p′, kη|S|Ψ⟩=ZdΦ(p)ϕ(p) eip·b/ℏˆ

δ3

+,⊥(p′+k−p)A3,

A3=−2ie

ℏ3/2Zy

εη·P(y) exp i

ℏZy

−∞

dzk·P(z)

p+−k+,(3.1)

where Ry:= R∞

−∞dyand ˆ

δ(x) := 2πδ(x). The ‘dressed’ momentum Pµ(y) is the classical

momentum of the particle in the background,

Pµ(y) = pµ−eaµ(y) + nµ

2ea(y)·p−e2a2(y)

2p+

,(3.2)

where aµ(y) = δ⊥

µa⊥(y), obeying P2(y) = m2. Only three components of overall momen-

tum are conserved in A3as the background breaks x−-translation symmetry.

Calculation of the waveform. We assemble the QED waveform in (2.3) from (3.1),

using the assumption of negligible memory eﬀects. We perform the sum over photon

helicities using the completeness relation in lightfront gauge. All gauge-dependent pieces

vanish by anti-symmetry or generate boundary terms which can be ignored [74], leaving

only a contribution from −ηµν . An immediate simpliﬁcation in the classical limit is that

the delta function sets p′=p, and thus the wavepacket appears as |ϕ(p)|2. This means

that the impact parameter bdrops out, and under the usual assumption that ϕis sharply

peaked around some classical momentum, we can integrate over p, localising the integrand

at the on-shell momentum of the incoming particle, which we continue to write as pfor

simplicity. This gives

Wµν (u, ˆx) = −ie

4π2p+Zy,ω

ωe−iω(u−ˆx·X(y)) ˆx[µPν](y),(3.3)

in which Xµ(y) is the classical particle orbit, obeying X′

µ(y) = Pµ(y)/p+. Performing the

frequency integral yields a very compact ﬁnal expression for the classical waveform:

Wµν (u, ˆx) = e

2πZy

δ(u−ˆx·X(y)) d

ˆx[µPν](y)

ˆx·P(y)

2πX

sols

ˆx·P

ˆx[µPν]

ˆx·P,

(3.4)

– 4 –

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AllordergravitationalwaveformsfromscatteringamplitudesTimAdamo,1AndreaCristofoli,1AntonIlderton2andSonjaKlisch11SchoolofMathematicsandMaxwellInstituteforMathematicalSciencesUniversityofEdinburgh,EH93FD,UK2HiggsCentre,SchoolofPhysicsandAstronomyUniversityofEdinburgh,EH93FD,UKE-mail:t.adamo@ed.ac.uk,a...

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All order gravitational waveforms from scattering amplitudes Tim Adamo1Andrea Cristofoli1Anton Ilderton2and Sonja Klisch1_2

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