Subtle and surprising aspects of Angular Momentum
At null infinity

Emphasizing compact binary coalescence

Abhay Ashtekat

the cosmos a PhysicsDepartment
Penn StateInstitute for Gravitation

Flat AsympopiaOIST

Q G unit

15th 18thMarch21

ORGANIZATION
1 Introduction a setting the stage
BMS Newman Penrose AA AA reubel Winicoot
Reviews AA centenaryvolume Biertia Yau ceds 20
AA Campiglia a 1addha FERG 20182

compact Binary coalescences

gSummary

The blackholekick
contrast with source free solutions
super translation ambiguity in Angular Momentum
toNotion of past 4 future stationarity as uTheory observations
surprising consequences

Discussion

Recent proposal to remove supestranslation ambiguity
A 70
YM theories
Sororise in Maxwel

Introduction

1aiii Iab that is asymptotically flat at nullos

space time
M gab

Selling the stage

conformal completion

It Future boundary

six

null genators x na

inb
equipped with flabhb
29ab.w
1Fa Hb O
Law
9ab i 0Ingab O

such thatO

This is the universal structure of I

subgroup of Diff IIT that preserves this universal

This provides an intrisic
The BMS groupcharacterization of TB w o ref to space time interiorstructureIlL

co dimBISI

semi direct productBI didAbelian normal subgroup of B
the G dim tolentz subgroup
costructure of B similar to that of Dr With X replacing the4 d group of translations of D
B does admit a unique Abelian 4 d normalsubgroup ofwell defined 4 momentum
translations I C Il
But while D admits a G parameterfamily of Lorentz
1B admits an infine parameterfamily related
subgroups Le
by super translations
supei translation ambiguity in the notion of angularmomentum

with Lnf oSupertranslationgenerators F fha
1f wff conformal weightflab ha wagab w naafunction
F Not justcan select a canonical 4 d subspace of translations T But nonatural notion of 1 Space of pure Supertranslations The 4pm
Chen Wang Yau
decomposition varies as we change go.b.ba

Radiative Modes in full GR

1BInvariant characterization gHijab completion Fagbc 0Da 9beSt

T1 induces DDantQ0nconformal freedom in gaboffDaKs

Tab hakeon

Tt0

EquivalenceclassesED5abhb o9ob5aTT2 Dots

perpoint offof curvature of ED is neatlycaptured in the440EsoKac Lim SI Kobedhand
asymptotic Weyl curvature
Inez
partscolat
InDf knows about radiative aspects of gray fieldB ut EDS has
Mab o CBondi news vanishes
Kab o0Re
no knowledge of the Coulombic aspects
INon trivial partMassclassical vacua EDDoclassical Vacuum if it has trivial curvatureison the basespace 2i e Dc
fully determined byDoQuTµgDo

Hel

No dynamical information

EachED's

Ibc 0an

EachEDis0,130 Im ITNabO

oomfpreserved preciselyby a D

subgroup of the BMS group Atsatacts simply and transitively on TI
space of Vaasa

02AngularMomentum

1B reduces toDDft

Me in absence of grayasaradiation

ios

any space time we Cain select two Poincare subgroupsof B but genetically DB
Do't Dot GabTutes is anobservable of GR Total gravitional memorysoinooI C Radiative phase space

fadIt

ED's ona

BHS Fluxes

sizes at

RtBMS

actionEg at

ab1czft Preserves D
Ba E Ib
Tz at
Epfa NabffLgDa Da2g lb
onhalof INas News

Ea Ln Gosymplectic structure on Bad

Covariant phasespace of GRIggy S ohir

FEWCharges
Affine space

EPI

8D's stab

Hamiltonians Fluxes
Linear in Sa
2 la Dio bCPIf o for superftanslations ARotations
Has all desired in variances

LgAab

2k Gabbe integrated to obtain 2 sphere charges
OH C UG
Eg DI
Q C J Ong CE
Cc through Bianchi identities
43 4,0 appear inFluxes

canORGANIZATION

I Introduction a setting the stage
2compact Binary coalescence

The blackholekick
contrast with source free solutions
Momentumintranslation
ambiguity
AngularsupertoNotion of past 4 future stationarity as uTheory Observations
consequences
surprisingDeLorenzo
Khera2020 PRD 2020
CGRGAA
MainlyMittman Iozzo Kheraetat PRD12021
Krishnan PRD 2021
Khera AA Krishnan in preparationcompere etat Elhashasha Nicols g
Alsog

Summary

Discussion

Recent proposal to remove supestranslation ambiguity
A 70
Surprise in Maxwell YM theories

Source free solutions

2Avs CBCs

Nonlinearstability of Minkowski space time All analyses

to date use source free sours with it 4124
Incoming radiation at g essential
only sold with no incoming radiation Minkowski ace
0 spherically symmetric at io and Zeta at it
ChristodoulouOsuperinomentum flux across JI
Klainnermann
initial conditions
vanishesor Linearordinarychrusciel Delay
Initial cords

samebutcontrast withCBCofNo incomino radiation

GeneticallyB smemory

hence Ang Mom Welldefined

Nsa BHtoa Kert BH

at f n SD's

subermomentom fluxacrossD reduction at ioa it

EDO bLorentz groups defining initial Fio
conceptually Meaningless to consider Fooao2Be

compact Binary coalescence

ED dand0distinct
a distinct

IitsettingFinal BH kick determinedby the flux of 3 momentum
radiated across f
Nablfy.mnDa Dafampleb
EamSyhmm41 NabNab 43J
GenericallytoFuture1Past test framesyield
af 95
ina w tha
9
Sunita sphere methw

raegroups
JI refers to bSOG subrest
pastframeD
determinedoffuture
restframe
determined
DbySt refers to soc3 subgroups ofThesesoC B

arerelated by a supertranslationooa boost2 C Angular Momentum Supertranslation ambiguity

choice of origin 0 e Lorentz subgroup L
Displacement
Mab 1 RadioMabFTa EabadMcd an
Boost momentum can be removed
Fi Refers to so 7 subgroups in the rest frame where p ospecial Relativity

GRI

Post test frame
E o selects Iab
But SO subgroups how have a supertranslation freedom
U const cuts
Rfk tandential select one SOBUtfco
UFreedomEIO

eliminate by demanding i shear Gab TF aDsU_ o asu 3 0selects 4 parameter family of SOG as in specialrelativity
precisely the family in the D C B selected by Edo's
canE times

JEBIThus

one canobtainQO

Ep tuff

EDistant Future
no'aAgain introduceoYieldsun'aDalek 1oCitasuGenetically BH kickyReed das

Eababpcii

Raaeliminate superttanslation ambiguity as u as
Total 4 mom a Ang momandJIPjoI bod4EpaImFuture test frame different
2GabwindStGabi

4 parameter family ofFF Ii

I lieL

in the future test frame

PTIwosotsnosas Usubgroups C Dt

Imei 8 Bii daioie ED's

EE turnRahF2EDL

E babeReto die

These sots subgroup
IIBut SOB subgroup c IF
1 to me so subgroups defining FFdefining i as relat
in special relativity and a supertranslation
boost
by a isfirst subtlety
ThistheFei Pei

Flux of some sob angular momentum
It also contains supermomentum fluxoo

Let us work withthe past testframe Iab Bd
is in X direction
since ofI
arerelatedb Uts coilbythe asymptotically shearfreeguts

More explicitly
suppose the kickPia

ThenSinha

RefR'IT Rfa
similarly for Riza

V KaossciIi

gzia

Lpii seal
Lai Seo eLkei Gab

Blue terms from specialrelativity black terms new

2KaiTabgu So 1 URs 05437

feature in the expression of flux ofconsequently supertromenta
angular momentum carriedbygravitional waves

EFiET'sEIn
astheaEEEsimilarly for the 3rd component
co2DF

Asymptotic stationarity asu

final state is kett expectation Asymptoticallyasalong ft fields approach those in Kerr spacetime

CBC

Borne out in numerics

it is assumed that the progenitors are so fat fromone another that there is stationarity In fact muchstronger
assumptions made in the waveformcommunity
As aA

Let usmake1

much weaker but still nontrivial assumptions

Bondi news Nstd assumption

Ei Eg IE

Nabimam's

JooaseasaIasNews a its derivatives
9 IE become
Cinexeryconformalframe flab haasymptotically stationaryzrest frame flag 159 as u s.es andtoo
If this were toin the future test frame 9 n 8 as a
fail ITwould diverge in the limit

UIooin the past

Even this weak notion of asymptoticstationarity hasinteresting

unforeseen consequencesaaconsequences

IfUI70

asInthis70
In

asthisthe past rest frame Iabnob
sphericallySimm
conformal frameGMLaingIfinasaw2Fab 19 wthabin the future rest frame
conformal frame Luim
GM 1
asu

If

If 4 UP 4GMi1

TFs.noFuture massaspect in thepast rest frame

GM

specification ofa

notspherical3

2

coffv tGsiIPO
3sincecosy V
KIcklpecoilvelocity

ZI

If
conformal Cor

Illustrates why the requirement

Cstheticallysymm

cannothold without
rest frameo

Iff dat 0
the fast frame we are working withAll supermomentum charges vanish
unless f is constant
IE dat Mio
except energy EIn2 E

subtlety and surprise

Recall Difference between initialEFiioEE

final angular moments

T'sFEEFsimilarly for the 3rd component

Fluxes of super momenta that feature are non Zet if only if
it are non zero
the correspondino final supermomentum chargesodat
CMioand Sosa
ga d21
because HE1 owhenarethese final supermomenta non F eto

special case
Then Salifo

Gert memory vanishes
Sci o
Ii

supermomentum terms

special casevanishGrail memory to

But kick 1recoil velocity

vanishes

Fixed conf frame but SOBiepig EDL
oatsin29 atrity

just as one expects
sae.ee nontrivial

flabFraSTI b i'a

related by a superttahslationin the past conf frame
However V oGMHE
2nd surprisesubtlety
supermomentum terms vanish
Ok to compare apples with oranges in practiceThus in CBCs because of asympolicstationatity as a too
the supertranslation ambiguity in angular momentum shows up
kick velocity 1 0
only when i Grail Memory 0 AND
But this is thegenetic casemomentum flux
Nonetheless the superFf

If1 Itftg

GMfuo.gl twill

Sei or Sci

is small for small V

I sincecosy V 1IElues

102103 kings Therefore theNR simulations of CBCs
super translation ambiguity in the notion of angular momentum is
not likely to be relevant for current 1 IGO detectors But
interesting for 3 g detectors that are being planned In thisrespect similar togravitational memory which has not beenIn

definitively measured

But memory and superinomenton and angularmomentum
balancelaws I discussed are already beingused as diagnostic
to test the goodness of modelwaveforms currentlytools
Ias pointers for further
used Phenom EOB surrogate
improvements

2 FTheoryObservations

Supermomentum balance lawIsoyu asiIaspillustrative Examples

CBC with a kick velocitya Mio 1Is du 1542

taxih

distributions

Liao Virgo provide posterior probability
for symbols in blue
ppps for memory

Memo'TAA DelorenzoKhera
GRG zoo
Khera Krishnan AA
DeLorenzo caozi

viewpoint weIntroduced

Mix 5
be used to improve waveform models Example GW170814

Gravitational memory iscanSEO BNR v3

aaninferred observable

IMR PhenomPV2 comparison

like

Need for improvementof highermodessecond Example
same two models examined with anotherinferred observable Mit calculated with the balance law
First LIGO Event Analysis from zois 16

to theWe found a bi modal Hump in SEOBNRV3 and traced
inprecession spin components 1 to I was included
way modelmodelthetoolAfor improving
thediagnosticAgain
Angular momentum considerationsc

3rd Examplec

Use of Angular momentum balance laws

Recent improvements inwaveform models
IMRPhenom Pilz
SEOBNRV3

IMRPhenomxpHat
SEOBNRK p

Diagnostic tool Balance

law for the E componentof

angularmomentum

clearly brings out theextent of improvement
the NR result isinstructiveBrings outsynergy

EvenKheraAA

Krishnanin preparatoryee

ORGANIZATION

it Introduction

a setting thestage

coalescences
Binarysource free solutions

2compactgSummary

The blackholekick
contrast withMomentum
translation
inambiguity
AngularsupertoNotion of past 4 future stationarity as u
Theory observations
Surprising consecuences

Discussion

Recent proposal to remove supestranslation ambiguity
A 70
surprise in Maxwell YM theories
2021
arXiv
KQG
12021
AAI Bonga
Chen Wang Yau
20187
2016
AA
CROPP
PRL
AA Bonga 2 Kesavano3A

summary

For every BMS vectorfield 5 C lb we have Fg Qf Q withthe balance lawEg and itslocalversions
Qijj Qi
linear in Saand 5 Eg
EQg
super translation ambiguity we do have preferred Poincare
subgroups D A D in any space time But gravitationaln two are related
thby a supertternslationDmemory toIf there is a non zero kick the asymptotic test frames also
HabhoaIab Fia
distinct 19 ima

57
Rotations R.fi defining F and Pia defining
related by a boost and a subertranslation

HenceEiFeo

areInn Eu'ffrffom

ftp.go.ufafffot.gln9udmtFakgxuia9Inbo9n9

No supermomentum termsupermomentum termobut if
If 17 1 D
Surpriseturns out to vanish because of asymptotic stationarity
But this assumption is not compelling should keepeyes open

IfforD

Dtmemory ocircumstances where it is violatedco

effect vanishes for vk.de O and expected BH
small Superttanslation term too small for theBecause thekicks

arepresent LIGOWingo detectors But likelyto besensitivity of the3G detectors
But the detailed balancelaw

non negligible foris already usefulasaas they are consequencesof exact GRDiscussion

JBlildiagnostic check on waveformmodels

Recent proposal to eliminate the supertranslationambiguity

in general

without reference to asymptoticstationatityie

propose a new formula for Qa for R Rotation
generators in 1B starting from quasi local expressions
Work in a fixed test frame C No kick for CBC
OTRCioit
FR has the property
ResultRC W2Yf Ffs

femYemialtranslationitfpure supertype ostandard supermomentum flux will not vanish
for all supertranslations Meaning of type
A pure supertranslation in one conformalframein anotherab ha has to 1 components
mboemea.innntIYuiaIagnwseiqintontne
sinew

Linearity
Tension05ii

ifFR 12Raasurprisein Maxwell

Broadbrushstrokesummary

YMtheoryAngular momentum

skipping important f nor pointsof radiative nodes
Aa on I Hana oin Mink space but not essential
AAma Radiative
nodes

phase space
canwork

GA 8 A

Gloria a IIF

D Action preserves 52

Translation

But forIn genealrotations

HgHznaHamiltonians HSr

Determine Iof Im I

Tab Ea Eb.mno

fgTab5ahbdud2S
Tab 3amb ducks only if total
charge 0

REIT thathence by Hz

The RHS has coulombic information

is not captured by radiative modesrecoverTo

itone

has to extend the radiative phaseappropriately
surprise A priori one would have thought that angularshouldn't
momentum carried by electromagnetic waves acrossBut it does
its

space

about total chargecareiii

1170
it

I

Case

iof

f1

Ittoo

the idea
Binary star system to illustrate
more complicatedfeatsalsoconsider CBC details

viewpoint

IEEEi

Ioc

space time

II

Iae

Pace

me that

Impose no incoming radiation condin
Rewire it to750,81,99172
on H Cio
2

Dragsymmetries from Ioc to 9

io Define fluxes of energy 4
angular momentum across
For linear gravity Einstein's quadrupole formula has been
extended in the PN Post desitter approximation

via