Gravitational Wave Memories and Asymptotic Charges in General Relativity

General Relativity and Gravitation: A Centennial Perspective Penn State 8 June 2015

Éanna Flanagan, Cornell

EF, D. Nichols, arxiv:1411.4599; EF, D. Nichols and A. Harte, in preparation

Motivations

• The Bondi-Metzner-Sachs asymptotic symmetry group (1960s) is a qualitatively new feature of general relativity.

• Implications for gravitational wave observations (memory) and numerical relativity.

• Relevant to quantum gravity, AdS/CFT etc

• Subject is not completely understood.

• This talk: highlights of some recent developments: ‣ New types of gravitational wave memory ‣ How to measure BMS charges

Gravitational Wave Memory: Review

• The permanent relative displacement of a pair of freely falling test masses caused the passage of a burst of gravitational waves (Zel’dovich and Polnarev 1974)

• Geodesic deviation: with⇠ i ! (�ij + hij)⇠jh ⇠

Zdt

Zdt0 (Riemann)

• Intuition: (non-radiative, non-stationary)

h ⇠ Q(t� r, ✓,')

r! Q(✓,')

r

Two Types of Memory

• Derived by Christoudoulu (1991), clarified by Bieri and Garfinkle (2014), who performed a gauge invariant analysis in linearized gravity using the Bianchi identity.

Ordinary Memory Null Memory

• Computable in terms of change in between lim

r!1r3Ctrtr

i0 and i+

• BMS “supermomentum"

• Computable in terms of dE

dtd2⌦(u, ✓,') at null infinity

• Formerly called nonlinear memory

Gravitational Wave Spin Memory

• Pasterski, Strominger and Zhiboedov (2015) discovered a new type of memory called “spin memory”, related to angular momentum flux to null infinity rather than energy flux

• Normal memory associated with an infinite set of conserved charges (BMS group); spin memory similarly related to new infinite set of conserved charges associated with conjectured extension of BMS group. (Barnich and Troessaert 2009)

• Measurement procedure:

Sagnac interferometer

Comoving observer in Bondi coordinates

Observable: Z

dt �tsagnac

Gravitational Wave Spin Memory

• Pasterski, Strominger and Zhiboedov (2015) discovered a new type of memory called “spin memory”, related to angular momentum flux to null infinity rather than energy flux

• Normal memory associated with an infinite set of conserved charges (BMS group); spin memory similarly related to new infinite set of conserved charges associated with conjectured extension of BMS group. (Barnich and Troessaert 2009)

• Measurement procedure:

Sagnac interferometer

Comoving observer in Bondi coordinates

Observable: Z

dt �tsagnac

• Goal: find a gauge invariant description of spin memory

• Stragegy: use Bieri-Garfinkle method and search for new observables

Four Gauge Invariant “Memories”

• Setup: 1. Fix an event on worldline of freely

falling observer , a spatial vector at , and a proper time interval

2. Parallel transport back by to 3. Exponentiate to get event 4. Parallel transport to get at 5. Solve for geodesic of observer 6. Find unique spatial vector at

whose exponential intersects ’s world line at an event

PA

P~⇠

�⌧A~⇠ �⌧A Q

~⇠ R~uA ~uB R

B~⇠0 P

BS

Four Gauge Invariant “Memories”

Displacement memory: at ~⇠ � ~⇠0 P

Proper time memory: �⌧A ��⌧B

(Strominger & Zhiboedov 2014)

Velocity memory: at uaA(P)� gaa0ua0

B (S) P (Grishchuk & Polnarev 1989, Tolish & Wald 2014, Harte 2015)

Frame Dragging memory: The net relative rotation of gyroscopes carried by and , compared by parallel transport:

A B�⌦i = Bij⇠j

Four Gauge Invariant “Memories”

Displacement memory: at ~⇠ � ~⇠0 P

Proper time memory: �⌧A ��⌧B

(Strominger & Zhiboedov 2014)

Velocity memory: at uaA(P)� gaa0ua0

B (S) P (Grishchuk & Polnarev 1989, Tolish & Wald 2014, Harte 2015)

Frame Dragging memory: The net relative rotation of gyroscopes carried by and , compared by parallel transport:

A B�⌦i = Bij⇠j

• These observables are captured by a generalized holonomy around the loop , defined by solving to give Poincare map

PQRS t

arab = t

b, t

a = dx

a/d�

a ! ⇤ab

b +�a

Bieri-Garfinkle Computational Method

• Linear perturbation theory, extend Bieri & Garfinkle to subleading order in . 1/r

• Bianchi identities in terms of electric and magnetic pieces of Weyl tensor:

Eij Bij

Bieri-Garfinkle Computational Method

• Linear perturbation theory, extend Bieri & Garfinkle to subleading order in . 1/r

• Bianchi identities in terms of electric and magnetic pieces of Weyl tensor:

• Coordinates and metric(u, r, ✓A)

ds2 = �du2 � 2dudr + r2⌦ABd✓Ad✓B

Eij Bij

• Expand all quantities as

• Displacement memory isZdu

Zdu0 EAB ⇠ O(1/r) +O(1/r2)

• Two pieces

Electric parity

Magnetic parity

f(u, r, ✓A) =X

k=0

1

rs+kf (k)(u, ✓A)

E(0)AB = (DADB � 1

2D2⌦AB)�+ ✏ C

(A DB)DC

Assumptions on Stress-Energy Falloff

Results for Memory Observables

• All new observables are subleading in , except one1/r

• E type displacement memory: with D = D2 +D4/2

Z Z� = D�1

�E(0)

rr +

ZT (0)uu

�+O

✓1

r

◆

• Proper time, velocity and frame dragging memories: all and all computable from T (0)

uu ,�E(0)rr ,�E(1)

rA

• B type displacement memory: T (0)uA = DA�+ ✏ABD

B�

O(1/r)

Z Z = D�1

h�B(0)

rr

i� 1

rD�1

ZB(0)rr + 2D2

Z� ��B(1)

rr

�+O

✓1

r2

◆

Results for Memory Observables

• All new observables are subleading in , except one1/r

• E type displacement memory: with D = D2 +D4/2

Z Z� = D�1

�E(0)

rr +

ZT (0)uu

�+O

✓1

r

◆

• Proper time, velocity and frame dragging memories: all and all computable from T (0)

uu ,�E(0)rr ,�E(1)

rA

• B type displacement memory: T (0)uA = DA�+ ✏ABD

B�

vanishes (Winicour)

• Strominger observable is reallyZ Z Z

= D�1

ZB(0)rr +O

✓1

r

◆

i.e. , measurable by LIGOZ

dt hij(t)

• Disagreement on contribution by angular momentum flux??

O(1/r)

Z Z = D�1

h�B(0)

rr

i� 1

rD�1

ZB(0)rr + 2D2

Z� ��B(1)

rr

�+O

✓1

r2

◆

Memory Effects: Summary

• Spin Memory: there is a leading order, non-vanishing magnetic parity effect; the observable is the time integral of the strain

• Other memory observables likely too small to be observationally important.

BMS Conserved Charges: Review• Spacetimes which are asymptotically flat at null infinity have a universal background structure there: a vector field and a metric such that and is a conformal Killing vector. Pairs , are equivalent if

na (0,+,+) hab nahab = 0na (hab, n

a) (hab, na)

hab = !2hab na = !�1na

• Diffeomorphisms of null infinity which preserve this structure give the BMS group.

• Explicit parameterization: ,(u, ✓,') = (u, ✓A)

ds2 = d✓2 + sin2 ✓d'2

z = cot(✓/2) exp[i']

~n = @u ad� bc = 1

w�1 =1 + |z|2

|az + b|2 + |cz + d|2z0 =az + b

cz + du0 =

1

w[u+ ↵(z, z)]

BMS Conserved Charges: Review• Spacetimes which are asymptotically flat at null infinity have a universal background structure there: a vector field and a metric such that and is a conformal Killing vector. Pairs , are equivalent if

na (0,+,+) hab nahab = 0na (hab, n

a) (hab, na)

hab = !2hab na = !�1na

• Diffeomorphisms of null infinity which preserve this structure give the BMS group.

• Explicit parameterization: ,(u, ✓,') = (u, ✓A)

ds2 = d✓2 + sin2 ✓d'2

z = cot(✓/2) exp[i']

~n = @u ad� bc = 1

w�1 =1 + |z|2

|az + b|2 + |cz + d|2

• There is a generalization of Noether’s theorem that allows the construction of conserved charges and fluxes for each generator (Wald 2000, Dray & Streubel 1984)

z0 =az + b

cz + d

charge = P ata �1

2Jab!ab +

X

l�2,m

P lm↵lm

u0 =1

w[u+ ↵(z, z)]

How to measure BMS Conserved Charges?

• What happens for observes who try to use methods of linearized general relativity to measure charges? Analogous to Newtonian observers making measurements in Special Relativity. There, inconsistencies (Lorentz contraction etc) arise due to unexpected dependence on observer’s Lorentz frame. So expect inconsistencies due to dependence on observer’s asymptotic Bondi frame. How does this work in detail?

• In principle can be measured by a set of observers on a 2-surface who measure the geometry (“surface integral”).

Local measurement algorithm: At an event , observer measures . From these she can compute which “work” in linearized, stationary spacetimes

PRabcd(P),raRbcde(P),rarbRcdef (P)

P a(P), Jab(P)

How can Observers Compare Charges?

• The inhomogeneous parallel transport law can be used to transport along curves. Equivalent to Associated generalized holonomy (i.e. memory) around measures obstruction to consistency of measurements:

taravb = ta

(P, J) r~t~P = 0, r~tJ = ~P ^ ~t

(P, J) ! (P 0, J 0) 6= (P, J)

C

• Not good enough: the generalized holonomy is not trivial near null infinity in stationary regions, where preferred charges exist. Improved transport law P a = �Ra

bcdtbJcd/2, Ja = �2P [atb]

(J. Vines; A. Harte)

Conjecture: For this law, transport around curves becomes trivial near null infinity in stationary regions.

• Framework: Observers measure charges locally and transport them to give consistent answers in stationary regions. Comparisons between different stationary regions yield inconsistencies as measured non-locally by holonomies.

BMS Charge Measurements: Summary

• We have a operational understanding of measurements for stationary to stationary transitions (assuming validity of a conjecture).

• We would like to generalize this to nowhere stationary spacetimes, i.e. to give an operational definition of asymptotic Bondi frame and an operational definition of supermomentum measurements.