arXiv:1411.5745v1 [hep-th] 21 Nov 2014

Gravitational Memory, BMS Supertranslations and Soft Theorems
Andrew Strominger and Alexander Zhiboedov
Center for the Fundamental Laws of Nature Harvard University, Cambridge, MA 02138 USA
Abstract The transit of a gravitating radiation pulse past arrays of detectors stationed near future null infinity in the vacuum is considered. It is shown that the relative positions and clock times of the detectors before and after the radiation transit differ by a BMS supertranslation. An explicit expression for the supertranslation in terms of moments of the radiation energy flux is given. The relative spatial displacement found for a pair of nearby detectors reproduces the well-known and potentially measurable gravitational memory effect. The displacement memory formula is shown to be equivalent to Weinberg’s formula for soft graviton production.

Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. BMS review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3. BMS vacuum transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4. Gravitational memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4.1. BMS detector memory . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4.2. Inertial detector memory . . . . . . . . . . . . . . . . . . . . . . . . . . 9 5. Memory and soft theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 6. Measuring black hole hair . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Appendix A. Corrections to geodesics and Fermi coordinates . . . . . . . . . . . . . . 12 Appendix B. Massive particle decay . . . . . . . . . . . . . . . . . . . . . . . . 14
1. Introduction
The passage of a finite pulse of radiation or other forms of energy through a region of spacetime produces a gravitational field which moves nearby detectors. The final positions of a pair of nearby detectors are generically displaced relative to the initial ones according to a simple and universal formula [1-11]. This effect is known as gravitational memory. Direct measurement of the gravitational memory effect may be possible in the coming decades, see e.g. the recent work [12,13].
According to Bondi, Metzner, van der Burg and Sachs (BMS)[14], the classical vacuum in general relativity is highly degenerate. The different vacua are related by the so-called ‘supertranslations’, which are spontaneously broken ‘BMS symmetries’. In quantum language, these vacua differ by the addition of soft (i.e zero-energy) gravitons. In this paper we will show that the passage of radiation through a region induces a transition from one such vacuum to another. An explicit formula (involving moments of the radiation energy flux) is derived for the BMS supertranslation which relates the initial and final vacua. Moreover, relative positions and clock times of a family of detectors stationed in the vacuum are shown to be related by the same supertranslation. This observation provides a concrete operational meaning to BMS transformations.
The relative spatial displacement of nearby detectors following from the radiationinduced BMS transformation is precisely the standard gravitational memory. We find that
1

certain families of nearby detectors undergo, in addition to the standard spatial memory displacement, a relative time delay. It would be of interest to investigate potential experimental consequences.
Recently it has been shown [15] that the the observable consequences of BMS symmetry are embodied in the soft graviton scattering amplitudes which they universally determine. Herein we show that the Weinberg formula [16] for soft graviton production is essentially a rewriting of the formula for gravitational memory, establishing compatibility of [15] with the current work. However, while it is quite difficult to imagine a real experiment which directly measures soft gravitons, there is already a sizable literature on observation of gravitational memory. Hence the memory effect provides both a conceptually and observationally useful reformulation of BMS symmetry.
This paper is organized as follows. Section 2 establishes notation and briefly reviews BMS supertranslations. In section 3 we show that a finite duration radiation pulse crossing null infinity can be viewed as a domain wall mediating a transition between two inequivalent vacua of the gravitational field. Our central formula (3.7) is derived, involving the convolution of the radiative energy flux with a Green function, for the specific supertranslation which relates the initial and final vacua. Section 4 considers the effects of this transition on two types (inertial and fixed-angle) of detectors, and show that it can be understood as a supertranslation acting on the detector worldlines and clocks. This is shown to reproduce the spatial gravitational memory effect. It further elucidates a clock desynchronization effect with potentially observable consequences for (fixed-angle) detectors. In section 5 we show that the gravitational memory formula, in the form given by Braginsky and Thorne [3] is, after a change of variables and notation, identical to Weinberg’s soft graviton formula. In section 6 we point out that black holes are not invariant under supertranslations and therefore, in tension with the standard lore, carry an infinite amount of hair which encodes memories of how they were formed. Appendix A contains details on subleading corrections to large-radius geodesics. Appendix B demonstrates compatibility of our results with the interesting recent analyses by Tolish et. al. [10,11] of a specific example of the memory effect.
The existence of a connection between gravitational memory and BMS symmetry is known and has been discussed periodically: see for example [17,18]. We expect the relation between asymptotic symmetries and memory to extend to other systems such as gauge theories. In particular in gauge theories the passage of charge through I+ should be remembered by angle-dependent gauge transformations on charged detectors.
2

2. BMS review

The metric of an asymptotically flat spacetime in retarded Bondi coordinates takes the asymptotic form

ds2 = −du2 − 2dudr + 2r2γzz¯dzdz¯

+

2

mB r

du2

+

rCzz

dz2

+

rCz¯z¯dz¯2

+

Dz

Czz dudz

+

Dz¯Cz¯z¯dudz¯

+

...

(2.1)

where

γzz¯

=

2 (1+zz¯)2

is

the

unit

metric

on

S2,

Dz

is

the

γ-covariant

derivative

and

sublead-

ing terms are suppressed by powers of r.1 The Bondi mass aspect mB and Czz are related

by the constraint equation Guu = 8πGTuMu on I+

∂umB

=

1 4

Dz2N zz + Dz2¯N z¯z¯

− Tuu,

Tuu

=

1 4

Nzz

N

zz

+

4πG rl→im∞[r2TuMu].

(2.2)

where Nzz = ∂uCzz is the Bondi news, T M is the matter stress tensor and Tuu is the total energy flux through a given point on I+. The asymptotic form of the metric (2.1) is preserved by infinitesimal supertranslations [14]

u → u − f, , r → r − DzDzf, z → z + 1 Dzf, z¯ → z¯ + 1 Dz¯f,
rr
whose generating vector fields we denote

f = f (z, z¯),

ζf

=

f ∂u

+

Dz Dzf ∂r

−

1 r

(Dz¯f

∂z¯

+

Dzf ∂z).

The Lie derivative action on the asymptotic data is

(2.3) (2.4)

Lf mB = f ∂umB, Lf Czz = f Nzz − 2Dz2f.

(2.5)

According to BMS [14] two spacetimes related by supertranslations should be regarded as physically inequivalent.

1

In particular we have corrections

1 4r2

Czz C zz dudr

+

γzz¯CzzCzz dzdz¯

which

contribute

to

the

Einstein equations at the same order.

3

3. BMS vacuum transitions
Consider spacetimes which, prior to some retarded time ui on I+, are asymptotically well-approximated by Schwarzschild with

mB = Mi = constant, Czz = 0, while for u > uf they are also nearly asymptotically Schwarzschild2

(3.1)

mB = Mf = constant, Czz = 0.

(3.2)

During the intermediate interval ui < u < uf the Bondi news and/or total radiation flux Tuu is nonzero on I+.3 Christodoulou and Klainerman [19] considered spacetimes of this type with Mf = 0, where ui and uf must be taken early and late enough to capture most of the long time tails. For nonzero Mf the late time geometry could for example be a stable star or black hole.
The initial and final regions of I+ before ui and after uf are in the vacuum in the sense that Nzz = 0: the radiative modes are unexcited. According to BMS, the vacuum is not unique. It is characterized by any u-independent Czz obeying

Dz2¯Czz − Dz2Cz¯z¯ = 0.

(3.3)

The general solution to this equation is

Czz = −2Dz2C(z, z¯).

(3.4)

Comparison with (2.5) implies that the different vacua are related by supertranslations

under which C → C + f . The supertranslation which relates the initial and final vacua can

be determined by integrating the constraint (2.2) over the transition interval ui < u < uf . Defining

∆Czz = Czz(uf ) − Czz(ui), ∆mB = Mf − Mi,

(3.5)

and using (3.3) one finds

uf

Dz2∆Czz = 2

du Tuu + 2∆mB.

ui

(3.6)

2 We exclude for simplicity cases with nonzero initial or final ADM momentum. 3 Generic spacetimes may have long time radiation tails outside this interval, but for our
purposes making the radiation flux outside the interval arbitrarily small is good enough.

4

Fig. 1: Remembrance of things passed. We consider a transit of radiation through a set of detectors in the vicinity of the future null infinity I+. Detectors are located at large r0 and inserted at different points on the sphere S2 separated by distance
L. Change in the vacuum state is detected by the net displacement ∆L. The new
vacuum is related to the old one by the supertranslation C(z, z¯).

Note that the second term just subtracts the constant zero mode of the first.4 The super-

translation ∆C which produces such a ∆Czz is obtained by inverting Dz2Dz2¯:

∆C(z, z¯) = 2

d2z′γz′z¯′ G(z, z¯; z′, z¯′)

uf
du Tuu(z′, z¯′) + ∆mB
ui

(3.7)

where

G(z,

z¯;

z′,

z¯′)

=

−

1 π

sin2

Θ 2

log

sin2

Θ 2,

Dz2Dz2¯G(z, z¯; z′, z¯′) = −γzz¯δ2(z − z′) + · · · .

sin2

Θ(z, z′) 2

≡

(1

|z − z′|2 + z′z¯′)(1 +

zz¯) ,

(3.8)

If we plug (3.8) into (3.7) and act with Dz2Dz2¯ using ∂z∂z¯ log |z − zi|2 = 2π δ(2)(z − zi) the delta function piece produces the RHS of (3.6) while the remaining terms integrate to zero

4 Had we allowed for net momentum loss as well as energy loss the right hand side would also contain a term subtracting the angular momentum ℓ = 1 mode.

5

due to the energy-momentum conservation. C(z, z¯) is unique up to the 4 global spacetime

translations

fglobal

=

c0

+

c1(1

−

zz¯)

+ c2z + 1 + zz¯

c3z¯

+

c¯3z

(3.9)

which do not affect Czz.

This discussion could be generalized to allow for initial and final momentum, or mul-

tiple vacuum transitions induced by multiple well-separated radiation intervals.

To summarize, the passage of radiation through I+ changes the vacuum by a BMS

transformation. The BMS transformation relating the initial and final vacuum is given in

(3.7) by an integral of the total radiation flux over the transition interval.

4. Gravitational memory

In this section we will relate the BMS transformation of the vacuum to the gravitational memory effect. Towards this end we introduce two families of observers or detectors at large r. The first, which we refer to as BMS (or fixed-angle) detectors, travel along worldlines at fixed radius and angle:

XBµMS (s) = (s, r0, z0, z¯0),

(4.1)

where r0 is large. The assertion that BMS diffeomporhisms are physically nontrivial is equivalent to the statement that it is meaningful to discuss observations at a fixed value of z near I+. Such observations are convenient as they behave simply under the action of BMS. The second family of detectors are inertial ones moving along geodesics

∂s2Xgµeo(s) + Γµνλ∂sXgνeo(s)Xgλeo(s) = 0.

(4.2)

At large r0 the BMS detectors are nearly inertial. One may readily check (see Appendix A) that

XBu,Mr S (s)

=

Xgue,ro(s)

+

O( 1 r0

),

XBz MS (s)

=

Xgzeo(s)

+

O(

1 r02

).

(4.3)

The truly inertial detectors however do not remain at fixed r or z, so over a long period

of time u > r0 the radius can become small. Hence we must consider only retarded time lapses which are parametrically less than r0.
The relevant type of detector - BMS or inertial - depends on the application in ques-

tion. For example the eLisa detectors move on geodesic orbits and so are perhaps best

modeled by inertial detectors. On the other hand the LIGO detector are at fixed separa-

tions on the earth and are not geodesic. It would be interesting to understand what type

of detector array is well-approximated by BMS detectors.

6

4.1. BMS detector memory

Let us now consider what happens to the BMS detector worldlines in the setup of the previous section when they encounter a pulse of radiation passing to I+. Let us denote the initial positions of a pair of nearby detectors, detector 1 and detector 2, by z1 and z2. They are initially separated by a finite distance

L

=

2r0|δz| 1 + z1z¯1

,

δz ≡ z1 − z2

(4.4)

where

we take δz

to

be order

1 r0

and subleading

corrections

to L are suppressed.

As z1,2

are fixed in (4.1), but the metric undergoes a transition described by (3.5), the radiation

induces a change in the proper distance between the detectors. Computing the new distance

between z1 and z2 using the metric (2.1) gives

∆L

=

r0 2L

∆Czz

(z1,

z¯1)δz2

+

c.c.

=

(1

+

z1z¯1)2 8

L r0

∆Czz

(z1,

z¯1)

δz δz¯

+

c.c

,

(4.5)

where ∆Czz(z1, z¯1) is given according to (3.5) in terms of the energy flux as

∆Czz(z, z¯)

=

4 π

d2 z ′ γz ′ z¯′

z¯ z

− −

z¯′ z′

(1

(1 + z′z¯)2 + z′z¯′)(1 + zz¯)3

uf
du Tuu(z′, z¯′) + ∆mB
ui
(4.6)

This is precisely the standard formula for gravitational memory [1,4,5,8] .

Not only will the distances between BMS detectors be shifted, but if they are equipped

with initially synchronized clocks they will no longer be synchronized after passage of the

radiation. This can be checked by sending a light ray from detector 1 to detector 2,

stamping it with the time at detector 2 and then returning it to detector 1. If the clocks

remain synchronized, the time stamp from detector 2 will be exactly midway between the

light emission and reception times at detector 1. A light ray emitted from z1 will travel to z2 in a retarded time interval δ12u obeying

r02γzz¯δzδz¯

+

r0 ∆Czz δzδz

+

Dz ∆Czz δ12 uδz

−

1 2

(δ12u)2

+

c.c.

=

0.

(4.7)

On the other hand, on the return trip, the change in z has the opposite sign so the retarded time interval δ21u obeys

r02γzz¯δzδz¯

+

r0 ∆Czz δzδz

−

Dz ∆Czz δ21 uδz

−

1 2

(δ21u)2

+

c.c.

=

0.

(4.8)

7

The difference is5

δ12u − δ21u = Dz∆Czzδz + c.c..

(4.9)

Since this is nonzero the clocks are not synchronized. An alternate way of computing the memory and clock desynchronization is as fol-
lows. The proper distance and time delay observed in the above mentioned experiments is invariant under all diffeomorphisms, including BMS transformations. We may therefore eliminate all ∆Czz terms in the late time metric by the inverse of the BMS transformation (3.7) which by construction obeys

2Dz2f = ∆Czz,

(4.10)

so that f = −∆C This will have the effect of resetting all the clocks and relabeling the positions of the family of BMS observers by (2.3). To see that this agrees with the previous analysis let ζ = δz∂z + δz¯∂z¯ denote the initial separation vector between the detectors. Using (2.4) the action of the supertranslation (2.3) on this separation is

Lf ζ

=

−

1 2

(Dzf δz

+

Dz¯f δz¯)

∂u

−

1 2

Dz2Dzf δz + Dz2¯Dz¯f δz¯

∂r

+

γ zz¯ r

Dz2¯f

δz¯

+

1

+ zz¯ 2r

[2z¯Dz¯f

+

(1

+

zz¯)DzDz¯f ]δz

∂z + c.c..

(4.11)

To compare to the original coordinate system we evaluate the norm of the vector at (z +

1 r

Dz

f

,

z¯

+

1 r

Dz¯f

).

The

proper

distance

changes

by

∆L

=

r0 2L

∆Czz

(z1

,

z¯1)δz2

+

c.c.

(4.12)

which agrees, as it must, with (4.5). The extra terms that appear in (4.11) cancel against

the change of the metric of the flat space evaluated at the shifted point.

To compare the time delay of the two detectors two effects must be taken into account.

First the transformation of u → u − f resets the clocks by a relative amount Dzf δz + c.c.. A second effect arises because the relative radius changes by

δr = −Dz2Dzf δz + c.c.

(4.13)

Due to the presence of the term 2dudr in the metric, this implies a difference proportional to δr in the time lapses for light rays traveling from detector 1 to detector 2 and the reverse. Adding these two effects, and using

[Dz, Dz¯]Dz¯f = −Dz f,
5 The total elapsed time is, to leading order in r0, δ12u + δ21u = 2L.

(4.14)

8

one finds

δ12u − δ21u = Dz∆Czzδz + c.c.,

(4.15)

as expected. In conclusion the effects of a radiation pulse passing through I+ on a family of BMS
observers is characterized by the induced supertranslation (3.7). They may be equivalently described as leaving the worldlines unchanged and supertranslating the metric, or leaving the metric unchanged and supertranslating the observers. In either case they imply the familiar gravitational memory effect as well as clock desynchronization.

4.2. Inertial detector memory
Most discussions of gravitational memory involve inertial (rather than BMS) detectors moving on geodesics (4.2) that are nearly, but not exactly, worldlines of constant (r, z) and varying u. According to (4.3), the difference between the two worldlines is suppressed by powers of r. It immediately follows that the spatial gravitational memory formula (4.5) applies equally at large r to either BMS or inertial detectors.
The situation is more subtle for the relative time delay. In that case, we found above that there are two contributions which cancel at leading order, and the final result (4.15) is the sum of the subleading terms for each contribution. These subleading terms are in fact sensitive to the difference between the BMS and inertial worldlines. Direct computation reveals that, for inertial observers, the relative time delay actually vanishes at the order (4.15) , as we show in appendix A. In Bondi coordinates, this cancelation looks miraculous. However it is in fact a consequence of the equivalence principle, which implies the existence of Fermi normal coordinates in which the connection vanishes everywhere along the worldlines of two neighboring geodesics. It follows there can be no discrepancy of order L in the proper times and (4.15) hence must be cancelled by subleading geodesic corrections.

5. Memory and soft theorems
Recently it has been shown [15,20] that Weinberg’s soft graviton theorem [16] is equivalent to - or more precisely is the Ward identity of - BMS invariance of the quantum gravity S-matrix. In the preceding we have seen that the gravitational memory effect captures the consequences of BMS symmetry. In this section we show how to directly understand
9

the relation between the memory effect and the soft theorem without an interpolating discussion of BMS symmetry.
Weinberg’s soft graviton theorem [16] is a universal relation between (n → m + 1)particle with one final soft graviton and (n → m)-particle quantum field theory scattering amplitudes given by

√

lim
ω→0

Am+n+1

p1, ...pn; p′1, ...p′m, (ωk, ǫµν)

=

8πGSµν ǫµν Am+n p1, ...pn; p′1, ...p′m +O(ω0),

(5.1)

where

 T T

Sµν

=

m

j=1

pjµpjν ωk · pj

n
−
j=1

p′jµp′jν ωk · p′j 

.

(5.2)

In this expression k = (ω, ωk) with k2 = 1 is the four-momentum and ǫµν the transverse-traceless polarization tensor of the graviton. The superscript T T denotes the transverse-traceless projection (as detailed in [21]) and µ, ν indices refer to asymptotically Minkowskian coordinates with flat metric ηµν .
Here we explicate the relation between memory and soft theorems in the general context considered by Braginsky and Thorne [3]. They analyzed the possible detection of “burst memory waves” produced by the collision and scattering of large massive objects such as stars or black holes. They found that such collisions resulted in a net difference in the transverse traceless part of the asymptotic metric at I+ given by6

∆hTµνT (k)

=

1 r0

 T T

G 2π

n

j=1

p′jµp′jν ωk · p′j

m
−
j=1

pjµpjν ωk · pj 

.

(5.3)

Here we have n (m) incoming (outgoing) objects with asymptotic momenta pjµ (p′jµ). k = (1, k) is the null vector pointing from the collision region to null infinity, and serves as a coordinate on the S2 at I+. Equation (5.3) was derived by solving the linearized Einstein equation with a retarded propagator. The gravitational memory of the collision is then simply constructed from (5.3) via (4.5).

6

This is equation (1) of [3] written with the normalization hµν

≡

1 √32πG

(gµν

−

ηµν ),

in

the

mostly plus (− + ...+) signature and in covariant gauge.

10

Evidently there are strong similarities between (5.3) and (5.2). To make it more manifest we note the Fourier transform of hTµνT (ω, k) on I+ can be written, using the stationary phase approximation at large r [22,20]

hTµνT

(ω,

k)

=

4πi

lim
r→∞

r

du eiωuhTµνT (u, rk),

(5.4)

Assuming that hTµνT (u, rk) approaches finite but different values at u → ±∞ and large r = r0 it then follows7 that (5.3) is proportional to the coefficient of the pole in ω

∆hTµνT (k)

=

1 4πir0

lim
ω→0

−iωhTµνT (ω, k)

.

(5.5)

Next we note that to linear order, the expectation value of the asymptotic metric fluctuation produced in the process of n → m scattering obeys

lim
ω→0

ωhTµνT

(ω

,

k)ǫµν

=

lim
ω→0
√

ωAm+n+1 p1, ...pn; p′1, ...p′m, (ωk, ǫµν) Am+n p1, ...pn; p′1, ...p′m

=

8π

Gǫµν

lim
ω→0

ωSµν

(ωk)

 T T

√m = 8πGǫµν 
j=1

pj µ pj ν k · pj

n
−
j=1

p′jµp′jν k · p′j 

.

(5.6)

Inserting this into (5.5) we then see that this is equivalent as claimed to the BraginskyThorne result (5.3).

6. Measuring black hole hair
One often hears that black holes have no hair. This statement does not take into account the subtleties associated with asymptotic structure at I. In particular, as we discussed in section 3, a supertranslation maps the Schwarzschild solution to a physically inequivalent configuration. Hence black holes have a lush infinite head of supertranslation hair. This may bear on the information puzzle.
7 In the formulas above we assume that ωr ≫ 1 when taking the limits.
11

Fig. 2: Supertranslation hair of black holes. We consider formation of a black hole by infalling matter. During the process radiation is necessarily created and leaks to future null infinity, where it mediates the transition to a new vacuum state. When the black hole settles down the state at I+ is characterized by the supertranslation hair C(z, z¯).

The present discussion clarifies the nature of supertranslation hair and how it can be measured classically. Let us consider, as a special case of the Braginsky-Thorne construction, N incoming stars which collide and collapse into a black hole. We station an array of evenly-spaced detectors near future null infinity. The relative positions of these detectors will shift due to the memory effect as given in (5.3) and (4.5). Hence the detector positions can record an infinite amount of data about how the black hole was formed. Black hole formed by different initial star configurations will carry different supertranslation hair.
Acknowledgements We are grateful to S. Gralla, D. Kapec, J. Maldacena, P. Mitra, S. Pasterski and A. Porfyriadis for useful conversations. This work was supported in part by NSF grant 1205550.

Appendix A. Corrections to geodesics and Fermi coordinates

Let us consider an asymptotic time-like observer with the velocity vµ in the Bondi

retarded coordinates. Its four-velocity is given by

vµ

=

(1,

mB (u, r

z,

z¯) ,

−

1 8

(1

+ zz¯)4 r2

Dz Cz¯z¯(u,

z,

z¯),

−

1 8

(1

+ zz¯)4 r2

Dz¯Czz (u,

z,

z¯)).

(A.1)

Notice

that

for

the

first

two

components

we

work

to

the

1 r

order

and

for

the

last

two

to

the

1 r2

.

The

reason

for

this

will

become

clear

below.

This

four-vector

describes

a

geodesic

vµ∇µvu,r

=

O(

1 r2

),

v µ ∇µ v z ,z¯

=

O(

1 r3

).

For

the

norm

we

have

vµvµ

=

−1

+

O(

1 r2

).

12

Equally well (by choosing different initial conditions) we could have chosen the fourvelocity to be

vµ = (1 + mB(u0, z, z¯) , mB(u, z, z¯) − mB(u0, z, z¯) , rr

−

1 8

(1

+ zz¯)4 r2

Dz [Cz¯z¯(u,

z,

z¯)

−

Cz¯z¯(u0,

z,

z¯)],

−

1 8

(1

+ zz¯)4 r2

[Dz¯Czz (u,

z,

z¯)

−

Dz¯Czz (u0 ,

z,

z¯)]).

(A.2)

In (A.1) we set u-independent initial values to zero. We do the same thing below since we

are interested in the vacuum-to-vacuum transitions described in the bulk of the paper.

If we integrate over u for a long enough time the corrections are not small since the

correspondent integrals diverge. Below we always assume r to be large enough (and the

measurement time to be small enough) so that the corrections are small.

We also consider the orthonormal spatial basis

nµ

=

(−1,

1

−

mB r

,

1 8

(1

+ zz¯)4 r2

Dz Cz¯z¯,

1 8

(1

+ zz¯)4 r2

Dz¯Czz ),

mµ

=

(0,

0,

1 r

1

+ 2

zz¯

,

−

1 r

1

+ zz¯ 2

(1

+

zz¯)2Czz 4r

),

m¯ µ

=

(0,

0,

−

1 r

1

+ zz¯ 2

(1

+

zz¯)2Cz¯z¯ , 4r

1 r

1

+ 2

z z¯ ).

(A.3)

All

of

these

are

parallel

transported

along

vµ

to

leading

order

in

1 r

so

that

we

have

vµ∇µeνi

=

O(

1 r2

)

and

are

orthogonal

to

vµ,

namely

vµeµi

=

O(

1 r2

).

They

are

also

normal-

ized

in

the

usual

way

n.n

=

1

+

O(

1 r2

),

m.m¯

=

1 2

+

O(

1 r2

),

n.m

=

n.m¯

=

m.m

=

m¯ .m¯

=

O(

1 r2

).

Physically,

these

vectors

describe

a

set

of

gyroscopes

that

is

carried

by

an

observer.

Based on this we can introduce Fermi normal coordinates which are the coordinates

that describe physics that that the observer experiences in the vicinity of his location.
Namely introducing eµ0 = vµ, eµ3 = nµ, eµ = mµ, e¯µ = m¯ µ we introduce corresponding coordinates xµi such that the metric takes the form

ds2 = −dx20 + dx23 + dxdx¯ + O(x2)

(A.4)

and the leading corrections are related to Rabcd = Rµνρσeµa eνb eρc eσd [23,24] where the Riemann tensor is evaluated along the geodesic γ(τ ). The equations for geodesics take the

form

x¨i = Ri00jxj .

(A.5)

13

For

the

case

in

hand

the

only

contribution

at

the

1 r

order

comes

from

Rx00x

=

(1+zz¯)2 8r

∂u2

Czz

and Rx¯00x¯

=

(1+zz¯)2 8r

∂u2

Cz¯z¯

which

describe

the

ordinary gravitational

memory.

The

same

analysis was done in [4].

It is clear in Fermi coordinates that there is no relative time shift linear in x unless

the original observer is accelerated. This is the case for BMS observers and it is the source

of time desynchronization linear order in x.

We can also analyze the nearby geodesic directly in Bondi coordinates as explained

in the main body of the paper. The result of course does not depend on the coordinate

system we used.

Appendix B. Massive particle decay

It is instructive to compare our results to those of Tolish et. el. [10], [11]. The starting

point of their work is the geodesic deviation equation for a small perturbation around the

flat space

d2Dµ dt2

= −Rtµtν Dν ,

∆Dµ = Mµν Dν .

(B.1)

For the cases considered in [11], symmetries imply that

Rtµtν = W (θµθν − φµφν )

(B.2)

where θµ and φµ are the unit vector fields on the sphere. In the Bondi coordinates we have for the tensor Mˆ µν = θµθν − φµφν

Mˆ zz

=

(1

2 + zz¯)2

zz¯,

Mˆ z¯z¯

=

2 (1 + zz¯)2

z z¯

.

(B.3)

Consider now the following situation. A particle at rest of mass M decays into a massless particle with energy E moving in the zˆ-direction and the particle of mass M ′ moving in the −zˆ-direction. On the sphere it corresponds to z = z¯ = 0 for zˆ and z = z¯ = ∞ for −zˆ. The contribution of the massless particle is [11]

Mµν

=

E (1 r

+

cos θ)Mˆ µν

=

E r

1

2 +

zz¯

Mˆ µν

(B.4)

whereas for the massive one [11]

Mµν

=

E2 Mr

1

−

sin θ2

E M

(1

−

cos θ) Mˆ µν

=

1 r

|p|2 sin θ2 p0 + |p| cos

θ

Mˆ µν

(B.5)

14

where p0 = M − E, |p| = E by energy-momentum conservation. In the formulae above we set G = 1.
We now would like to reproduce the same formulas using the soft theorem which states that the memory is given by the soft factor8

 T T

Mµν

=

1 2

√ 32π

GhTµνT

=

2G r



piµpiν (pi.n)

−

pfµ pfν (pf .n) 

,

if

(B.7)

where

we

adopted

field

theoretical

normalization

hµν

=

√1 32πG

(gµν

−

ηµν )

and

n

=

(1, n)

is

the unit four-vector in the direction of observation. Plugging the momenta in the formula

above reproduces the result of [11]. For massless particles in the (z, z¯) coordinates it

becomes

Mzz

=

4G r

i

Ei

z¯i zi

− −

z¯ z

(1

(1 + ziz¯)2 + ziz¯i)(1 + zz¯)3

.

(B.8)

In the example above we have zi = z¯i = 0. Notice also that the (z, z¯)-dependent kernel that appeared in (B.8) is identical to the one that appeared in (4.6). Indeed, as pointed out

in [7] the Christodoulou memory effect can be thought as a generalization of the usual soft

factor where instead of a finite set of particles approaching infinity we imagine arbitrary

energy flux of gravitational radiation. Any fixed energy scattering can produce at most

finite number of massive particles. It means that a generic final state can be thought

as the finite number of massive particles plus arbitrary complicated profile of radiation.

The memory due to the radiation is captured by (4.6), whereas for massive particles the

contribution to the memory is simply given by (B.7).

8 The prescription for taking transverse-traceless part is thoroughly reviewed in [21]

hTµνT = hµν − nµhνλnλ − nν hµλnλ + nµnν (hλdnλnd)

−

1 2

(δµν

−

nµnν )hλd

δλd − nλnd

.

(B.6)

15

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17