New gravitational memories

Abstract

Abstract

The conventional gravitational memory effect is a relative displacement in the position of two detectors induced by radiative energy flux. We find a new type of gravitational 'spin memory' in which beams on clockwise and counterclockwise orbits acquire a relative delay induced by radiative angular momentum flux. It has recently been shown that the displacement memory formula is a Fourier transform in time of Weinberg's soft graviton theorem. Here we see that the spin memory formula is a Fourier transform in time of the recently-discovered subleading soft graviton theorem.

Published for SISSA by Springer Received: November 10, 2016

Accepted: December 9, 2016 Published: December 14, 2016

JHEP12(2016)053

New gravitational memories

Sabrina Pasterski, Andrew Strominger and Alexander ZhiboedovCenter for the Fundamental Laws of Nature, Harvard University,

Cambridge, MA 02138, U.S.A.

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: The conventional gravitational memory e ect is a relative displacement in the position of two detectors induced by radiative energy ux. We nd a new type of gravitational spin memory in which beams on clockwise and counterclockwise orbits acquire a relative delay induced by radiative angular momentum ux. It has recently been shown that the displacement memory formula is a Fourier transform in time of Weinbergs soft graviton theorem. Here we see that the spin memory formula is a Fourier transform in time of the recently-discovered subleading soft graviton theorem.

Keywords: Classical Theories of Gravity, Gauge Symmetry, Space-Time Symmetries

ArXiv ePrint: 1502.06120

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP12(2016)053

Web End =10.1007/JHEP12(2016)053

Contents

1 Introduction 1

2 Asymptotically at metrics 2

3 Displacement memory e ect 5

4 Spin memory e ect 5

5 Spin memory and angular momentum ux 6

6 Equivalence to subleading soft theorem 8

7 An in nity of conserved charges 10

A Massless particle stress-energy tensor 12

1 Introduction

The passage of gravitational radiation past a pair of nearby inertial detectors produces oscillations in their relative positions. After the waves have passed, and spacetime locally reverts to the vacuum, the detectors in general do not return to their initial relative positions. The resulting displacement, discovered in 1974 [1{11], is known as the gravitational memory e ect. Direct measurement of the gravitational memory may be possible in the coming years, see e.g. [12, 13]. The e ect is a consequence [14] of the fact that the radiation induces transitions among the many BMS-degenerate [15, 16] vacua in general relativity. The initial and nal spacetime geometries, although both at, di er by a BMS super-translation. The displacement is proportional to the BMS-induced shift in the spacetime metric, which in turn is given by a universal formula involving moments of the asymptotic energy ux.

Since the initial and nal metrics di er, the Fourier transform in time must have a pole at zero energy. A universal formula for this pole was found in 1965 [17] and is known as Weinbergs soft graviton theorem. The complete equivalence of the soft graviton and displacement memory formulae was demonstrated in [14].

Recently, a new universal soft graviton formula was discovered [18] (see also [19, 20]) that governs not the pole but the nite piece in the expansion of soft graviton scattering about zero energy. This was shown [21] to be a consequence of the BMS super-rotations1 of [22] in the same sense that Weinbergs pole formula is a consequence of BMS supertranslations.

1The associated symmetry group is the familiar Virasoro symmetry of euclidean two-dimensional conformal eld theory [22]. This may be usefully embedded in a larger group of all di eomorphisms of the sphere [23, 24].

{ 1 {

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This discovery immediately raises the question: is there a new kind of gravitational memory associated with superrotations and the subleading soft theorem? In this paper we show that the answer is yes. While displacement memory is sourced by moments of the energy ux through null in nity (I), the new memory is sourced by moments of the angular

momentum ux. Accordingly we call it spin memory. The spin memory e ect provides a cogent operational meaning to the superrotational symmetry of gravitational scattering.

Spin memory has a chiral structure and cannot be measured by inertial detectors. Instead, we consider light rays which repeatedly orbit (with the help of ber optics or mirrors) clockwise or counterclockwise around a xed circle in the asymptotic region. The passage of angular-momentum-carrying radiation will induce a relative time delay between the counter-orbiting light rays, resulting for example in a shift in the interference fringe. A universal formula for this delay is given in terms of moments of the angular momentum ux through in nity. The relative time delay for counter-orbiting light rays is the spin memory e ect. It is a new kind of gravitational memory.

This paper is organized as follows. Section 2 outlines the metric and constraint equations for asymptotically at spacetimes. Section 3 reviews the displacement memory e ect. Section 4 introduces the spin memory e ect. Section 5 uses the constraint equations and boundary conditions to relate this new memory e ect to angular momentum ux. Section 6 demonstrates that the spin memory formula is the Fourier transform in time of the subleading soft graviton formula of [18]. Finally, in section 7, we discuss the in nite family of conserved charges associated to the in nite superrotational symmetries. Measurements verifying the conservation laws are described. We close with a short comment on implications for black hole information. The appendix derives several formulae concerning the angular momentum of spinning particles on null geodesics.

2 Asymptotically at metrics

The expansion of an asymptotically at spacetime metric near I+ in retarded Bondi coor

dinates takes the form2

ds2 = du2 2dudr + 2r2 zzdzd

z + 2mB

2(1+zz)2 is the unit metric on S2 (used to raise and lower z and

z

indices), Dz is the -covariant derivative, and subleading terms are suppressed by powers of r. The Bondi mass aspect mB, the angular momentum aspect Nz, and Czz are functions of (u; z;

z), not r. They are related by the I+ constraint equations Guu = 8GT Muu

@umB = 14

D2zNzz + D2zN zz

Tuu;

14NzzNzz + 4G lim

r!1

2Our de nition of Nz, which is proportional to the Weyl tensor (see below) is related to NBTz of [22] by 4Nz = 4NBTz + CzzDzCzz + 34 @z(CzzCzz).

{ 2 {

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r du2

43Nz 14@z(CzzCzz) dudz + c:c: + : : :(2.1)

where u = t r, zz =

+

rCzzdz2 + DzCzzdudz + 1 r

(2.2)

Tuu

r2T Muu ;

and Guz = 8GT Muz

@uNz = 14@z

D2zCzz D2zCzz + @zmB Tuz;

(2.3)

where Nzz = @uCzz is the Bondi news, T M is the matter stress tensor and Tuu (Tuz) is the total energy (angular momentum) ux through a given point on I+. The angular

momentum aspect is related to the Weyl tensor component 01 on I+ by

Nz = lim

r!1

r3Czrru: (2.4)

We also note that

Im 02 = Im lim

r!1

Tuz 8G lim

r!1

r2T Muz

14Dz [CzzNzz] 1 2CzzDzNzz;

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r zzCuzzr = Im

1

2D2zCzz +

1 4CzzNzz

: (2.5)

Most of our discussion will concern I+, but the metric expansion near I in the

retarded Bondi coordinate v = t + r is

ds2 = dv2 + 2dvdr + 2r2 zzdzd

z + 2mB

r dv2

43Nz +14@z(CzzCzz) dvdz + c:c: + : : :(2.6)

where here the metric perturbations are functions of (v; z;

z). The z coordinate on I is

de ned so that (v; z;

z) is the P T conjugate of (u; z;

z): hence they lie on the same null generator of I and are antipodally located relative to the origin of the spacetime. The I

constraint equations become Gvv = 8GT Mvv

@vmB =

1 4

+ rCzzdz2 + DzCzzdvdz +1 r

D2zNzz + D2zN zz + Tvv;

(2.7)

Tvv

14NzzNzz + 4G lim

r!1

r2T Mvv ;

and Gvz = 8GT Mvz

@vNz =

1 4@z

D2zCzz D2zCzz + @zmB + Tvz;

(2.8)

We use the symbol I+ (I++) to denote the past and future S2 boundaries of I+, and I for those of I. In this paper, we consider spacetimes which decay to the vacuum in

the far past and future I and I++. (The more general case requires an analysis of extra

past and future boundary terms.) In particular, we require

Nz[notdef]I

+

Tvz 8G lim

r!1

r2T Mvz

14Dz [CzzNzz] 1 2CzzDzNzz:

+ = Nz[notdef]I

= mB[notdef]I

+

+ = mB[notdef]I

= 0: (2.9)

{ 3 {

Moreover near all the boundaries I of I, the geometry is in the vacuum in the sense that

the radiative modes are unexcited:

Nzz[notdef]I

= 0: (2.10)

More precisely, following Christodoulou and Klainerman [25], we take Nzz [notdef]u[notdef]3/2 ([notdef]v[notdef]3/2) on I+ (I) as well as a corresponding fallo of the stress tensor.

These conditions do not imply Czz[notdef]I

= Im 02[notdef]I

= 0. Rather, the general solution of (2.10) is

(see e.g [26])

Czz = 2D2zC; (2.11) where C is any (u-independent) function of (z;

z). These solutions are mapped to one another by BMS supertranslations and exhibit the large vacuum degeneracy in general relativity.

As described in [26], to de ne gravitational scattering one must specify boundary or continuity conditions on mB and Czz near where I+ and I meet. The unique Lorentz,

PT and BMS-invariant choice is simply

Czz[notdef]I

+ = Czz[notdef]I

+ : (2.12)

In this paper Nz plays an important role and its determination from the constraint equation (2.3) also requires a continuity condition. This is a bit tricky because outside the center-of-mass frame, Nz may grow linearly with u or v near I. It follows from (2.3)

and (2.10) that the divergent term is exact: Nz u@zmB. Fortunately for us, such exact

terms are irrelevant for our purposes (see section 5). We will need a continuity condition for the curl of Nz. (2.10), (2.12) and the Bianchi identity imply

@[zNz][notdef]I

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+ ; mB[notdef]I

+ = mB[notdef]I

+ = @[zNz][notdef]I

+ : (2.13)

We are interested in the di erence between the initial and nal functions C in (2.11) on I. This can be determined by integrating the constraint (2.2) as follows (see [14] for

more details). De ning

+Czz = Czz[notdef]I

+

+ Czz[notdef]I

+ ; +mB = mB[notdef]I

+

+ mB[notdef]I

+ = mB[notdef]I

+ ; (2.14)

and using (2.2) one nds

D2z +Czz = 2

Z

du Tuu + 2 +mB: (2.15)

The +C which produces such a +Czz is obtained by inverting D2zD2z:

+C(z;

z) =

Z

d2w w wG(z; w)

Z

du Tuu(w) + mB

(2.16)

where the Greens function is given by

G(z; w) =

1 sin2

2 log sin2

2 ; sin2

(z; w)

2 [notdef]

z w[notdef]2

(1 + w

w)(1 + z

z): (2.17)

{ 4 {

An equation similar to (2.16) may be derived for the shift of C on I. Adding the two

equations, using the boundary condition (2.12), and de ning

C = +C C; (2.18)

one arrives at the simple relation

C(z;

z) = Z

d2w w wG(z; w)

Z

du Tuu(w)

Z

dv Tvv(w)

: (2.19)

3 Displacement memory e ect

In this section, we brie y review the standard gravitational memory e ect. The passage of a nite pulse of radiation or other form of energy through a region of spacetime produces a gravitational eld which moves inertial detectors. The nal positions of a pair of nearby detectors are generically displaced relative to the initial ones according to a simple and universal formula [1{11], which we now review brie y.

Consider two nearby inertial detectors with proper worldline tangent vectors t and relative displacement vector s. We take the worldlines to be at large r and extend for in nite retarded time near I+. s evolves according to the geodesic deviation equation

@2s = R t ts ; (3.1)

where is the detectors proper time. At large r in the geometry (2.1), we may approximate u, t @ = @u and

Rzuzu =

1

2r@2uCzz: (3.2)

It follows that

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2r @2uCzzsz: (3.3)

Integrating twice one nds, to leading order in 1

r , a net change in the displacement

+sz = zz

2r +Czzsz; (3.4)

where +Czz is given in term of moments of the asymptotic energy ux by the second derivative of (2.16). (3.4) is the standard displacement memory formula.

4 Spin memory e ect

This section describes the new spin memory e ect, which a ects orbiting objects such as protons in the LHC, or signals exchanged by eLISA detectors. Consider a circle C of radius

L near I+ centered around a point z0 on a sphere of large xed r = r0, where L r0.

This is described by

Z() = z0

1 + Lei2r01 + z0 z0 pz0 z0 + O

@2usz = zz

L2 r20

(4.1)

{ 5 {

where + 2. A light ray in either a clockwise or counterclockwise orbit (aided by

mirrors or ber optics) along C starting at (0) = 0, has a trajectory (u) that obeys ds2 = 0

= 1 2r20 zz@uZ@u

Z 2

mBr0 r0Czz (@uZ)2 r0Czz @u

Z

2

(4.2)

where it is here and hereafter assumed that Czz does not change signi cantly over a single period. To this order, only the term in square brackets in (4.2) is odd under @uZ ! @uZ.

If two light rays are simultaneously set in orbit in opposite directions, the times at which they return to = 03 will di er by the u-integral of this odd term

P =IC DzC

zzdz + DzCzzd

z

DzCzz@uZ + DzCzz@u

Z + : : :

: (4.3)

This formula in fact applies to any contour C, circular or not.

At rst glance, it appears from (4.3) that the returns of the counter-orbiting light rays are desynchronized, even in the vacuum, as long as Czz is nonzero. In fact this is not the case. For Czz of the vacuum form (2.11), one readily nds that

Pvacuum = 2

IC

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d(DzDzC + C) = 0: (4.4)

Hence desynchronization occurs only during the passage of radiation through I+. The total relative time delay, integrated over all orbits is

+u = 1

2L

Z

du

IC DzC

zzdz + DzCzzd

z

: (4.5)

This leads to a shift in the interference pattern between counter-orbiting light pulses. The shift is an infrared e ect proportional to the u-zero mode of Czz. This is the spin memory e ect.

5 Spin memory and angular momentum ux

Displacement memory (3.4) can be expressed as an integral of the net local asymptotic energy ux convoluted with the Greens function (2.19) on the sphere. In this section we derive an analogous formulae for spin memory as a convoluted integral involving the net local asymptotic angular momentum ux.

Taking @z of the Guz constraint in (2.3) and @z of the complex conjugate Guz constraint gives:

@z@zmB = Re [@u@zNz + @zTuz] (5.1)

Im

@zD3zCzz = 2Im [@u@zNz + @zTuz] : (5.2) 3The line = 0 is a geodesic in the induced geometry of the ring (4.2) only in the limit r ! 1. Detectors

at xed are BMS detectors of the type discussed in [14]. A nite-r geodesic detector will be boosted and observe a di erent P .

{ 6 {

Multiplying (5.2) by the Greens function

G(z; w) = log sin2 2 ; sin2

(z; w)

2 [notdef]

z w[notdef]2

(1 + w

w)(1 + z

z) (5.3)

which obeys

@z@zG(z; w) = 2 2(z w)

1

2 zz; (5.4)

and integrating over d2z gives:

Im

D2wCww = Im

Z

d2z@zG (z; w) [@uNz + Tuz] : (5.5)

Note that the right hand side of (5.5) is invariant under shifts Nz ! Nz + @zX for any real X, so only the curl part of Nz contributes. Integrating both sides over the disk DC whose

boundary is C and using Stokes theorem, (5.5) leads to

IC

(DwCwwdw + D wC w wd

w) = 2Im

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ZDCd2w w wZd2z@zG(z; w)[@uNz + Tuz]: (5.6)

Multiplying by 1

22L and integrating over u then yields

+u =

1 2LIm

ZDCd2w w wZd2z@zG(z; w) +Nz +ZduTuz

: (5.7)

where +Nz Nz[notdef]I

+

+ Nz[notdef]I

+ = Nz[notdef]I

+ is the shift in the angular momentum as-

pect. In many applications | for example geometries which are initially asymptotically Schwarzschild through subleading order | this term will vanish. Moreover, if +Nz is

exact, i.e. +Nz = @zX for any real X, no contribution to the imaginary part appears in (5.7). Hence (5.7) depends only on the curl of +Nz.

A similar analysis near I leads to the formula

v =

1 2LIm

ZDCd2w w wZd2z@zG(z; w) Nz +ZdvTvz

: (5.8)

where the contour C and disk DC on I is de ned by the curve de ned in equation (4.1)

on I. This means it will lie in the antipodal spatial direction from the origin. Using the

continuity condition (2.13) on Nz one nds, in analogy to (2.19), an expression relating time delays and uxes

+u v =

1 2LIm

ZDCd2w w wZd2z@zG(z; w) ZduTuz

ZdvTvz

: (5.9)

The right hand side of (5.9) is related to the local angular momentum ux through I.

Consider the case when Tuz arises from massless particles or localized wave packets which puncture I+ at points (uk; zk). Then, as we show in the appendix,

Tuz = 8G

Xk (u uk)

Luz(zk) i2hk@z 2(z zk) zz ; (5.10)

{ 7 {

together with a similar formula for Tvz. Here Luz(zk); (hk) is the orbital (spin) angular momentum of the kth particle associated to a rotation around (boost towards) the point zk on the sphere. The leading contribution from such particles to the time delay is

=

8G L

Xk

zkzkIm

ZDCd2w w wLuz(zk)@zkG(zk; w) + hk2C : (5.11)

The second term in (5.11) has the simple interpretation that an object of spin hk passing through C at (or near) lightspeed induces a time delay of order

hk

L , with no factors of r0.

This can be understood as the frame-dragging e ect. If C lies a distance of order L from

zk, the rst term is typically a factor of L

r0 smaller than the second.

Another interesting case is outgoing quadrupole radiation, with no incoming news or

Tvz on I.4 The displacement memory e ect for con gurations of this type is of potential

astrophysical interest and was analyzed in [6]. This is described by the news tensor on I+ Nzz = NY izY jz (5.12)

for some N(u). As an example, take i = j, Y zi = z and

N =

(2)1/4 @ueu2+i!u: (5.13)

The resulting angular momentum ux obeys

Z

duTuz = 3i

2 2!@z

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z2

z2 (1 + z

z)4 ; (5.14)

so that

Im

Z

du

Z

d2z@zG(z; w)Tuz = 3 2!

130

w2

w2 (1 + w

w)4

; (5.15)

using (5.4). The quadrupole contribution to the time delay around a contour C becomes

+u = 3 2!

L

ZDCd2w w w

w2

w2 (1 + w

w)4

1 30

: (5.16)

For typical choices of C such that the area of DC is order L2=r20, we have +u 2Lr20 .

6 Equivalence to subleading soft theorem

In the sixties, Weinberg [17] showed that scattering amplitudes in any theory with gravity exhibit universal poles as the energy ! of any external graviton is taken to zero. Recently [18{21, 23, 24, 27, 28] it has been shown that the nite, subleading term in the ! ! 0 expansion also exhibits universal behavior. The coe cient of the leading pole

was shown in [14] to be related by a timeline Fourier transform of the expression for displacement memory. We now show that the subleading term is a Fourier transform of the expression for spin memory.

4Since we are taking mB[notdef]I

= 0 here, the initial energy would have to enter in a spherically symmetric

wave from I.

{ 8 {

The subleading soft graviton theorem is a universal relation between (n + 1)-particle (with one soft graviton) and n-particle tree-level quantum scattering amplitudes [18]

1

2

lim

!!0+

+ lim

!!0

An+1 (p1; : : : pn; (!q; )) = S(1) An (p1; : : : pn) ; (6.1)

where

S(1) = i

2

n

Xk=1 pk(Jk ) q q [notdef] pk

(6.2)

and = p32G. In this expression, the parentheses denote ( ; ) symmetrization, q = (!; !^

q) with ^

q2 = 1 is the four-momentum, and is the transverse-traceless polarization tensor of the graviton. We de ne incoming particles to have negative p0 and take ! positive for an outgoing graviton. ; indices refer to asymptotically Minkowskian coordinates given in terms of retarded coordinates as

x0 = u + r;

x1 + ix2 = 2rz

1 + z

z ;

x3 = r(1 z

z) 1 + z

z ;

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(6.3)

or in terms of advance coordinates as

x0 = v r;

x1 + ix2 =

2rz1 + z

z ;

(6.4)

The symmetrized limit in (6.1) projects out the leading Weinberg pole, leaving the sub-leading nite term of interest here. The linearized expectation value of the asymptotic metric uctuation produced in the n-particle scattering process obeys the semiclassical momentum space formula

lim

!!0+

x3 =

r(1 z

z) 1 + z

z :

+ lim

!!0

h (!; q) = lim!!0++ lim !!0

An+1 (p1; : : : pn; (!q; ))

An (p1; : : : pn)

= i

n

(6.5)

Xk=1 pkJk q q [notdef] pk:

In the last line, and in similar expressions below, an expectation value of the expression involving the di erential operator Jk acting in the matrix element in An is implicit.

Expression (6.5) characterizes linearized elds by their momenta whereas the new memory formula (5.9) is given in terms of I values of elds. These are simply related.

Using the large-r stationary phase approximation as in [21, 29]

Z

duCzz (u; ^

q)

Z

dvCzz (v; ^

q) =

lim

!!0+

+ lim

!!0

i 8 @zX@zX h (!; ^

q) ; (6.6)

{ 9 {

where X lim

r!1

xr and the unit vector ^

q is viewed as a coordinate on I. The hard particle

momenta pk, the soft graviton momentum q, and complex polarization = are given in terms of the points zk and z at which they arrive on the on the asymptotic S2 and their energies Ek; !

pk =

Ek1 + zk

zk (1 + zk

zk;

zk + zk; i(

zk zk); 1 zk

zk) ;

q = !

1 + z

z (1 + z

z;

z + z; i(

z z); 1 z

z) ;

(6.7)

= 1p2(z; 1; i; z):

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Rewriting the soft formula (6.5) in terms of the variables in (6.6) and (6.7), de ning ^

S(1)zz

@zX@zX S(1) , and acting with D2z gives [21]

Im Z

duD2zCzz Z

dvD2zCzz

hD2z ^

S(1)zz D2z^

S(1)zzi: (6.8)

The formulae for the angular momentum and stress energy of a particle emerging at zk in

the appendix enables this to be rewritten:

Im

Z

duD2zCzz Z

dvD2zCzz

=

8

= 8G

Xk zkzkIm

Luz(zk)@zkG(zk; z) +i 2hk@zk@zkG(zk; z)

(6.9)

where total angular momentum conservation has been used. To summarize, the subleading soft graviton theorem [18], after some di erentiation, change of notation and Fourier transform, becomes the formula for the contour integrals characterizing spin memory.

7 An in nity of conserved charges

A conserved charge enables one to determine the outcome of a measurement on I+ from

a measurement on I. For example, for a process which begins and ends in a vacuum,

the total integrated outgoing energy ux across I+ equals the incoming energy ux across I. In this section we describe an infinite set of I+ measurements | one for every

contour C | whose outcome is determined by a measurement on I.

Conserved charges on I+ may be obtained from any moment of the curl @[zNz] on I+.

Consider for example

Q(z;

z) = i@[zNz](z;

z)[notdef]I

+ : (7.1)

This can be written using the constraints as an integral over a null generator of I+. Inte

grating by parts then gives the I+ expression

Q (z;

z) = i

Z

du

=

1 Im

Z

d2w@ wG(w; z) Z

duTuw

Z

dvTvw

:

1 4@z@z

D2zCzz D2zCzz

@[zTz]u

: (7.2)

{ 10 {

On the other hand using the continuity condition (2.13) and integrating by parts, one nds the I expression

Q(z;

z) = i

Z

dv

1 4@z@z

D2zCzz D2zCzz

@[zTz]v

: (7.3)

Equating (7.2) and (7.3) gives the conservation law:

Z

du

1 4@z@z

D2zCzz D2zCzz

@[zTz]u

(7.4)

This conservation law equates the stress energy ux through and a zero mode of the metric uctuations along a null generator of I+ to the same quantities on the P T -conjugate

generator of I. There is one such law for every null generator.

The meaning of the conservation law (7.4) is a bit obscured by the fact that zero modes of metric uctuations are hard to measure. However, in the preceding we have found that the time delay e ectively measures a particular combination of the zero modes. Our main formula (5.9) can be rephrased as a conservation laws for the charge

QC

1 2LIm

=

Z

dv

1 4@z@z

D2zCzz D2zCzz

@[zTz]v

:

ZDCd2w w wZd2z@zG(z; w)Nz[notdef]I+ ; (7.5)

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which, as noted above, involves only the curl of Nz. Using (2.13) QC may be rewritten as

a I charge

QC

1 2LIm

ZDCd2w w wZd2z@zG(z; w)Nz[notdef]I+ : (7.6)

Integrating by parts, using the constraints to express (7.5) and (7.6) as integrals over I, and equating the two expressions yields

+u + 1

2LIm

ZDCd2w w wZd2z@zG(z; w) ZduTuz

= v + 1

2LIm

ZDCd2w w wZd2z@zG(z; w) ZdvTvz:(7.7)

Thus if we measure the component Tvz of the radiative stress-energy ux and the time delay on C at past null in nity, we can determine a moment of Tuz and the time

delay of an antipodally-located contour at future null in nity. There are in nitely many such conservation laws | one for every contour C | which in nitely constrain the scatte

ring process.

Should it persist to the quantum theory, this in nity of conservation laws has considerable implications for the black hole information puzzle. The output of the black hole evaporation process, as originally computed by Hawking, is constrained only by energy-momentum, angular momentum and charge conservation. Imposing the in nity of conservation laws (7.7) (together with a second in nity arising from BMS invariance [26]) will greatly constrain the outgoing Hawking radiation. These constraints follow solely from low-energy symmetry considerations, and do not invoke any microphysics. It would be interesting to understand how the semiclassical computation of black hole evaporation must be modi ed to remain consistent with these symmetries.

{ 11 {

Acknowledgments

We are grateful to D. Christodoulou, S. Gralla, T. He, D. Kapec and P. Mitra for useful conversations. This work was supported in part by NSF grant 1205550.

A Massless particle stress-energy tensor

We start with the trajectory of a massless point particle:

x() = p

E + b (A.1) where p is the particles four momentum, with pp = 0. b = (0; bi) = x(0) describes the impact parameter of the straight-line trajectory relative to the spacetime origin. The orbital angular momentum of this trajectory is:

L = bp pb ; (A.2)

which implies

b = 1E L0: (A.3)

The total angular momentum is

J = L + S (A.4)

where S is the intrinsic spin. The large behavior of the trajectory (A.1) is:

r() = + 1

E2 pLu + O(1)

u() =

1E2 pLu + O(1)

z() = p1 + ip2E + p3 +

JHEP12(2016)053

(A.5)

1E L zu + O(2)

where we have used L0 = Lu and 1

E L zu is O(1). The matter stress-energy tensor of

the point particle is [30]:

T M (y) = E Z

d _

x _

x 4(y x())

pg r Z

dS( _

x ) 4(y x())

pg

: (A.6)

Using Szz = ir2 zzh near a particle with helicity h, a collection of point particles obeys:

lim

r!1

r2T Muu =

XkEk (u uk) 2(z zk) zz

lim

r!1

(A.7)

r2T Muz =

Xk (u uk)

Luz(zk) i2hk@z 2(z zk) zz

where uk 1E2k pkLku and

Luz(zk) lim

r!1

1 r

@xk @uk

@x k

@zk Lk

= b1k(1

z2k) ib2k(1 +

z2k) 2b3k

zk

(1 + zk

zk)2 Ek:

(A.8)

{ 12 {

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/

Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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