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New gravitational memories
Received: November
gravitational memories
Sabrina Pasterski 0 1
Andrew Strominger 0 1
Alexander Zhiboedov 0 1
0 ux. It has recently been shown
1 Center for the Fundamental Laws of Nature, Harvard University
The conventional gravitational memory e ect is a relative displacement in the position of two detectors induced by radiative energy ux. We tational `spin memory' in which beams on clockwise and counterclockwise orbits acquire a relative delay induced by radiative angular momentum 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.
Classical Theories of Gravity; Gauge Symmetry; Space-Time Symmetries
Contents
1 Introduction 2 3 4
Asymptotically
at metrics
Displacement memory e ect
Spin memory e ect
Spin memory and angular momentum
Equivalence to subleading soft theorem
An in nity of conserved charges
Massless particle stress-energy tensor
Introduction
The initial and
nal spacetime geometries, although both
at, di er by a BMS
super
Since the initial and
nal metrics di er, the Fourier transform in time must have a
displacement memory formulae was demonstrated in [14].
that governs not the pole but the
nite piece in the expansion of soft graviton
scatteBMS supertranslations.
sphere [23, 24].
ux. Accordingly we call it spin memory. The spin memory e ect provides a
Instead, we consider light rays which repeatedly orbit (with the help of
ber optics or
memory e ect. It is a new kind of gravitational memory.
boundary conditions to relate this new memory e ect to angular momentum
ux.
Secangular momentum of spinning particles on null geodesics.
Asymptotically
at metrics
dinates takes the form2
The expansion of an asymptotically at spacetime metric near I
ds2 =
2dudr + 2r2 zzdzdz + 2
rCzzdz2 + DzCzzdudz +
4 @z(CzzCzz) dudz + c:c: + : : :
+ in retarded Bondi
coorwhere u = t
of (u; z; z), not r. They are related by the I
+ constraint equations Guu = 8 GTuMu
@umB =
Dz2N zz + Dz2N zz
NzzN zz + 4 G lim r2TuMu ;
4Nz = 4NzBT + CzzDzCzz + 34 @z(CzzCzz).
and Guz = 8 GTuMz
@uNz =
8 G lim r2TuMz
Dz [CzzN zz]
2 CzzDzN zz;
the total energy (angular momentum)
momentum aspect is related to the Weyl tensor component
ux through a given point on I+. The angular
We also note that
Nz = lim r3Czrru:
Im 20 = Im lim r zzCuzzr =
4 CzzN zz :
Most of our discussion will concern I+, but the metric expansion near I
ds2 =
dv2 + 2dvdr + 2r2 zzdzdz + 2
rCzzdz2 + DzCzzdvdz +
4 @z(CzzCzz) dvdz + c:c: + : : :
where here the metric perturbations are functions of (v; z; z). The z coordinate on I
@vmB =
Dz2N zz + Dz2N zz + Tvv;
NzzN zz + 4 G lim r2TvMv ;
and Gvz = 8 GTvMz
@vNz =
8 G lim r2TvMz
Dz2Czz + @zmB + Tvz;
Dz [CzzN zz]
2 CzzDzN zz:
We use the symbol I
+ (I++) to denote the past and future S2 boundaries of I+, and
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
NzjI++ = NzjI
= mBjI++ = mBjI
= 0:
Moreover near all the boundaries I
the radiative modes are unexcited:
of I, the geometry is in the vacuum in the sense that
= Im 20jI
= 0:
More precisely, following Christodoulou and Klainerman [25], we take Nzz
(jvj 3=2) on I
I ) as well as a corresponding fallo
of the stress tensor.
(see e.g [26])
These conditions do not imply CzzjI
Czz =
ral relativity.
continuity conditions on mB and Czz near where I
PT and BMS-invariant choice is simply
meet. The unique Lorentz,
u 3=2
CzzjI+ = CzzjI+ ;
mBjI+ = mBjI+ :
and (2.10) that the divergent term is exact: Nz
u@zmB. Fortunately for us, such exact
for the curl of Nz. (2.10), (2.12) and the Bianchi identity imply
= @[zNz]jI+ :
more details). De ning
+Czz = CzzjI++
+mB = mBjI++
mBjI+ =
and using (2.2) one nds
Dz2 +Czz = 2
+C which produces such a
+Czz is obtained by inverting Dz2Dz2:
+C(z; z) =
d2w wwG(z; w)
du Tuu(w) +
where the Green's function is given by
G(z; w) =
(1 + ww)(1 + zz)
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 =
one arrives at the simple relation
C(z; z) =
d2w wwG(z; w)
du Tuu(w)
dv Tvv(w) :
Displacement memory e ect
universal formula [1{11], which we now review brie y.
u, t @ = @u and
It follows that
@2s = R
Rzuzu =
+sz =
Z( ) = z0 1 +
Integrating twice one nds, to leading order in 1r , a net change in the displacement
+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.
Spin memory e ect
This is described by
period. To this order, only the term in square brackets in (4.2) is odd under @uZ !
they return to
P =
DzCzzdz + DzCzzdz :
This formula in fact applies to any contour C, circular or (...truncated)