New gravitational memories

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.


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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 flux through null infinity (I), the new memory is sourced by moments of the angular momentum flux. Accordingly we call it spin memory. The spin memory effect 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 fiber optics or mirrors) clockwise or counterclockwise around a fixed 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 flux through infinity. The relative time delay for counter-orbiting light rays is the spin memory effect. It is a new kind of gravitational memory.
This paper is organized as follows. Section 2 outlines the metric and constraint equations for asymptotically flat spacetimes. Section 3 reviews the displacement memory effect. Section 4 introduces the spin memory effect. Section 5 uses the constraint equations and boundary conditions to relate this new memory effect to angular momentum flux. 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 infinite family of conserved charges associated to the infinite 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.

Asymptotically flat metrics
The expansion of an asymptotically flat spacetime metric near I + in retarded Bondi coordinates takes the form 2 where u = t − r, γ zz = 2 (1+zz) 2 is the unit metric on S 2 (used to raise and lower z andz indices), D z is the γ-covariant derivative, and subleading terms are suppressed by powers of r. The Bondi mass aspect m B , the angular momentum aspect N z , and C zz are functions of (u, z,z), not r. They are related by the

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where N zz = ∂ u C zz is the Bondi news, T M is the matter stress tensor and T uu (T uz ) is the total energy (angular momentum) flux through a given point on I + . The angular momentum aspect is related to the Weyl tensor component Ψ 0 1 on I + by We also note that Most of our discussion will concern I + , but the metric expansion near I − in the retarded Bondi coordinate v = t + r is where here the metric perturbations are functions of (v, z,z). The z coordinate on I − is defined 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 and (2.8) We use the symbol I + − (I + + ) to denote the past and future S 2 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 (2.9)

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Moreover near all the boundaries I ± ± of I, the geometry is in the vacuum in the sense that the radiative modes are unexcited: More precisely, following Christodoulou and Klainerman [25], we take N zz ∼ |u| −3/2 (|v| −3/2 ) on I + (I − ) as well as a corresponding falloff of the stress tensor. These conditions do not imply C zz | I ± ± = 0. Rather, the general solution of (2.10) is (see e.g [26]) 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 define gravitational scattering one must specify boundary or continuity conditions on m B and C zz near where I + and I − meet. The unique Lorentz, PT and BMS-invariant choice is simply (2.12) In this paper N z 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, N z may grow linearly with u or v near I ± ± . It follows from (2.3) and (2.10) that the divergent term is exact: N z ∼ u∂ z m B . Fortunately for us, such exact terms are irrelevant for our purposes (see section 5). We will need a continuity condition for the curl of N z . (2.10), (2.12) and the Bianchi identity imply (2.13) We are interested in the difference between the initial and final functions C in (2.11) on I. This can be determined by integrating the constraint (2.2) as follows (see [14] for more details). Defining and using (2.2) one finds The ∆ + C which produces such a ∆ + C zz is obtained by inverting D 2 z D 2 z : where the Green's function is given by . (2.17)

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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 defining (2.18) one arrives at the simple relation

Displacement memory effect
In this section, we briefly review the standard gravitational memory effect. The passage of a finite pulse of radiation or other form of energy through a region of spacetime produces a gravitational field which moves inertial 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][2][3][4][5][6][7][8][9][10][11], which we now review briefly. Consider two nearby inertial detectors with proper worldline tangent vectors t µ and relative displacement vector s µ . We take the worldlines to be at large r and extend for infinite retarded time near I + . s µ evolves according to the geodesic deviation equation where τ is the detector's proper time. At large r in the geometry (2.1), we may approximate τ ∼ u, t λ ∂ λ = ∂ u and It follows that Integrating twice one finds, to leading order in 1 r , a net change in the displacement where ∆ + C zz is given in term of moments of the asymptotic energy flux by the second derivative of (2.16). (3.4) is the standard displacement memory formula.

Spin memory effect
This section describes the new spin memory effect, which affects 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 z 0 on a sphere of large fixed r = r 0 , where L r 0 . This is described by

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where φ ∼ φ + 2π. A light ray in either a clockwise or counterclockwise orbit (aided by mirrors or fiber optics) along C starting at φ(0) = 0, has a trajectory φ(u) that obeys where it is here and hereafter assumed that C zz does not change significantly over a single period. To this order, only the term in square brackets in (4.2) is odd under ∂ u Z → −∂ u Z.
If two light rays are simultaneously set in orbit in opposite directions, the times at which they return to φ = 0 3 will differ by the u-integral of this odd term This formula in fact applies to any contour C, circular or not. At first glance, it appears from (4.3) that the returns of the counter-orbiting light rays are desynchronized, even in the vacuum, as long as C zz is nonzero. In fact this is not the case. For C zz of the vacuum form (2.11), one readily finds that Hence desynchronization occurs only during the passage of radiation through I + . The total relative time delay, integrated over all orbits is This leads to a shift in the interference pattern between counter-orbiting light pulses. The shift is an infrared effect proportional to the u-zero mode of C zz . This is the spin memory effect.

Spin memory and angular momentum flux
Displacement memory (3.4) can be expressed as an integral of the net local asymptotic energy flux convoluted with the Green's 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 flux. Taking ∂z of the G uz constraint in (2.3) and ∂ z of the complex conjugate G uz constraint gives:

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Multiplying (5.2) by the Green's function and integrating over d 2 z gives: Note that the right hand side of (5.5) is invariant under shifts N z → N z + ∂ z X for any real X, so only the curl part of N z contributes. Integrating both sides over the disk D C whose boundary is C and using Stokes' theorem, (5.5) leads to Multiplying by 1 2π 2 L and integrating over u then yields is the shift in the angular momentum aspect. In many applications -for example geometries which are initially asymptotically Schwarzschild through subleading order -this term will vanish. Moreover, if ∆ + N z is exact, i.e. ∆ + N z = ∂ z X for any real X, no contribution to the imaginary part appears in (5.7). Hence (5.7) depends only on the curl of ∆ + N z .
A similar analysis near I − leads to the formula where the contour C and disk D C on I − is defined by the curve defined 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 N z one finds, in analogy to (2.19), an expression relating time delays and fluxes The right hand side of (5.9) is related to the local angular momentum flux through I. Consider the case when T uz arises from massless particles or localized wave packets which puncture I + at points (u k , z k ). Then, as we show in the appendix,

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together with a similar formula for T vz . Here L uz (z k ), (h k ) is the orbital (spin) angular momentum of the k th particle associated to a rotation around (boost towards) the point z k on the sphere. The leading contribution from such particles to the time delay is The second term in (5.11) has the simple interpretation that an object of spin h k passing through C at (or near) lightspeed induces a time delay of order h k L , with no factors of r 0 . This can be understood as the frame-dragging effect. If C lies a distance of order L from z k , the first term is typically a factor of L r 0 smaller than the second. Another interesting case is outgoing quadrupole radiation, with no incoming news or T vz on I − . 4 The displacement memory effect for configurations of this type is of potential astrophysical interest and was analyzed in [6]. This is described by the news tensor on I + for some N (u). As an example, take i = j, Y z i = z and The resulting angular momentum flux obeys so that Im du d 2 z∂zG(z; w)T uz = 3πα 2 ω 1 30 − w 2w2 (1 + ww) 4 , (5.15) using (5.4). The quadrupole contribution to the time delay around a contour C becomes .
For typical choices of C such that the area of D C is order L 2 /r 2 0 , we have ∆ + u ∼ α 2 L r 2 0 .

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 ω of any external graviton is taken to zero. Recently [18-21, 23, 24, 27, 28] it has been shown that the finite, subleading term in the ω → 0 expansion also exhibits universal behavior. The coefficient 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.

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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] A n+1 (p 1 , . . . p n ; (ωq, µν )) = S (1) µν µν A n (p 1 , . . . p n ) , and κ = √ 32πG. In this expression, the parentheses denote (µ, ν) symmetrization, q = (ω, ωq) withq 2 = 1 is the four-momentum, and µν is the transverse-traceless polarization tensor of the graviton. We define incoming particles to have negative p 0 and take ω positive for an outgoing graviton. µ, ν indices refer to asymptotically Minkowskian coordinates given in terms of retarded coordinates as 3) or in terms of advance coordinates as (6.4) The symmetrized limit in (6.1) projects out the leading Weinberg pole, leaving the subleading finite term of interest here. The linearized expectation value of the asymptotic metric fluctuation produced in the n-particle scattering process obeys the semiclassical momentum space formula In the last line, and in similar expressions below, an expectation value of the expression involving the differential operator J kνλ acting in the matrix element in A n is implicit. Expression (6.5) characterizes linearized fields by their momenta whereas the new memory formula (5.9) is given in terms of I values of fields. These are simply related. Using the large-r stationary phase approximation as in [21,29] duC zz (u,q) − dvC zz (v,q) = − lim

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On the other hand using the continuity condition (2.13) and integrating by parts, one finds the I − expression Equating (7.2) and (7.3) gives the conservation law: This conservation law equates the stress energy flux through and a zero mode of the metric fluctuations 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 fluctuations are hard to measure. However, in the preceding we have found that the time delay effectively measures a particular combination of the zero modes. Our main formula (5.9) can be rephrased as a conservation laws for the charge which, as noted above, involves only the curl of N z . Using (2.13) Q C may be rewritten as a I − charge Integrating by parts, using the constraints to express (7.5) and (7.6) as integrals over I ± , and equating the two expressions yields Thus if we measure the component T vz of the radiative stress-energy flux and the time delay on C at past null infinity, we can determine a moment of T uz and the time delay of an antipodally-located contour at future null infinity. There are infinitely many such conservation laws -one for every contour C -which infinitely constrain the scattering process.
Should it persist to the quantum theory, this infinity 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 energymomentum, angular momentum and charge conservation. Imposing the infinity of conservation laws (7.7) (together with a second infinity 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 modified to remain consistent with these symmetries.

A Massless particle stress-energy tensor
We start with the trajectory of a massless point particle: where p µ is the particle's four momentum, with p µ p µ = 0. b µ = (0, b i ) = x µ (0) describes the impact parameter of the straight-line trajectory relative to the spacetime origin. The orbital angular momentum of this trajectory is: The total angular momentum is where S µν is the intrinsic spin. The large τ behavior of the trajectory (A.1) is: where we have used L 0µ = L uµ and 1 E L z u is O(τ −1 ). The matter stress-energy tensor of the point particle is [30]: T M µν (y ρ ) = E dτẋ µẋν δ 4 (y ρ − x ρ (τ )) √ −g − ∇ ρ dτ S ρ(µẋν) δ 4 (y ρ − x ρ (τ )) √ −g . (A.6) Using S zz = ir 2 γ zz h near a particle with helicity h, a collection of point particles obeys: where u k ≡ − 1 (1 + z kzk ) 2 E k . (A.8)

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