New gravitational memories

Journal of High Energy Physics, Dec 2016

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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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)


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Sabrina Pasterski, Andrew Strominger, Alexander Zhiboedov. New gravitational memories, Journal of High Energy Physics, 2016, pp. 53, Volume 2016, Issue 12, DOI: 10.1007/JHEP12(2016)053