Asymptotic behavior of massless fields and the memory effect

Abstract

We investigate the behavior of massless scalar, electromagnetic, and linearized gravitational perturbations near null infinity in $d\ensuremath{\ge}4$ dimensional Minkowski spacetime (of both even and odd dimension) under the assumption that these fields admit a suitable expansion in $1/r$. For even $d$ with $d>4$, our $1/r$ expansion ansatz is equivalent to smoothness at ${\mathcal{I}}^{+}$, whereas for $d=4$ it is slightly weaker, so all solutions that are smooth at ${\mathcal{I}}^{+}$ are encompassed by our analysis. We also analyze the solutions to the full nonlinear Einstein equation in $d\ensuremath{\ge}4$ dimensions near null infinity, assuming a similar $1/r$ expansion. We show that for $d>4$ the Lorenz gauge condition can be imposed for electromagnetic and gravitational perturbations in a manner compatible with our assumed $1/r$ expansion. However, for $d=4$ the Lorenz gauge can be imposed if and only if there is no flux of charge-current (in the electromagnetic case) or stress-energy (in the linearized gravitational case) to null infinity. Similarly, in the nonlinear gravitational case, the harmonic gauge condition can be imposed for $d>4$ but cannot be imposed for $d=4$ if there either is a flux of stress-energy at null infinity or if the Bondi news is nonvanishing. We explicitly obtain the recursion relations on the coefficients of the $1/r$ expansion implied by the wave equation as well as the ``constraints'' in the electromagnetic and gravitational cases arising from the Lorenz/harmonic gauge condition. We also characterize the ``free data'' needed to determine a solution. We then consider the memory effect in fully nonlinear general relativity, i.e., the permanent displacement of test particles near null infinity following a burst of gravitational radiation. We show that in even dimensions, the memory effect first arises at Coulombic order---i.e., order $1/{r}^{d\ensuremath{-}3}$---and can naturally be decomposed into ``null memory'' and ``ordinary memory.'' Null memory is associated with an energy flux to null infinity. We show that ordinary memory is associated with the metric failing to be stationary at one order faster fall-off than Coulombic in the past and/or future, as will typically be the case if matter (on timelike inertial trajectories) comes in or goes out to infinity. In odd dimensions, we show that the total memory effect at Coulombic order and slower fall-off always vanishes. It is easily seen that null memory is always of ``scalar type'' with regard to its behavior on spheres, but the ordinary memory can be of any (i.e., scalar, vector, or tensor) type. In 4-spacetime dimensions, we give an explicit example in linearized gravity of an expanding shell with vector stresses which gives rise to a nontrivial vector (i.e., magnetic parity) ordinary memory effect at order $1/r$. We show that scalar memory is described by a diffeomorphism, which is an asymptotic symmetry (a supertranslation) in $d=4$ and a gauge transformation for $d>4$. Vector and tensor memory cannot be described by diffeomorphisms. In $d=4$ dimensions, we show that there is a close relationship between memory and the charge and flux expressions associated with supertranslations. Similar formulas are given in higher dimensions. We analyze the behavior of solutions that are stationary at Coulombic order and show how these suggest ``antipodal matching'' between future and past null infinity, which gives rise to conservation laws. The relationship between memory and infrared divergences of the ``out'' state in quantum gravity is analyzed, and the nature of the ``soft theorems'' is explained.