The Asymptotic Behavior of Massless Fields and the Memory Effect*

We investigate the behavior of perturbations of in $d \geq 4$ Minkowski spacetime  (in both even and odd dimensions) near null infinity in full, nonlinear General Relativity under the assumption that the perturbations admit a suitable expansion in $1/r$. We explicitly obtain the recursion relations on the coefficients of the $1/r$ expansion implied by the field equations (in Harmonic gauge) as well as the ``constraints''. We then consider the memory effect in fully nonlinear general relativity.  We show that in even dimensions, the memory first arises at Coulombic order---i.e., order $1/r^{d-3}$---and can naturally be decomposed into ``null memory'' and ``ordinary memory.'' In odd dimensions, the memory effect vanishes near null infinity at Coulombic order and slower fall-off. The 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. 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.