The not-so-nonlinear nonlinearity of Einstein's equation

Many of the technical complications associated with the general theory of relativity ultimately stem from the nonlinearity of Einstein's equation. It is shown here that an appropriate choice of dynamical variables may be used to eliminate all such nonlinearities beyond a particular order: Both Landau-Lifshitz and tetrad formulations of Einstein's equation are obtained which involve only finite products of the unknowns and their derivatives. Considerable additional simplifications arise in physically-interesting cases where metrics becomes approximately Kerr or, e.g., plane waves, suggesting that the variables described here can be used to efficiently reformulate perturbation theory in a variety of contexts. In all cases, these variables are shown to have simple geometrical interpretations which directly relate the local causal structure associated with the metric of interest to the causal structure associated with a prescribed background. A new method to search for exact solutions is outlined as well.Comment: 5 pages, added some additional details, accepted by PR

Optics in a nonlinear gravitational plane wave

Gravitational waves can act like gravitational lenses, affecting the observed positions, brightnesses, and redshifts of distant objects. Exact expressions for such effects are derived here in general relativity, allowing for arbitrarily-moving sources and observers in the presence of plane-symmetric gravitational waves. At least for freely falling sources and observers, it is shown that the commonly-used predictions of linear perturbation theory can be generically overshadowed by nonlinear effects; even for very weak gravitational waves, higher-order perturbative corrections involve secularly-growing terms which cannot necessarily be neglected when considering observations of sufficiently distant sources. Even on more moderate scales where linear effects remain at least marginally dominant, nonlinear corrections are qualitatively different from their linear counterparts. There is a sense in which they can, for example, mimic the existence of a third type of gravitational wave polarization.Comment: 32 pages, minor additional explanation

Tails of plane wave spacetimes: Wave-wave scattering in general relativity

One of the most important characteristics of light in flat spacetime is that it satisfies Huygens' principle: Initial data for the vacuum Maxwell equations evolves sharply along null (and not timelike) geodesics. In flat spacetime, there are no tails which linger behind expanding wavefronts. Tails generically do exist, however, if the background spacetime is curved. The only non-flat vacuum geometries where electromagnetic fields satisfy Huygens' principle are known to be those associated with gravitational plane waves. This paper investigates whether perturbations to the plane wave geometry itself also propagate without tails. First-order perturbations to all locally-constructed curvature scalars are indeed found to satisfy Huygens' principles. Despite this, gravitational tails do exist. Locally, they can only perturb one plane wave spacetime into another plane wave spacetime. A weak localized beam of gravitational radiation passing through an arbitrarily-strong plane wave therefore leaves behind only a slight perturbation to the waveform of the background plane wave. The planar symmetry of that wave cannot be disturbed by any linear tail. These results are obtained by first deriving the retarded Green function for Lorenz-gauge metric perturbations and then analyzing its consequences for generic initial-value problems.Comment: 13 pages, 1 figure, minor typos correcte

Effective stress-energy tensors, self-force, and broken symmetry

Deriving the motion of a compact mass or charge can be complicated by the presence of large self-fields. Simplifications are known to arise when these fields are split into two parts in the so-called Detweiler-Whiting decomposition. One component satisfies vacuum field equations, while the other does not. The force and torque exerted by the (often ignored) inhomogeneous "S-type" portion is analyzed here for extended scalar charges in curved spacetimes. If the geometry is sufficiently smooth, it is found to introduce effective shifts in all multipole moments of the body's stress-energy tensor. This greatly expands the validity of statements that the homogeneous R field determines the self-force and self-torque up to renormalization effects. The forces and torques exerted by the S field directly measure the degree to which a spacetime fails to admit Killing vectors inside the body. A number of mathematical results related to the use of generalized Killing fields are therefore derived, and may be of wider interest. As an example of their application, the effective shift in the quadrupole moment of a charge's stress-energy tensor is explicitly computed to lowest nontrivial order.Comment: 22 pages, fixed typos and simplified discussio