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General Relativity and Quantum Cosmology, gr-qc,Mathematical Physics, math-ph,Mathematics, Mathematical Physics, math.MP
Abstract:
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.
