Electromagnetic and gravitational waves: the third dimension

Plane electromagnetic and gravitational waves interact with particles in such a way as to cause them to oscillate not only in the transverse direction but also along the direction of propagation. The electromagnetic case is usually shown by use of the Hamilton-Jacobi equation and the gravitational by a transformation to a local inertial frame. Here, the covariant Lorentz force equation and the second-order equation of geodesic deviation followed by the introduction of a local inertial frame are, respectively, used. It is often said that there is an analogy between the motion of charged particles in the field of an electromagnetic wave and the motion of test particles in the field of a gravitational wave. This analogy is examined and found to be rather limited. It is also shown that a simple special relativistic relation leads to an integral of the motion, characteristic of plane waves, which is satisfied in both cases.