Gravitomagnetic and Gravitoelectric Waves in General Relativity

Abstract

Lower-order terms in expansions of the equations of General Relativity in powers of v/c (post-Newtonian approximations) have long been a source of analogies with em theory. A classic textbook example is the steadily spinning sphere generating a constant dipole gravitomagnetic field, with its associated vector potential B^* ₀ =∇ × ζ (analog of the magnetic field B of a spinning charged sphere). In the nonsteady case there are associated gravitoelectric fields E* = − ζ_t − ∇φ* also, where φ* is the gravitational Coulomb potential. The case of a rigid sphere spun up from rest by an external (nongravitational) torque at t = 0 is enlightening, as it demonstrates the generation of B* and E* wave fields propagating outward with the velocity of light c: for large t, B* → B^* ₀. In a coordinate system for which the metric tensor is nearly equal to the Minkowski tensor, the three-vector potential ζ obeys an equation isomorphic to the electrodynamic equation, that is, □²ζ = −μ*j* with j* = −ρv, where ρ is the mass density, v the three-velocity, and μ* = 16πGc⁻² = 3.7 × 10⁻²⁶ mksu, G being the gravitational constant. Significantly, one can construct a gauge invariant four-vector potential F* = (ic⁻¹4φ*, ζ), obeying field equations isomorphic to Maxwell's in the Lorentz gauge F^α _,α = 0. The traveling transient dipole field exerts torques on matter in its path, setting up shear strains that may be measurable for very large momentum transfers, for example, between massive astronomical bodies. A rough calculation suggests that such strains are in principle observable.