Gravito-electromagnetism

Abstract

Abstract.We develop and apply a fully covariant 1 + 3 electromagnetic analogy for gravity. The free gravitational field is covariantly characterized by the Weyl gravito-electric and gravito-magnetic spatial tensor fields, whose dynamical equations are the Bianchi identities. Using a covariant generalization of spatial vector algebra and calculus to spatial tensor fields, we exhibit the covariant analogy between the tensor Bianchi equations and the vector Maxwell equations. We identify gravitational source terms, couplings and potentials with and without electromagnetic analogues. The nonlinear vacuum Bianchi equations are shown to be invariant under covariant spatial duality rotation of the gravito-electric and gravito-magnetic tensor fields. We construct the super-energy density and super-Poynting vector of the gravitational field as natural U (1) group invariants, and derive their super-energy conservation equation. A covariant approach to gravito-electric/magnetic monopoles is also presented.

Microwave background anisotropies from gravitational waves: the 1 + 3 covariant approach

“…Making use of the linearised identities [39], valid for an arbitrary first-order PSTF tensor S ab , 10) it is straightforward to verify that the constraints in equations (3.1) and (3.2) are consistent with the linearised propagation equations (3.3-3.7). Inhomogeneous wave equations for the shear and the magnetic part of the Weyl tensor follow from differentiating equations (3.5) and (3.7) along the flow lines of u a :…”

“…Indeed, the non-vanishing of the super-energy flux [39] P a ≡ ǫ ac d E bc H bd for any timelike u a forms the basis of Bel's first condition for the existence of gravitational radiation in an (exact) solution of the field equations (see e.g. [40]).…”

Complete classification of purely magnetic, nonrotating, nonaccelerating perfect fluids

“…For covariant differential operations orthogonal to u a , the streamlined notation of [4,21,45] is the most transparent. The covariant spatial derivative D and the associated curl and divergence (div) operators, acting on vectors and 2-tensors, are defined as:…”

“…Here and in general, a dot denotes covariant ('time') differentiation along u a . For scalar functions f and vectors v a one has [4,45] …”

“…They were first introduced (for the vacuum Riemann tensor) by Matte [1] when searching for gravitational quantities playing an analogous role to the electric and magnetic field in classical electromagnetism. Using the decomposition (1), the Bianchi identities take a form analogous to Maxwell's equations for the electromagnetic field [4]. A non-conformally flat spacetime for which E ab , resp.…”