Differential operators in terms of Clebsch-Gordon (C-G) coefficients and the wave equation of massless tensor fields

Abstract

The quantum theory of angular momentum affords a treatment of tensors and vectors in a spherical basis. By using this theory we define the tensor differential operators: divergence, curl and gradient which act on a tensor of any rank, in terms of C-G coefficients. With these definitions we obtain a matrix representation and useful properties for those operators. An interesting application of this formalism is to find the wave equation of a tensor of any rank in a linear theory. This provides a new common way to look at the wave equations associated with both Maxwell's equations and the Maxwell-like equations for the linearized Weyl curvature tensor in gravitoelectromagnetism describing gravitational radiation on a Minkowski spacetime background.

On a Lagrangian formulation of gravitoelectromagnetism

“…Note that ϕ is the GEM vector counterpart of the EM scalar potential φ. As forÃ, we observe here that curl grad ϕ = 0 [40,46,47], since curl grad ϕ = 1 2 grad curl ϕ.…”

“…Henceforth, we work with Gaussian units, c is the speed of light in vacuum, and the metric of flat Minkowski space is given by η μν , is (−1, +1, +1, +1). In the presence of sources [40], the Maxwell-like equations become…”

“…Modern approaches to this analogy and some applications are described in [32][33][34][35][36][37][38]. This description of gravity in analogy with electromagnetism has also been studied with group theoretical methods [39,40]. An extensive list of references is provided in [41].…”

On a group-theoretical approach to the curl operator

“…The matrix elements which represent the differential operators are given by (see Ref. [14] and appendix) i) Divergence:…”

“…This paper is organized as follows. In Section 2, we recall [14] the matrix representations of various differential operators (divergence, gradient and curl) which were obtained by using the Clebsch-Gordan coefficients, and display some of their properties. In Section 3, we give a new definition and physical motivation of the curl operator.…”