Field Equations and Lagrangian for the Kaluza Metric Evaluated with Tensor Algebra Software

Abstract

This paper calculates the Kaluza field equations with the aid of a computer package for tensor algebra, xAct. The xAct file is provided with this paper. We find that Thiry's field equations are correct, but only under limited circumstances. The full five-dimensional field equations under the cylinder condition are provided here, and we see that most of the other references miss at least some terms from them. We go on to establish the remarkable Kaluza Lagrangian, and verify that the field equations calculated from it match those calculated with xAct, thereby demonstrating self-consistency of these results. Many of these results can be found scattered throughout the literature, and we provide some pointers for historical purposes. But our intent is to provide a definitive exposition of the field equations of the classical, five-dimensional metric ansatz of Kaluza, along with the computer algebra data file to verify them, and then to recover the unique Lagrangian for the theory. In common terms, the Kaluza theory is an " = 0" scalar field theory, but with unique electrodynamic couplings.

Field Equations and Lagrangian of the Kaluza Energy-Momentum Tensor

“…To resolve apparent contradictions among expressions for the field equations found in the English language literature,G ab was evaluated using tensor algebra software [10], and the associated Kaluza Lagrangian was established from the 5D Hilbert Lagrangian density to be…”

“…where R μν is the Ricci tensor, F μν ≡ ∂ μ A ν − ∂ ν A μ is the electromagnetic field strength tensor, g is the determinant of g μν , and the gravitational field equations are obtained from variation with respect to g μν . A review of the literature based on [10] shows that few authors recover this Lagrangian, and the literature is littered with algebraic errors in either the 5D curvature tensorR ab or the 5D Einstein tensorG ab . Only Ref.…”

Geometrical unification of gravitation and electromagnetism

“…The electric charge conservation equationQ = 0, and the particles equations of motion (11) and (12) may be derived by using the (matrix) Lagrangian (13) which are written in terms of the Pauli matrices, by varying with respect to φ and x α .…”

Electrodynamics and Time Orientability

“…And we can calculate the Einstein tensor in five dimensions directly from the metric [11]. We use Roman a,b etc for indices {t,x,y,z,5} and Greek μν for the spacetime indices {t,x,y,z} :…”

“…After all,"no interaction with matter" seems to be a plausible theoretical outcome. Implicit in this approach is to equate the constant e in (22) with the charge of a particle that appears in (11). The identity of the two is fundamental to classical electromagnetism, but is absent in these approaches to deriving the Lorentz force law.…”