Abstract
First, we remind that derivation of continuum physics equations, namely the formulation of constitutive laws and conservation laws with respect to a given spacetime requires the identification of physical measurable quantities with geometrical variables (metric, torsion, and curvature on the material manifold).
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- 1.
Again, \(\hat {\nabla }_\mu \) denotes the four-dimensional connection in the Minkowski spacetime.
- 2.
Hodge star operator The Hodge star operator is the unique linear map on a semi-Riemannian manifold, say \(\mathcal {M}\), from r-forms to n − r-forms defined by: \( \star : \varOmega ^r (\mathcal {M}) \to \varOmega ^{n-r} (\mathcal {M}) \), such that for all \((\omega , \omega ') \in \varOmega ^r (\mathcal {M})\), we have ω ∧ ω∗′ := 〈ω, ω′〉 where 〈, 〉 is an interior product on \(\mathcal {M}\) e.g. Nakahara (1996).
- 3.
In the remaining part of this subsection the symbol \(\overline {\nabla }\) represents the Levi-Civita connection associated to the nonuniform metric tensor of the continuum manifold.
- 4.
Care should be taken since suggesting the coordinate dependence \(e^{\kappa _\sigma x^\sigma }\) implicitly assumes that time and space are separated (method of variable separation). The covariance of the formulation is expected to allow us to derive the dispersion equation in any inertial frame in the framework of special relativity.
- 5.
Physics background: This continuum version is the extension of Lagrangian for particles within Minkowskian spacetime. For physical particles with relativistic speeds, the action of a charged particle e moving within such a spacetime with electromagnetic field A μ takes the form of e.g. Kovetz (2000):
$$\displaystyle \begin{aligned} &\mathcal{S} := \int \left[ - mc^2 - e \ u^\mu A\mu \right] d\tau, \quad d\tau := dt \sqrt{1 - v^2 / c^2}, \quad u^\mu : (1, v^i) \left( 1 - v^2 / c^2\right)^{-1},\\ &A_\mu : (- \phi, A_i) \end{aligned} $$where the Lagrangian function in terms of three dimensional variables (integration with respect to dt), with its Euler–Lagrange equation (Heaviside-Lorentz equation) hold:
(6.82)modelling the charged particle e motion under given electromagnetic field (E, B).
- 6.
In astrophysics, most planets as stars being electrically neutral, the Reissner-Nordström may be considered as only an academic exercise, although interesting, rather than a realistic and relevant field of gravitation.
- 7.
For instance in the equation G 00 = 0, we define U(r) := e −2μ(r) to arrive to the differential equation U′(r) + U(r)∕r = 1∕r. We remark that U(r) := 1 is a particular solution of the non homogeneous equation. The second solution may be found by a change of variable V (r) := U(r) − 1.
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R. Rakotomanana, L. (2018). Topics in Gravitation and Electromagnetism. In: Covariance and Gauge Invariance in Continuum Physics. Progress in Mathematical Physics, vol 73. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-91782-5_6
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