Maxwell Equations and Riemann Geometry
A straightforward presentation of the minimal coupling principle, and its application to the classical electromagnetic theory. The modifications of the Maxwell equations induced by the space–time geometry are compared with those due to the presence of an
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Maxwell Equations and Riemann Geometry
If we adopt a model of space–time based on the Riemann geometry, we have to face the problem of how to transfer to such a generalized context the old, standard results of relativistic physics obtained in the context of the Minkowski geometry. The equivalence principle tell us that the equations of special relativity still hold in a suitable inertial chart, but only locally, over a space–time region of small enough (infinitesimal) extension (see Sect. 2.2). In order to be globally extended on a general Riemann manifold, such equations are be suitably generalized. The correct generalization procedure is provided by the so-called minimal coupling principle, which will be introduced in the next section and which will be applied, in this chapter, to the theory of the electromagnetic interactions. The validity of such a procedure is not restricted to the electromagnetic phenomena, however, and can be generally extended to all known physical systems and interactions. In the following chapters the minimal coupling principle will be indeed applied to many (and in largely different) physical situations.
4.1 The Minimal Coupling Principle The generalized relativity principle introduced in Chap. 2 imposes on the physical laws to respect an exact covariance property under the action of general coordinate transformations. If the physical system we are considering is described by equations which are already covariant in the context of a Minkowski geometric structure, then such a system can be easily embedded into a more general Riemann structure—namely, its equations can be lifted to a general-covariant form—by applying a standard procedure called “minimal coupling principle”. Such a procedure, in practice, amounts to the following set of operations: • replace the Minkowski metric with the Riemann metric, ημν → gμν , for all scalar products and all raising (and lowering) indices operations; M. Gasperini, Theory of Gravitational Interactions, Undergraduate Lecture Notes in Physics, DOI 10.1007/978-88-470-2691-9_4, © Springer-Verlag Italia 2013
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Maxwell Equations and Riemann Geometry
• replace all partial (and total) derivatives with the corresponding covariant derivatives, ∂μ → ∇μ ; √ • use the appropriate powers of −g to saturate to zero the weights of all tensor densities. In the action integral, in particular, the covariant measure is given by √ the prescription d 4 x → d 4 x −g. By applying the above procedure to the equations of motion or—better—to the action of the physical system, the system turns out to be “coupled” to the geometry of the given Riemann manifold. The coupling is “minimal”, in the sense that it depends only on the metric and on its first derivatives (the connection), and thus it can be locally neutralized in the limit in which g → η and Γ → 0 (in agreement with the principle of equivalence). A coupling procedure introducing higher-order derivatives of the metric would involve the space–time curvature (see Chap. 6), and it would be impossible to eliminate
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