![]() ![]() ![]() ![]() For these reasons, it may be useful to think of Maxwell's equations in Minkowski space as a special case of the general formulation. For example, the equations in this article can be used to write Maxwell's equations in spherical coordinates. Also, the same modifications are made to the equations of flat Minkowski space when using local coordinates that are not rectilinear. The electromagnetic field admits a coordinate-independent geometric description, and Maxwell's equations expressed in terms of these geometric objects are the same in any spacetime, curved or not. Without this distinction, they are sometimes called the "microscopic" Maxwell's equations for contrast. When the distinction is made, they are called the macroscopic Maxwell's equations. When working in the presence of bulk matter, distinguishing between free and bound electric charges may facilitate analysis. But because general relativity dictates that the presence of electromagnetic fields (or energy/ matter in general) induce curvature in spacetime, Maxwell's equations in flat spacetime should be viewed as a convenient approximation. By contrast, the conservation-law weak form of the Vlasov equation in cylindrical phase space coordinates is largely unexplored, and to the authors knowledge. These equations can be viewed as a generalization of the vacuum Maxwell's equations which are normally formulated in the local coordinates of flat spacetime. In physics, Maxwell's equations in curved spacetime govern the dynamics of the electromagnetic field in curved spacetime (where the metric may not be the Minkowski metric) or where one uses an arbitrary (not necessarily Cartesian) coordinate system. ![]()
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