Electromagnetic Wave Equation
The nonhomogeneous electromagnetic wave equation in terms of the field tensor is modified from the special relativity form to
where Racbd is the covariant form of the Riemann tensor and is a generalization of the d'Alembertian operator for covariant derivatives. Using
Maxwell's source equations can be written in terms of the 4-potential as,
or, assuming the generalization of the Lorenz gauge in curved spacetime
where is the Ricci curvature tensor.
This the same form of the wave equation as in flat spacetime, except that the derivatives are replaced by covariant derivatives and there is an additional term proportional to the curvature. The wave equation in this form also bears some resemblance to the Lorentz force in curved spacetime where Aa plays the role of the 4-position.
Read more about this topic: Maxwell's Equations In Curved Spacetime
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