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This question already has an answer here:

Classical physics assumes that spacetime is evenly distributed in the sense that Coulomb's Law predicts that a charged particle will create a spherically symmetric electric field around its location. Coulomb's Law also predicts that the magnitude of the electric field's strength is dependent purely upon a point's distance to the location of the charged particle. However, these ideas do not seem to take into account relativistic effects due to gravity and velocities approaching the speed of light. Here are my questions:

  1. Is there a version of Coulomb's Law (and possibly Maxwell's Equations) that takes relativistic effects into consideration?
  2. How does spacetime curvature affect the propagation of electromagnetic fields? It seems to me intuitively that if there was a 'denser' region of spacetime, that an electromagnetic field passing through that region would act as if it had traveled through a greater length of non-warped spacetime. Is this correct?
  3. How does an electromagnetic field theoretically behave around the singularity of a black hole? If an electromagnetic field's propagation is influenced by spacetime distortion due to gravity, how does the field behave as spacetime curvature/distortion approaches infinity?
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marked as duplicate by ACuriousMind, Kyle Kanos, John Rennie general-relativity May 25 '15 at 6:45

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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I'll answer to 1

Maxwell equations are already relativistic, but - in a flat spacetime. You can write Maxwell equations for a general metric $ g_{\mu \nu} $ (The original Maxwell equations are formulated for a flat spacetime - $ g_{\mu \nu} = \eta _{\mu \nu}$ ). One way this can be done is by the following algorithm:

  1. Transform coordinates to a local coordinates around arbitrary point (this can always be done for non singular point in spacetime). In those local coordinates - the metric is flat at the point (up to a second order corrections in coordinates)
  2. Write the Maxwell equations (as you know them) but replace the regular derivatives with covariant derivatives (i.e. - make the equations tensorial equations, and thus covariant under coordinates transformations).
  3. Return to your original coordinates.

There exists a tensorial formulation of the Maxwell equations (for flat spacetime) which makes it even easier to transform them to a general metric - check this .

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  • $\begingroup$ Thanks! Is there a version of Maxwell's Equations that accounts for some mathematically defined non-flat spacetime? $\endgroup$ – Alekxos May 24 '15 at 21:03
  • $\begingroup$ Look at the link I posted under Tensor calculus, "any-spacetime". All the curvature is "hidden" in the metric tensor, and all the derivatives there are covariant. It is possible but very messy to expand this tensorial expression in regular derivatives of the electric and magnetic fields. $\endgroup$ – Alexander May 24 '15 at 21:11

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