# Influence of spacetime curvature on electromagnetic wave propagation [duplicate]

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?

## marked as duplicate by ACuriousMind♦, Kyle Kanos, John Rennie general-relativity StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); May 25 '15 at 6:45

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: