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Unformatted text preview: Maxwells Equations In a Gravitational Field by Andrew E. Blechman PHY413 Term Paper December 13, 2000 Revised: April 11, 2003 By the turn of the twentieth century, physics had come to a turning point. Maxwell had successfully united the electric and magnetic force, and Einstein had begun to make a breakthrough in the theory of relativity. He had used Maxwells theory to suggest that there was no absolute rest frame, and that all motion was relative. Einstein had also begun to see how gravity might play a part in this picture in 1907, when he published his equivalence principle. He suggested that there was no difference between an object being accelerated and it being in freefall in a gravitational field. This duality led him to assume that gravity was nothing but the bending of spacetime, and the theory of gravity became a question of semiRiemannian Geometry. But with the final formulation of General Relativity, Einstein noticed a nontrivial connection between Maxwells electrodynamics and his theory of gravity. One sees that by solving Maxwells equations in a gravitational field, not only does the electromagnetic field generate gravity, which is certainly believable from the massenergy relation, but gravity can enhance a background electromagnetic field given the proper conditions. This duality was one of the key features of physics that led Einstein and his followers to propose that there is a Grand Unified Theory of all the forces. In this paper, we shall derive this duality, and show how it might help us to find gravity waves. * * * 1 Start with Maxwells equations: J t E B t B E B E 4 4 =  = + = = We can write the two homogeneous equations in terms of potentials: A B t A E =   = Now lets introduce a new fourvector A, and construct the rank2 tensor F: ( 29  =  , x y z x z y y z x z y x B B E B B E B B E E E E A A F A A F is called the field strength tensor . In terms of this tensor, we can write Maxwells Equations as: ( 29 J v j F F F j F , 4 , = = = + + = where v is the 4velocity. Maxwells equations written in this form are said to be in covariant form . Now that we have Maxwells Equations in covariant form, we can add gravity to the picture. When we add gravity, we are allowing for curved space; therefore all 2 derivatives must be replaced by covariant derivatives. Firstly, we note that when this happens, the definition of F does not change: A A A A A A A A F  = +   = ; ; Similarly, the homogeneous equation is unchanged. We therefore only have to worry about the inhomogeneous equation....
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