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Grav12 - GRAVITATION F10 S G RAJEEV Lecture 12 1 Maxwells...

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GRAVITATION F10 S. G. RAJEEV Lecture 12 1. Maxwell’s Equations in Curved Space-Time 1.1. Recall that Maxwell equations in Lorentz covariant form are. μ F μ ν = j ν , F μ ν = μ A ν ν A μ . 1.2. They follow from the variational principle. S = 1 4 ˆ F μ ν F μ ν dx + ˆ j μ A μ dx First, δ S = ˆ F μ ν μ δ A ν dx + ˆ j ν δ A ν dx Now integrate by parts the first term. 1.3. This leads to a wave equation with source for the electromagnetic potential. μ μ A ν ν [ μ A μ ] = j ν It is common to impose the condition μ A μ = 0 ,(the Lorentz gauge) taking advantage of the gauge invariance A μ A μ + μ Λ . Then each component of A μ satisfies the wave equation μ μ A ν = j ν 1.4. The generally covariant form of Maxwel’s equations is. D μ F μ ν = j ν , F μ ν = μ A ν ν A μ Recall that the Christo ff el symbols cancel out in the antisymmetric derivative of a covariant vector. 1.5. In terms of potentials. D μ D μ A ν D μ D ν A μ = j μ We cannot interchange the derivatives in the second term without introducing some terms involving curvature. 1.6. An equivalent form of the curved space Maxwell’s equations is. 1 det g μ det gg μ ρ g νσ F ρσ = j ν 1
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GRAVITATION F10 2 1.7. This follows from the covariant variational principle. S = 1 4 ˆ F μ ν F ρσ g μ ρ g νσ det gdx + ˆ j μ A μ det gdx 1.8. These equations tell us how the gravitational field a ff ects the propa- gation of light. For example it can tell us how light is di ff racted and refracted by a gravitational field. Spectacular phenomena such as gravitational lensing follow from this. More later.
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