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I am trying to derive geodesic motion for photons from the Lagrangian of electromagnetism coupled to General Relativity.

I tried to use the covariant conservation of the Stress energy tensor: $$\nabla_aT^{ab}=0$$ where the stress energy tensor is: $$T^{ab}=F^{ac}F^{b}_{c}-\frac{1}{4}F_{cd}F^{cd}g^{ab}$$ In this way, supposing $F_{cd}F^{cd}=\,\text{constant}\;$ and using the equation of motion $\nabla_aF^{ab}=0$ I get to the equation: $$g_{cd}F^{ac}\nabla_{a}F^{bd}=0$$ which looks fairly similar to the geodesic equation but is not, due to the contraction between the two $F$.

I would like to know if there is any way to interpret geometrically the equation obtained for $F^{ab}$ or the corresponding equation for $A_b$. Maybe one can write $A_b$ as plane waves with non constant wave vector and show that this vector follows a geodesic and that the polarization is always orthogonal to it?

AoZora
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  • The premises seem a little weird. Why are you expecting to get the geodesic equation (which it wouldn't really be, since $F_{\mu\nu}$ is not a vector), and why do you suppose $F_{cd}F^{cd}$ is constant? – Javier Sep 05 '19 at 23:47
  • Related: Do light waves precisely follow null geodesic paths in General Relativity? on Physics SE. This book might be helpful, too: Ray Optics, Fermat’s Principle, and Applications to General Relativity, https://books.google.com/books?id=0NDzCAAAQBAJ – Chiral Anomaly Sep 06 '19 at 00:17
  • @Javier well it all started by trying to motivate geodesics from a qft point of view. Clearly the fact that F is a tensor makes things not straightforward, but maybe one can show that from the equation for F it follows that some vectors (like wave vector) follow geodesics.. The assumption on F^2 is just to get to a simpler equation and because I could expect having F^2=constant in the equivalent of a plane wave. – AoZora Sep 06 '19 at 08:57
  • @ChiralAnomaly thanks for the reference! – AoZora Sep 06 '19 at 08:57
  • I'll have to look at that book myself! It's generally not true that particles with spin follow geodesics. For massive particle their motion is governed by the Matthisson-Papapetrou equations. Massless particles have problems. – mike stone Sep 06 '19 at 15:08

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