Spacetime and Geometry · Problem 13

Manifolds — Sean Carroll

Physics — General Relativity · electromagnetism-topological-lagrangian-term

Problem

Consider adding to the Lagrangian for electromagnetism an additional term of the form

\mathcal{L}' = \tilde{\epsilon}_{\mu\nu\rho\sigma} F^{\mu\nu} F^{\rho\sigma}

. (a) Express

\mathcal{L}'

in terms of

\mathbf{E}

and

\mathbf{B}

. (b) Show that including

\mathcal{L}'

does not affect Maxwell's equations. Can you think of a deep reason for this?