Editing Openai/6897769e-4ee4-800f-aba5-69cca34f701c (section)

=== Vary SEMS_{\rm EM}SEM in the usual way and integrate by parts: ===

δSEM=∫Vd4x  (−∂νFνμ) δAμ  +  ∫Σd3y  nνFνμ δAμ .\delta S_{\rm EM}
= \int_V d^4x\; \big(-\partial_\nu F^{\nu\mu}\big)\,\delta A_\mu
\;+\;
\int_{\Sigma} d^3y\; n_\nu F^{\nu\mu}\,\delta A_\mu \,.δSEM=∫Vd4x(−∂νFνμ)δAμ+∫Σd3ynνFνμδAμ.
The first term gives the bulk Maxwell equation, the second is the boundary term that must be canceled or matched by the boundary action variation.

Variation of the boundary action under δAa\delta A_aδAa produces

δSbdy=∫Σd3y  ja(y) δAa(y).\delta S_{\rm bdy} = \int_{\Sigma} d^3y\; j^a(y)\,\delta A_a(y).δSbdy=∫Σd3yja(y)δAa(y).
Identifying δAa=eμaδAμ\delta A_a = e^\mu{}_a \delta A_\muδAa=eμaδAμ, the stationary action condition δStot=0\delta S_{\rm tot}=0δStot=0 for arbitrary δAμ\delta A_\muδAμ yields:

Bulk Maxwell equation

∂νFνμ(x)=0in V(or =Jbulkμ if bulk sources)\partial_\nu F^{\nu\mu}(x) = 0\qquad\text{in }V\quad(\text{or }=J^\mu_{\rm bulk}\text{ if bulk sources})∂νFνμ(x)=0in V(or =Jbulkμ if bulk sources)
Boundary (matching) condition (pullback onto Σ\SigmaΣ):

nνFνμ∣Σ  =  jsurfμ(y),where jsurfμ≡eμa jan_\nu F^{\nu\mu}\Big|_{\Sigma}
\;=\; j^\mu_{\rm surf}(y),
\qquad\text{where } j^\mu_{\rm surf}\equiv e^\mu{}_a\, j^anνFνμΣ=jsurfμ(y),where jsurfμ≡eμaja
(i.e. the normal projection of the field strength at the surface equals the surface current pulled back to the embedding spacetime).

This single condition is the Lagrangian origin of the usual Maxwell boundary conditions.