2.2.3 Electrodynamics in Curved Spacetime
Stationary action principle .
The Lagrangian density for a point particle of charge e and bare mass mo,
interacting with an electromagnetic field Fab in a spacetime with metric gab, can be
written as
C C source + interaction + re.m.
-mo -T ( Ia' b' )1/2 (4)dT + e Aa,a'(4)T (167)-1 g12Fbab
= Lo(4 T + Aaj (167)-1 g1/2FabFab, (2-94)
where
Fab Ab;a Aab, (2-95)
Lo = -mo -ga'b' b) 12 (2-96)
ja e / a2ga' (4) dr. (2-97)
Here, the world-line of the particle is described by a set of functions za'('r), with
7 representing an arbitrary parameter, and the dot over z denotes differentiation
with respect to 7. Multiple dots will be used to denote repeated absolute covariant
differentiation with respect to 7,
a' = dza/d-, (2-98)
a-' = ad'"/d+r,,, bc', (2-99)
"d = ac'/dT+r,/b', (2-100)
The action for the system is given by