[ home ] Variation (least action) principle General Relativity. Note: « 6 »
Unnamed Web page The differential dg of the determinant of the metric tensor gab is dg=ggabdgab=−ggabdgab

Variational principle (= least action principle)

The laws of physics can be formulated as a vanishing variation, δ S=0, of some action S=∫ LdΩ, where L is called Lagrangian (density). Examples:
  1. Nonrelativistic particle: L=T−V, where T= 1/2 mv2 is the particle's kinetic energy and V is its potential energy.
  2. Free relativistic particle: S = −mc∫ ds.
  3. Electromagnetic field: Lem = − 1/16π FabFab ∝ − 1/8π Aa,bAa,b.
  4. Relativistic particle with charge q in the electromagnetic field Aa: = −mc∫ ds − q∫ Aadxa +Lem

The action for the matter in the presence of a gravitational field

If the action S for a physical system in special relativity is Sm = ∫ L dΩ, then in a gravitational field it must have basically the same form, Sm = ∫ L[g]√[−g]dΩ, where the notation L[g] indicates that all contractions in the Lagrangian L must be written explicitely through the metric tensor gab.

Variation of the action with respect to gab and the energy−momentum tensor of matter

The variation δ Sm of the matter action due to a variation δ gab is given in terms of a symmetric tensor Tab,
δ Sm = 1/2 ∫ Tab δ gab √[−g] dΩ = − 1/2 ∫ Tab δ gab √[−g] dΩ,

where

1/2 √[−g]Tab = δ(√[−g]L)/δgab .

This tensor satisfies the equation Tab;b=0 which in a flat space turns into the energy−momentum conservation equation Tab,b=0 and we thus deduce that it must be the energy−momentum tensor.

Exercises

  1. Prove, that Γaba=(ln√[−g]),b (where ,b= ∂/∂ xb ).
  2. Prove, that √[−g]Aa;a=(√[−g]Aa),a (where ;b= D/∂ xb ).
  3. The lagrangian density for the electromagnetic field is L = − 1/16π FabFab. Calculate the corresponding energy−momentum tensor using 1/2 √[−g] Tab = ∂√[−g]L/∂gab . Answer: Tab = 1/4π (−FacFbc + 1/4 FcdFcdgab)
Copyleft © 2005 D.V.Fedorov (fedorov @ phys au dk)