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Conditions under which the stress tensor for a point particle is conserved We take the energy-momentum of the point particle to be given by: ∫ T µν (x) = m Then δ 4 (xα − z α (τ )) µ ν √ ż ż dτ. −g(z) ∂T µν (x) µ + Γµσ (x)T σν (x) + Γνµσ (x)T µσ (x) µ ∂x [ 4 α ∫ ∂δ (x − z α (τ )) ż µ ż ν √ =m dτ ∂xµ −g(z) ) δ 4 (xα − z α (τ )) ] ( µ √ + Γµσ (x)ż ν + Γνµσ (x)ż µ ż σ −g(z) [ ∫ dδ 4 (xα − z α (τ )) ż ν √ =m dτ − dτ −g(z) ( ) δ 4 (xα − z α (τ )) ] µ √ + Γµσ (x)ż ν + Γνµσ (x)ż µ ż σ −g(z) [ ( ∫ ) ν d ż √ =m dτ δ 4 (xα − z α (τ )) dτ −g(z) ( ) ż σ ] µ + Γµσ (x)ż ν + Γνµσ (x)ż µ √ −g(z) [ ∫ ( ) µ µ =m dτ ż σ ∇σ ż ν + Γµσ (x) − Γµσ (z) ż ν ( ) ] δ 4 (xα − z α (τ )) ν ν √ + Γµσ (x) − Γµσ (z) ż µ −g(z) ∇µ T µν (x) = The last two terms in square brackets are zero because the δ-function multiplies terms which vanish at coincidence — for any orbit z α (τ ). Thus, if z(τ ) is a geodesic, ∇µ T µν (x) = 0. Note: • The second equality holds because the derivative acts only on the δ-function. • The third equality holds by introducing a change of sign upon swapping the arguments of the derivative on the δ-function and using ż µ ∂/∂xµ = d/dτ . • The fourth equality holds from performing integration by parts. • The fifth equality holds by rearrangement of terms after completing the covariant deriva√ µ tive on ż ν and using d ln −g(z)/dτ = Γµσ (z)ż σ . • The result given holds, and somewhat simpler, even if, from the beginning, g(z) ⇒ g(x).