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).