# Conditions under which the stress tensor for a point particle... We take the energy-momentum of the point particle to be... ∫ x ```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 &micro;ν (x) = m
Then
δ 4 (xα − z α (τ )) &micro; ν
√
ż ż dτ.
−g(z)
∂T &micro;ν (x)
&micro;
+ Γ&micro;σ (x)T σν (x) + Γν&micro;σ (x)T &micro;σ (x)
&micro;
∂x
[ 4 α
∫
∂δ (x − z α (τ )) ż &micro; ż ν
√
=m dτ
∂x&micro;
−g(z)
) δ 4 (xα − z α (τ )) ]
(
&micro;
√
+ Γ&micro;σ (x)ż ν + Γν&micro;σ (x)ż &micro;
ż σ
−g(z)
[
∫
dδ 4 (xα − z α (τ )) ż ν
√
=m dτ −
dτ
−g(z)
(
) δ 4 (xα − z α (τ )) ]
&micro;
√
+ Γ&micro;σ (x)ż ν + Γν&micro;σ (x)ż &micro;
ż σ
−g(z)
[ (
∫
)
ν
d
ż
√
=m dτ δ 4 (xα − z α (τ ))
dτ
−g(z)
(
) ż σ ]
&micro;
+ Γ&micro;σ (x)ż ν + Γν&micro;σ (x)ż &micro; √
−g(z)
[
∫
(
)
&micro;
&micro;
=m dτ ż σ ∇σ ż ν + Γ&micro;σ (x) − Γ&micro;σ (z) ż ν
(
) ] δ 4 (xα − z α (τ ))
ν
ν
√
+ Γ&micro;σ (x) − Γ&micro;σ (z) ż &micro;
−g(z)
∇&micro; T &micro;ν (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, ∇&micro; T &micro;ν (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 &micro; ∂/∂x&micro; = d/dτ .
• The fourth equality holds from performing integration by parts.
• The fifth equality holds by rearrangement of terms after completing the covariant deriva√
&micro;
tive on ż ν and using d ln −g(z)/dτ = Γ&micro;σ (z)ż σ .
• The result given holds, and somewhat simpler, even if, from the beginning, g(z) ⇒ g(x).
```