CONSERVATION OF FOUR-MOMENTUM IMPLIES THE
GEODESIC EQUATION
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Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) - Chapter 22, Box 22.1.
The stress-energy tensor obeys the conservation of four-momentum
∇j T ij = 0
(1)
We can show that the geodesic equation actually follows from this conservation condition. For the case of ’dust’ (a fluid whose constituent particles
are locally at rest with one another), the stress-energy tensor is
T ij = ρ0 ui uj
(2)
where ρ0 is the dust’s density in its own rest frame and ui is its fourvelocity measured in the observer’s frame. In this case
∇j T ij = ∇j ρ0 ui uj
(3)
j
(4)
0 = ui ∇j ρ0 u + ρ0 uj ∇j ui
Note that ρ0 is not necessarily a constant so its gradient will, in general
be non-zero.
From the equation
gij ui uj = −1
(5)
we get
∇k gij ui uj
= 0
(6)
gij ui ∇k uj + gij uj ∇k ui = 0
(7)
2gij ui ∇k uj = 0
(8)
i
j
gij u ∇k u
= 0
(9)
The second line follows from the fact that the absolute gradient of the metric
tensor is zero (so there’s no term ui uj ∇k gij in the product rule expansion).
1
CONSERVATION OF FOUR-MOMENTUM IMPLIES THE GEODESIC EQUATION
2
The third line comes from swapping the bound indices i and j in the second
term in line 2, and then using the symmetry of the metric tensor (gij = gji ).
Returning to 4, we can multiply through by gil ul and get
l
i
gil u u ∇j ρ0 u + ρ0 gil u u ∇j u = −∇j ρ0 u + ρ0 u gil u ∇j u(10)
0 = −∇j ρ0 uj
(11)
l i
j
l j
i
j
j
where we used 5 on the first term on the LHS of the first line, and 9 on the
second term of the RHS of the first line (with indices suitably relabelled).
Therefore
∇j ρ0 uj = 0
Substituting this back into 4 we get
(12)
uj ∇j ui = 0
(13)
The absolute gradient of a four-vector can be written in terms of Christoffel symbols as
∇j ui = ∂j ui + Γikj uk
(14)
uj ∇j ui = uj ∂j ui + Γikj uk
(15)
so we get
∂xj ∂ui
+ Γikj uk uj
(16)
∂τ ∂xj
dui
0 =
+ Γikj uk uj
(17)
dτ
j
k
d 2 xi
i dx dx
0 =
+
Γ
(18)
kj
dτ 2
dτ dτ
This is just the geodesic equation, so we see that (for dust, anyway; the
result is generally true for fluids but is harder to prove) conservation of
four-momentum implies the geodesic equation.
0 =