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
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
where is the dust’s density in its own rest frame and is its four-velocity measured in the observer’s frame. In this case
Note that is not necessarily a constant so its gradient will, in general be non-zero.
From the equation
The second line follows from the fact that the absolute gradient of the metric tensor is zero (so there’s no term in the product rule expansion). The third line comes from swapping the bound indices and in the second term in line 2, and then using the symmetry of the metric tensor ().
Returning to 4, we can multiply through by and get
Substituting this back into 4 we get
The absolute gradient of a four-vector can be written in terms of Christoffel symbols as
so we get
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.