Second order expansion of metric in free-fall

[InsertName]

Hello,

I have read that, in a freely-falling frame, the metric/ interval will be of the form:

ds² = -c²dt²(1 + R_0i0jxⁱx^j) - 2cdtdxⁱ(\frac{2}{3} R_0jikx^jx^k) + (dxⁱdx^j(δ_ij - \frac{1}{3} R_ikjlx^kx^l)

to second order.

Does anyone know where I could find a derivation of this result?

 

[Bill_K]

InsertName, That's an interesting formula! It's derivation is almost immediate, except I have doubts about the absence of the time coordinate, and the factors of 1/3.

Expand the metric in a Taylor's series:
g_μν = A_μν + B_μνσx^σ + C_μσντx^σx^τ + ...
It's always possible to choose coordinates such that A_μν = η_μν and B_μνσ = 0, and in these coordinates the Christoffel symbols vanish. Then the formula for the Riemann tensor reduces to
R_μσντ = ½(g_μτ,σν + g_σν,μτ - g_μν,στ - g_στ,μν) = C_μστν + C_τνμσ - C_μσντ - C_σμτν.
If you assume C to have the same symmetry as the Riemann tensor, then this is 4C_μστν, showing that C_μστν = (1/4)R_μσντ