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Cool - this partial solution should be sufficient to define the path of the geodesic, because the the exact nature of the affine parameterization is irrelevant to the path the particle actually follows. Hopefully I should be able to use it as a double check of my previous calculations.Hurkyl said:So, in the end, one family of solutions (I don't think the only one -- I should be missing a class of solutions to that second equation) is:

x(λ) = K + (log |λ|) / (2H)

t(λ) = √(2 H C λ)

(Of course, I've just looked at the x-t slice, assuming y' and z' were both zero. This should suggest the approach for the 4-D case)