Reference: Moore, Thomas A., *A General Relativity Workbook*, University Science Books (2013) – Chapter 8; Problem 8.4.

The geodesic equation is

We can write this in terms of the four-velocity if we expand the first derivative:

Therefore, the geodesic equation is

We can express this in a different form by multiplying by and summing:

The geodesic equation thus confirms that along a geodesic, which we knew beforehand, since is an invariant.

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DanI believe your first geodesic equation is incorrect.

DanIt seems like you’ve used “j” as a replacement index for Thomas Moore’s “beta” and “mu.” It looks like an additional index needs to be added to your geodesic equation to make it work.

DanCould you clarify which indices correspond to the indices in the book?

gwrowePost authorEquation 1 is correct. Remember that repeated indices are just dummy indices since they are summed, so it doesn’t matter what you call them. In my previous posts I was lazy and used Roman letters for indices since they are easier to type than Greek letters, which Moore uses. [Actually about half the textbooks on general relativity use Roman indices anyway, so I’m not that far out of line.] Moore uses for my in the first term, and for my and for my in the second term, though he could have used instead of in the first term as well and still have a correct equation. The only index that isn’t summed is (Moore uses ).

DanThank you! I appreciate it.