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In particular, I am referring to the last part of question 3 on this sheet --

http://www.hep.man.ac.uk/u/pilaftsi/GR/example3.pdf

about the freely falling massive bodies.

My thoughts: The gravitational effects would be significant since for a massive body, the geodesic is timelike. There woud thus be a [tex]\eta^{\mu\delta}\partial_\delta \phi \dot x^\beta \dot x_\beta[/tex] is not of the form [tex]f(\lambda)\dot x^\mu[/tex] so the affine parametrization does not eliminate this term containing the gravitational potential [itex]\phi[/itex].

Does this argument make any sense at all? Also, what more can I say about the geodesics of such massive particles?

Thanks.