Understanding Nordstrom Metric & Freely Falling Massive Bodies

[c299792458]

Could somebody please explain something regarding the Nordstrom metric?

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 $$\eta^{\mu\delta}\partial_\delta \phi \dot x^\beta \dot x_\beta$$ is not of the form $$f(\lambda)\dot x^\mu$$ so the affine parametrization does not eliminate this term containing the gravitational potential $\phi$.

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

Thanks.

 

[member 553083]

Yes, your argument does make sense. The Nordstrom metric describes the spacetime geometry produced by a static, spherically symmetric source of mass or energy. When a massive body moves through this geometry, it will experience an effective gravitational force due to the curvature of spacetime, as described by the metric. This force can be calculated from the Christoffel symbols associated with the metric. The geodesic equation for the massive body then takes the form of a second-order differential equation, with the gravitational force represented by terms containing derivatives of the gravitational potential.