# Nordström's second gravitational theory

1. Jan 6, 2010

### Altabeh

A sort of simple question (equation of motion)

1. The problem statement, all variables and given/known data

Hello everybody
I'm new to these forums, so wish to be clear enough in my first post to not ask again.

In the Nordström's second theory of gravitation, the field equation is $$\varphi \,\square \left( \varphi \right) =4\,\pi { \it GT}_{{m}}$$ where $$\square$$ is the D'Alembertian operator defined in the Minkowskian spacetime with metric (+,-,-,-), T_m is the trace of the material contribution to the total stress-energy-momentum tensor $$T_{{\mu \nu }}$$ and finally we have $$\varphi$$ implying the potential.

This field is said to have the following Lagrangian proposed by Einstein: $$L={\frac {{\eta}^{\mu \nu }\partial _{{\mu}} \left( \varphi \right) \partial _{{\nu}} \left( \varphi \right) }{8\pi }}-\rho \varphi$$ where $$\rho=\varphi \,T_{{m}}$$ is the density of matter.

Now my question is that how can one proceed to use the above Lagrangian to show that the equation of motion of a test particle moving in the field under discussion is $$\varphi \,d_{{\tau}} \left( u_{{\mu}} \right) =-\partial _{{\mu}} \left( \varphi \right) -d_{{\tau}} \left( \varphi \right) u_{{\mu}}$$ where $$\tau$$ is the proper time, $$u_{{\mu}}$$ is the 4-velocity of the moving particle and $$d_{{\tau}}(..)$$ refers to the derivative of (..) with respect to $$\tau$$??