- #1
beans73
- 12
- 0
Homework Statement
Question: "A theory of gravity devised by physicist G. Nordstrom, relates [itex]g_{μ\nu}[/itex] to [itex]T^{μ\nu}[/itex] by the equation:
[itex]R=κg_{μ\nu}T^{μ\nu}[/itex]
where the metric has the form [itex]g_{μ\nu}=e^{2\Phi}[/itex] with [itex]\Phi=\Phi(x^{μ})[/itex] a function of the spacetime coordinates (the special form of the metric follows from requiring the vanishing of the Weyl curvature tensor [itex]_{αβγδ}=0[/itex]
a) Show that in the Newtonian limit [itex]\Phi^{2}<<1[/itex] the geodesic equation for a test body moving slowly in this spacetime reproduces the kinematics of Newtonian gravity.
b) Calculate the ricci scalar R in the Newtonian limit showing that it is just a second order differential operator acting on [itex]\Phi[/itex].
The Attempt at a Solution
My first question is really just about the metric i need to use. from a couple of things I've read on the internet, it seems i should just use the simple flat-space minkowski metric (ct, x, y, z), but i am not really sure how to calculate the christoffel coefficients in this metric as all [itex]g_{μ\nu}[/itex] must be a function of r (from [itex]\Phi=2GM/r[/itex]). wouldn't that mean that the christoffel coefficients all vanish (ie. [itex]∂_{t}g_{xx}=0[/itex])? does this mean i need to use the shwarzschild metric??