Nordstrom Gravity: Exploring R w/ Minkowski & Schwarzschild Metrics

[beans73]

Homework Statement

Question: "A theory of gravity devised by physicist G. Nordstrom, relates g_{μ\nu} to T^{μ\nu} by the equation:

R=κg_{μ\nu}T^{μ\nu}

where the metric has the form g_{μ\nu}=e^{2\Phi} with \Phi=\Phi(x^{μ}) a function of the spacetime coordinates (the special form of the metric follows from requiring the vanishing of the Weyl curvature tensor _{αβγδ}=0

a) Show that in the Newtonian limit \Phi^{2}<<1 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 \Phi.

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 g_{μ\nu} must be a function of r (from \Phi=2GM/r). wouldn't that mean that the christoffel coefficients all vanish (ie. ∂_{t}g_{xx}=0)? does this mean i need to use the shwarzschild metric??

 

[fzero]

Homework Statement

Question: "A theory of gravity devised by physicist G. Nordstrom, relates g_{μ\nu} to T^{μ\nu} by the equation:

R=κg_{μ\nu}T^{μ\nu}

where the metric has the form g_{μ\nu}=e^{2\Phi} with \Phi=\Phi(x^{μ}) a function of the spacetime coordinates (the special form of the metric follows from requiring the vanishing of the Weyl curvature tensor _{αβγδ}=0

a) Show that in the Newtonian limit \Phi^{2}<<1 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 \Phi.

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 g_{μ\nu} must be a function of r (from \Phi=2GM/r). wouldn't that mean that the christoffel coefficients all vanish (ie. ∂_{t}g_{xx}=0)? does this mean i need to use the shwarzschild metric??

You're missing some things, g_{μ\nu}=e^{2\Phi} doesn't make sense, since the left-hand side is a tensor and the right-hand side is a scalar. What you want is (referring to http://en.wikipedia.org/wiki/Nordström's_theory_of_gravitation#Features_of_Nordstr.C3.B6m.27s_theory)

$$ g_{\mu\nu} = e^{2\Phi} \eta_{\mu\nu},$$

where $\eta_{\mu\nu}$ is indeed the flat Minkowski metric. Since $\Phi$ is a function of the $x^\mu$, the Christoffel symbols for $g_{\mu\nu}$ will not vanish, but because of the high amount of symmetry, they will have somewhat simple expressions.

It is not the case that ∂_{t}g_{xx}=0, rather ∂_{t}g_{xx}=2e^{2\Phi}(∂_{t}\Phi).