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 $∂_{t}g_{xx}=2e^{2\Phi}(∂_{t}\Phi)$.