I have never been formally trained in GR, and have a question regarding the basics of how to calculate the energy density from a metric.(adsbygoogle = window.adsbygoogle || []).push({});

This question arises from thought experiments involving a field with a negative energy density. This is important only because I expect the energy density of the metric in question to be negative, but still conform to the EFE.

Suppose one has the following metric, dependent on a length scale of [itex]r_p[/itex] (the Prather radius of course :) ) :

[itex]c^2 d\tau^2=(1-\frac{r_p}{2r})^4 c^2dt^2 - (1-\frac{r_p}{2r})^{-4} dr^2 -r^2d\Omega^2[/itex]

Based on how I cooked up this metric, I suspect this metric would be generated by an energy density of:

[itex]\eta(r)=\frac{-c^4r_p^2}{8\pi Gr^4}(1-\frac{r_p}{2r})^{-2}[/itex]

1) How would I show this is correct?

2) If this is not correct, how would I find the actual metric for the suggested energy distribution?

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# Energy Density from Metric

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