I've been learning GR in my sparetime, and occationally I run into a conceptual problem that stalls my progress. Here is a question that has come up. I expect that this is a stupid question, but it's really bugging me, and an explanation will help me move forward more efficiently.(adsbygoogle = window.adsbygoogle || []).push({});

If we wish to find the metric in the region outside a spherically symmetric source, the approach is to note that the energy-momentum tensor vanishes in the exterior and the Einstein field equation becomes:

[tex]R_{\mu \nu}=0[/tex]

So, I understand that a vanishing Ricci tensor does not imply a vanishing Rieman tensor ([tex]R_{\mu \nu \alpha \beta}[/tex]), and the exterior space need not be flat. However, it's not clear to me how the Einstein field equation can "know" about the source mass and generate the correct metric. It seems to me that the metric would be different between a nonrotating neutron star vs. a nonrotating planet, for example.

So, my question is, what is the correct way for me to interpret this situation? If I solve for the metric with an equation that makes no reference to the source, how do I get the correct metric for a massive object that warps space, vs. a small object that barely warps space. My instinct says that the correct approach would be to match an exterior solution to the interior solution with boundary conditions, but I see no mention of this in the books I'm studying. I do see where a Newtonian limit is taken to bring the source mass back into the fold, so perhaps this is just an alternative way to get the right answer? Is this the correct interpretation, or am I missing something important here?

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# Metric outside a spherically symmetric source.

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