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In Misner, Thorne and Wheeler's "Gravitation" chapter 39 section 3, they write the following:

"In general relativity theory, the metric is generated directly by the stress-energy of matter and of nongravitational fields".

My question is then if stress-energy is responsible for the generation of the metric, to what extent can the metric be defined prior to introducing stress energy? Furthermore, the Einstein equations can be derived from an action principle whereby the stress energy tensor is defined according to the variation of the matter action with respect to the metric and so to me it seems more as though the metric works to generate the stress-energy content, as opposed to the other way around. This however feels nonsensical, as stress-energy is responsible for curvature, and the metric itself captures this.

However, one can consider vacuum solutions to the Einstein equations whereby the metric (e.g. Schwartzschild) is naturally defined without the need for stress-energy. So what is the content of Misner, Thorne and Wheeler's statement?

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Ignoring stress energy, you still have a metric, namely Minkowski metric. So I think "generate" is probably a poor choice of word. It would be better to say that stress energy (and $\Lambda$, and boundary conditions) determines the metric.

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  • $\begingroup$ Charles, this is an excellent point. I’m still a little stunted by something though. As I said, one can define the stress-energy tensor as the variation of the matter content with respect to the metric. But stress-energy as you say determines the metric. So which comes first, the stress-energy itself or the metric? Thanks and best, Jack. $\endgroup$ – Jack Hughes Apr 11 '20 at 17:46
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    $\begingroup$ ... or say that stress-energy curves the metric $\endgroup$ – Thomas Fritsch Apr 11 '20 at 17:52
  • $\begingroup$ In my view, the metric comes first, in the form of Minkowski metric for local inertial reference frames. Einstein's original argument for SR should be imported into GR at a fundamental level (unfortunately I see few texts of GR which actually do this. It is something I sought to correct in my own book). $\endgroup$ – Charles Francis Apr 11 '20 at 17:53
  • $\begingroup$ Another possible point of view is that neither comes first, rather both are related by Einstein equations. $\endgroup$ – Prof. Legolasov Apr 12 '20 at 14:46

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