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If we defined spacetime as a purely geometrical (not physical) structure of the kind that is in general relativity (a 4-dimensional Lorentzian manifold), would it automatically have properties that would behave like energy and momentum in Einstein field equations?

I am wondering whether the purely geometrical properties of a 4D Lorentzian manifold impose existence of matter (that is, properties that behave like energy and momentum).

From what I have read, it seems that the answer is no, and so energy and momentum seem to be encoded in the points of the manifold rather than in its geometry.

Qmechanic
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glidingforward
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1 Answers1

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Given a Lorentzian manifold, one can calculate $$ R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu}$$ If you want, you can declare that this quantity represents energy and momentum, and then the Einstein field equations are satisfied. Is this what you are asking?

Daniel
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  • No, I am asking whether this manifold necessarily has the properties of energy and momentum. It seems that energy and momentum are just added to the manifold by choice. – glidingforward Jul 12 '20 at 22:37
  • Which properties? This has the property "satisfies Einstein's field equations." What other equations do you want satisfied? – Daniel Jul 12 '20 at 22:39
  • Does this manifold necessarily satisfy Einstein's field equations? Do Einstein's field equations logically follow from the geometrical properties of this manifold? – glidingforward Jul 12 '20 at 23:05
  • All the manifold gives you is topological structure and $g$. The point is that for any $g$, there necessarily exists a $T$ that satisfies the field equations - the manifold doesn't come with this $T$ built-in, but you are free to define it. When people say "geometrical properties", they usually mean equations only involving $g$. So no, in that sense Einstein's equations don't follow from geometrical properties. – Daniel Jul 13 '20 at 00:30
  • I was told that the manifold has energy and momentum also when T=0 because there are gravitational waves defined purely by geometrical properties of the manifold. So it seems that for this special case, the manifold indeed has energy and momentum necessarily? – glidingforward Jul 13 '20 at 09:09
  • I don't know what you mean by "has energy and momentum necessarily" - can you formalize that? Or give examples of what kind of property you are looking for? – Daniel Jul 13 '20 at 17:43
  • Sorry, apparently I misunderstood about the necessity of energy and momentum when T=0, so the manifold does not necessarily have energy or momentum when T=0. – glidingforward Jul 14 '20 at 21:56