# Are energy and momentum imposed by purely geometrical properties of spacetime?

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.

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?
• 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. Jul 13, 2020 at 0:30