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Matter and [non-gravitational] energy via the stress tensor can cause spacetime curvature because the stress tensor is algebraically related to the Ricci curvature tensor, according to Einstein's field equation. So energy in a region of spacetime can be associated with some curvature in that region of spacetime. I am confused about causation.

Spacetime curvature (more precisely, Riemann curvature) is composed of a part determined by Ricci curvature and the rest determined by Weyl (conformal) curvature (see Ricci decomposition). That is to say, one can have spacetime curvature (namely, Weyl-type curvature) without the presence of matter and [non-gravitational] energy. Can we say it is possible to have curvature of space time without energy just because of mathematics (like have only three phases of matter just because of central limit theorem)

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  • $\begingroup$ When you say energy, do you mean only the non-gravitational part of total energy? $\endgroup$
    – paul230_x
    Commented Jun 29, 2022 at 20:16
  • $\begingroup$ There are infinitely many phases of matter and classifying them requires much more than the central limit theorem. $\endgroup$ Commented Jun 29, 2022 at 20:31
  • $\begingroup$ You should quote each section, followed by your question about that section. Ref: (my answer to your earlier question) physics.stackexchange.com/questions/714940/… $\endgroup$
    – robphy
    Commented Jun 30, 2022 at 2:11
  • $\begingroup$ You cannot have the Weyl curvature without matter, only the Ricci curvature. $\endgroup$
    – safesphere
    Commented Jun 30, 2022 at 15:18

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You can have curvature without having a non-zero energy momentum tensor, indeed. In fact, these are special solutions to Einstein Equations, called the Vacuum solutions, where you set $T_{\mu \nu}=0$.

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