Question

Einstein field equations have vacuum solutions that (probably) assumes the source of curvature (either energy-momentum tensor or the cosmological constant term or both) is elsewhere. Like, in Schwarzschild solutions we assume that the source is a star or a planet or even black hole singularity - but it is there somewhere. Now - can we prove that there must be a source somewhere in 4D space-time? In 2d the Einstein tensor is zero - thought we can have non-zero Riemann and therefore curvature - and a zero Einstein Tensor necessarily means no source is allowed, the sum of energy-momentum tensor and cosmological constant term must be zero.

Answer 1

In Einstein's theory of general relativity, the presence of matter and energy dictates spacetime curvature and allows for vacuum solutions implying distant sources. The theory is experimentally confirmed and includes predictions like black holes, influenced by terms like the cosmological constant.

Step-by-step explanation:

The question you're asking essentially deals with whether we can prove the necessity of a source for spacetime curvature within the context of Einstein's theory of general relativity.

The Einstein field equations allow for vacuum solutions and illustrate how mass and energy determine the curvature of spacetime. When these equations yield nonzero solutions in the absence of matter or energy, we interpret this to imply that the source of curvature is not necessarily localized; it may be 'elsewhere' as in the Schwarzschild solutions. However, in lower-dimensional cases such as 2D spacetime, the Einstein tensor vanishes, indicating no source for curvature. This shows a striking contrast with our 4D spacetime where sources can exist at a distance from the observed curvature.

Einstein's theory of general relativity describes how matter and energy influence the curvature of spacetime. The theory has been confirmed by multiple experiments and observations, such as gravitational lensing and the prediction of black holes. The cosmological constant is a term introduced by Einstein that acts as a repulsive force to explain a static universe, which was later found to be unnecessary when the expansion of the universe was discovered.