Is source of space-time curvature necessary?

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

The fact that we have Ricci-flat solutions means we have solutions to the field equations $G_{ab}=8\pi T_{ab}$ with $T_{ab}$ identically vanishing. In other words, we truly have non-trivial vacuum solutions to the field equations (even if we assume all the nice initial and boundary conditions, like a smooth initial hypersurface, asymptotically flat etc). This partly has to do with the non-linear dependence of the Ricci tensor on the metric and its derivatives. I’m not sure what you mean by “assumes the source of curvature is elsewhere”.

For example, thinking of the curvature in Schwarzschild as being sourced by the blackhole region doesn’t make sense, because the blackhole region is still perfectly smooth and the Ricci tensor vanishes identically there. Also, trying to ascribe the non-zero curvature to the $r=0$ singularity, doesn’t really seem to be a good solution. Note that Schwarzschild solutions are a 1-parameter family labelled by $M$; which we interpret as mass. This is a geometric feature of the spacetime itself, and this is the reason why it is curved. Note that when $M=0$ we get Minkowski, where $r=0$ is suddenly no longer a singularity of spacetime (it’s simply a breakdown of polar coordinates) and this is a nice perfectly flat spacetime.

Another reason why trying to ascribe the source of curvature to $r=0$ is bad is that if for example you look at the (maximal globally hyperbolic developments, not the maximal analytic extensions of) Kerr spacetimes, then they do not possess such a singularity. They have so-called Cauchy-horizons which is the boundary of the region where the spacetime is globally hyperbolic (roughly speaking, the region of spacetime that is uniquely determined from the initial data), but as a Lorentzian manifold itself, one can smooth extend beyond these Cauchy-horizons. Again, the Kerr-spacetimes are Ricci-flat everywhere, and sicne there are no singularities like in Schwarzschild, we now have to really confront the fact that the curvature is due to the spacetime itself (particularly the two parameters $M,a$ of mass and specific angular momentum).

Answer 2

Is source of space-time curvature necessary?

This is largely a matter of definitions: What kind of “sources” do we admit and where exactly is this “somewhere” can be?

When we are looking at the metric of a star, the stress–energy tensor of the star's matter would be the “source” of curvature. But what about spacetimes containing only gravitational waves? An example of such spacetime would have zero stress–energy tensor everywhere inside the spacetime yet such a spacetime would satisfy nontrivial boundary conditions at infinity. Can we call these boundary conditions “source of curvature”? If we answer this question in the affirmative we can then recast specific boundary conditions as a form of “boundary matter” and associate some boundary stress–energy tensor to them.

This program is outlined in paper:

• Khoury, J., & Parikh, M. (2009). Mach’s holographic principle. Physical Review D, 80(8), 084004, DOI:10.1103/PhysRevD.80.084004, arXiv:hep-th/0612117.

Note, that when we have such boundary matter as “sources” then we would realize a version of Mach's principle very close to original formulation by Einstein, where metric (and thus curvature) is uniquely specified by the stress–energy tensor, now understood as containing both “bulk” and “boundary” parts.

While “boundary matter” was quite uncommon during the first half-century of GR development there are many examples nowadays: black hole membrane paradigm, Brown–York boundary stress–energy, brane–world models, Hořava–Witten end-of-the-world brane, holographically dual matter …

Answer 3

There must be a "source," i.e. a mass or the metric reduces to the 3+1 flat-space metric.