# Angular momentum of vacuum solution in Einstein gravity

In Strominger's "Lecture Notes on Infrared Structure of Gravity", page 38, he mentioned about how part of this whole mess about "vacuum degeneracy" (classically, i.e. in the sense of Minkowski space being vacuum solution of Einstein equation) has to do with angular momentum in general relativity.

My (naive) understanding of this is that he claims there can be initial vacuum configuration that decays to flat space after emitting gravitational radiation (which can carry angular momentum), and this final vacuum configuration has nonzero angular momentum. However, this makes no sense to me as I thought that Minkowski space is the unique vacuum solution to Einstein solution with zero ADM mass and ADM angular momentum. It is effectively the one you "naively" get from Kerr-Newman solution by setting $$M,J,Q=0$$. There are stories (?) about how there are different versions of angular momentum (defined by Komar integrals, Noether, ADM, or Killing symmetries, all of which may be related to one another), but none of these definitions matter: if you claim it is angular momentum, it should be zero (or can some of these be zero but not others?).

There are some subtler issue that may resolve my confusion, but the more I read the worse it gets:

1. Initial data set: in this paper by Yau et al., they explicitly constructed initial data set (spacelike hypersurface) with zero energy-momentum vector but nonzero angular momentum. Strominger's page 38 footnote 10 makes this sound like this refers to vacua with nonzero $$\vec{J}$$, but how is this possible? What the paper did was to show that there exists nonzero angular momentum spacelike slice that you can embed in Minkowski space. Does being able to find this slice imply Minkowski space has nonzero angular momentum? Does this mean that Minkowski space is really just defined by zero energy-momentum but anything else is okay?
2. Supertranslation ambiguity: Suppose I take it for granted that vacuum with nonzero $$\vec{J}$$ is fine. According to Bondi-Metzner-Sachs (BMS) group calculations, then, these different vacua are related to each other by (conjugate action of) supertranslation (Strominger, page 68). Now this, according to Strominger, implies that there is no BMS-invariant definition of angular momentum. If so, what is the "angular momentum" we always spoke about in Kerr geometry?

If there is anything that comes close to a resolution to me right now, it looks like "vacuum" here just means flat initial data set, not the full four-dimensional geometry. But that seems to me an abuse of what is called a vacuum in GR.

In short, I am really just asking how to make sense classically that vacuum is degenerate in that it can have nonzero angular momentum (which angular momentum?), and how all these are mapped to each other by supertranslation action are well-defined if there is no BMS-invariant definition anyway. If I misunderstand something very basic about classical GR, I would be very happy to be corrected about these.

• About 99.8% of mass of solar system is in the Sun, yet planets carry 98% of solar system's angular momentum. Taken to the limit, is it really surprising that “soft gravitons” can have zero energy yet nonzero angular momentum? Oct 20 at 4:19
• @A.V.S. not surprising, if you tell me that Minkowski space as full 4D vacuum solution can have nonzero angular momentum. In that case, solar system as a whole still has the same angular momentum as the planets alone (which are effectively massless). What it looked like to me was the vacua Strominger spoke of is more about 3D initial data sets, not the 4D vacuum of Einstein equations. Oct 20 at 4:32