Prescriptions for measuring and transporting local angular momenta in general relativity

I, along with my collaborators David Nichols, Justin Vines, and Éanna Flanagan, posted our latest preprint, Prescriptions for measuring and transporting local angular momenta in general relativity, on the arXiv tonight. After I tweeted the link, George informed me that I was now obligated to blog about it:

@Vagelford Let me guess, blog about it on the RemarXiv?

— Leo C. Stein (@duetosymmetry) February 8, 2016

Ok, so here goes a little self-promotion!

This paper is a refinement on David and Éanna's earlier paper. The topic at hand is angular momentum in general relativity.

So, how is angular momentum defined in GR? One quantity that everyone agrees upon is the ADM angular momentum, which is precisely the Noether charge associated with the rotational symmetry of asymptotically flat spacetime. Masha recently touched upon the ADM mass in this post.

Angular momentum is related to a symmetry, so of course we must care about it. From Emmy Noether's theorem, we know that this total angular momentum is conserved. In a sense we can say that the ADM angular momentum (and mass) "live at $$i^0$$," the point at spacelike infinity.


In this paper, we were studying a different type of angular momentum. Instead of living at $$i^0$$, we are interested in a quantity near $$\mathcal{I}^+$$ (pronounced "scri plus"), the region of future null infinity. Here we're talking in the language of conformally compactifying an asymptotically flat spacetime, so we can squeeze the whole thing into a Penrose diagram like this one. Every spacetime which is asymptotically flat has the same structure "far away" from all the curvy bits of spacetime.

Why do we want to study angular momentum near scri? One reason: it's very natural to think of angular momentum decreasing when e.g. gravitational radiation carries it away from a system—like a black hole binary, allowing the two black holes to eventually merge! But the ADM angular momentum is conserved, so what gives? To understand this, you have recognize that the ADM quantity comes from integrating along a "Cauchy surface," a hypersurface that's everywhere spacelike and makes it out to $$i^0$$, like any of the $$\Sigma_{1,2,3}$$ in the Penrose diagram.

Spacelike infinity ($$i^0$$) is not the right setting to discuss things like how much angular momentum is carried away by gravitational waves, because the ADM angular momentum can't change! Instead of talking about quantities that "live at" $$i^0$$, we want to talk about quantities that "live at" $$\mathcal{I}^+$$.

Future null infinity ($$\mathcal{I}^+$$) is much more complicated than $$i^0$$. Spacelike infinity is kind of "rigid," while $$\mathcal{I}^+$$ is comparatively "floppy," though less floppy than the interior (bulk) of the spacetime. These statements can be made mathematically precise in terms of the symmetry groups of these spaces. The symmetry group of $$\mathcal{I}^+$$ is the famous BMS group, named for Bondi, van der Burg, Metzner, and Sachs, who studied its structure in the 60s. The BMS group has enjoyed a renewed interest in recent years, when it was discovered that BMS symmetries, Weinbergesque soft theorems, and long-ranged "memories" are different faces of the same underlying physics.

So, if you want to discuss angular momentum at $$\mathcal{I}^+$$, the BMS group is going to tell you the mathematical rules you have to follow (technically: quantities must "live in" representations of the group). For a while, the literature referred to something called the BMS angular momentum "ambiguity." However, ambiguity is not really the right word. Angular momentum is not ambiguous, it just transforms in a much more complicated way (under BMS transformations) than angular momentum does in flat (Minkowski) spacetime).

Ok, enough with the longwinded background. What did we actually do in our paper?

First of all, we worked in a simplified setting: when the region of interest of $$\mathcal{I}^+$$ is approximately stationary. In this setting, angular momentum has a much simpler transformation law when you move from point to point than in the fully dynamical setting.

Let's consider what happens when you expand around stationary $$\mathcal{I}^+$$ in powers of $$1/r$$. To leading order, the only property of spacetime is the Bondi mass. When spacetime is stationary, this is constant, and every stationary asymptotically flat vacuum spacetime is identical to Schwarzschild expanded to this order. When you go to next-to-leading order, you learn one additional property of the spacetime, which is like an angular momentum of the spacetime. Here is a crucial fact we make use of: every stationary asymptotically flat spacetime vacuum is identical to the Kerr spacetime expanded to this order.

Now, in the Kerr spacetime, it just so happens that there's a pair of tensors $$(\xi^a,{}^{*}\!f^{ab})$$ which satisfy the pair of differential equations:
\[
\nabla_a \xi^b &=  -\frac{1}{4} R^{b}{}_{acd} {}^{*}\!f^{cd} \, , \\
\nabla_a {}^{*}\!f^{bc} &= -2 \xi^{[b} {\delta^{c]}}_a \, .
\]
We can interpret these equations as a rule for how to transport a linear momentum $$\xi$$ and an angular momentum $${}^{*}\!f$$ from point to point in a consistent fashion (the quantity $${}^{*}\!f$$ is not really an angular momentum, but that's another topic).

So here's hope that a bunch of observers hanging out in spacetime can locally measure what they think are the linear momentum $$P^a$$ and angular momentum $$J^{ab}$$ of the spacetime about them. And moreover, if Alice measures her quantities $$(P,J)_A$$, and Bob measures his quantities $$(P,J)_B$$; and then Bob drags his quantities over to Alice in a certain well-defined way, then their two quantities will agree, up to the required accuracy!

David Nichols has laid out the 8-step procedure for locally measuring $$(P,J)$$ up to the required accuracy, which is really impressive to me.

My contribution to the calculation was the fiber bundle approach, which really makes it quite easy to compute the holonomy of this transport law, and makes clearer what are the necessary and sufficient conditions for the existence of consistent solutions to the transport differential equations. The fiber bundle idea is represented pictorially in the first image at the top of this post.

Read the paper for all the gory details! It's only 9 pages long. Satisfaction guaranteed or your money back.