Mass and Angular Momentum, Left Ambiguous by Einstein, Get Defined

Mass and Angular Momentum, Left Ambiguous by Einstein, Get Defined

In 2008, the mathematicians Mu-Tao Wang of Columbia University and Shing-Tung Yau, now a professor at Tsinghua University in China and an emeritus professor at Harvard University, advanced a definition of quasilocal mass that has proved particularly fruitful. In 2015, it enabled them and a collaborator to define quasilocal angular momentum. And this spring, those authors and a fourth collaborator published the first-ever, long-sought definition of angular momentum that is “supertranslation invariant,” meaning it does not depend on where an observer is located or what coordinate system he or she chooses. With such a definition, observers can, in principle, take measurements of ripples in space-time generated by a rotating object and calculate the exact amount of angular momentum carried away from the object by these ripples, which are known as gravitational waves.

“It’s a great result,” Lydia Bieri, a mathematician and general relativity expert at the University of Michigan, said about the March 2022 paper, “and a culmination of intricate mathematical investigations over several years.” Indeed, the development of these facets of general relativity took not just years but many decades.

Staying Quasilocal

In the 1960s, Stephen Hawking came up with a definition of quasilocal mass that is still favored today in some circumstances owing to its simplicity. Seeking to calculate the mass enclosed by a black hole’s event horizon — its invisible spherical boundary — Hawking showed that you can calculate the mass inside any sphere by determining the extent to which incoming and outgoing light rays are bent by the matter and energy contained within. While “Hawking mass” has the virtue of being relatively easy to compute, the definition only works either in a space-time that is spherically symmetric (an idealized condition, as nothing in the real world is perfectly round) or in a “static” (and rather boring) space-time where nothing changes in time.

The search for a more versatile definition continued. In a lecture at Princeton University in 1979, the British mathematical physicist Roger Penrose, another pioneer of black hole physics, identified the task of characterizing quasilocal mass — “where one does not need to go ‘all the way to infinity’ in order for the concept to be meaningfully defined” — as the number-one unsolved problem in general relativity. A definition of quasilocal angular momentum ranked second on Penrose’s list.

Earlier that year, Yau and his former student Richard Schoen, who is now an emeritus professor at Stanford University, proved a major prerequisite for establishing these quasilocal definitions. Namely, they showed that the ADM mass of an isolated physical system — its mass as measured from infinitely far away — can never be negative. The Schoen-Yau “positive mass theorem” constituted an essential first step for defining quasilocal mass and other physical quantities, because space-time and everything in it will be unstable if its energy has no floor but instead can turn negative and keep dropping without limit. (In 1982, Yau won a Fields Medal, the highest honor in mathematics, in part for his work on the positive mass theorem.)

In 1989, the Australian mathematician Robert Bartnik offered a new definition of quasilocal mass that relied on that theorem. Bartnik’s idea was to take a region of finite size enclosed by a surface and then, by enveloping it with many layers of surfaces of ever-larger area, extend the finite region to one of infinite size so that its ADM mass can be computed. But the region can be extended in many ways, just as a balloon’s surface area could be blown up uniformly or stretched in various directions, each yielding a different ADM mass. The lowest value of ADM mass that can be obtained is, according to Bartnik, the quasilocal mass. “The argument would not have been possible before the positive mass theorem,” explained Wang, “because otherwise the mass could have gone to negative infinity,” and a minimum mass could never be ascertained.

Bartnik mass has been an important concept in mathematics, said the University of Connecticut mathematician Lan-Hsuan Huang, but its main drawback is a practical one: Finding the minimum is extremely difficult. “It is almost impossible to compute an actual number for the quasilocal mass.”

The physicists David Brown and James York came up with an entirely different strategy in the 1990s. They wrapped a physical system in a two-dimensional surface and then tried to determine the mass within that surface based on its curvature. One problem with the Brown-York method, however, is that it can give the wrong answer in a completely flat space-time: The quasilocal mass might turn out positive even when it should be zero.

Still, the approach was utilized in the 2008 paper by Wang and Yau. Building upon Brown and York’s work, as well as on research that Yau had carried out with the Columbia mathematician Melissa Liu, Wang and Yau found a way to bypass the problem of positive mass in totally flat space. They measured the curvature of the surface in two different settings: the “natural” setting, a space-time representative of our universe (where curvature can be rather complex), and a “reference” space-time called Minkowski space that is perfectly flat because it is devoid of matter. Any difference in the curvature between these two settings, they surmised, must be due to the mass confined within the surface — the quasilocal mass, in other words.


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