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

Einstein's theory of gravity has passed every experiment it has been subjected to. The way space and time curve in the presence of mass and energy has been shown to transform our understanding of gravity. The theory has achieved stunning successes, from the confirmation in 1919 that light bends in the sun's gravity to the discovery of a black hole in 2019. General relativity is still a work in progress.

The theory doesn't offer a simple or standard way of determining the mass of an object. It is difficult to define a concept using a measure of an object's rotation.

Some of the difficulties are related to a feedback loop. The matter and energy curve the space-time continuum, but this curvature becomes a source of energy itself, which can cause additional curvature. There is no way to separate an object's mass from its energy. One must have a firm grip on mass in order to define momentum.

Einstein didn't fully explain what mass is or how it can be measured. The first rigorous definition was proposed in the 1960's. The physicists defined the mass of an isolated object, such as a black hole, as viewed from almost infinitely far away, where space-time is almost flat.

This method of calculating mass doesn't allow physicists to quantify the mass within a finite region. They want to know the mass of each individual black hole prior to the merging of the two black holes, as opposed to the system as a whole. Quasilocal mass is defined as the mass enclosed within any individual region, which is measured from the surface of that region.

The definition of quasilocal mass was advanced by Mu-TAo Wang of Columbia University and Shing-Tung Yau of Tsinghua University. They and a partner were able to define quasilocal. The first-ever, long-sought definition of angular momentum that is "supertranslation invariant meaning" was published this spring by those authors and four others. Observers can use this definition to measure ripples in space-time generated by a rotating object and calculate the amount of inertia carried away from the object by these ripples, which are known as gravitational waves.

Lydia Bieri, a mathematician and general relativity expert at the University of Michigan, said that the paper was a culmination of several years of hard work. It took a long time for these aspects of general relativity to be developed.

Staying Quasilocal

Stephen Hawking came up with a definition of quasilocal mass in the 1960's that is still used today. In order to calculate the mass enclosed by a black hole's event horizon, you need to know the extent to which incoming and outgoing light rays are bent by the matter and energy within. The definition of "hawking mass" only works in a space-time that is spherically symmetric, or in a "static" space, which is boring.

A more versatile definition was being searched for. One of the pioneers of black hole physics, Roger Penrose, identified the task of defining quasilocal mass in a 1979 lecture. The second definition on Penrose's list was a definition of quasilocalangular momentum.

The founding of these quasilocal definitions was made possible by Yau and his former student Richard Schoen. They demonstrated that the mass of an isolated physical system can never be negative. The positive mass theorem is an essential first step for defining quasilocal mass and other physical quantities because space-time and everything in it will be unstable if it doesn't have a floor. The Fields Medal, the highest honor in mathematics, was won by Yau.

Robert Bartnik came up with a new definition of quasilocal mass. Bartnik wanted to extend a region of finite size enclosed by a surface to one of infinite size so that it could be computed. Just as a balloon can be blown up uniformly or stretched in various directions, the region can be extended in manyTrademarkiaTrademarkiaTrademarkias. According to Bartnik, the quasilocal mass is the lowest value of the mass. Wang said that the argument wouldn't have been possible if the mass could have gone to negative infinite and a minimum mass couldn't be determined.

The main disadvantage of Bartnik mass is that finding the minimum is very difficult. It is not possible to calculate an actual number for the mass.

In the 1990s, physicists David Brown and James York came up with a new strategy. They tried to determine the mass of the surface by wrapping it in a two-dimensional surface. The problem with the Brown-York method is that it can give the wrong answer in a flat space-time.

The approach was used by Wang and Yau. Wang and Yau were able to find a solution to the problem of positive mass in flat space thanks to the work of Brown and York. The two settings they used to measure the surface's curve were the natural setting and a space-time representative of our universe. The quasilocal mass of the surface must be the reason for the difference between the settings.

They stated in the paper that their definition fulfilled all the requirements needed for a valid definition. Wang said that their approach suffers from one feature that restricts its applicability. The approach is tiring in practice.

Ambiguous Angles

In 2015, Wang and Yau collaborated with Po-Ning Chen of the University of California, Irvine to define quasilocal angular momentum. The inertia of an object moving in a circle is given by its mass times the circle's radius. It is a useful quantity because it is not created or destroyed. Physicists can track the exchange of momentum between objects and the environment to understand a system's dynamics.

Wang, Yau and Chen needed a definition of quasilocal mass and detailed knowledge of how rotation works in space-time in order to define the quasilocal angular momentum. They chose Minkowski space-time because it is unerringly flat and therefore has the property of rotation symmetry. The researchers were able to define quasilocal angular momentum in a way that doesn't depend on where the x, y, and t axes intersect. They established a one-to-one correspondence between points on the surface in Minkowski space-time and points on the same surface when placed in its natural space-time.

Physicists prefer things that are easy to compute.

The trio joined forces with Ye-Kai Wang of National Cheng Kung University to tackle a problem that had remained unsolved for about 60 years: how to characterize the angular momentum swept away by gravitational waves. The measurement needs to be far from the maelstrom rather than in close proximity to the black hole merger. The ultimate destination of radiation is referred to as null infinite, a concept invented by Penrose.

As often happens in general relativity, there is a new problem that arises, which is the effect of the direction of the observer's coordinate system. When the waves travel through space-time, they leave a permanent imprint. Waves will expand space-time in one direction and contract it in the other, but space-time never goes back to its original state. Eanna Flanagan is a general relativist at Cornell University. Observers won't know that they've been moved by the waves

Even if different observers agree on where the coordinate system came from, they won't agree after the waves jiggled things. Supertranslations are the result of uncertainty in their assessments. One way to understand supertranslations is that the mass of an object and its speed will not be affected by a passing wave. Depending on the orientation of the radius relative to one's coordinate system it might seem to be stretched out.

Conserving physical quantities shouldn't change based on how we label them. The situation that Chen, Wang, Wang, and Yau wanted to change was that. They used their definition of a region of finite radius as a starting point for their calculation. They took the limit of that quantity and turned it into a supertranslation invariant quantity. The first ever supertranslation invariant definition of angular momentum was published in March in Advances in Theoretical and Mathematical Physics.

"This is a wonderful paper and a wonderful result, but the question is, how useful is it?" said Marcus Khuri, a mathematician. He said that the new definition is hard to compute and that physicists don't like things that are hard to compute.

A Unique Choice

It is hard to compute in general relativity. Most of the time it is not possible to solve the equations that Einstein created in 1915. Researchers rely on the power of computers to come up with solutions. They break space-time into small grids and estimate the curve of each grid at different times. Adding more grids can make their approximations even better.

Researchers can use these approximations to calculate the mass and momenta of merging black holes. Current observations are not accurate enough for the subtle differences caused by supertranslations to be seen. When the accuracy of our observations improves, those considerations will be more important. He said that the order could be improved as soon as the 20th century.

Supertranslations are not a problem that needs to be fixed but rather are inevitable properties of general relativity that we need to live with.

Robert Wald, a general relativity specialist at the University of Chicago, said that supertranslations are more of an "inconvenience" than a real problem. He carefully reviewed the Chen, Wang, Wang and Yau paper and concluded that the proof holds up. Wald said that it was nice to have a "unique choice" to pick out.

Yau has been working on defining these quantities for over 40 years. He said that it can take a long time for ideas from mathematics to physics to come to fruition. Even if the new definition of angular momentum isn't used anymore, scientists at LIGO and Virgo are still calculating. It is good to know what you are attempting to approximate.

Po-Ning Chen is supported by the Simons Foundation and gets funding from them.