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


General relativity, Albert Einstein's grand theory of gravity, has stood the test of time more than a century after it was first proposed. Our understanding of gravity has changed as a result of general relativity, which shows gravity as a result of how space and time curve in the presence of mass and energy rather than as an attracting force between enormous objects as was previously believed. From 1919 measurements that proved light bends in the gravitational field of the sun to 2019 observations that showed the silhouette of a black hole, the theory has gained astounding victories. The fact that general relativity is still being developed may surprise some people.

The theory does not provide a straightforward or uniform method of calculating an object's mass, despite the fact that the equations Einstein proposed in 1915 deal with the curvature caused by large objects. Even more difficult to define is the idea of angular momentum, which is a gauge of an object's rotating motion in space-time.

General relativity has a feedback loop, which contributes to some of the problems. The space-time continuum is curved by matter and energy, yet this bending generates energy on its own and can lead to more bending, a phenomena known as the "gravity of gravity." Furthermore, it is impossible to distinguish between an object's inherent mass and the additional energy brought on by this nonlinear effect. Furthermore, without a good understanding of mass, it is impossible to define momentum or angular momentum.

Although he acknowledged the difficulties in defining and measuring mass, Einstein never went into great detail. The first precise definition wasn't put forward until the late 1950s and early 1960s. The mass of an isolated object, such as a black hole, as seen from a nearly infinite distance, when space-time is almost flat and the object's gravitational pull is essentially nil, was defined by physicists Richard Arnowitt, Stanley Deser, and Charles Misner.

This method of determining mass, called "ADM mass" after its creators, has been beneficial, however it does not enable scientists to measure the mass inside a limited space. Imagine, for example, that they are investigating the merger of two black holes and would like to know the mass of each black hole separately rather than the mass of the system as a whole. "Quasilocal mass" refers to the mass contained inside any specific location as measured from that region's surface, where gravity and space-time curvature may be quite high.

The notion of quasilocal mass first out in 2008 by the mathematicians Mu-Tao Wang of Columbia University and Shing-Tung Yau, who is currently a professor at Tsinghua University in China and an emeritus professor at Harvard University, has shown to be particularly useful. It allowed them and a partner to develop quasilocal angular momentum in 2015. The first ever, long-awaited definition of angular momentum that is "supertranslation invariant," or independent of an observer's location or choice of coordinate system, was published this spring by those authors and a fourth collaborator. With such a definition, observers might theoretically measure the gravitational waves, which are ripples in space-time caused by spinning objects, and determine the precise amount of angular momentum transferred away from the rotating object.

The March 2022 study, according to mathematician and general relativity specialist Lydia Bieri of the University of Michigan, "is a wonderful result and the conclusion of detailed mathematical research spanning several years." It is true that it took several decades, rather than simply years, to establish certain aspects of general relativity.

keeping it local
Due to its clarity, Stephen Hawking's description of quasilocal mass from the 1960s is still used in some situations today. Hawking demonstrated that you can calculate the mass inside any sphere by figuring out the degree to which incoming and outgoing light rays are bent by the matter and energy contained therein. This was done in an effort to calculate the mass enclosed by a black hole's event horizon—its invisible spherical boundary. Even though the concept of "Hawking mass" is rather simple to calculate, it only holds true in two idealized scenarios: a space-time that is spherically symmetric (because nothing in the real world is exactly round) or a "static" (and quite dull) space-time where nothing changes over time.

The hunt for a more adaptable definition persisted. The first unsolved problem in general relativity, according to British mathematical physicist and pioneer of black hole physics Roger Penrose, is to characterize quasilocal mass, where "one need not go 'all the way to infinity' in order for the concept to be meaningfully defined." Penrose made this claim in a 1979 lecture at Princeton University. On Penrose's list, a definition of quasilocal angular momentum came in second.

Yau and Richard Schoen, a former student who is now an emeritus professor at Stanford University, demonstrated a crucial condition for creating these quasilocal concepts earlier that year. In particular, they demonstrated that an isolated physical system's ADM mass, or mass as measured from an infinite distance away, is always positive. The Schoen-Yau "positive mass theorem" was a crucial first step in establishing quasilocal mass and other physical quantities because without a floor for its energy, which might turn negative and continue falling without end, space-time and everything inside it will be unstable. (Yau received the Fields Medal, the top award in mathematics, in 1982 in part for his contributions to the positive mass theorem.)
The theory was used as the foundation for a new definition of quasilocal mass proposed by the Australian mathematician Robert Bartnik in 1989. In order to calculate the ADM mass of a region with an infinite extent, Bartnik proposed to start with a region of finite size contained by a surface and then to extend it by numerous layers of surfaces of ever-larger area. However, the region may be expanded in several ways, much like a balloon's surface area can be stretched in multiple directions or inflated equally, each producing a distinct ADM mass. According to Bartnik, the quasilocal mass is the lowest possible value for the ADM mass. Before the positive mass theorem, Wang said, "the argument would not have been feasible since otherwise the mass may have gone to negative infinity," making it impossible to determine a minimum value.

According to mathematician Lan-Hsuan Huang of the University of Connecticut, the Bartnik mass has been a significant mathematical notion, but it has one major practical flaw: Finding the minimal is quite challenging. The quasilocal mass is difficult to calculate in precise terms.

In the 1990s, the physicists David Brown and James York developed a completely different approach. They attempted to calculate the mass contained within a two-dimensional surface that encased a physical system using the surface's curvature. The quasilocal mass could be positive even though it should be zero, which is a drawback of the Brown-York technique in a space-time that is fully flat.

However, Wang and Yau used the strategy in their 2008 work. Wang and Yau discovered a solution to get around the issue of positive mass in completely flat space by building on the work of Brown and York as well as the study that Yau had conducted with the Columbia mathematician Melissa Liu. Both the "natural" environment, a space-time representative of our world (where curvature can be rather complicated), and a "reference" space-time termed Minkowski space, which is completely flat since it is devoid of matter, were used to quantify the surface's curvature. They reasoned that any variation in curvature between these two settings must be caused by the mass contained inside the surface, or the quasilocal mass.

According to their publication, their concept met "all the parameters essential for a legitimate definition of quasilocal mass." However, their strategy has a drawback that limits its applicability: Despite being extremely exact, Wang noted that "it invariably entails solving numerous extremely challenging nonlinear equations." The strategy is sound in principle, but difficult to implement.

Uncertain Angles
In 2015, Wang and Yau set out to establish quasilocal angular momentum with the help of Po-Ning Chen from the University of California, Riverside. According to classical physics, an object's angular momentum is simply equal to its mass times its speed times the radius of the circle. Because it is conserved, or travels between objects without ever being generated or destroyed, it is a valuable number to measure. To understand the dynamics of a system, physicists can observe how angular momentum is transferred between objects and the surrounding space.

Wang, Yau, and Chen required two things in order to describe the quasilocal angular momentum contained inside a surface: a definition of quasilocal mass, which they already possessed, and in-depth understanding of how rotation functions in space-time. As previously, they started by embedding their surface in the most basic environment possible: Minkowski space-time. This environment was selected because it is perfectly flat and has the feature of rotational symmetry, which means that every direction seems the same. Researchers were able to define quasilocal angular momentum thanks to rotational symmetry in a way that is independent of the location of the coordinate system's origin while measuring velocities and distances (the origin is the point where the x, y, z, and t axes intersect). Then, to guarantee coordinate independence in both settings, they constructed a one-to-one correspondence between points on the surface in Minkowski space-time and points on the same surface when put in its original (natural) space-time.

In order to characterize the angular momentum carried away by gravitational waves, such as those released as two black holes spiral together and violently coalesce, the trio then teamed up with Ye-Kai Wang of National Cheng Kung University to work on a problem that had remained unsolved for about 60 years. The measurement would have to be made distant from the maelstrom rather than right next to the merging black holes, therefore their notion of quasilocal angular momentum would not be applicable. The correct vantage point is "null infinity," a term Penrose used to describe the final location of radiation that is moving externally, including gravitational and electromagnetic radiation.

A fresh issue surfaces, as is typical in general relativity: Even if observed at null infinity (or a distance distant enough to be a plausible replica), the angular momentum carried by gravitational waves may appear to fluctuate depending on the origin and direction of the observer's coordinate system. The challenge is caused by the "gravitational wave memory effect," which refers to the fact that gravitational waves leave a lasting impression as they move across space-time. Waves cause space-time to enlarge in one direction and compress in the other direction (this is the signal that gravitational-wave observatories like LIGO and Virgo have been able to detect), but space-time never returns to its original condition exactly. Eanna Flanagan, a general relativist from Cornell University, stated that passing gravitational waves alter the distance between things. The waves might also slightly jolt spectators. They won't be aware that they have been transferred, though.

It follows that even if several observers initially agree on the location of their coordinate system's origin, they won't once gravitational waves have moved everything about. This ambiguity in turn causes "supertranslations," or ambiguities, in the estimates of angular momentum made for each system. The fact that the radius of an object's rotational motion will be warped by a passing gravitational wave but its mass and velocity won't is another approach to interpret supertranslations. Different alternative estimations of angular momentum can be made depending on how the radius seems to be stretched out or shortened by gravitational radiation in relation to one's coordinate system.

Physical quantities that have been conserved shouldn't change or appear to change dependent on our labeling preferences. Chen, Wang, Wang, and Yau wanted to change such scenario. They calculated the angular momentum contained in an area of limited radius beginning with their concept of quasilocal angular momentum from 2015. The coordinate-independent quasilocal definition was then transformed into a supertranslation invariant quantity at null infinity by taking that quantity's limit as the radius approached infinity. In theory, one might ascertain the angular momentum transported away by the gravitational waves released during a black hole merger using this first-ever supertranslation invariant definition of angular momentum, which was published in March in Advances in Theoretical and Mathematical Physics.

Marcus Khuri, a mathematician at Stony Brook University in New York, remarked, "This is a fantastic study and a wonderful discovery, but the issue is, how practical is it?" According to him, "generally speaking, physicists don't like things that are hard to compute," and the new definition is abstract and difficult to compute.

A Different Option
However, one essentially inevitable aspect of general relativity is its difficulty in computation. Except in extremely symmetrical circumstances, it is typically not even possible to precisely solve the nonlinear equations that Einstein proposed in 1915. Instead, to get approximations of the answers, researchers use supercomputers. By segmenting space-time into tiny grids and calculating the curvature of each grid independently and at different points in time, they are able to solve the problem. Additional grids can improve their estimates, similar to more pixels on a high-definition television.

Based on the gravitational-wave signals discovered by the LIGO and Virgo observatories, researchers may approximate the masses and angular moments of merging black holes or neutron stars. According to Vijay Varma, a scientist with the LIGO team at the Max Planck Institute for Gravitational Physics in Potsdam, Germany, present studies of gravitational waves are not precise enough to pick up on the minute variations brought on by supertranslations. But those factors will become more significant as our observations' precision increases by a factor of ten, Varma added. He emphasized that advancements of such magnitude may be made as early as the 2030s.

Supertranslations, according to Flanagan, are "not a problem that has to be rectified" rather, they are unavoidable characteristics of angular momentum in general relativity that we must accept.

Supertranslations are more of a "inconvenience" than a real issue, according to University of Chicago general relativity expert Robert Wald, who shares some of Flanagan's views. But after closely examining the Chen, Wang, Wang, and Yau work, he comes to the conclusion that the evidence is still valid. Wald stated, "It truly is resolving the supertranslation ambiguity," adding, "In general relativity, it is wonderful to have a "unique choice" to select out from when you have all these other meanings to choose from."

Yau, who has been attempting to define these terms since the 1970s, adopts a long-term perspective. "Ideas from mathematics can take a long time to infiltrate physics," he remarked. He said that even if the new definition of angular momentum isn't employed right away, LIGO and Virgo researchers "are always approximating something. But in the end, it's important to understand what you're attempting to mimic.
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