Robert McNees
Mathematician Shing-Tung Yau (丘成桐), Fields Medalist who proved Calabi's conjecture on the existence of unique metrics with certain properties on compact, complex Kähler manifolds, and who proved the positivity of ADM mass in GR, was born #OTD in 1949.
Image: D. Ferus, Berlin
Robert McNees
Yau's proof of Calabi's conjecture came in 1977-78. An immediate consequence is the existence of Calabi-Yau manifolds, compact 6 dimensional (or 3 complex-dimensional) manifolds that admit a unique Ricci-flat metric Kähler metric.
https://www.pnas.org/doi/10.1073/pnas.74.5.1798
https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.3160310304
Robert McNees
Calabi-Yau manifolds are an essential ingredient of compactification scenarios in ten-dimensional string theories, where they describe the wrapped up shape of 6 of the 10 spacetime dimensions.
More info on this particular visualization, which shows a "slice” through a six-dimensional manifold, can be found here: http://members.wolfram.com/jeffb/visualization/stringtheory2.shtml
Visualization: Jeff Bryant
Robert McNees
(Also, the article that Bryant quotes, with its ancient Mathematica code, is available here starting on page 1156: https://www.ams.org/journals/notices/199411/199411FullIssue.pdf )
Robert McNees
It's a neat trick! A 10-dimensional string theory looks 4-dimensional at low energies, and the resulting physics inherits structures reminiscent of the standard model of particle physics.
Unfortunately, it introduces other problems. These include (broken) supersymmetry, which we have no evidence for, and – absent a mechanism for stabilizing them – light or massless scalar fields called "moduli" that pose problems for 5th force experiments and tests of the equivalence principle.
Robert McNees
In 1978, Yau and Schoen used variational methods to give the first proof of the Positive Energy Conjecture in general relativity. The paper was submitted in September, and appeared in print February of the following year.
Robert McNees
(Two years later, a young up-and-comer named Edward Witten would give a separate proof that used entirely different methods.)
https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-80/issue-3/A-new-proof-of-the-positive-energy-theorem/cmp/1103919981.full
Robert McNees
So what exactly did Schoen and Yau do in their paper?
In general relativity, there is no meaningful local notion of the energy density of the gravitational field. You can't just visit every point in a region, measure some quantity and add it to a running total, then call that result the total energy of a gravitating system.
Robert McNees
This is important. Statements like "the total energy of this configuration of a physical system has less energy than any other configuration of the system" are one of the ways we sort out stable configurations from unstable ones.
There is, however, a *global* notion of the total energy of an asymptotically flat gravitating system, introduced by Arnowitt, Deser, and Misner. Roughly speaking, asymptotically flat means there is a notion of "far away" where the spacetime looks like Minkowski space.
Robert McNees
The ADM result shows how the total energy of the spacetime is encoded in the diminishing strength of the gravitational field as you approach this far-away region. You just need the right mathematical procedure to tease that information out.
Now, Minkowski spacetime is empty and there's nothing happening there gravitationally. So you would expect it to be the lowest energy configuration in general relativity. Indeed, modulo assumptions, the ADM procedure assigns zero energy to Minkowski space.
Robert McNees
For Minkowski space to be stable, other configurations of the gravitational field should have positive energy. Otherwise, a nice flat spacetime could quantum mechanically transition to some weird and exotic configuration with lower energy!
But it's not clear from the ADM result that the energy for other asymptotically flat spacetimes is positive! It *seems* like that should be true in a sensible theory, but it's not obvious. What if there was a spacetime with less energy than Minkowski space?
Robert McNees
What if there was a spacetime with ~less~ energy than Minkowski space?
This isn't as outlandish as it sounds. When we do the accounting, contributions from matter (with certain assumptions) tend to increase the total energy. But gravitational effects can decrease it. Which contributions win out? How do we know the total has to be positive?
Could there be some weird gravitational configuration that's energetically more stable than empty space?
Robert McNees
Schoen and Yau showed that there is not. They proved that an asymptotically flat spacetime containing matter that satisfies the dominant energy condition (matter with a non-negative energy density) always has positive total energy. Only Minkowski space has zero total energy.
So our intuition is right. Modulo some physically reasonable assumptions, Minkowski spacetime is the lowest energy configuration among asymptotically flat spacetimes.
Robert McNees
(Schoen and Yau didn't know it at the time, but it looks like our Universe probably isn't an asymptotically flat spacetime. That's a topic for another thread.)
Robert McNees
Anyway, the (classical) positivity of the total energy for asymptotically flat gravitating systems and the uniqueness of Minkowski space as the lowest energy configuration were long-standing problems, which Schoen and Yau solved.
Robert McNees
Now here is an aside for physicists of a certain age.
Near the top of the thread I linked to an article cited by Bryant, who made that Calabi Yau gif. The article, on computer visualization of complex curves, is by A.J. Hanson.
This is the Hanson of "Gravitation, Gauge Theories, and Differential Geometry" by Eguchi, Gilkey, and Hanson.
Robert McNees
His interest in visualization makes me wonder if he also drew the famous cartoon that was Figure 3.1 in Eguchi, Gilkey, and Hanson.
Robert McNees
In any case, Hanson is emeritus at Indiana and you can find all sorts of wonderful visualizations on his webpage.(Though some of the formats and tech are outdated.)
https://legacy.cs.indiana.edu/~hanson/
Images: A.J. Hanson
Robert McNees
@SchnittGetsReal It's a classic reference, and I feel like folks who came of age (mathematical physics-wise) in the 80s often used it on us 90s and early 00s folks.
Jeremy Schnittman
@mcnees
beautiful! I'm embarrassed to say that I was not familiar with it, so I'm either not "of a certain age" or maybe a little further to the spacetime engineering (Weinberg, index-heavy) end of the GR spectrum
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