Category Archives: math

Maths Prove Slowly Rotating Black Holes Are Stable

Black holes. They're massive, they're mysterious, and they sound really eerie.

And up until now, we didn't know how stable they might be. Fortunately, 2022 has seen a monster achievement in mathematical physics. Quanta Magazine's Steve Nadis sets the intergalactic stage for a remarkable mathematical proof:

In 1963, the mathematician Roy Kerr found a solution to Einstein’s equations that precisely described the space-time outside what we now call a rotating black hole. (The term wouldn’t be coined for a few more years.) In the nearly six decades since his achievement, researchers have tried to show that these so-called Kerr black holes are stable. What that means, explained Jérémie Szeftel, a mathematician at Sorbonne University, “is that if I start with something that looks like a Kerr black hole and give it a little bump” — by throwing some gravitational waves at it, for instance — “what you expect, far into the future, is that everything will settle down, and it will once again look exactly like a Kerr solution.”

The opposite situation — a mathematical instability — “would have posed a deep conundrum to theoretical physicists and would have suggested the need to modify, at some fundamental level, Einstein’s theory of gravitation,” said Thibault Damour, a physicist at the Institute of Advanced Scientific Studies in France.

Would Einstein be proven wrong? As a wise man once said, "bet on Big Al and give the points". In this case, Team Einstein has chalked up a new win in the form of a mathematical proof that such slowly rotating black holes are stable. And all it took was a team of three mathematical physicists working for years to invent the needed mathematical tools before finally producing a single 912 page paper to prove it.

In a 912-page paper posted online on May 30, Szeftel, Elena Giorgi of Columbia University and Sergiu Klainerman of Princeton University have proved that slowly rotating Kerr black holes are indeed stable. The work is the product of a multiyear effort. The entire proof — consisting of the new work, an 800-page paper by Klainerman and Szeftel from 2021, plus three background papers that established various mathematical tools — totals roughly 2,100 pages in all.

We may be understating the amount of work that resulted in the proof by calling it a monster achievement. What we found particularly interesting was the path by which they reached the proof:

The three mathematicians relied on a strategy — called proof by contradiction — that had been previously employed in related work. The argument goes roughly like this: First, the researchers assume the opposite of what they’re trying to prove, namely that the solution does not exist forever — that there is, instead, a maximum time after which the Kerr solution breaks down. They then use some “mathematical trickery,” said Giorgi — an analysis of partial differential equations, which lie at the heart of general relativity — to extend the solution beyond the purported maximum time. In other words, they show that no matter what value is chosen for the maximum time, it can always be extended. Their initial assumption is thus contradicted, implying that the conjecture itself must be true.

Proof by contradiction is one of the oldest tools in maths. The mathematician Euclid used it to prove there are an infinite number of prime numbers roughly 2,300 years ago. Because it's a great introduction into how the tool of proof by contradiction works, here is a seven minute video of Trefor Bazett working through the basic steps Euclid followed to prove there are an infinite number of primes in Book IX of Elements:

One could reasonably argue that the multiyear effort to prove that slowly rotating Kerr black holes are stable really took hundreds, if not thousands, of years to reach.

We are however putting the cart before the horse at this point. These are preprint papers that are now going through peer review phase, so until it passes muster, the Giorgi-Klainerman-Szeftel proof is under challenge. That will almost certainly add some additional years to the total assuming it all comes together and the proof is confirmed.

References

Elena Giorgi, Sergiu Klainerman, Jeremie Szeftel. Kerr stability for small angular momentum. Arxiv. [PDF Document]. DOI: 10.48550/arXiv.2205.14808. 30 May 2022.

Sergiu Klainerman, Jeremie Szeftel. Wave equations estimates and the nonlinear stability of slowly rotating Kerr black holes. Arxiv. [PDF Document]. DOI: 10.48550/arXiv.2104.11857. 30 May 2022.

Image credit: Stable Diffusion Demo: "A rotating black hole in space". Generated 28 August 2022.

A rotating black hole in space - Generated by Stable Diffusion Demo

The Lurking Chaos in Ecosystems

There's been a remarkable development in the field of ecology. An established belief that chaotic dynamics are a relatively rare factor in shaping ecosystems has turned out to not be true.

The story of how that discovery was made is just as interesting as the discovery itself. Here's how Quanta Magazine's Joanna Thompson tells it:

Tanya Rogers was looking back through the scientific literature for recent studies on chaos in ecosystems when she discovered something unexpected: No one had published a quantitative analysis of it in over 25 years. “It was kind of surprising,” said Rogers, a research ecologist at the University of California, Santa Cruz and the new study’s first author. “Like, ‘I can’t believe no one’s done this.’”

So she decided to do it herself. Analyzing more than 170 sets of time-dependent ecosystem data, Rogers and her colleagues found that chaos was present in a third of them — nearly three times more than the estimates in previous studies. What’s more, they discovered that certain groups of organisms, like plankton, insects and algae, were far more prone to chaos than larger organisms like wolves and birds.

The signs of chaos had been there all along, lurking within the reams of accumulated ecological data. But earlier researchers had missed it because their models were too simple. It wasn't until Rogers and her fellow researchers applied more complex models that the telltale signs of chaotic influences could be teased out.

The new results from Rogers, Munch and their Santa Cruz mathematician colleague Bethany Johnson, however, suggest that the older work missed where the chaos was hiding. To detect chaos, the earlier studies used models with a single dimension — the population size of one species over time. They didn’t consider corresponding changes in messy real-world factors like temperature, sunlight, rainfall and interactions with other species that might affect populations. Their one-dimensional models captured how the populations changed, but not why they changed.

But Rogers and Munch “went looking for [chaos] in a more sensible way,” said Aaron King, a professor of ecology and evolutionary biology at the University of Michigan who was not involved in the study. Using three different complex algorithms, they analyzed 172 time series of different organisms’ populations as models with as many as six dimensions rather than just one, leaving room for the potential influence of unspecified environmental factors. In this way, they could check whether unnoticed chaotic patterns might be embedded within the one-dimensional representation of the population shifts. For example, more rainfall might be chaotically linked to population increases or decreases, but only after a delay of several years.

In the population data for about 34% of the species, Rogers, Johnson and Munch discovered, the signatures of nonlinear interactions were indeed present, which was significantly more chaos than was previously detected. In most of those data sets, the population changes for the species did not appear chaotic at first, but the relationship of the numbers to underlying factors was. They could not say precisely which environmental factors were responsible for the chaos, but whatever they were, their fingerprints were on the data.

This is exactly the kind of study that spawns new research. The effort to find out what factors are at play and how they interact with each other will shape generations of work in the now understood to be underdeveloped field of ecological population growth dynamics.

We would expect the initial phase of that new work to resemble the equivalent of a design of experiments in statistics to verify which factors are most influential, followed by more detailed studies into the effects of their interactions over time. It's an exciting development for a field that's now coming out of a period of stagnation no one knew it was in as a result of the discovery.

More Information

For some basic information on how chaos can influence ecological population dynamics, we found Numberphile's 19-minute video on the Feigenbaum constant provides a nice introduction to the surprisingly simple population modeling math that produces chaotic outcomes:

For more background, we've also built a tool to model the chaotic growth of the population of a species over time. Our post also features Veratiseum's video exploring the logistic map and its role in the emergence of complexity.

We can also point you to HHMI Bioactive's Population Dynamics simulator, which features a good primer on the simpler logistic growth model math that wasn't capturing the extent of chaotic influences found by Rogers, Johnson and Munch.

References

Rogers, T.L., Johnson, B.J. & Munch, S.B. Chaos is not rare in natural ecosystems. Nature Ecology & Evolution. Volume 6, pp 1105–1111. (2022). DOI: 10.1038/s41559-022-01787-y.

Maxwell’s Equations

In 1985, Caltech's David Goodstein introduced The Mechanical Universe, which stands as one of the best explanatory series on physics ever committed to video. The following 28-minute video telling the story of James Maxwell's mathematical insights linking electricity and magnetism represents one of the highlights of the 52-part series.

Maxwell's equations opened the door to great practical applications, which we find all around us today in the electronic devices that make the world of today so different from how it was when Maxwell first formulated them.

HT: Steven Strogatz, who also points to YouTube's playlist for the entire series.

Words for Describing Probabilities

When you try to describe how likely something that will happen in the future is to someone, how do they interpret what you mean?

For instance, if you say something will a "slam dunk", will they interpret that as having 100% odds of happening? Or will they assign a lower chance to whatever that is occurring?

What if you're the person on the receiving end of the probabilitistic statement? Would you say something you're told has a "real possibility" of occurring is more or less likely to happen than if the same person told you something had a "serious possibility" of occurring?

Words mean things, and when it comes to describing probabilities, they come with their own probability distributions. That's the finding of Andrew Mauboussin and Michael J. Mauboussin from their 2018 paper, If You Say Something Is “Likely,” How Likely Do People Think It Is?, in which they presented the results of their study into that topic. Better yet, they provided the following chart to illustrate the probability distributions the participants in their study helped them develop for various common words and phrases that American English speakers use in everyday language.

Here's how the Mauboussins describe their findings:

The wide variation of likelihood people attach to certain words immediately jumps out. While some are construed quite narrowly, others are broadly interpreted. Most — but not all — people think “always” means “100% of the time,” for example, but the probability range that most attribute to an event with a “real possibility” of happening spans about 20% to 80%. In general, we found that the word “possible” and its variations have wide ranges and invite confusion.

The cool thing about this chart is that if you are searching for words to describe the likelihood of something that will happen in the future, you now have a useful guide to help you convey the odds you're trying to communicate.

More often than not, with a high probability of getting your intended message across.

Homer Simpson Is Made of Maths

Let's start with two parametric equations, x(t) and y(t), as given by the following formulations:

Homer Simpson-like Curve Parametric Equations x(t) and y(t)

If you plot the results of these equations for values of t from 0 through 40π, using an online application like Wolfram Alpha, you'll get the following surprising output:

Homer Simpson-like Curve Parametric Equations x(t) and y(t)

This, of course, is just another unique intersection between maths and The Simpsons, which runs very deep throughout the animated show's 31+ year run.