Hobgoblins in Our Equations

Gödel’s Lost Letter and P=NP 2019-11-01

With consequences for physics

Britannica source

Paul Painlevé was a French mathematician who specialized in classical mechanics. He is known for a conjecture in 1895 about anomalies in Isaac Newton’s equations that was proved in 1988.

Tonight, Ken and I celebrate Halloween with some mathematical horrors that may have real physical consequences.

Painlevé is also known for having been Prime Minister of France for two short stints in 1917 and 1925. He was also Minister of War under President Raymond Poincaré, a cousin of Henri Poincaré. He was the first Frenchman in powered flight—alongside Wilbur Wright in 1908—and at the end of his life headed the French Air Force.

A basic fact about Newtonian gravity is that the gravitational potential energy of two point masses ${m}$ and ${M}$ is proportional to

$\displaystyle -\frac{mM}{R},$

where ${R}$ is the distance between them. We usually think of how gravity becomes weaker as ${R}$ grows but when ${R}$ is tiny it becomes quite strong. Since the potential is negative, it is possible for an individual particle in a finite ${n}$ -body system to accelerate to arbitrary speed without violating the conservation of energy. But can it happen in a finite amount of time—and without ${R}$ actually becoming zero in a collision?

Painlevé proved a ‘no’ answer for ${n = 3}$ but suspected ‘yes’ for ${n \geq 4}$ . Zhihong Xia proved in 1988 it can happen for ${n = 5}$ , extending earlier advances by his thesis advisor, Donald Saari. The case ${n = 4}$ is still open; in all cases, the initial conditions for it to happen form a measure-zero subset of the configuration space.

The difference from ${n = 3}$ is shown by this diagram from their 1995 AMS Notices paper, which has a greatly understandable telling of the whole story. We, however, envision a fantasy story about what could have happened much earlier.

A Fantasy

We want to imagine that Xia’s result was proved not near the end of the 20th century but near the start—in particular, before Albert Einstein’s creation of General Relativity, but after Special Relativity. Say the result was obtained in 1908 by Painlevé after his flight with Wright. That same year, Hugo von Zeipold proved a startling consequence of Painlevé’s conjecture, which we now know to be a theorem:

Theorem 1 Under Newtonian gravity, a ${n}$ -body set of point masses can eject a particle to infinity in finite time.

That is, without collisions, a Newtonian ${n}$ -body problem of point masses can create separations ${R'}$ between particles that grow to infinity within a finite time. Saari and Xia say that the effect can be partially seen if you do the following:

Place a tennis ball on top of a basketball and drop the pair from chest height. The tennis ball rebound is pretty dramatic—enough so that it should be done outside.

Ken tried it and it works. It does not work so well if the tennis ball is replaced by a piece of Halloween candy. If the basketball is replaced by a pumpkin, it definitely will not work.

We know already, however, from Einstein’s theory of special relativity that it cannot work. We would have an instant before the singular time at which the point mass has just passed the speed of light. At that time it has acquired infinite energy according to special relativity, but cannot have withdrawn it from the potential by then.

In 1908 this would have been an internal indication that Newton’s theory of gravity must break down. There were of course external indicators, such as the theory’s incorrect prediction of the orbit of Mercury. Maybe internal ones were known, but this seems most glaring in retrospect. Are we right in this interpretation? The result by Xia is shocking enough as it stands, and makes us wonder what other surprises lurk in equations.

It Could End With a Bang

Other possibilities of singularities happening within finite time have not been ruled out by any physical theories. The New York Times Magazine in July 2015 ran a profile titled “The Singular Mind of Terry Tao” with a great opening sentence:

This April, as undergraduates strolled along the street outside his modest office on the campus of the University of California, Los Angeles, the mathematician Terence Tao mused about the possibility that water could spontaneously explode.

Indeed, Tao proved it can happen under a plausible modification of the Navier-Stokes equations of fluid dynamics. He writes there:

Intriguingly, the method of proof in fact hints at a possible route to establishing blowup for the true Navier-Stokes equations, which I am now increasingly inclined to believe is the case (albeit for a very small set of initial data).

As with Painlevé’s conjecture—Xia’s theorem—the point would be not that the initial conditions could happen with any perceptible probability, but that our world is capable of their happening at all. At least, that is, with the equations by which we describe our world. Our own mention at the start of 2015 alluded to the possibility of Tao’s blowup applying to fluid-like fields in cosmological theories.

Just last week, a column on the Starts With a Bang alerted us to an issue with equations that could be skewing cosmological theories today. The blog is written by Ethan Siegel for Forbes and its items are linked regularly on RealClear Science. Siegel draws an analogy to a phenomenon with Fourier series that feeds into things Dick and I (Ken writing this part) have already been thinking about. Things used all the time in theory…

Hobgoblins You Can Hear

The Fourier phenomenon is named for the American physicist Josiah Gibbs, but Gibbs was not the original discoverer. We could add this to our old post on a law named for Stephen Stigler, who did not discover it, that no scientific law is named for its original discoverer. The first discoverer of the Gibbs Phenomenon was—evidently—Henry Wilbraham in 1848. Through the magic of the Internet we can convey it by lifting Wilbraham’s original figure straight from his paper:

What this shows is the convergence of a sum of sine waves to a square wave. The convergence is pointwise except at the jump discontinuities ${x = t\pi}$ , but it is not uniform. The ${y}$ -values do not converge on any interval crossing ${t\pi}$ but instead rise about ${18\%}$ above the square wave function value ${f(x)}$ in the vicinity of ${t\pi}$ , no matter how many terms are summed in the approximations ${f_n}$ . Wilbraham’s middle drawing depicts this in finer detail than any other rendering I have found. The persistent overshoot is physically real—it is the cause of ringing artifacts in signal processing.

Now the convergence does satisfy a criterion of ${\epsilon}$ -approximation of a function ${f}$ that we use all the time in theory: for any ${\epsilon > 0}$ and large enough ${n}$ , ${|f_n(x) - f(x)| < \epsilon}$ except for an ${\epsilon}$ fraction of ${x}$ in the domain. This kind of convergence is used internally in the proofs of quantum algorithms for linear algebra which we recently discussed. If the value ${f(x)}$ is explicitly what you’re after, this is fine. But if you use the value only implicitly while composing the approximation with some other function ${g}$ , you must beware that the compositions ${g_n = g\circ f_n}$ are not thrown off in a constant way by the overshoots.

Dark Matter in Shadows?

Siegel draws attention to what is alleged as something similar actually happening to current physical theories that use a well-known class of algorithmic simulations. The details are in a new paper by Anton Baushev and Sergey Pilipenko, titled “The central cusps in dark matter halos: fact or fiction?” To show the relation between this and the opening example in this post, we need only quote the paper (page 2)—

However, the present state-of-art of ${N}$ -body simulation tests … can hardly be named adequate. The commonly-used criterion of the convergence of ${N}$ -body simulations in the halo center is solely the density profile stability [per which] the central cusp (close to ${\rho \propto r^{-1}}$ ) is formed quite rapidly ( ${t < \tau_r}$ ).

—and then quote Siegel’s own description of the core-cusp problem and the allegation:

In theory, matter should fall into a gravitationally bound structure and undergo what’s known as violent relaxation, where a large number of interactions cause the heaviest-mass objects to fall towards the center (becoming more tightly bound) while the lower-mass ones get exiled to the outskirts (becoming more loosely bound) and can even get ejected entirely.

Since similar phenomena to the expectations of violent relaxation were seen in the simulations, and all the different simulations had these features, we assumed that they were representative of real physics. However, it’s also possible that they don’t represent real physics, but rather represent a numerical artifact inherent to the simulation itself.

Open Problems

Is all this real physics? Or is it artifacts showing that current theories and/or algorithms are flawed? Whichever is the truth, our equations have tricks that may not lead to treats.

For a relevant postscript, Painlevé did not abandon physics when he rose in politics. In 1921, he effectively removed an apparent singularity at the event horizons of black holes in general relativity. He and colleagues discussed this with Einstein the following year, but apparently not the ${n}$ -body conjecture. How far are we right in our interpretation that the (proved) conjecture plus special relativity suffices to disprove Newtonian gravity by itself?