“Resilience rises! Not even gravity contains humanity when we explore as one for all.”
My eyes were glued to NASA-TV last weekend as I followed the flight of the SpaceX Dragon “Resilience” to the International Space Station. Ferrying a diverse Crew 1 of Mike Hopkins, Victor Glover, Shannon Walker, and Soichi Noguchi for a six-month stay on the ISS, this was the first commercial FAA certified human spaceflight.
When I was a kid I used to read Scientific American at the local library. I loved Martin Gardner‘s Mathematical Games column, and I vividly remember his description of Roger Penrose‘s then recent discovery of two shapes that force a nonperiodic tiling of the plane, an aperiodic tiling, a kind of visual music.
Only later did I learn of Penrose’s important contributions to General Relativity, including topological arguments to demonstrate the inevitably of gravitational collapse leading to astrophysical black holes. And earlier this month I was excited to hear that Penrose — in his 90th year — shares the 2020 Nobel Prize in physics!
So I got out my Penrose tiles (thanks Woody) and assembled a small pattern. It’s not easy, but a combination of local edge and vertex rules (or a global inflation rule) can extend the aperiodic pattern to infinity.
Cookie dough in a cookie factory moves on a conveyor belt at a constant relativistic speed. A circular cutter stamps out cookies as the dough rushes by beneath it. In the factory frame, the dough is length contracted along the conveyor belt, and when the conveyor belt stops and the cookies are packaged for eating, they are stretched elliptically along the belt.
But the cookie dough riding the belt observes the circular stamp moving toward it and length contracted along the belt. How can the cutter contracted along the belt cut cookies stretched along the belt?
The cut is simultaneous in the factory frame but not simultaneous in the cookie dough frame! Clocks synchronized in their own frame are not synchronized in a moving frame, and the rear clock is ahead of the front clock. In the looping animation, the top panel illustrates the cut in the factory frame, where the cutter moves up & down simultaneously, while the bottom panel illustrates the cut in the dough frame, where the cutter rear moves up & down first and the cutter front moves up & down later. The wave of up-&-down motion spreads the cut over time lengthening it.
The relativistic effects of time dilation, length contraction, clock desynchronization, and the relativity of rigidity are significant only near the billion-kilometers-per-hour invariant speed of light and gravitational waves.
Yesterday, SpaceX successfully flew a full-sized Starship tank-section prototype at its launch facility in Boca Chica, Texas. Standing thirty meters tall without its nosecone, weighing one to two hundred tons with methalox propellant, and made from stainless steel, Starship was powered by a single Raptor engine, the first full-flow staged-combustion engine to fly. Operational Starships will have six Raptors in two pairs of three. For this test, a single engine was necessarily installed off-center. In the photo below from a SpaceX drone, the thrust of the Raptor, traced by a standing wave of shock diamonds, passes through the tilted Starship’s center-of-mass. The heavy mass on top simulates the nosecone, minimizes the tilt, and helps land (as the Raptor can’t throttle below 50% of thrust). Note the construction tractors parked in the lower left.
As I kid, I used to help my dad with electrical wiring projects (among other things). I learned that home electricity was “hot & cold”, like water in pipes — or at least, that’s how I understood the explanation. Later I learned that home electricity uses alternating current (AC) rather than direct current (DC), and the plumbing analogy seemed confusing, even if I substituted “hot & neutral” for “hot & cold”. I was especially baffled by polarized outlets, with a smaller “hot” slot and a larger “neutral” slot. What possible difference could that make, given that AC was simply sloshing the current back and forth (with electrons oscillating only ~ 0.2 µm in the Drude model)?
I understood how Maxwell’s equations describe electromagnetism long before I understood polarized outlets. The key is that north American homes tap transformers in three places to get a “neutral” and two “hots” that alternate but oppositely!
Starting at the top of the figure, a power plant wire loop rotates in a magnetic field inducing AC according to Faraday’s law of induction (one of Maxwell’s equations). The first transformer increases the voltage and lowers the current for energy efficient long distance transmission (again as described by Faraday). The second transformer decreases the voltage and increases the current for safe residential use. In north American homes, which use split-phase power, three wires enter the home: a neutral wire from the center of the final transformer coil, which is zeroed or grounded, and two hot wires from the ends of the coil, where current oscillates sinusoidally but 180° out of phase. Connecting a lamp across a hot and neutral wire harnesses a root-mean-square 120 volts of energy per charge, while connecting an oven across the two hots harnesses RMS 240 V. The polarized outlet forces appliances to connect their loads to neutral and their switches to one of the hots, so no current flows across their loads when the appliances are off, which is both safe and efficient.
Newton wrote, “My brain never hurt more than in my studies of the moon [and Earth and Sun]”. Unsurprising sentiment, as the seemingly simple three-body problem is intrinsically intractable and practically unpredictable. … If chaos is a nonlinear “super power”, enabling deterministic dynamics to be practically unpredictable, then the Hamiltonian is a neural network “secret sauce”, a special ingredient that enables learning and forecasting order and chaos.
So begins an article I co-authored with NAIL, the Nonlinear Artificial Intelligence Lab at NCSU, which appears today in Physical Review E.
Inspired by how brains work, artificial neural networks are powerful computational tools. Natural neurons exchange electrical impulses according to the strengths of their connections. Artificial neural networks mimic this behavior by adjusting numerical weights and biases during training sessions to minimize the difference between their actual and desired outputs.
From cancer diagnoses to self-driving cars to game playing, neural networks are revolutionizing our world. But although they are universal approximators, their approximations may require exponentially many neurons. In particular, they can be confounded by the mix of order and chaos in natural and artificial phenomena.
NAIL’s solution to this “chaos blindness” exploits an elegant and deep structure to everyday movement discovered by William Rowan Hamilton, who remarkably re-imagined Isaac Newton’s laws of motion as an incompressible energy-conserving flow in an abstract, higher-dimensional space of positions and momenta.
In this phase space, any motion is a unique trajectory confined to a constant-energy surface, and regular motion is further confined to a donut-like hypertorus. This structure constrains our Hamiltonian neural networks to properly forecast systems that mix order and chaos.
Since the mid 1990s, a highlight of my year has been the Physics Department’s National Science Foundation Research Experience for Undergraduates summer program. Our research assistants come from Wooster and from all over the United States, as detailed in the accompanying bubble chart (where the bubble diameters code number of participants per institution). To date, the program has gathered and meshed people from 55 colleges and universities for research, education, fellowship, and play.
Like the Tokyo Olympics, the 2020 pandemic has postponed — but not cancelled — this summer’s NSF REU program until next summer.
A second ago, I posted this blog entry. A kilosecond ago, I wrote it. A megasecond ago, I isolated myself against the 2020 pandemic. A gigasecond ago, I began my career at The College of Wooster, which I celebrate today, 31.7 years later.
Although almost all ordinary mass effectively arises from the kinetic and binding energy of quarks and gluons bound to protons and neutrons in atomic nuclei, the Higgs mechanism does endow some particles like quarks and weakons with intrinsic masses. Here I gently introduce the Higgs mechanism without using loose analogies like vacuum field viscosity.
For simple systems, the Lagrangian is the difference between the kinetic and potential energies,
L = T – V.Demand that the action
S = \int dt \,Lbe stationary and integrate by parts
0 = \delta S = \int dt \left(\frac{\partial L}{\partial x} \delta x + \frac{\partial L}{\partial \dot x} \delta \dot x \right) = \int dt \left(\frac{\partial L}{\partial x}- \frac{d}{d t}\frac{\partial L}{\partial \dot x} \right)\delta x,for all variations \delta x, to find the Euler-Lagrange motion equation
\frac{\partial L}{\partial x} = \frac{d}{d t}\frac{\partial L}{\partial \dot x}.For the simple harmonic oscillator, the Lagrangian
L[x,\dot x] = \frac{1}{2} m \dot x^2 – \frac{1}{2} k x^2implies the Euler-Lagrange equation
-kx = m \ddot x,which recovers familiar laws of Hooke and Newton.
In 1+1 dimensional spacetime, represent a spinless particle by the quantum of the field \phi[t,x]. In units where c = 1 and \hbar = 1, the Lagrangian
L = \int dx\, \mathcal L = \int dx \left( \mathcal{T} – \mathcal{V} \right)and the Lagrangian density
\mathcal{L}[\phi, \partial_t \phi, \partial_x \phi] = \frac{1}{2} \left(\partial_t \phi \right)^2 -\frac{1}{2} \left( \partial_x \phi \right)^2 -\frac{1}{2} m^2 \phi^2,where the signs of the kinetic energy terms reflect the signs of the spacetime interval d\tau^2 = dt^2 – dx^2. The corresponding Euler-Lagrange equation
\frac{\partial \mathcal{L}}{\partial \phi} = \frac{\partial}{\partial t}\frac{\partial \mathcal{L}}{\partial (\partial_t\phi)} + \frac{\partial}{\partial x}\frac{\partial \mathcal{L}}{\partial (\partial_x\phi)}reduces to the Klein-Gordon equation
-m^2 \phi = \partial_t^2 \phi – \partial_x^2 \phior
\left(\partial_t^2 – \partial_x^2 + m^2 \right) \phi = 0.Assuming wave-particle duality, seek plane wave solutions
\phi = e^{i(-Et+ px)} = e^{i(-Et+ px)/\hbar}= e^{i(-\omega t+ kx)}to find
\left( (-iE)^2 – (i p)^2 + m^2 \right) \phi = 0or
E^2 = p^2 + m^2,where m is the particle mass (and E = m = m c^2 for p = 0).
Nonzero mass determines the simple harmonic oscillator potential energy \mathcal V = m^2 \phi^2 / 2 > 0 and the curvature of the corresponding upright parabola. Zero mass collapses the Klein-Gordon equation to the electromagnetic wave equation.
Consider the massless Lagrangian density
\mathcal{L}=\mathcal{T} – \mathcal{V} =\frac{1}{2} \left(\partial_t \phi \right)^\dagger \left(\partial_t \phi \right) -\frac{1}{2} \left( \partial_x \phi \right)^\dagger \left( \partial_x \phi \right) + \frac{a}{2} \phi^\dagger \phi – \frac{b}{4} \left( \phi^\dagger \phi \right)^2,where \phi = \phi_m e^{\phi_a} = \phi_r + i \phi_i , the adjoint \phi^\dagger = \phi^* reduces to complex conjugation, and the parameters a,b > 0. For all complex arguments \phi_a, the potential energy density
\mathcal{V} = – \frac{a}{2} \phi_m^2 + \frac{b}{4} \phi_m^4depends only on the complex magnitude \phi_m. The vanishing derivative
0 = \frac{d \mathcal{V}}{d \phi_m} = – a \phi_m + b \phi_m^3= \phi_m(- a + b \phi_m^2)implies local maximum and minimum at \phi_m = 0 and \phi_m = \sqrt{a/b} = \phi_0, as in the figure, where the circular symmetry reflects the invariance of the Lagrangian under global phase (or gauge) transformations \phi \rightarrow \phi \, e^{i \Lambda} = \phi_m e^{i( \phi_a + \Lambda )}.
If the potential is a sombrero, the circular brim are ground or vacua states of nonzero field
\phi_r^2 + \phi_i^2 = \phi_0^2.Break the circular symmetry by fixing \phi_r = \phi_0 and \phi_i = 0, but allow the field to oscillate radially and shift circularly
\phi[t, x] = \phi_0 + \phi_1[t, x] + i \phi_2[t, x] .The Lagrangian density becomes
\begin{array}{c}\displaystyle\mathcal{L}= \frac{1}{2} \left(\partial_t \phi_1 \right)^2 -\frac{1}{2} \left( \partial_x \phi_1 \right)^2 – a \phi_1^2 \\ \displaystyle+ \frac{1}{2} \left(\partial_t \phi_2 \right)^2 -\frac{1}{2} \left( \partial_x \phi_2 \right)^2 \\ \displaystyle- \sqrt{a b}\,\phi_1 \left(\phi_1 ^2+\phi_2 ^2\right) -\frac{b}{4} \left(\phi_1 ^2+\phi_2^2\right)^2\\ \displaystyle+ \frac{a^2}{4 b}\end{array}or
\begin{array}{c}\displaystyle\mathcal{L}= \frac{1}{2} \left(\partial_t \phi_1 \right)^2 -\frac{1}{2} \left( \partial_x \phi_1 \right)^2 – m_1^2 \phi_1^2 \\ \displaystyle+ \frac{1}{2} \left(\partial_t \phi_2 \right)^2 -\frac{1}{2} \left( \partial_x \phi_2 \right)^2 – m_2^2 \phi_2^2 \\ \displaystyle+\text{interaction} + \text{constant},\end{array}where m_1 = \sqrt{a} > 0 and m_2 = 0.
Starting with a massless field \phi with nonzero vacuum states, symmetry breaking creates a field \phi_1, corresponding to parabolic radial motion and a massive quantum m_1 > 0, and a field \phi_2, corresponding to constant circular motion and a massless quantum m_2 = 0 (a Goldstone boson). This is the Higgs mechanism of mass endowment.
I met two scientists for my BZ-history project with a combined age of 177 years.
It was a great pleasure and honor to talk to them.
Nice post, John! Thanks for writing these. I always enjoy them.
David Smith’s ein stein is an awesome shape!
Thanks, Mark! I enjoy reading your posts as well.