Constant acceleration in Newtonian mechanics is **parabolic**, while constant acceleration in Einsteinian mechanics is **hyperbolic** and asymptotic to light speed c = 1 in natural units. For 1+1-dimensional **Minkowski spacetime**, the *difference* in squared space and time displacements is the square of **proper time** displacement,

For *constant* **proper acceleration**, this has the solution

with velocity

v = \frac{dx}{dt} = \tanh a\tau \le 1,where for small times v \sim a \tau \sim a t. If t = 0 and a x = 1 at \tau = 0, then integration gives

a t = \sinh a \tau,\\a x = \cosh a\tau.Hyperbolic identities then imply

a^2 x^2 - a^2 t^2 = 1and

Due to **Heisenberg indeterminacy**, electromagnetic fluctuations fill the vacuum. Consider a single such sinusoidal wave of angular frequency \omega_0 = k_0 in natural units. If you move at constant velocity, you observe the wave **doppler-shifted** to a different frequency. But if you move at constant acceleration, you observe the wave doppler-shifted to *a range* of frequencies corresponding to your range of velocities. For an accelerated observer at time \tau,

\phi[x,t] = \exp\left[i(k_0 x + \omega_0 t)\right] = \exp\left[i\omega_0(x + t)\right] = \exp\left[{i \frac{\omega_0}{a} e^{a \tau}} \right] = \phi[\tau].

Expand this waveform as a sum of **harmonics**

where the **Fourier components**

To** regularize** this divergent integral, subtract a tiny imaginary part i \epsilon from the angular frequency \omega to incorporate a decaying exponential factor e^{-\epsilon \tau} in the integrand, and zero it after integrating. Find

where \Gamma[n+1] = n! analytically continues the factorial function to the complex plane. The spectrum is the absolute square of the **Fourier transform**,

where the **Planck factor **suggests **Bose-Einstein statistics** and a thermal **photon** bath of **temperature** kT = a \hbar / 2\pi. In SI units,

where g_E is** Earth’s** **surface gravity**, and a **zeptokelvin** is very cool.

Just prior to the 1970s work of Fulling, Davies, and Unruh, **Stephen Hawking** famously predicted that despite their reputations **black holes **should radiate with an effective temperature

where \kappa is the black hole’s surface gravity (observed at infinity). The Unruh and Hawking results may be linked by the **equivalence principle**, which equates acceleration and gravity, and by **event horizons**. The black hole horizon is a boundary that **causally** disconnects the interior from the exterior. Similarly, when you accelerate, a **Rindler horizon** appears a distance c^2/a \sim 1~\text{ly} \left(g_E / a \right) behind you, causally disconnecting you from a region of spacetime whose photons you can outrun (so long as your acceleration continues).

By **relativistically** conserving **spacetime** **momentum**, as in the diagram below, and treating light as particles now called **photons**, Compton discovered that deflecting an electron through an angle \theta stretches the light’s wavelength by

where \lambda_c = h / mc \sim 1/40 \textup{~\AA} is the **Compton wavelength of the electron**. Compton’s experiment helped convince the physics community of **wave-particle duality**. Today, sophomores Compton scatter in our **Modern Physics** lab.

Christina holds the record for longest space flight by a woman, during which she participated in the first all-woman spacewalk. She is now scheduled to become the first woman to fly around Moon. Along with two **electrical engineer** degrees, Christina obtained a bachelor’s degree in **physics** from NC State, and I often think of her as I walk to my present office in NC State’s physics building (although I understand that physics moved into the renovated **Riddick** shortly *after* she graduated).

Last week, a preprint at arxiv.org by David Smith et al. announced an “ein Stein”, or one stone, a shape that forces a non periodic tiling of the plane, ending a half-century quest by many researchers, including me. A retiree and tiling enthusiast, Smith discovered the shape, which he calls “the hat”, and his professional colleagues provided mathematical and computational rigor.

In the 1960s, Robert Berger found a set of 20 426 tile-types that tile a plane but only non periodically — a wonderful mix of the expected and the surprising, a kind of visual music. By the 1970s, Roger Penrose has reduced that to just 2 tile-types, but a further reduction to just 1 tile-type proved elusive. Of Smith’s hat, the 91-year-old Roger Penrose reportedly observed, “It’s a really good shape, strikingly simple”, and I agree.

In the grayscale figure below, an underlying hexagonal grid emphasizes the hat’s construction. Given the prior theoretical and computational work that failed to find it, the hat is a surprisingly simple polykite formed from 8 kites or 4 double kites. In the color figure below, I color the hats according to their orientations (brightnesses) and reflections (hues), something I’ve dreamed of doing for many years.

You could order many flat paving stones shaped like the hat and translate, rotate, and flip them to non periodically tile your patio. I just might.

]]>Suppose a source charge q^\prime is at position \vec r^{\,\prime} with velocity \vec v^{\,\prime} and acceleration \vec{a}^{\,\prime}, and a test charge q is at position \vec r with velocity \vec v and acceleration \vec{a}. Let their separation \vec{\mathcal{R}}= \vec r - \vec r^{\,\prime}. Since electromagnetic “news” travels at light speed \color{blue}c\color{black} = 1 (in natural units), define the earlier **retarded time** s < t implicitly by t - s = \mathcal{R}[s] / \color{blue}c\color{black} = \mathcal{R}_{\color{red}s}.

**Liénard-Wiechert** electric and magnetic potentials at test charge q due to source charge q^\prime are

where the source charge q^\prime is “smeared” to an **effective charge** q^\prime_e by its motion, and the subscripts indicate evaluation times. If the source charge velocity \vec v^{\,\prime}_{\color{red}s} = 0, then the potentials simplify to the electrostatics limit. Differentiate the potentials to find Maxwell’s electric and magnetic fields

and the **Lorentz force law**

generalizes **Coulomb’s law **to

If the test charge velocity \vec v = \vec 0, then the **magnetic** terms vanish; if the source charge acceleration \vec{a}^{\,\prime} = \vec 0, then the **radiation** terms vanish; if, in addition, the source charge velocity \vec v\,^\prime = \vec 0, then the generalized Coulomb’s law reduces to the familiar electrostatic limit. Finally, **Newton’s** second law with **Einstein’s** momentum

generates the motion equations using **3-vectors** and **lab time** (instead of **4-vectors** and **proper time**), succinctly summarizing all of **electromagnetism**.

Using **Mathematica**, I eliminated present time t from the motion equations with the substitution t = s + \mathcal{R}_{\color{red}s} and numerically integrated them with respect to the retarded time s for various initial conditions, as in the figure below. Positive and negative charge pairs tugged by non-zero net forces spiral into one another as they radiate away energy and momentum, graphically demonstrating the instability of **classical** atoms.

See more information over at the NEWS page on the Wooster Physics website!

]]>Replace the **map** with a **graph** of V vertices, E edges, and F faces. Represent the 5 colors by the numbers 1, 2, 3, 4, 5.

For connected planar graphs, the **Euler characteristic** \chi = V-E+F=2 is a **topological invariant**. The proof is by **induction** on the number of vertices. An isolated vertex V = 1 connects to no edges E = 0 but is surrounded by a single face F = 1 extending to infinity, so \chi = 1-0+1=2. Assume \chi = 2 for a graph of V vertices, and extend it in one of two ways: add an edge and a vertex, so \Delta \chi =1 - 1 + 0 = 0 , or add an edge connecting two existing vertices, so \Delta \chi =0 - 1 + 1 = 0 . Either way, the Euler characteristic is invariably \chi = 2.

For simple planar graphs (with no loops or repeated edges), double the number of edges is at least triple the number of faces. To prove this, first consider a maximally triangulated graph where each face (including the exterior one) is bounded by 3 edges. Tripling the faces counts each edge twice, so 2E = 3F. To reach a generic case, deleting any edge reduces this equality’s left side by 2 and reduces its right side by 3, so 2E \ge 3F.

In simple planar graphs, there exists a vertex with 5 or fewer edges. The proof is by contradiction. If each vertex connects to 6 edges, then sextupling the vertices counts each edge twice, so 2E = 6V. For the generic case, where each vertex connects to *at least* 6 edges, adding edges increase this equality’s left side without changing its right side, so 2E \ge 6V and E \ge 3V.

Use this inequality to eliminate vertices V from the tripled Euler characteristic 3\chi = 3V-3E+3F = 6 and get E-3E+3F\ge 6, so 3F - 6\ge 2E \ge 3F by the above twice-thrice inequality. But 3F - 6\ge 3F implies - 6\ge 0, which is a contradiction.

Every map is 5-colorable. The proof is by induction on the number of vertices of the corresponding graphs. A single V = 1 vertex is trivially colored by a single color. Assume all V > 1 graphs are 5-colorable and consider a V+1 graph. Delete the vertex with the fewest edges, which must be 5 or less, and the resulting graph is 5-colorable. Restore the vertex. If it is connected to 4 or less edges, color it differently than its neighbors.

If the restored vertex is connected by 5 edges, consider nonadjacent neighbor vertices V_1 and V_4 colored by colors 1 and 4. If the subgraph of vertices colored 1 and 4 is disconnected and V_1 and V_4 are in different components, interchange the 1 and 4 colors in the component including V_1 and color the restored vertex 1.

Else if V_1 and V_4 are in *the same* component of the subgraph, a **Kempe chain **barrier separates the subgraph connecting V_3 and V_5, so interchange the 3 and 5 colors in one component and color the restored vertex 5.

Compton reported his results in a manuscript submitted to the journal *Science* on 1913 January 13 and published as “A Laboratory Method of Demonstrating the Earth’s Rotation”, Arthur Holly Compton, **Physical Laboratory**, **University of Wooster**, ** Science**, 1913 May 23, volume 37, issue 960, pages 803-806.

Compton’s generator is nowadays often used as a text book example of the \vec F_C = 2 m\, \vec v \times \vec \omega **Coriolis** pseudo-force deflecting the fluid in a circular tube as it is flipped in Earth’s **rotating** reference frame, which is analogous to the \vec F_B = q\, \vec v \times \vec B magnetic force deflecting a moving charge. However, an **inertial** perspective is simpler. After **viscosity** has damped the fluid motion relative to Earth, the fluid farther from the rotation axis is moving faster relative to distant stars, so quickly flipping the ring reverses the speed gradient and induces transient circulation.

Flipping a wire loop in a magnetic field generates transient electric current like flipping Compton’s ring in a spinning reference frame generates transient fluid flow, where the magnetic field plays the role of the angular velocity; indeed, the former is an **electric generator** and the latter is a **motion generator**!

As in the above figure, assume Earth has radius R and angular speed \omega. Assume the ring is at co-latitude \theta and subtends an angle 2\delta from Earth’s center. Then the spread in the fluid’s inertial speeds is

\delta v = \omega R \sin[ \theta + \delta] - \omega R \sin[\theta - \delta] = 2\omega R \cos \theta\sin \delta.If the ring’s radius r \ll R , then

1 \gg r / R \approx \delta \approx \sin\delta,and so

\delta v = 2 \omega r \cos \theta = 2 \omega r \sin \lambda,where \lambda is the latitude. As checks, \delta v = 0 when \lambda = 0 (straddling the equator), and \delta v = 2\omega r when \lambda = \pi/2 (straddling the north pole).

For Earth’s \omega = 2\pi/\text{day} angular speed, flipping a quiescent r = 1~\text{m} ring at \lambda = \pi/4 mid latitude generates an initial fluid speed \delta v = 0.1~\text{mm/s}. Compton **amplified** this motion by using a **microscope** to view the fluid through a window in a **constriction** of the tube.

The 2D animation below illustrates the formation of a 1D back-and-forth analemma for Planet whose spin axis is perpendicular to its orbital plane. Planet’s year divides into 7 equal **sidereal days **(when red arrow observer points left), 6 equal **mean** **solar days** (when white ray from Sun crosses yellow lines), and 6 unequal **apparent solar days** (when red arrow points along white ray toward Sun).

Planet’s** rotation** (spin), which is constant, and **eccentric revolution** (orbit), which is faster nearer Sun where gravity is stronger, cause mean solar days to be nonuniformly distributed about the orbit, dense near **aphelion** (far point at left) and sparse at **perihelion** (near point at right). For an equatorial observer who experiences noon at perihelion, mean noon (when yellow dots record Sun’s direction) precedes apparent noon during the orbital top half and succeeds apparent noon during the orbital bottom half, with the constant rotation “falling behind” during the fast orbital right half and “getting ahead” during the slow orbital left half.

The 3D animation below (with a wide-angle perspective) illustrates the formation of a 2D figure-8 analemma for an **oblique** Planet whose spin axis is tilted 45° downward. Planet’s year divides into 40 equal **sidereal days,** 39 equal **mean** **solar days**, and 39 unequal **apparent solar days**. The constant tilt** **and elliptical orbit cause an equatorial observer to oscillate above and below Sun. For an equatorial observer who experiences noon at perihelion, mean noon (when yellow dots record Sun’s direction) precedes apparent noon, with Sun above the equator, during the orbital top half and succeeds apparent noon, with Sun below the equator, during the orbital bottom half, forming a figure 8. (In practice, the precise shape and orientation of analemmas vary with time of day and latitude.)