Recent content by thomsj4

Because ##U_\theta(r)## can be extended evenly around ##r=0## (##U_\theta(-r) = U_\theta(r)##), a Fourier cosine series is used. This means sine terms are zero (##B_n = 0##). We work on ##[0, L]##, with ##L## significantly larger than ##R## (e.g., ##L = 4R##, ##5R##, or ##10R##). The Fourier...

Consider a four-dimensional spacetime with coordinates ##t x y z##. For any massive particle, the interval is given by $$ ds^2 = -c^2\,dt^2 + dx^2 + dy^2 + dz^2. $$ To move faster than light in ordinary special relativity, we would require a timelike worldline to become spacelike, meaning the...

Let ##y(t)## be a quantity that grows from a lower bound ##d## to an upper bound ##a##. Set ##z(t) = y(t) - d## so that ##z## increases from ##0## to ##M = a - d##. Suppose we introduce a parameter ##g≠1## and impose the following governing principle: at each time ##t##, the ratio of ##dz/dt##...

Suppose we construct a real vector space endowed with the Minkowski bilinear form $$ \langle u, v \rangle = u^0 v^0 - u^1 v^1 - u^2 v^2 - u^3 v^3, $$ so that for two points ##p_1## and ##p_2##, one defines $$ \Delta s^2 = \langle p_2 - p_1, p_2 - p_1 \rangle. $$ If ##\Delta s^2 < 0##, one might...

Begin by letting the standard Minkowski metric in ##(t,x,y,z)## be $$ ds^2 = c^2\,dt^2 \;-\; dx^2 \;-\; dy^2 \;-\; dz^2. $$ Define a vector field ##X## on ##\mathbb{R}^4## by $$ X = \omega\,\bigl(y\,\partial_x \;-\; x\,\partial_y\bigr), $$ where ##\omega## is a constant. Observe that ##X##...

We analyze the Doppler shift by tracking individual wave crests emitted by a source orbiting with angular frequency ##\Omega## at a radius ##R##. The source's position at emission time ##t## is ##\bigl(R\cos(\Omega t),\,R\sin(\Omega t)\bigr)##. The emitted wave has an angular frequency...

Consider four photons (##P1, P2, P3, P4##) whose polarizations live in a four-fold tensor product of two-dimensional Hilbert spaces. The initial state is $$ \lvert \Psi \rangle_{1234} = \lvert \Phi \rangle_{12} \otimes \lvert \Phi \rangle_{34}$$ where $$ \lvert \Phi \rangle =...

Let ##G = (V, E)##. A node in the graph represents a distinct ephemeral position. Formally, an ephemeral position is a tuple ##P = (B, C, E, H)## where ##B## is an 8x8 matrix representing the board layout, with each element ##B_{ij} \in \{ \text{pieces} \} \cup \{ \text{empty} \}##. ##C \in \{...

Let ##\text{Hom}(\pi_1(S^3 \setminus K), G)## be the set of group homomorphisms from the fundamental group of a knot ##K##'s complement into a finite group ##G##. We define the "information content" of ##K## with respect to ##G## as $$I(K;G) = \ln |\text{Hom}(\pi_1(S^3 \setminus K), G)|.$$...

Consider a timelike worldline parameterized in flat Minkowski spacetime by coordinates ##(ct(\lambda), x(\lambda), y(\lambda), z(\lambda))## with the metric signature chosen as ##(+,-,-,-)##. Proper time is defined by integrating the interval ##d\tau## along the worldline, where ##d\tau^2 =...

Define a sequence of nested open sets that approximate ##E## from the outside, and then exploit a summation-by-parts style argument with an auxiliary function to pass to the limit. First, write the complex measure ##\mu_F## as ##\mu_F = \mu_1 + i\mu_2##, where ##\mu_1## and ##\mu_2## are real...

Imagine enclosing the entire setup in a spacetime “drum” whose boundary is formed by all possible light paths between the midpoint and the two mirrors. Let the distance between the mirrors be ##L##, and place them so that in the rest frame ##S## the left mirror is at ##-\tfrac{L}{2}## and the...

Consider a 4-dimensional manifold with metric signature ##- + + +##, and let ##(t,x,y,z)## be coordinates such that the spacetime interval is given by ##ds^2 = -c^2,dt^2 + dx^2 + dy^2 + dz^2##. Define a scalar field ##\Phi(t,x,y,z)## that vanishes everywhere except along two distinct...

Let us choose an operator $$ U(t) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & e^{i\omega t} & 0 \\ 0 & 0 & e^{i\omega t} \end{pmatrix} $$ where $$ \omega = \frac{\omega_1 + \omega_2}{2}. $$ Transform the state vector as $$ |\tilde{\psi}(t)\rangle = U^\dagger(t)|\psi(t)\rangle $$ and the Hamiltonian...