Spacetime, Fields, and Light
A connected walk through core physics ideas with live canvases and the equations that define them.
Hover here for connections to the Primer/Toolkit; look for green dotted hints throughout.
Where to go for the ideas that power each section:
- Spacetime → Primer: Metric hyperbolas; Toolkit: Basis mixing.
- Spin‑1/2 → Primer: Two‑state (Bloch) and Rotations.
- Gauge → Primer: Phase & Covariant derivative; this page: U(1) demo.
- Higgs → Primer: Potentials & curvature.
- Oscillations → Primer: Two‑state; Toolkit: Fourier builder.
Before We Begin
We keep the math light. You will see a few symbols: $\gamma=1/\sqrt{1-v^2}$ tells you how strong a boost is (we set $c=1$). The phase $e^{i\alpha}$ is a relabeling that does not change measurements. A “potential” is a simple hill or valley that pushes things toward lower values. Every canvas below draws exactly what the equation says.
Spacetime
Einstein kept one rule: everyone who moves at a constant speed measures the same speed of light. That rule forces space and time to mix when you change your speed. Minkowski showed it is easier to think of one thing called spacetime. Different observers take different “slices”, but the physics stays the same.
In our simple 1+1 picture (time and one space direction), a boost along $x$ mixes coordinates as $x' = \gamma(x-vt)$ and $t' = \gamma(t-vx)$ with $\gamma = 1/\sqrt{1-v^2}$. In the canvas, the light cone stays fixed. The blue $x'$ and $t'$ axes tilt as you change $v$. The gray curves show the same quantity $t^2-x^2$ staying constant.
This picture shows a Lorentz boost: $x' = \gamma(x- v t)$ and $t' = \gamma(t- v x)$; the light cone stays fixed.
Terms in the formula: $x,t$ are the original space and time; $v$ is speed as a fraction of $c$; $\gamma=1/\sqrt{1-v^2}$ stretches time and squeezes length; the lines $t=\pm x$ are light rays and do not change under boosts; $t^2-x^2$ stays the same for all inertial observers.
Spin‑1/2
Electrons have spin. That means they behave like tiny magnets and come in two basic states we call “up” and “down”. When you rotate a spin‑1/2 particle by 360°, its wave picks up a minus sign. That does not change measured probabilities, but it matters when waves add together. After 720°, the wave comes back to itself.
A rotation about $z$ looks like $R_z(\theta)=\exp\!\big(-\tfrac{i}{2}\,\theta\,\sigma_z\big)$. The factor $1/2$ explains the 720° return. In the canvas, the blue arrow shows the physical direction (period $2\pi$). The green hand shows the spinor phase (period $4\pi$).
This picture shows a spin‑1/2 rotation: $R_z(\theta)=\exp\!\big(-i\,\theta\,\sigma_z/2\big)$; $2\pi$ flips sign, $4\pi$ returns.
Terms in the formula: $\theta$ is the rotation angle; $\sigma_z=\begin{bmatrix}1&0\\0&-1\end{bmatrix}$ is the Pauli matrix that distinguishes up/down; the exponential makes a unitary rotation; the factor $1/2$ causes the 720° return; an overall minus sign leaves probabilities unchanged.
Gauge Symmetry
A complex wave has a phase. If you change only the phase, measurements do not change. If you let that phase depend on position, ordinary derivatives create extra terms. We fix this by using the covariant derivative $D_\mu=\partial_\mu+i g A_\mu$. The new field $A_\mu$ is the gauge field. It keeps the math consistent when phases vary in space and time.
The field strength is $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$ (and plus a commutator for non‑Abelian groups). A simple Lagrangian is $\mathcal L=-\tfrac14 F^a_{\mu\nu}F^{a\,\mu\nu}+ i\bar\psi\gamma^\mu D_\mu\psi$. In the canvas, we pick $A_\mu=-(1/g)\,\partial_\mu\alpha$. That choice cancels the extra gradient from a pure phase so $\|D\psi\|$ is small when $|\psi|$ is constant.
This picture shows gauge cancelation with the covariant derivative: $D_\mu=\partial_\mu+i g A_\mu$, choosing $A_\mu=-(1/g)\,\partial_\mu\alpha$.
Terms in the formula: $\psi$ is the complex field (amplitude + phase); $\alpha(x)$ is the local phase relabeling; $\partial_\mu$ is the ordinary derivative; $A_\mu$ is the gauge field; $g$ is the coupling strength; $D_\mu$ adjusts derivatives so relabeling does not change physics; $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$ is the field’s curl‑like strength.
The Higgs and Mass
Gauge symmetry blocks you from just writing a mass for many fields. A way around this is a scalar field with a potential that prefers a nonzero value everywhere. We call that the vacuum value. Small waves around that value behave like particles. One of them is the Higgs boson. Gauge fields also pick up mass from this background.
A simple model is $\mathcal L_\phi=|D_\mu\phi|^2-\tfrac{\lambda}{4}(|\phi|^2-v^2)^2$. Around $|\phi|=v$, the radial mode has $m_H^2=2\lambda v^2$. The angular mode is “eaten”, giving a gauge boson $m_A=g v$ (Abelian case). A fermion that couples with $\mathcal L_Y=-y\,\bar\psi\phi\psi$ gets $m_f= y v/\sqrt{2}$. In the canvas, a point rolls downhill and the readouts show the masses from your settings.
This picture shows the Higgs setup: $V=\tfrac{\lambda}{4}(|\phi|^2- v^2)^2$, with $m_H^2=2\lambda v^2$, $m_A=g v$, and $m_f= y v/\sqrt{2}$.
Terms in the formula: $\phi$ is the scalar field; $|\phi|$ is its size; $v$ is the vacuum value picked by the minimum; $\lambda$ sets how stiff the potential is; $m_H$ is the mass of the radial ripple; $g$ controls the gauge boson mass $m_A$; $y$ (Yukawa) sets the fermion mass $m_f$.
Neutrino Oscillations
Detectors on Earth saw fewer electron neutrinos from the Sun than expected. The reason: neutrinos can change flavor as they travel. The states that interact are not the same as the states with definite mass, so the flavor content varies with distance and energy.
In a two‑flavor model, $\lvert \nu_e\rangle=\cos\theta\,\lvert \nu_1\rangle+\sin\theta\,\lvert \nu_2\rangle$. Different masses pick up different phases, $\Delta\phi\sim \Delta m^2 L/(2E)$. The survival probability is $P_{ee}=1-\sin^2(2\theta)\,\sin^2\!\big(1.267\,\Delta m^2 L/E\big)$ with $\Delta m^2$ in eV$^2$, $L$ in km, and $E$ in GeV. Move the sliders to see how $L/E$ shifts the pattern.
This picture shows two‑flavor oscillations: $P_{ee}=1-\sin^2(2\theta)\,\sin^2\!\big(1.267\,\Delta m^2 L/E\big)$.
Terms in the formula: $\theta$ is the mixing angle between flavor and mass states; $\Delta m^2$ is the mass‑squared difference; $L$ is travel distance (km); $E$ is energy (GeV); the factor $1.267$ makes the units work; $P_{ee}$ is the chance to remain electron flavor and oscillates with $L/E$.
Gravitational Lensing
In general relativity, gravity is the shape of spacetime. Light follows the straightest possible path in that shape. Near a massive object the path bends. On the sky this looks like one object making multiple images of a more distant source. If the alignment is close, the image turns into a ring.
In the thin‑lens, small‑angle limit a point mass deflects by $\alpha=\tfrac{4GM}{c^2 b}$. In angles on the sky, $\beta=\theta-\theta_E^2\,\theta/|\theta|^2$. The solutions are $\theta_{\pm}=\tfrac12\big(\beta\pm\sqrt{\beta^2+4\theta_E^2}\big)$. We draw in units where the Einstein radius $\theta_E$ is the green circle. Slide the source and watch the two images move and merge.
This picture shows thin‑lens gravity: $\beta=\theta-\theta_E^2\,\theta/|\theta|^2$, images $\theta_{\pm}=\tfrac12(\beta\pm\sqrt{\beta^2+4\theta_E^2})$.
Terms in the formula: $\theta$ is the apparent image angle; $\beta$ is the source angle without lensing; $\theta_E$ is the Einstein radius set by mass and distances; the two solutions $\theta_\pm$ are the bright images; at $\beta=0$ they form a ring at $\theta=\theta_E$.
Putting It Together
These pieces fit well. Spacetime says how to relate moving observers. Gauge symmetry turns a relabeling rule into a force field. The Higgs field lets particles and some force carriers have mass without breaking the rules. Neutrino mixing shows how a change of basis can have clear effects in experiments. Gravity bends light because spacetime has curvature. The equations here are short, and the pictures let you see them act.
Try It Yourself
Each canvas is a small experiment. Change a slider, watch what moves, and connect it to the line of math beside it. If something feels unclear, try a slower change and look for patterns. Plain rules, seen clearly, go a long way.