Theory 01 · Established · Einstein, 1905
Special Relativity
When you move fast, space and time trade places.
Light is strange. No matter how fast you chase a beam, it always races away from you at the very same speed — about 300,000 kilometres every second. Hold onto that one fact and everything familiar begins to bend.
For that speed to stay fixed for everyone, something has to give. Moving clocks tick slow. Moving objects shrink along their direction of travel. Two events that look simultaneous to you happen at different moments for someone gliding past. None of this is an illusion or a trick of measurement — it is how space and time actually fit together. And from the same idea falls the most famous equation in science, E = mc², the discovery that mass is a vast store of frozen energy.
Special relativity rests on two postulates: the laws of physics are the same in every uniformly moving (inertial) frame, and the speed of light \(c\) is the same in all of them. The second is the radical one — it flatly contradicts the everyday rule that velocities simply add — and keeping it forces a wholesale revision of space and time.
The replacement for "velocities add" is the set of Lorentz transformations, which relate one observer's coordinates \((t,x)\) to another's \((t',x')\). Everything follows from them, governed by a single quantity, the Lorentz factor \(\gamma = 1/\sqrt{1 - v^2/c^2}\), which is \(1\) at rest and grows without bound as \(v\to c\).
Its consequences are the famous effects. Moving clocks run slow by a factor of \(\gamma\) (time dilation); moving objects contract by \(1/\gamma\) along their motion (length contraction); and simultaneity is relative — whether two separated events count as "at the same time" depends on who is asking. A concrete check: cosmic-ray muons, created high in the atmosphere and decaying in microseconds, should never reach the ground, yet they do — their internal clocks, time-dilated, run slow in our frame.
What all observers do agree on is the spacetime interval, a new "distance" that blends time and space and stays fixed under any boost. And buried in the same relation between energy, momentum and mass is the most famous equation in science, \(E = mc^2\): mass is a vast reservoir of frozen energy.
Special relativity is the geometry of Minkowski spacetime: \(\mathbb{R}^4\) with the indefinite metric \(\eta_{\mu\nu} = \operatorname{diag}(-1, 1, 1, 1)\). The invariant interval between events,
\[ ds^2 = -c^2\,dt^2 + dx^2 + dy^2 + dz^2, \]
is preserved by the Poincaré group (the Lorentz group \(O(1,3)\) together with translations). A boost of rapidity \(\varphi\), with \(\tanh\varphi = v/c\), acts as a hyperbolic rotation mixing \(t\) and \(x\) — which is why velocities combine by adding rapidities, and never exceed \(c\). Causal structure is encoded by the sign of \(ds^2\): timelike, lightlike or spacelike separation.
Proper time \(d\tau^2 = -ds^2/c^2\) parametrises worldlines; the four-velocity \(u^\mu = dx^\mu/d\tau\) and four-momentum \(p^\mu = m\,u^\mu\) package energy and momentum into single objects, giving the dispersion relation \(E^2 = (pc)^2 + (mc^2)^2\) — which reduces to \(E = mc^2\) at rest and \(E = pc\) for massless light. Maxwell's equations are already Lorentz-invariant, so electromagnetism needs no modification; mechanics does.
Status: established. A cornerstone of physics for over a century, confirmed to extraordinary precision — particle accelerators, the lifetimes of fast-moving particles, GPS timing — and the flat, local spacetime that general relativity goes on to curve.
The light cone: every worldline stays inside it, because nothing outpaces light. Tick-marks count each clock's ticks — the moving clock's are farther apart, so it runs slow. Drag the diagram to boost the frame; it also sweeps on its own. A Minkowski diagram: the 45° cone is invariant, but a moving frame's axes \((ct', x')\) tilt toward it. Drag to boost the frame. The invariant hyperbola (constant proper time) and a tilted line of simultaneity — moving clocks run slow by \(\gamma\). Drag to boost β.
Theory 02 · Established · Einstein, 1915
General Relativity
Mass tells spacetime how to curve; curved spacetime tells mass how to move.
Newton said gravity is a force reaching across space, pulling apples and planets alike. Einstein offered a stranger, deeper picture: gravity is not a force at all. Mass and energy bend the very stage of space and time, and everything simply coasts along the straightest path available through that warped geometry. A planet orbits the Sun for the same reason a ball circles a funnel — not because something tugs it, but because the surface beneath it is curved.
This sounds like poetry until you notice that it makes hard predictions which turned out to be true: light bends as it grazes the Sun, clocks tick slower deeper in gravity (your phone's GPS corrects for it every second), the universe is strewn with black holes, and colliding ones ring spacetime like a bell — ripples we have now directly heard.
General relativity begins with the equivalence principle: inside a sealed lift you cannot tell whether you are standing on Earth or accelerating through empty space — gravity and acceleration are locally the same thing. Since acceleration is about geometry (how your path bends), this hint says gravity must be geometry too.
So spacetime stops being a fixed flat stage and becomes a curved four-dimensional surface, its shape recorded by a metric \(g_{\mu\nu}\) that measures distances and times. Free-falling bodies and light rays are not pushed by a force; they simply follow geodesics — the straightest available paths through the curved geometry. A planet orbits the Sun for the same reason a ball circles a funnel.
The heart of the theory is Einstein's field equation, which in words reads: the curvature of spacetime equals the density of energy and momentum within it. Matter tells spacetime how to curve; the curvature, in turn, tells matter how to move — a two-way loop with no force acting at a distance.
The classic confirmations are exact and varied: Mercury's slowly precessing orbit, starlight bent during the 1919 eclipse, gravitational redshift (which your phone's GPS corrects for every second), the 2015 detection of gravitational waves from colliding black holes, and the 2019 image of a black hole's shadow — all match its predictions.
General relativity models spacetime as a four-dimensional Lorentzian manifold \((M, g)\); the equivalence principle becomes the statement that \(g\) is locally Minkowskian. The dynamics are Einstein's field equations,
\[ G_{\mu\nu} + \Lambda\, g_{\mu\nu} = \frac{8\pi G}{c^4}\, T_{\mu\nu}, \]
where \(G_{\mu\nu} = R_{\mu\nu} - \tfrac12 R\, g_{\mu\nu}\) is the Einstein tensor (built from the Ricci curvature of \(g\)), \(T_{\mu\nu}\) is the stress–energy tensor and \(\Lambda\) the cosmological constant. Free test bodies obey the geodesic equation \(\ddot{x}^\mu + \Gamma^\mu_{\alpha\beta}\,\dot{x}^\alpha \dot{x}^\beta = 0\), with the Christoffel symbols \(\Gamma\) derived from \(g\).
Exact solutions organise the theory: the Schwarzschild and rotating Kerr black holes, with their event horizons; the Friedmann–Lemaître–Robertson–Walker cosmologies behind the expanding universe; and linearised waves, whose detection by LIGO confirmed a century-old prediction. The Hawking–Penrose theorems show singularities are generic, not artefacts.
Status: established. The standard theory of gravity, agreeing with every precision test to date. The open frontier is not whether it holds but how to reconcile it with quantum mechanics at the Planck scale — a quantum theory of gravity remains unfinished.
Mass dents spacetime into a well; a free body just follows the curve — that is gravity. The mass breathes on a loop, deepening the well. Top-down, the grid warps near the mass and light bends past it — and GR bends it twice as much as Newton (1.75″ vs 0.87″, Eddington 1919). The sheet shows space curvature, but felt gravity is mostly time curvature. The mass cycles; watch the rays curve. A gravitational wave is transverse and quadrupolar — it stretches a ring of test particles one way while squeezing the perpendicular way (LIGO, 2015). Toggle + / × polarization.