Relativity
Spacetime, Lorentz. Every formula below opens into a live, hands-on simulation.
Lorentz Factor
Quantifies how much time, length, and mass distort as velocity approaches c.
Time Dilation
A moving clock ticks slower than a stationary one, as seen by an outside observer.
Length Contraction
Objects in motion appear shorter along their direction of travel.
Relativistic Momentum
Momentum diverges as velocity approaches c, preventing massive objects from reaching light speed.
Energy–Momentum Relation
The full relativistic relation linking total energy, momentum, and rest mass.
Relativistic Velocity Addition
Velocities don't simply add at high speeds; the result never exceeds c.
Relativistic Doppler Effect
Receding sources redshift, approaching sources blueshift — with a relativistic gamma correction.
Schwarzschild Radius
The radius at which escape velocity equals c — the event horizon of a non-rotating black hole.
Mass-Energy Equivalence
Mass and energy are interchangeable currencies; any object at rest carries an enormous reservoir of energy because c^2 is huge.
Relativistic Kinetic Energy
Kinetic energy is the extra energy a body gains by moving — it diverges as v approaches c, not 1/2 mv^2.
Lorentz Transformation
Space and time mix between observers in relative motion — the geometry of spacetime is a hyperbolic rotation, not a Galilean shear.
Gravitational Time Dilation
Gravity slows time. The deeper you are in a gravitational potential, the slower your clock runs compared to a distant observer.
Spacetime Interval
Spacetime has its own Pythagorean theorem — but with one minus sign. The interval is the only true 'distance' all observers agree on.
Proper Time
Proper time is what you'd read on a wristwatch you carry with you — the invariant 'age' of any worldline.
Gravitational Redshift
Photons climbing out of a gravity well lose energy — their wavelength stretches and clocks at the bottom appear to tick slower.
Hawking Temperature
Quantum effects at a black hole's event horizon make it radiate like a blackbody — the temperature is inversely proportional to its mass.
The Twin Paradox
Send one twin on a fast round trip and bring her home: she returns younger than the twin who stayed. There's no paradox — the situations aren't symmetric. The traveling twin must turn around, and that acceleration breaks the symmetry, so it is unambiguously she who logs less proper time along her bent path through spacetime.
Relativistic Aberration of Light
Run fast enough and the sky rearranges itself: light that arrived from your sides crowds into the direction you're heading, like rain on a windshield slanting forward as you accelerate. At near-light speed almost the entire sky compresses into a bright spot dead ahead — the relativistic headlight effect.
Relativistic Beaming (Doppler Boosting)
A source moving toward you doesn't just blue-shift — it gets dramatically brighter, because aberration funnels its photons forward, time dilation packs more of them per second, and the Doppler shift lifts each photon's energy. These three effects multiply into a steep δ⁴ dependence, so a jet pointed at you can outshine an identical one pointed away by factors of thousands.
Energy–Momentum Four-Vector
Energy and momentum aren't separate bookkeeping — they're the time and space components of one four-vector, just as duration and length are facets of spacetime. Different observers disagree on E and on p, but they all agree on the length of that four-vector, and that invariant length is the particle's rest mass. The mass is the part of energy-momentum nobody can boost away.
Gravitational-Wave Strain
Accelerating masses ripple spacetime itself, and those ripples stretch and squeeze every length they pass through by a fractional amount h — the strain. The catch is the factor c⁴ in the denominator: it makes h staggeringly tiny. Two merging black holes a billion light-years away wobble LIGO's 4 km arms by less than a thousandth the width of a proton, which is exactly what it detected.
Frame Dragging (Lense–Thirring Effect)
A spinning mass doesn't just curve spacetime — it drags it around, winding the very fabric of space into a slow vortex like a spoon stirring honey. A gyroscope held nearby is gently twisted by the rotation even though no force touches it; near a spinning black hole the dragging becomes so violent that, inside the ergosphere, standing still is physically impossible.
General Relativity: Light Bending & Curved Spacetime
Mass curves spacetime, and light follows the straightest possible path through that curved geometry — so starlight grazing the Sun bends by 1.75 arcseconds, exactly twice what Newton's gravity-on-light would give. The same curvature, turned up, gives gravitational lensing, Einstein rings, ripples in spacetime itself (gravitational waves), and at the extreme, black holes from which no path leads out.
Einstein Equivalence Principle
Inside a sealed elevator you cannot tell gravity from acceleration: a falling lab and a coasting rocket give identical physics. Because gravitational and inertial mass are the same number, everything falls at the same rate — so 'gravity' is locally just the geometry of an accelerating frame.
Rapidity (Additive Velocity Parameter)
Velocities don't add in relativity — but rapidity does. Define phi = arctanh(v/c) and successive boosts just sum like ordinary angles. A Lorentz boost is literally a hyperbolic rotation of spacetime, and rapidity is its rotation angle.
Four-Velocity
Everything moves through spacetime at the speed of light. At rest, all your motion points along time; speed up and you trade time-motion for space-motion. The four-velocity is that fixed-length arrow tangent to your worldline, with magnitude always c.
Relativity of Simultaneity
Two events that happen 'at the same time' for you happen at different times for someone moving past you. Simultaneity is not absolute — it tilts with velocity. In a moving frame, the leading clock reads earlier (leading clocks lag), which is the real root of the twin and ladder paradoxes.
Invariant Mass (Energy-Momentum Relation)
Energy and momentum each change between frames, but the combination E^2 - (pc)^2 does not — it is a fixed invariant equal to (mc^2)^2. Mass is the 'length' of the energy-momentum four-vector, the same for every observer, and for a system it can exceed the sum of its parts' masses.
Why GPS Needs Both Relativities
A GPS satellite clock runs fast by 45 microseconds/day because it sits higher in Earth's gravity well (GR), and slow by 7 microseconds/day because it orbits fast (SR). The net +38 microseconds/day, uncorrected, would shoot your position off by ~11 km per day. GPS only works because engineers pre-tune the clocks for relativity.
Gravitational Lensing & the Einstein Radius
Mass bends light, so a galaxy acts like a lens: a background source directly behind it smears into a glowing ring, and slightly off-axis into multiple arcs and images. The ring's radius — the Einstein radius — measures the lens's total mass, including dark matter you cannot otherwise see.
Shapiro Time Delay (Fourth Test of GR)
Light slows down (in coordinate time) as it passes deep in a gravity well — not because its local speed changes, but because spacetime is curved. A radar pulse grazing the Sun on its way to Venus and back returns up to ~240 microseconds late. Irwin Shapiro proposed this as the 'fourth classical test' of general relativity.
Geodetic Precession (Gravity Probe B)
Carry a spinning gyroscope around a massive body and its axis slowly tips — not from any torque, but because the space it moves through is curved. After one orbit the gyroscope points in a slightly new direction. Gravity Probe B measured Earth's geodetic precession at 6.6 arcseconds per year, exactly as general relativity predicts.
Physics of the 1919 Eclipse
During the total solar eclipse of 29 May 1919, Eddington photographed stars near the darkened Sun and found their apparent positions shifted outward by ~1.75 arcseconds — twice the Newtonian value. Einstein's general relativity predicted exactly that doubling. Overnight, the result made Einstein world-famous.
How LIGO Detected Gravitational Waves
Two black holes spiralling together stretch and squeeze spacetime, sending out gravitational waves. LIGO's 4 km laser arms change length by less than one-thousandth of a proton's width (strain ~10^-21). On 14 Sept 2015 LIGO caught the 'chirp' of two ~30-solar-mass black holes merging a billion light-years away.
Kerr Metric for Rotating Black Holes
A spinning black hole is not just a deeper well — it drags spacetime into a vortex. The Kerr solution has two horizons and an outer 'ergosphere' where nothing can stand still. Its spin sets a maximum (a_* = 1); push past it and the horizon vanishes, exposing a forbidden naked singularity.
Minkowski Metric
The single rule that replaces Pythagoras in spacetime: time enters with a minus sign, so the 'distance' between two events can be negative, zero, or positive. That one sign is the whole of special relativity.
Four-Vectors and Invariants
Package the four numbers of an event, a velocity or a momentum into one object whose 'length' every observer agrees on. The bookkeeping of relativity collapses into a single dot product with one minus sign.
Light Cone & Causal Structure
Every event sits at the tip of a double cone of light. Only what lies inside the past cone can have caused it, and only what lies inside the future cone can it affect. Everything else is 'elsewhere' — forever causally disconnected.
Relativistic Total Energy
A moving object's total energy is its rest energy scaled up by the Lorentz factor. As v approaches c, gamma runs to infinity, so no finite energy can ever push a massive body to light speed.
Schwarzschild Metric
The exact geometry of empty space around any non-spinning mass. The (1 - r_s/r) factor squeezes time and stretches radial distance more and more as you approach the horizon, where it hits zero.
Event Horizon
A surface of no return, not a wall. At the horizon the outward escape speed equals light speed, so even light aimed straight out just hovers — everything inside is causally sealed off from the rest of the universe.
Hubble-Lemaître Law
Every galaxy is receding, and the farther one is, the faster it flees — not because we are special, but because space itself is stretching uniformly. The proportionality constant is the current expansion rate.
Cosmological Redshift
Light does not lose energy climbing out of anything — the space it travels through simply stretches, and the wave stretches with it. The factor by which wavelengths grow equals the factor by which the universe grew during the light's journey.
Geodesic Equation
Gravity is not a force — it is the shape of spacetime. A freely falling body just coasts along the straightest possible path, and the Christoffel symbols encode how 'straight' bends when the geometry is curved.
The Metric Tensor
The metric is the ruler-and-clock field of the universe: at every point it tells you how to convert coordinate steps into real measured distances and times. In flat space it is Minkowski; where mass is present it warps, and that warping IS gravity.
Ladder (Pole-Barn) Paradox
The 'paradox' is really a lesson about simultaneity. Length contraction lets a long ladder fit inside a short barn in one frame; in the ladder's frame it never fits, because the two doors do not close at the same time. Nothing is contradictory — the frames just slice spacetime differently.
Longitudinal Doppler Shift
When a source moves straight toward or away from you, two effects stack: the classical bunching/stretching of wavefronts AND time dilation of the source's own clock. The square-root form is what you get when both are included exactly.
Why Cosmic Muons Reach the Ground
Muons are the most tangible proof of time dilation. Without relativity almost none would survive the fall; because their internal clock runs slow (or, in their frame, the atmosphere is contracted), a large fraction make it to sea level. The exponential decay law does the counting.
Transverse Doppler Effect
Even when a source flies straight ACROSS your line of sight — with zero radial velocity — you still see it redshifted, purely because its clock is time-dilated. There is no classical counterpart; this shift is relativity, naked.