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23 formulas

Relativity

Spacetime, Lorentz. Every formula below opens into a live, hands-on simulation.

special relativity
γ=11−v2/c2\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}

Lorentz Factor

Quantifies how much time, length, and mass distort as velocity approaches c.

special relativity
Δt=γΔt0\Delta t = \gamma \Delta t_0

Time Dilation

A moving clock ticks slower than a stationary one, as seen by an outside observer.

special relativity
L=L0/γL = L_0 / \gamma

Length Contraction

Objects in motion appear shorter along their direction of travel.

special relativity
p=γmvp = \gamma m v

Relativistic Momentum

Momentum diverges as velocity approaches c, preventing massive objects from reaching light speed.

special relativity
E2=(pc)2+(mc2)2E^2 = (pc)^2 + (mc^2)^2

Energy–Momentum Relation

The full relativistic relation linking total energy, momentum, and rest mass.

special relativity
u=u′+v1+u′v/c2u = \frac{u' + v}{1 + u'v/c^2}

Relativistic Velocity Addition

Velocities don't simply add at high speeds; the result never exceeds c.

special relativity
f=f01−β1+βf = f_0 \sqrt{\frac{1 - \beta}{1 + \beta}}

Relativistic Doppler Effect

Receding sources redshift, approaching sources blueshift — with a relativistic gamma correction.

general relativity
rs=2GMc2r_s = \frac{2GM}{c^2}

Schwarzschild Radius

The radius at which escape velocity equals c — the event horizon of a non-rotating black hole.

special relativity
E=mc2E = mc^2

Mass-Energy Equivalence

Mass and energy are interchangeable currencies; any object at rest carries an enormous reservoir of energy because c^2 is huge.

special relativity
K=(γ−1)mc2K = (\gamma - 1) m c^2

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.

special relativity
x′=γ(x−vt),t′=γ ⁣(t−vxc2)x' = \gamma(x - vt), \quad t' = \gamma\!\left(t - \frac{vx}{c^2}\right)

Lorentz Transformation

Space and time mix between observers in relative motion — the geometry of spacetime is a hyperbolic rotation, not a Galilean shear.

general relativity
dτdt=1−2GMrc2\frac{d\tau}{dt} = \sqrt{1 - \frac{2GM}{rc^2}}

Gravitational Time Dilation

Gravity slows time. The deeper you are in a gravitational potential, the slower your clock runs compared to a distant observer.

special relativity
s2=−c2Δt2+Δx2+Δy2+Δz2s^2 = -c^2 \Delta t^2 + \Delta x^2 + \Delta y^2 + \Delta z^2

Spacetime Interval

Spacetime has its own Pythagorean theorem — but with one minus sign. The interval is the only true 'distance' all observers agree on.

special relativity
dτ=dt1−v2/c2=dtγd\tau = dt\sqrt{1 - v^2/c^2} = \frac{dt}{\gamma}

Proper Time

Proper time is what you'd read on a wristwatch you carry with you — the invariant 'age' of any worldline.

general relativity
Δλλ=GMrc2\frac{\Delta\lambda}{\lambda} = \frac{GM}{rc^2}

Gravitational Redshift

Photons climbing out of a gravity well lose energy — their wavelength stretches and clocks at the bottom appear to tick slower.

black hole thermodynamics
TH=ℏc38πGMkBT_H = \frac{\hbar c^3}{8\pi G M k_B}

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.

special relativity
τtraveler=τEarth1−v2c2\tau_{\text{traveler}} = \tau_{\text{Earth}}\sqrt{1 - \frac{v^2}{c^2}}

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.

special relativity
cos⁡θ′=cos⁡θ+β1+βcos⁡θ\cos\theta' = \frac{\cos\theta + \beta}{1 + \beta\cos\theta}

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.

special relativity
Iobs=δ 3+α Iemit,δ=1γ(1−βcos⁡θ)I_{\text{obs}} = \delta^{\,3+\alpha}\,I_{\text{emit}},\qquad \delta = \frac{1}{\gamma(1 - \beta\cos\theta)}

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.

special relativity
E2=(pc)2+(mc2)2E^2 = (pc)^2 + (mc^2)^2

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.

general relativity
h=2Gc4d Q¨h = \frac{2G}{c^4 d}\,\ddot{Q}

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.

general relativity
ΩLT=2GJc2r3\Omega_{\text{LT}} = \frac{2GJ}{c^2 r^3}

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
θ=4GMc2b\theta = \frac{4GM}{c^{2}b}

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.