Relativity
Special and general relativity: spacetime, time dilation and gravity as geometry. 43 topics, from beginner to advanced. Open the interactive version with labs and quizzes
Relativity changes how space and time are understood. Special relativity starts from two postulates, the constancy of the speed of light and the equivalence of inertial frames, and leads to time dilation, length contraction, relativity of simultaneity and the equivalence of mass and energy. General relativity extends these ideas, describing gravity as the curvature of spacetime.
How the topics connect
Relativity changes how space and time are understood. Special relativity links time, length and energy to speed, while general relativity explains gravity as curved spacetime. Both are essential for GPS, particle accelerators and our picture of black holes and the cosmos.
Real-world uses
Correct satellite clocks for both the speed effect and the gravitational effect of relativity, otherwise positions would drift by kilometres each day.
Use the relativistic energy and momentum formulas to calculate the speeds and energies of particles, which cannot be found from classical physics.
Explain why a small mass loss in fission or fusion releases a huge amount of energy, using E = mc squared.
Interpret gravitational lensing, the bending of light around massive objects, and the signals from black hole mergers.
GPS clocks must be corrected for special and general relativistic effects to give accurate positions.
Accelerator designers use relativistic energy and momentum, since the particles travel close to the speed of light.
Michelson–Morley experiment
Found no evidence of an ether wind.
In 1887 Michelson and Morley split light into two perpendicular beams to detect Earth's motion through the supposed ether. They saw no fringe shift, to high precision. The null result undermined ether theories and prepared the ground for relativity.
Expected shift ≈ 0.4 fringe; observed < 0.01.
The result concerns the speed of light being the same in all directions.
Postulates of special relativity
Physics is the same in all inertial frames and c is constant.
Einstein's two postulates: (1) the laws of physics are identical in all inertial frames; (2) the speed of light in vacuum is the same for all inertial observers. All of special relativity follows.
Light from a moving lamp doesn't travel faster than c.
Postulate 2 contradicts Galilean addition of velocities.
Relativity of simultaneity
Events simultaneous in one frame are not in another.
Two events simultaneous in one frame generally happen at different times in a frame moving relative to it. There's no absolute 'now' across space.
Lightning strikes at both ends of a moving train are simultaneous for the ground observer, but not for passengers.
Events at the same place and time are simultaneous for all.
Time dilation
Moving clocks run slow.
Δt = γΔt₀A moving clock runs slow by γ: Δt = γΔt₀, where Δt₀ is the proper time on the clock itself. Observed with muons and atomic clocks on jets.
At 0.8c, γ = 1.67: 3 s on the clock is 5 s for you.
Symmetric between frames; each sees the other's clock run slow.
Length contraction
Moving objects shorten along the direction of motion.
L = L₀/γLengths along the direction of motion shorten: L = L₀/γ, where L₀ is the proper length in the object's rest frame. Perpendicular lengths are unchanged.
A 100 m ship at 0.8c measures 60 m.
Appears only in the frame where the object moves.
Lorentz factor
Measures how significant relativistic effects are.
γ = 1/√(1−v²/c²)γ = 1/√(1 − v²/c²) is 1 at rest, 1.005 at 0.1c, 2 at 0.866c, 7.1 at 0.99c, and diverges as v → c. It governs all relativistic corrections.
At v = 0.6c, γ = 1.25.
Use β = v/c to simplify.
Lorentz transformations
Relate coordinates between inertial frames.
x' = γ(x−vt)x' = γ(x − vt), t' = γ(t − vx/c²) connect coordinates in two frames, reducing to Galilean transformations at low speed. They keep the interval invariant.
At low speeds γ ≈ 1 and vx/c² ≈ 0.
Events, not objects, transform.
Velocity addition
Speeds never add to exceed c.
u' = (u+v)/(1+uv/c²)u' = (u + v)/(1 + uv/c²): speeds never exceed c. 0.9c + 0.9c = 0.994c.
Light seen from a moving frame still has speed c.
At low speeds reduces to u + v.
Muon decay evidence
Muons reach the ground due to time dilation.
Cosmic-ray muons (T½ 2.2 μs at rest) are created ~15 km up and reach sea level because at 0.998c, γ ≈ 16 extends their lifetime to 35 μs.
From the muon's frame the distance contracts instead.
Both pictures agree.
Twin paradox
The travelling twin ages less.
One twin travels to a star and returns, while the other stays home; the traveller ages less. The asymmetry is that the traveller accelerates (changes inertial frame).
At 0.8c for a 8-light-year round trip: 10 years at home, 6 for the traveller.
Verified with atomic clocks on aircraft.
Spacetime diagrams
Plots of ct versus x show light cones and worldlines.
Plot ct versus x: worldlines trace motion, light moves at 45°, and the lines of constant time of a moving frame tilt. They make simultaneity and causality visual.
A stationary object has a vertical worldline.
Slopes steeper than 45° are physical for matter.
Spacetime interval
Invariant between events.
s² = c²t² − x²s² = c²Δt² − Δx² − Δy² − Δz² is the same in all frames. Timelike (s² > 0), spacelike (< 0) or lightlike (= 0) classify event separation.
Two events 5 s apart and 1 light-second away have s² > 0.
Time and space mix, but the interval doesn't.
Light cones and causality
Only events inside the cone can influence each other.
The light cone from an event separates the future, past and elsewhere; only events inside can be causally connected. It guarantees effects follow causes in all frames.
The Sun's current state can't affect you for 8 minutes.
Faster-than-light signals would break causality.
Proper time
Time measured by a clock moving with the object.
The time measured by a clock along its own worldline, dτ = dt/γ. All observers agree on it and it is the 'wristwatch time'.
Astronaut's own clock.
Maximised along the straight (inertial) path.
Relativistic momentum
Momentum grows without bound as v approaches c.
p = γmvp = γmv grows without limit as v → c, so accelerating to c needs infinite energy. At low speeds p ≈ mv.
At 0.8c: p = 1.33 mc.
Conservation of this p holds in all frames.
Relativistic energy
Total energy includes rest energy.
E = γmc²E = γmc² includes rest energy. Total energy grows without bound as v → c.
An electron at 0.99c has E = 3.6 MeV (rest energy 0.511 MeV).
Differences matter for kinetic energy.
Rest energy
Mass is energy.
E₀ = mc²E₀ = mc² means mass is concentrated energy: 1 kg = 9×10¹⁶ J. In reactions, mass changes appear as released energy.
Electron: 0.511 MeV; proton: 938 MeV.
Mass isn't converted into something else; it is the energy content in the rest frame.
Energy–momentum relation
Links energy, momentum and mass.
E² = p²c² + m²c⁴E² = (pc)² + (mc²)² applies to all particles; for photons m = 0 and E = pc.
An electron with p = 1 MeV/c: E = 1.12 MeV.
Valid at any speed.
Relativistic kinetic energy
Difference between total and rest energies.
K = (γ−1)mc²K = (γ − 1)mc², which approaches ½mv² at low speeds. At high speeds the energy needed to speed up grows rapidly.
Accelerating from 0.9c to 0.99c needs far more energy than from 0 to 0.9c.
Don't use ½mv² at high v.
Massless particles
Photons have E = pc.
Photons (and gluons) have no rest mass; they always travel at c with E = pc and zero rest frame.
A photon of 1 eV has p = 5.3×10⁻²⁸ kg·m/s.
They have no rest frame to transform into.
Relativistic Doppler effect
Frequency shift for light.
f' = f√((1−β)/(1+β))f' = f√((1 − β)/(1 + β)) for approach/recession along the line of sight. A transverse Doppler shift exists, due to time dilation alone.
A galaxy at 0.1c shifts wavelengths by 10.5%.
Can't be explained by classical Doppler.
Aberration of light
Direction of starlight shifts due to Earth's motion.
The apparent direction of starlight shifts due to Earth's orbital velocity, about 20.5″ (tan α ≈ v/c). Relativistic aberration pushes images forward at high speeds.
Bradley discovered it in 1725.
Not the same as parallax.
Four-vectors
Combined spacetime vectors invariant under boosts.
Four-component vectors (ct, x, y, z), (E/c, p) whose lengths are invariant under Lorentz transformations. They make relativistic equations covariant.
The four-momentum norm gives the mass.
Used in particle physics.
Relativistic collisions
Energy and momentum conservation at high speed.
Energy and momentum are conserved but not rest mass: kinetic energy can convert to new particles. The invariant mass √(E_tot² − p_tot²c²) sets the available energy.
In collider, two protons at 6.8 TeV each give 13.6 TeV.
Colliders beat fixed targets as more energy is available.
Particle accelerators and relativity
Particles near c need relativistic treatment.
Protons at the LHC have γ ≈ 7000, speed within 3 m/s of c. Synchrotron frequency must follow γ, and radiation losses grow as γ⁴.
Cyclotrons fail above ~20 MeV for protons.
Time dilation extends the lifetime of unstable particles.
Mass–energy in nuclear reactions
Small mass changes give large energy.
Reactions change rest masses; the difference Δm appears as kinetic energy. In fission of U-235 about 0.1% of the mass is released.
Δm = 0.2 u gives 200 MeV.
Energy per kg is millions of times chemical energy.
Relativistic electromagnetism
Magnetism as a relativistic effect of electricity.
Magnetic force can be seen as the relativistic effect of electric force between moving charges: a wire carrying current has length-contracted charge densities in another frame. E and B mix under boosts.
A neutral wire looks charged to a moving charge.
Maxwell's equations are already relativistic.
Tachyons
Hypothetical faster-than-light particles with imaginary mass.
Hypothetical particles with imaginary mass that always move faster than c. They'd allow causality violations and none have been observed; imaginary mass in field theory signals instability instead.
Used as a concept in string theory.
Relativity forbids ordinary matter from reaching c, not hypothetical particles that were always faster.
Equivalence principle
Gravity locally equals acceleration.
Locally, a uniformly accelerating frame is indistinguishable from a uniform gravitational field. This implies light bends and clocks slow in gravity.
Dropping a ball inside a lift in free fall.
Holds only for small regions.
Gravitational time dilation
Clocks run slower in stronger gravity.
Δt = Δt₀/√(1−2GM/rc²)Clocks run slower deeper in a gravitational potential: Δt = Δt₀/√(1 − 2GM/rc²). On Earth's surface vs high altitude it's a few parts in 10¹⁶ per metre.
GPS clocks gain 45 μs/day from weaker gravity.
Seen with atomic clocks separated by 1 m.
Gravitational redshift
Light climbing out of a gravity well loses energy.
Photons climbing out of a gravity well lose energy: Δf/f = −GM/(rc²) = −gh/c². Verified by Pound–Rebka (1959).
Light from white dwarfs.
Same effect as gravitational time dilation.
GPS and relativity
Satellites must correct for both special and general relativistic effects.
GPS satellites orbit at 14,000 km/h; special relativity slows their clocks by 7 μs/day while weaker gravity speeds them by 45 μs/day. Net +38 μs/day would accumulate 10 km position error daily without corrections.
Corrected by offsetting the clock frequency.
Both effects matter.
Curved spacetime
Mass tells spacetime how to curve; spacetime tells mass how to move.
In general relativity gravity is the curvature of spacetime caused by energy and momentum. Free-falling objects follow geodesics. Wheeler: 'matter tells spacetime how to curve; spacetime tells matter how to move'.
The Moon follows a straight path in curved spacetime.
Space and time are curved, not just space.
Einstein field equations
Relate geometry to energy and momentum.
G_μν = 8πG T_μν/c⁴G_μν = 8πG T_μν/c⁴ relates curvature (left) to the energy-momentum distribution (right). Their solutions include black holes, gravitational waves and expanding universes.
Ten coupled nonlinear equations.
Highly nonlinear, so exact solutions are rare.
Geodesics
Free-falling bodies follow straightest paths in spacetime.
The straightest possible paths in curved spacetime; free-falling objects and light follow them. They maximise proper time for timelike paths.
Orbits of planets are geodesics.
Not straight lines in Euclidean sense.
Schwarzschild metric
Spacetime around a spherical mass.
The exact solution for the spacetime outside a spherical mass: ds² = −(1 − r_s/r)c²dt² + dr²/(1 − r_s/r) + r²dΩ². It predicts perihelion precession, light bending and black holes.
At r = r_s the coordinates are singular (horizon).
Only a coordinate singularity at r_s, real singularity at r = 0.
Black hole event horizon
The surface beyond which nothing escapes.
The boundary r = 2GM/c² beyond which no signal can escape to outside observers. Falling in takes finite proper time, but outside observers see the infall freeze.
Crossing the horizon of a large black hole feels unremarkable.
Tidal forces differ with mass.
Hawking radiation
Black holes emit thermal radiation.
T = ħc³/8πGMkQuantum effects near the horizon give black holes a temperature T = ħc³/(8πGMk_B) and slow evaporation. A solar-mass hole has 60 nK.
Smaller holes are hotter and evaporate faster.
Not yet observed.
Gravitational waves detection
LIGO detected mergers of black holes.
LIGO detected the merger of two black holes (GW150914) in 2015 by a differential arm-length change of 10⁻¹⁸ m; later events include neutron star mergers seen in light too.
Waves took 1.3 billion years to arrive.
Opened gravitational-wave astronomy.
Frame dragging
Rotating masses drag spacetime.
A rotating mass drags spacetime around with it (Lense–Thirring effect), measured by Gravity Probe B in Earth orbit at 39 milliarcseconds/year.
Near a Kerr black hole nothing can remain at rest.
Effect is tiny for Earth.
Light bending by the Sun
Eddington's 1919 eclipse confirmed general relativity.
Starlight grazing the Sun bends by 1.75″, twice Newton's prediction. Eddington's 1919 eclipse observations supported Einstein.
Measured with radio telescopes to 0.01%.
Basis of gravitational lensing.
Cosmological redshift
Space expansion stretches light.
In an expanding universe light's wavelength stretches with the scale factor: 1 + z = a_now/a_then. It's not a Doppler shift through space.
z = 1 means the universe was half its present size.
Used to date distant objects.
Wormholes
Hypothetical spacetime shortcuts.
Solutions of general relativity (Einstein–Rosen bridges) connecting distant regions. They'd need exotic negative energy to stay open and are not known to exist.
Fiction like 'Interstellar' is based on them.
Speculative.
Frequently asked questions
Does relativity mean everything is relative?
No. The speed of light is the same for all observers, and the laws of physics take the same form in all inertial frames. What is relative is measurements of time and length, which depend on the observer's motion.
Why can nothing travel faster than light?
As an object speeds up, its relativistic mass and energy increase, and the energy needed to accelerate it further grows without limit as it approaches the speed of light. Only massless particles such as photons travel at light speed.
Why does time run slower near Earth's surface?
General relativity predicts that clocks in stronger gravity run more slowly than clocks farther away. The effect is tiny at everyday heights, but it is measurable and must be corrected for GPS.
Does time really slow down?
Yes. Moving clocks run slow relative to observers at rest, and clocks in stronger gravity run slow too. Both effects have been measured precisely.
Is Einstein's theory of gravity needed for everyday life?
For most daily experiences, Newtonian gravity is accurate enough. General relativity is needed for GPS, black holes, gravitational waves and cosmology.
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