Relativity — Laboratorium

Galileo's ship

The ball falls straight down, right below where it was released.

In 1632 Galileo described an experiment in the closed cabin of a ship sailing smoothly and steadily: neither a falling ball, nor flying flies, nor dripping water will reveal whether the ship is moving or standing still. From the shore the ball's path is an arc; from the cabin — a straight line. Uniform motion can't be detected from the inside.

Michelson and Morley, 1887

19th-century physicists believed light was a wave in an invisible “ether”, and that Earth rushes through it at about 30 km/s. If so, light in the arm pointing “into the wind” should return a little later than in the perpendicular arm, and rotating the apparatus should shift the fringes by about 0.4 of a fringe width. The measured shift was many times smaller — practically zero.

Einstein, 1905: two postulates

The principle of relativityThe laws of physics are the same in every inertial frame (one moving without acceleration). No experiment inside a closed room can show whether it is “really” moving.
The constant speed of lightLight in a vacuum moves at the same speed c relative to every observer — regardless of how the source and the receiver move.
c = 299,792,458 m/s
Exactly — since 1983 it has been a definition: a meter is the distance light travels in 1/299,792,458 of a second.

Adding velocities: a ball and a flashlight on a train

Train speed v0.50c

Throw speed u (on the train)0.50c

Galileo: w = u + v = 1.00cA good approximation at low speeds.

Einstein: w = u + v1 + uv/c² = 0.80cThe result never exceeds c.

At everyday speeds the denominator 1 + uv/c² differs from 1 by less than one billionth — that's why adding velocities “Galileo's way” works so well in daily life. The formula has been tested in experiments with particles in accelerators and with light in flowing water (Fizeau, 1851).

The light clock — time dilation

Clock speed v0.600c

Presets

Lorentz factor γ: 1.250
A tick in motion is longer: 1.250 ×
Ticks: clock at rest: 0
Ticks: moving clock: 0

In each clock the photon travels at speed c. In the moving clock it has to cover a longer, diagonal path — so each of its “ticks” lasts longer. For an observer riding along with that clock, everything is normal: to them it is our clock that seems slow. Time dilation works both ways.

Where does γ come from? The Pythagorean triangle

Half of one tick: the photon travels the diagonal c·Δt/2, while the clock moves by v·Δt/2. The mirror separation is L = c·Δt₀/2.

The math

The Pythagorean theorem: (cΔt/2)² = L² + (vΔt/2)²
Substitute L = cΔt₀/2 and multiply by 4: c²Δt² = c²Δt₀² + v²Δt²
Rearrange: Δt²(c² − v²) = c²Δt₀²

Δt = γ·Δt₀, γ = 1√1 − v²/c²Δt₀ — proper time (measured by the moving clock), Δt — the time we measure

Pythagoras in action — more in Geometry.

Muons from cosmic rays

Muon speed0.9c

A muon lives on average τ ≈ 2.2 µs (in its own frame). It is created about 15 km above the ground when cosmic rays collide with air molecules.

γ: 2.29
Mean range without relativity: 590 m
Mean range with relativity: 1.36 km
Atmosphere in the muon's frame: 6.5 km
Reaching the ground (theory): 0 %
Produced: 0
Reached the detector: 0
Would arrive without relativity: 0 %

From Earth: the muon's clock ticks γ times slower, so the muon lives longer and flies farther. From the muon's point of view: its clock is normal, but the atmosphere is rushing toward it and is γ times thinner (Lorentz contraction). Two descriptions, one result — the same number of muons in the detector. Simplified model: all muons are created at the same altitude and fly straight down at the same speed.

The γ curve and everyday speeds

Both axes are logarithmic. At ordinary speeds γ − 1 ≈ v²/(2c²) — the points lie on a straight line, far down below. Only just short of c does the curve shoot straight up.

Einstein's train and two lightning bolts

t = 0

Train speed0.60c

On the platform the lightning bolts strike simultaneously: observer M stands exactly in the middle, and both flashes reach him at the same moment. Observer M′, in the middle of the train, is riding toward the flash at the front — for her, the front bolt struck first. Who is right? Both. Simultaneity depends on the frame of reference. In both frames light travels at the same speed — notice that the flash circles are always centered on the strike point in the given frame.

Lorentz contraction and the ladder in the barn

t = 0

Ladder speed0.900c · γ = 2.29

L = L₀γA moving body is shorter along its direction of motion (but not across it). L₀ — the rest length. The ladder is 20 m long, the barn 10 m. This is what a measurement in a given frame shows; in a photo of a fast object, light delays distort the image further (the Terrell–Penrose effect).

The Minkowski diagram — spacetime

Speed of frame O′0.00c

Δt (O): B − A: 1.00
Δx (O): 3.00
Δt′ (O′): B − A: 1.00
Interval (cΔt)² − Δx²: −8.00

O axes and gridO′ axeslight coneconstant-interval hyperbola

Drag A, B or the orange ct′ axis handle. Units: years and light-years (c = 1).

Every point is an event: where (x) and when (ct). Light lines run at 45° in every frame. The lines of simultaneity of frame O′ (dashed, orange) are tilted — that's why moving observers order events differently. The order can be reversed only when the events lie outside each other's light cones (a spacelike interval) — such events cannot influence each other, so cause always comes before effect.

A trip to the stars and back

Distance4.24 ly

Rocket speed0.800c · γ = 1.667

Time on Earth: 10.60 years
Time on the rocket: 6.36 years
Brother on Earth, on return: 30.6 years
The traveler is then: 26.4 years

Simplified model: the rocket instantly flies at constant speed and turns around in the blink of an eye (in reality, accelerating would take months or years). The twins set off at age 20.

Why isn't it a paradox?

Each twin sees the other's clock as slowed down — so who ends up younger? The situation is not symmetric: the twin on Earth stays in one inertial frame the whole time, while the traveler turns around — she changes frames. On the diagram her path is bent, and in spacetime (unlike on paper!) a bent path between two events has a shorter proper time than a straight one.

Honest bookkeeping: let each twin send a flash once a year of their own time. On the way out, the traveler receives flashes from Earth less often; on the way back — more often (the relativistic Doppler effect). Counting flashes doesn't depend on any convention about simultaneity.

f_rec = f · √(1 ∓ β)/(1 ± β)β = v/c; upper sign — receding, lower — approaching

Flashes from Earth received on the rocket	count
on the way out (one every 3.00 years)	1.06
on the way back (3.00 per year)	9.54
total = years that passed on Earth	10.60

Doppler effect

It has really been measured

Atomic clocks on airplanes (1971)

Joseph Hafele and Richard Keating took cesium atomic clocks around the world on ordinary passenger flights — once eastward, once westward — and compared them with clocks at the U.S. Naval Observatory. The result matched the predictions (motion + flight altitude) within the measurement uncertainty:

Flight	predicted	measured
eastward	−40 ± 23 ns	−59 ± 10 ns
westward	+275 ± 21 ns	+273 ± 7 ns

A year on the space station

On the ISS a clock falls behind by about 25 µs per day relative to Earth: the fast motion (7.7 km/s) slows it by about 28 µs, while the weaker gravity speeds it up by about 3–4 µs. Scott Kelly spent 340 days on the station (2015–2016); a calculation like this gives a few milliseconds of “rejuvenation” relative to his twin brother Mark — the media usually quoted about 5 ms (Mark also flew to space, for less time). Milliseconds, not years — but measurable.

And the muons from the previous mode are the same effect in nature's laboratory — also confirmed in accelerators (in the 1970s, muons circling in a ring at CERN lived about 29 times longer — in line with the γ factor).

The equivalence principle

Shut inside a cabin, you can't tell whether you're standing on Earth or flying in a rocket accelerating at 9.81 m/s². In the rocket the light beam curves because the floor “chases” it upward — so gravity must bend light too. The bending in the picture is hugely exaggerated: across a 3-meter room, light drops by about 5·10⁻¹⁶ m, less than the diameter of a proton.

The rubber sheet — only an analogy

The popular picture: a mass bends the “sheet” of space, and balls roll along the dip. Its limits: the sheet works thanks to Earth's gravity (it explains gravity with gravity), it shows only 2 dimensions of space, and for freely moving bodies the most important thing is the curvature of time — that's what makes the apple fall.

Higher up = time runs faster

Clock height1 m

Faster by (fraction): 1.1·10⁻¹⁶
Per day: 9.4 ps
Over 80 years: 0.27 µs

For small heights: Δf/f ≈ gh/c² ≈ 1.1·10⁻¹⁶ per meter. The best optical clocks detected this difference with a lift of only about 33 cm (NIST, 2010), and in 2022 — even within a tiny cloud of atoms about 1 mm tall (JILA). We count only gravity; the clock's motion (e.g., Earth's rotation) is ignored.

Fix the GPS clocks

Satellite clock correction0.0 µs/day

Atomic clock Black hole: time at the horizon

Evidence for general relativity

Gravity and orbits Laser (the LIGO interferometer)Black hole Scale of the Universe

A brief history

1887Michelson and Morley fail to detect the “ether wind”.
1905The annus mirabilis of Einstein, a patent office clerk in Bern: photons, Brownian motion, special relativity and E = mc².
1908Hermann Minkowski: space and time merge into spacetime.
1915General relativity: gravity is the curvature of spacetime. It explains the shift of Mercury's perihelion.
1919Eddington and Dyson's solar eclipse expeditions confirm the bending of starlight.
1938Leopold Infeld, a physicist from Kraków, works with Einstein in Princeton: together they write the popular book “The Evolution of Physics”, and with Banesh Hoffmann — a paper on the motion of bodies in GR (the Einstein–Infeld–Hoffmann equations).
1959Pound and Rebka measure the gravitational frequency shift of gamma rays in a 22.5 m tower.
1971Hafele and Keating: atomic clocks on airplanes.
1978Launch of the first GPS satellite; the satellites' clocks are deliberately “detuned” at the factory to compensate for relativistic effects.
2015LIGO detects gravitational waves for the first time — from the merger of two black holes (Nobel Prize 2017).
2019The Event Horizon Telescope reveals the first image of a black hole's shadow (M87*).

Relativitytime that flows differently

Galileo's ship

Michelson and Morley, 1887

Einstein, 1905: two postulates

Adding velocities: a ball and a flashlight on a train

The light clock — time dilation

Where does γ come from? The Pythagorean triangle

The math

Muons from cosmic rays

The γ curve and everyday speeds

Einstein's train and two lightning bolts

Lorentz contraction and the ladder in the barn

The Minkowski diagram — spacetime

A trip to the stars and back

Why isn't it a paradox?

It has really been measured

Atomic clocks on airplanes (1971)

A year on the space station

E = mc² calculator

How much mass really “disappears”?

The Sun is slimming down

Mass defect: helium-4

Why can't you catch up with light?

The equivalence principle

The rubber sheet — only an analogy

Higher up = time runs faster

Fix the GPS clocks

Evidence for general relativity

A brief history

How does it work?

The Lorentz factor

Time dilation and Lorentz contraction

Adding velocities

The Lorentz transformation and simultaneity

Mass is energy

General relativity (GR)

Challenges

Did you know…?

Keys