Time Dilation Explained: The Equations and Their Derivation

The two ideas everything rests on

Special relativity needs just two assumptions, both well tested:

1. The laws of physics are the same for everyone moving at a steady speed. There is no experiment that can tell you that you are "really" at rest.
2. Light moves at the same speed \(c\) for every observer (\(c = 299{,}792{,}458\) m/s), however fast the source or the observer is moving.

The second idea is the strange one. If you run at a beam of light it still passes you at \(c\). Something has to give so that this stays true, and what gives is time (and distance). Time dilation is the consequence you can work out with nothing more than Pythagoras.

Derivation 1: the light clock

Build the simplest clock imaginable: a pulse of light bouncing between two mirrors a distance \(L\) apart. One tick is one round trip. In the clock's own frame the light travels \(2L\) at speed \(c\), so

This is the clock's proper time: the time measured by a clock that is at rest next to the events. Now set the clock moving sideways at speed \(v\) (the mirrors are perpendicular to the motion). A friend who is not moving watches it go past. During one tick, which lasts \(\Delta t\) for her, the clock slides along by \(v\,\Delta t\), so the light follows a slanted path: up to the far mirror, then back down.

Light still travels at \(c\) in her frame, so each slanted half is \(c\,\Delta t/2\) long. Pythagoras on the right triangle gives

Collect the \(\Delta t\) terms:

The numerator is exactly the proper time \(\Delta\tau = 2L/c\). So

The moving clock ticks slower by the factor \(\gamma\). Because every process in nature can be compared with a light clock (otherwise you could tell who is really moving, breaking the first idea), all clocks, hearts and chemical reactions slow down together.

What the Lorentz factor looks like

\(\gamma\) is 1 at rest and grows slowly at first, then explodes as \(v\) approaches \(c\):

Everyday speeds are tiny compared with \(c\): a jet at 900 km/h has \(\gamma - 1 \approx 3\times10^{-13}\), which is why we never notice. Airborne atomic clocks can, though (Hafele and Keating, 1971).

Proper time: the clock you carry

The quantity \(\Delta\tau\) has a name: the proper time between two events is the time shown on a clock that travels from one to the other. For a speed that changes, add up tiny pieces:

It is the relativistic version of "distance travelled along a path": different paths between the same two events have different proper times, and the straight, steady path takes the longest. That one fact explains the twin paradox.

Derivation 2: length contraction

Take a trip of distance \(d\) (measured by the people at home, e.g. the Earth and a star at rest relative to each other). At speed \(v\) it takes \(t = d/v\) for them. The traveller's clock runs slow, so she experiences

From her point of view the star approaches at speed \(v\) and arrives after \(\tau\). So she measures the distance as \(v\tau = d/\gamma\):

Time dilation and length contraction are the same effect seen from the two sides. This also resolves the muon puzzle: muons made 15 km up live only 2.2 microseconds, enough to travel 0.66 km at \(0.995c\), yet they reach the ground. In our frame their clocks run slow (\(\gamma \approx 10\): 22 microseconds, 6.6 km). In the muon's frame the atmosphere is squashed to a tenth of its height.

Worked example: Proxima Centauri at 0.9 c

Proxima is \(d = 4.25\) light-years away. At \(v = 0.9c\) (so \(\gamma = 2.294\)):

• Time on Earth: \(t = d/v = 4.25/0.9 = 4.72\) years.
• Time on the ship: \(\tau = t/\gamma = 4.72/2.294 = 2.06\) years.
• Distance the traveller measures: \(d/\gamma = 1.85\) light-years (and \(1.85/0.9 = 2.06\) years: consistent).

You can check any of these in the Calculator tab of the simulator.

The twin paradox

Each twin sees the other's clock run slow, so who is younger when they meet again? The answer is the traveller, because the situation is not symmetric: to come back she must turn round, which means accelerating, and her path through space-time is not the steady straight one. Using the proper-time integral above, a path with turns always has less proper time than the straight path between the same two events (the stay-at-home twin's). Equivalently, she switches from one inertial frame to another, and "now" at home jumps ahead in her bookkeeping at the turn-round.

Derivation 3: travelling at a constant 1 g

A ship can accelerate at \(a = 1\,g \approx 9.8\) m/s² for years, which gives Earth-like gravity and is the usual dream of fast interstellar travel. At large speeds you cannot just add \(a\,t\) to the speed (it would pass \(c\)). Describe the motion with rapidity \(\varphi\), defined by \(v = c\tanh\varphi\) (so \(\gamma = \cosh\varphi\)). Constant proper acceleration means the rapidity grows steadily with the ship's own clock:

Integrating \(dx = v\,dt\) and \(dt = \gamma\,d\tau\) with \(\gamma = \cosh\varphi\) and \(v\gamma = c\sinh\varphi\) gives the time and distance covered from rest:

For a trip of distance \(d\) that accelerates for the first half and brakes for the second half, the ship covers \(d/2\) while speeding up. Solve \(\tfrac{d}{2} = \tfrac{c^2}{a}(\cosh\varphi - 1)\) for the rapidity \(\varphi\) reached at the half-way point:

This is the "constant 1 g" equation in the simulator's Twin trip tab. Numbers: Proxima (4.25 light-years) takes 3.5 years on the ship; the Andromeda galaxy (2.5 million light-years) takes about 28.6 years on the ship (while about 2.5 million years pass on Earth).

Why nothing with mass can reach c

The energy of a moving object of mass \(m\) is \(E = \gamma m c^2\); at rest it is \(mc^2\). The extra is its kinetic energy:

For small speeds the series expansion \(\gamma \approx 1 + v^2/2c^2\) gives back the familiar \(\tfrac12 m v^2\). But as \(v \to c\), \(\gamma \to \infty\) and so does the energy needed. For one kilogram at \(0.9c\) you need \((\gamma - 1)c^2 = 1.16\times10^{17}\) J, about the energy a large power station makes in a year; at \(0.999c\) it is 17 times more.

Common misunderstandings

• "The clock only looks slow because of the time light takes to arrive." No: that Doppler effect is a separate, easily corrected effect. Time dilation is what remains after correcting for it.
• "It is only a measurement trick." No: muon lifetimes, particle accelerators, atomic clocks on aircraft and GPS satellites all show the effect. GPS corrects for it (and for gravity, which acts the opposite way) every day.
• "Then I can go back in time by going fast." No: speed lets you reach the future faster (fewer of your own ticks), never the past.
• "Nobody feels time slow down." Right: on the ship everything runs together, so nothing feels different. You only notice by comparing with a clock that stayed home.

Try it yourself

Answers are hidden: work them out first, then check, or use the simulator's calculator.