Face two mirrors at each other, one above the other, and let light bounce between them. Each round trip of the light advances the clock by one tick; this is called a light clock.
If the mirrors are apart, one round trip covers . Dividing by the speed of light , one tick takes .
Unlike a clock driven by a spring or a battery, a light clock runs on the speed of light alone, so the constancy of the speed of light applies to it directly.
Now move the whole light clock sideways at speed . To someone moving along with it, the light still goes straight up and down.
To someone watching it go past, the clock moves to the right while the light climbs from the lower mirror to the upper one, so the path of the light is a slanted line.
The slanted path is longer than the straight one. The speed of light is still for this observer too, so the round trip takes longer.
One tick of the moving clock is measured as longer by the observer watching it. This is time dilation.
It is not only light clocks that slow down. If an ordinary clock carried alongside a light clock ran at a different rate, comparing the two would reveal your motion, against the principle of relativity. So every clock slows by the same amount as the light clock.
Let be one tick as measured by the watching observer. In that time the clock moves sideways and the light travels along the slant.
Take the half of the path that climbs to the upper mirror and it forms a right triangle: the vertical side is , the horizontal side , the hypotenuse . By Pythagoras,
Multiply both sides by and use to eliminate .
Move to the left, divide by , and write .
Taking the square root gives . The factor is written and called the Lorentz factor. One tick of a moving clock is stretched by the factor for the watching observer.
is at . As approaches , the denominator approaches , so grows without limit. At the denominator is and the formula has no meaning.
The point in the figure stops at , , , and in turn. At , ; at it is about ; at , about .
For small , . Even a bullet train at km/h has , so is only about .
At m/s, is about , and in time dilation of this size was measured with atomic clocks using aluminium ions.
Newtonian mechanics works so well in everyday life because it only deals with speeds at which can hardly be told apart from .
Let observers A and B move relative to each other at speed . For A it is B who moves, and B's clock runs at only of the rate. For B it is A who moves, and now A's clock is the slow one.
The clocks in the figure move at , so . In the upper row, B's clock ticks times while A's ticks ; in the lower row it is the other way round.
By the principle of relativity, neither A nor B can be declared the one really at rest. Each account is correct in its own frame.
No contradiction arises because comparing clocks that are far apart requires deciding what is simultaneous. A and B pair up different moments, so each can say the other's clock is slow without disagreeing about anything that actually happens.
The time read on a clock by someone moving with it is called proper time. The of the light clock is a proper time, and it comes out the same whichever frame it is calculated from.