Two mirrors. A photon bouncing between them. One bounce is one tick. The clock on the left is standing still; the one on the right is flying past you at the speed above. Both photons travel at exactly the same speed — that is the one rule — but the moving photon has further to go, because the mirrors move out from under it. So its tick takes longer. That is time dilation, and there is nothing else to it.
Unfold that diagonal into a triangle. The upright is the distance light covers in a stationary tick. The base is how far the clock slid sideways. The hypotenuse is the path the moving photon actually took. Because all three are travelled at the same speed c, the sides are just times — and Pythagoras hands you the whole of special relativity.
Time stretches by γ and length squashes by exactly the same γ. Here is a 100-metre train at rest, and the same train as you measure it going past.
This is the strange one, and it is the heart of the theory. Lightning strikes both ends of a moving train. Watch it twice: once from the platform, once from inside the train. Same events, same light, different answer to which happened first.
Up is time, across is space. A light ray is a 45° line, and because light must stay at 45° for everyone, moving to a different speed tilts both axes towards the light line like a pair of scissors. Your “now” is the flat line of dots — watch it tip as you change speed. That tipping is simultaneity coming apart.
Fire a rocket forward from a rocket. Common sense adds the speeds. Reality does not — and the result never reaches c, no matter what you put in.
Relativity is not a curiosity in a textbook. Satellite navigation is a clock comparison, and it only works because both halves of the theory are corrected for. The satellites are moving, which slows their clocks; they are also higher up the gravity well, which speeds them up by more.
Accelerate at a comfortable 1 g — ordinary Earth gravity, so the ship has a floor — flip over halfway and slow down. This is the twin paradox with an engine that at least obeys mechanics.
Newton is not wrong at everyday speeds. He is indistinguishable from right. Here is how much relativity actually changes things at speeds people have reached.
| Speed | m/s | v/c | γ − 1 | Clock drift in a year |
|---|
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Everybody measures light at the same speed: 299,792,458 m/s, exactly. If that cannot change, the things we use to measure speed, distance and time, must.
A photon bouncing between mirrors has to travel a diagonal once the clock moves, and cannot go faster to compensate. Pythagoras gives t = t₀/√(1 − v²/c²).
1/√(1−β²): 1 at rest, infinity at c. Every other number on the page is it in a different hat.
Two events simultaneous for you are not for someone moving: there is no universal “now”. Order is fixed only for pairs light could connect, the ones that could be cause and effect.
That light’s speed is the same for everyone is not obvious or intuitive, but it is measured, repeatedly, to absurd precision. Once you accept it, everything else on the page is forced: if the speed of light cannot change between observers, then distance and time must. Relativity is not a claim about light: it is what happens to everything else to keep light’s speed fixed. Time dilation is strange but swallowable; simultaneity is the one that hurts, and the trade is that relativity protects causality and abandons simultaneity.
A moving clock is not damaged, dragged or running badly: there is no mechanism acting on it, and it genuinely records less elapsed time. Each observer sees the other’s clock as slow, and both are right. What stops that being a contradiction is that they no longer agree on which events are simultaneous, and once “now” differs there is nothing left to disagree about.
Hold a comfortable 1 g and flip at the midpoint: the galactic centre is under twenty years of ship time, Andromeda under thirty. The crew could do it in a career. The catch is behind them: 26,000 years pass on Earth for the first trip and 2.5 million for the second. Relativity makes the journey survivable and the return pointless.
Speeds do not stack: 0.9c added to 0.9c gives 0.994c, and no chain of platforms each fast relative to the last ever gets there. It is a property of how space and time fit together, not of how strong the engine is.
GPS is a clock comparison. The satellites lose about 7 microseconds a day to their speed and gain about 46 to being higher in Earth’s gravity, a net gain near 38 microseconds a day. Uncorrected, that is a position error growing by around 11 kilometres every day. The correction is designed into the satellites’ clocks before launch.
Proper time: the time measured by a clock carried along the journey: every observer agrees on it, the one honest answer to “how long did it take”. Length contraction: a moving object measures shorter along its direction of travel by the same γ, and not sideways. β: speed as a fraction of c: walking is about 0.000000005, the fastest spacecraft 0.00064. c: exact because the metre is now defined from it, not the other way round.
Light cone: everywhere light could reach from an event; inside it is cause and effect, outside it no influence of any kind can pass. Rest energy: E = mc², the energy something has purely for existing: a gram is about 21 kilotons of TNT. Schwarzschild radius: squeeze a mass inside it and light can no longer escape: 9 mm for Earth.
No. It is not damaged, dragged or running badly, and there is no mechanism acting on it. Each observer sees the other clock as the slow one, and both are right; what stops that being a contradiction is that they no longer agree on which events happen at the same time.
Faster than you would think. Half the speed of light buys a time dilation of only 15%; to make clocks run at half speed you need 86.6% of c. Almost the whole effect is crammed into the last few per cent, which is why ordinary life shows no trace of it.
Because speeds do not stack the way they seem to. 0.9c added to 0.9c is 0.994c, not 1.8c, and no chain of platforms each fast relative to the last ever gets there. The barrier is a property of how space and time fit together, not of how strong the engine is.
Yes, every day. Each satellite loses about 7 microseconds a day to its speed and gains about 46 to being higher in Earth’s gravity, for a net gain near 38 microseconds a day. GPS is a clock comparison, so uncorrected that is a position error that grows by kilometres.