A train runs at speed , and someone inside throws a ball forward at speed . Measured from the ground, the ball moves at .
The speeds add because we take a second on the train and a second on the ground to be the same length, and a metre stick to be a metre in both. Newtonian mechanics is built on the idea that time and length come out the same whichever frame measures them.
Now send light forward from the same train. Adding as we did for the ball, the ground should measure .
In a experiment at CERN, neutral pions moving at more than percent of the speed of light decayed and gave off light in the form of gamma rays, and the speed of that light was measured. Simple addition would give nearly ; the result could not be told apart from .
However the source or the observer moves, light in vacuum travels at . This is the principle of the constancy of the speed of light.
Since the metre has been defined as the distance light travels in vacuum in of a second. So m/s is not a measured value but one fixed by that definition.
Two laboratories, A and B, move past each other at constant velocity. With the windows shut, no experiment inside either one can decide which of them is moving.
On a smoothly sailing ship, drops fall straight down and fish swim equally in every direction in their bowl, just as when the ship is at rest. Galileo gave this example in his Dialogue of .
The upper row is drawn as seen from A, with B moving to the right. The lower row is drawn as seen from B, and now A moves to the left. Neither drawing is more correct than the other.
Every frame in uniform motion is as good as any other, and the laws of physics take the same form in each. This is the principle of relativity.
If the speed of light differed from frame to frame, you could measure it with the windows shut and learn how you are moving. Michelson and Morley tried exactly this in to detect the motion of the Earth, and found no difference in the speed of light from one direction to another.
Speed is the distance travelled, , divided by the time taken, . For light this ratio is the same in every frame.
Inside the car, send light from the back wall to the front wall. For someone on the train it covers the car's length and takes .
Seen from the ground, the front wall moves forward while the light is on its way. The light covers more than , so if its speed is still , it also takes longer than .
The time from sending the light to its arrival comes out different on the train and on the ground. And once time differs between frames, so does length, because a length is found by reading the positions of both ends at the same moment.
Special relativity starts from just two assumptions, the principle of relativity and the constancy of the speed of light, and brings in no new force or particle. The relativity of simultaneity, time dilation and length contraction all follow from these two.
For example, if a flash goes off at the middle of the car, it reaches the front and back walls together for someone on the train, but reaches the back wall first for someone on the ground. With light moving at in both frames, nothing else is possible.
At everyday speeds all of these effects are far too small to notice with ordinary clocks and rulers. That is why adding the speeds of a train and a ball works in daily life.