Two spacetug captains are sitting in a bar arguing about who has the faster ship. They decide to have a contest to settle the dispute so set out for a space station near Alpha Centauri, 4 light years away, having agreed with the pilots of two slower ships that they would monitor the race. They move some distance away from the Earth then accelerate up to their maximum speed (using about half their fuel). They pass the monitoring ships near Earth simultaneously and coast to the station near Alpha Centauri arriving exactly together.
Having agreed to turn of their tractor beams, they return to Earth and are surprised to find the slower ships' pilots arguing with the barman over the duration and distance of the race. The pilot from Ship A is saying that he had checked carefully and the ships covered the 2.62 light year course in 3.98 years. The pilot of Ship B meanwhile is insisting that the course was only 1.73 light years long and they made the trip in 3.46 years.
At this point, in walks a physicist looking for a relatively cool beer. He points out that actually they are all correct since the monitoring ships were travelling at different speeds. By using SR, he shows that the reality they are both describing is that the separation between the departure and arrival events was 3 years.
- : - : - : -
The readings from Earth, the monitoring ships and the tugs were:
| Observer: | Earth | Ship A | Ship B | Spacetugs |
|---|---|---|---|---|
| Speed | 0.00 | 0.30 | 0.50 | 0.80 |
| x | 4.00 | 2.62 | 1.73 | 0.00 |
| t | 5.00 | 3.98 | 3.46 | 3.00 |
| x2 - t2 | -9.00 | -9.00 | -9.00 | -9.00 |
Both the temporal and spatial separations vary with speed just as the radial and lateral separations varied with position in the case of the Tugboats, but if we calculate the quantity:
sqrt(Spatial2 - Temporal2)
we always get a value of -9 representing a time-like separation of 3 years.
Duration and distance vary with the observer's speed for the same reason that the radial and lateral separation measured by the radar stations varied in the tugboat example. They are not fundamental aspects of the universe but are measurements of a deeper reality. This variation of measured duration with observer speed is known as 'time dilation'.