How can a simple equation for the separation of two events lead to a limit on the speed of objects?
To understand this, we need a simple piece of test equipment, a brick fitted with two flash bulbs and a timer. The timer is arranged to fire the first flash bulb after a short delay and the second exactly one year after the first. The initial delay is simply to allow us to accelerate the brick to some speed then switch off the engines and let the brick coast along without further acceleration during the test.
We have seen that the separation between the events obeys the rule:
In this case, s is fixed by the timer. When the brick stays at rest throughout, we have the trivial result x=0, y=0, z=0, t=1 hence s2=-1. We can use the equation to determine when the second flash will occur. In general:
hence in this example:
If the brick is thrown hard enough to be 0.25 light years away when the second flash occurs, it will happen 1.03 years after the first flash. This is exactly the same as the variation in radial and lateral components for the Tugboats but in this case affects the spatial and temporal components.
Here is a table showing the distance (in light years) and time (in years) between the flashes as measured by the space station:
| Distance | Time | Speed |
|---|---|---|
| 0.00 | 1.00 | 0.00 |
| 0.25 | 1.03 | 0.24 |
| 0.50 | 1.12 | 0.45 |
| 1.00 | 1.41 | 0.71 |
| 2.00 | 2.24 | 0.89 |
| 4.00 | 4.12 | 0.97 |
| 8.00 | 8.06 | 0.99 |
Note that if we through the brick hard enough, it can travel any desired distance before the timer sets off the second flash. Alternatively, we can travel to any part of our universe in as little experienced elapsed time as we wish (given suitable rocket technology). In this sense, the speed of light is not really a limit. As the distance increases however, the time also increases. Clearly, no matter how hard the brick is thrown, the speed measured from the launch site will never exceed 'c'. Although the analogy is only approximate, saying that we are limited because we can never exceed c is like saying hot air ballooning is limited because we can never exceed 90 degrees angle of elevation.
It is also important to realise that we could launch the brick in any way we like, the method is irrelevant. It is the relationship between distance and time imposed by the spacetime the brick passes through which creates the limiting effect, not the brick launching technology.