Game Development Reference
In-Depth Information
Now suppose that you are above the mirrors as they speed past you to the right. Then
the clock would look something like Figure 1-13 .
Figure 1-13. Stationary with respect to the clock
One tick of the clock is now defined as twice the distance of the hypotenuse over the
speed of light. Clearly H must be larger than L , so we see that the clock with the relative
velocity will take longer to tick than if you were moving with the clock.
If this isn't clear, we can also come to the same conclusion a different way. If we define
the speed of light as the amount of time it takes for the light beam to travel the distance
between the mirrors divided by the time it took to travel that distance, we see that:
c = 2L/Δt
but because the speed of light must be held constant in all frames of reference via locality,
we also have:
c = 2H/Δt
For the two preceding equations to be equivalent, Δt must be different for each system.
This means that if I were in a rocket moving at high velocity past you as you read this
book, you would look at the clock on my rocket wall and see it ticking more slowly than
your clock. Now, it may seem as though I would look out of my rocket and see your
clock running fast, but in fact the opposite is true. I would consider myself at rest and
you speeding past me such that I would say your clock is running slowly. This may seem
counterintuitive, but think of it in the same way as visual perspective. If you are at a
great distance from me, then you appear to be small. That doesn't imply that I would
appear to be huge to you!
Now the amount of dilation for a given velocity v is given by the Lorentz transformation :