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
: