So in the last episode, I explained very basically what relativity means. We saw that the same event could appear differently to two observers depending each man’s frame of reference. That was bizarre, and from there we could begin to ask other questions. For instance, what other measurements depend on our frame of reference?

Let’s start with one of the more basic measurements-time. We’re going to use another example to explain what’s happening, but this time we’re also going to use some algebra and a triangle. Bear with me, the math isn’t too bad.

Alright, so imagine that we’ve got a space ship that’s traveling at speed v, and there’s a flash of light that is bouncing up and down inside of the space ship. The speed of the flash of light (and all light for that matter) is c, and that doesn’t change because the speed of light is a constant. So far so good? Great.

For an observer on the ship, we’re at rest with respect to the ship. Another way of saying that is that the ship is his rest frame. If we call the height of the ship *L*, then the total time it takes the light to travel from the floor to the ceiling is

But looking up at the ship from Earth is a different story because we’re dealing with a different frame of reference. While the light is bouncing up and down, it also has to travel horizontally with the rest of the ship. The total distance it travels is the diagonal, *D*. As a result, the time it takes in this alternate frame is.

Note that the speed of light, *c*, is the same for both of these equations. *D*, however, is not the same as *L*; *D* is longer than *L.*

**Let’s stop right here to drive home that concept. **The light travels less distance in the rest frame than it does in the moving frame. As a result, less time passes in the rest frame than in the moving frame. A clock on the space ship would go slower than a clock on Earth. That’s time dilation, and it’s not just a theory; It’s a fact!

… Now, if you wanted to find out how much slower, you’d have to do some math. We can find *D* using the Pythagorean theorem

At this point the math get’s really elegant (don’t worry, I won’t show it). Suffice it to say, you plug the *D *into the Δt’ equation, square the whole expression, collect the Δt’ terms, isolate the Δt’, square root the whole expression, and you end up with this equation

Which is cool, because then we can substitute in Δt to get

…the equation for how time in a moving frame relates to time in a rest frame. Boom.

So that’s time travel. Questions are totally appropriate at this time. Stay tuned for the next episode where I’ll show you how to shrink objects.

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