## Invariant Spacetime Interval

This title contains a lot of fancy words. No need to worry.

"Invariant" means "not changing." In this context, it refers to what doesn't change about the interval between two events when you look at that interval from different points of view?

In relativity, the point of view has to do with how fast the observer of these events is moving. If you are standing beside the road watching a car go by, the driver looks like she is moving. But if you are a passenger in the car, you would hope the driver appears to be sitting still.

In this short introduction to the idea of invariance, we will settle for a relatively simple example: Just two points on a blank wall. Call them P and Q.

To describe where those points are on the wall, use graph paper with x and y axes. Imagine the graph paper to be painted on a large glass plate.

As long as everyone knows where the origin (the 0,0 point) of the graph paper is, and which way the x and y axes run, any way you want to put that glass plate over the wall is just fine

For example, you can put the "graph glass" in the standard way, with the x-direction horizontal and the y-direction vertical, as it is at the right. But you don't have to.

If you choose to rotate the glass about 27 degrees clockwise (below), an interesting thing happens. The new y axis goes right through the point Q, so that the x-coordinate of the point Q suddenly becomes zero. Cool, right?

The point is it makes no difference. The x and y values of Q are now different, but Q is still where it was all along, up there on the wall.

Are we making a point? Sure we are. The numbers that describe where P and Q are depend on where we put the axes. But there is a number that doesn't change, that is "invariant."

The number that remains invariant is the distance between P and Q. That can't change just because you move the glass.

Pythagoras' theorem (c² = a² + b²) tells you what that distance is.

If instead of two points, there are two events, both time and distance are involved as variables in what is called the interval between the events.

Suppose an event took place here at x=0 at the time t=0, and a second event took place 3meters in the x-direction from here 5 minutes later. The two events can be described this way:

Event 1 at x=0 and t=0, and event 2 at x=3meters and t=5minutes. But those are the coordinates, and they are not invariably true. As seen by someone who is moving, both the time and the distance between those events might be shorter.

But while the time and the distance between the events change, there is a "space-time interval" that, like the distance between P and Q remains the same no matter how you choose to observe these events.

It turns out that this space-time interval is also a Pythagorean-type quantity, that we can call "I" for space-time Interval. We'd find it like this: I² = {ct}² - x². In this equation, c is the speed of light.

The idea is not complicated. How it comes about is something you'll understand if you read the book.

CAVENDISHSCIENCE.ORG