## Friday, 29 June 2012

### Dimensionally Challenged

Prepare yourselves, as I take you into ... The Fourth Dimension! <Twilight Zone Music>

All done!  While you've been reading this post, time---the fourth dimension---has passed.  What I really want to talk about here is the nature of dimensions, and why Einstein forced us to put time and space on the same footing.

So what is a dimension?  A better question is probably: how do we know how many dimensions there are?  The answer to that question is quite easy.  The number of dimensions of a thing is how many numbers we need to give to locate objects within that thing.  For example, the location of a pixel on your computer screen requires two numbers; say, one giving the distance from the left side and one the distance from the top.  A moment's thought makes it clear that
1. All points are  uniquely defined by those two numbers; and
2. Different values for those numbers always correspond to different points, and vice versa.
So the computer screen is a two-dimensional surface.  The numbers we use to label positions are called coordinates.
 A toy example of two-dimensional Cartesian coordinates

As an aside, there are in general many different possible choices of coordinates; indeed, usually infinitely many.  A couple more examples: the surface of the Earth is two dimensional, because I can identify places using latitude and longitude; but if I need to include altitude, then we have three dimensions.

In this sense, time is an obvious fourth dimension.  After locating something in space, I need one more number to tell me when it happened (or is going to happen).  But there is something different about how time was handled before and after Einstein, that is quite interesting.  Imagine that you are standing by a straight railway.  Your friend is on a train travelling along the tracks at a constant speed.  You both agree to time how long it takes for the train to pass between where you are standing, and a tree some moderate distance away.

To nobody's surprise, when you compare the times you measured you both agree.  You both measured the difference in time between two events to be the same.  This is not true for the distance between the two events; from your point of view, the train travelled along the tracks while from your friend's perspective, he didn't move and the tree approached him.  In other words, you agree on the time coordinate difference between two events, but not the space difference.  This is the usual Newtonian result, that is in full agreement with everything we are used to experiencing.

Now imagine that in addition to everything above, at the moment that the train passes you, you shine a flashlight at the tree.  If you could measure the time it takes for the light to reach the tree, you could compute the speed of light using distance over time; elementary stuff.  But what about your friend?  If he did the same measurement, he'd get a smaller value for the speed of light, because from his point of view the light has not travelled as far.

There's nothing intrinsically wrong with that, of course.  That's how speeds normally work; a bunch of cars travelling down the highway will seem to not move much relative to each other, even if they are doing fifty mph.  The problem is that Einstein tells us (and experiments confirm) that all observers should measure the same speed of light.  The only way that this can happen is if your friend thinks the light reached the tree in less time than you thought.  And this is actually what happens.

This is the essence of the Einsteinian situation: what one person thinks is a difference in time, another will measure as a difference in distance.  Time can no longer be held separate from the other three dimensions of the world, but truly mixes in as a fourth dimension.