romansland

The difference between time and space

January 4th, 2009 Posted in Science

In this post I would like to explain the way nature seems to choose the most simple and elegant way, but it is often thought to be complex due to the limited minds of us humans. For this I will talk about the relation between space and time, why they are similiar, and why they are different. I hope you will find it interesting.

Any event can be labeled by simple coordinates: there are three space dimensions, so we need three coordinates x, y, and z to denote where it happened, and there is one time dimension so we need one coordinate t to denote when it happened. This makes it tempting to just see events as points in a 4-dimensional space (which is generally called Minkowski space).

So here it is: our event a has got a place in the four-dimensional time-space. We can denote it’s position with a vector . In the illustration I did not draw the third space dimension z, because I cannot draw a 4-D cube, but that is not important for my purpose.  What is important is that in this way, I can represent time and space as absolutely equivalent members of the space-time, and there is also no prefered direction like up or down, of backward and forward in time. But we feel that time and space are not alike in our experience, so something is missing. We should investigate what is the difference, and what is the most elegant way to include it.

In physics, we have learned that it is often instructive to look at symmetries. This is because if the laws of physics are symmetric under a certain transormation, then there exists no measurement that can distinguish between the situation before, and the situation after the transformation, and thus they are equivalent. In this case there are symmetries in the space and time dimensions. If I translate the event a and other events that are caused by it to a different location in space and time, all laws of physics remain the same and the principles of cause and effect work in the same way. This is called translational invariance. Alternatively, we say that the universe is observed by an observer (a physicist, of course), who stands at the point t = x = y= z = 0. Now moving all events is equivalent to moving the observer.

There is another symmetry, which is rotational symmetry. It is equivalent to the observer tilting its head a little and seeing the universe a bit rotated. Let’s look in detail how this works for two events in a two-dimensional space of x and y.

Here you can see we have rotated two events a and b around the origin, over a certain angle θ. The new coordinates are given by

Allright, this seems very well. Now lets try the same thing when we mix space and time dimensions. For simplicity, we’ll take a slice of space-time, and rotate in the x and t coordinates.

I have used an arrow to denote that event a may be the cause of event b, because in the left figure event a happens before event b.  An object may be travelling from a to b. But you see in the right figure, after we have done something to the observer and rotated the universe around him, b surprisingly happens before a! An object travelling from a to b is now going back through time!

Now I don’t completely understand why natures refuses this type of symmetry. but it is certainly strange that one observer finds a to be the cause of b, and another may find the opposite, even though they are watching the same events. So indeed, time and space are different after all.

Now it would make sense to dismiss the idea that space and time are related after all. It would seem simple to just say that time and space do not rotate into each other, and we need other ways of describing a transformation of space-time that affects both space and time. This would be the Galilean transformation. But would this really be the simplest and most elegant way of describing space-time?

In fact, nature was so wise to make rules that are so elegant that humans did not discover them until the 19th century. It works like this: whenever we want to rotate through space and time, we should not rotate over an angle θ, but over an angle iθ. And we must replace all time coordinates t with the time coordinate ict:

The i is the well known imaginary unit, defined by The c is the speed of light, which serves as a conversion factor between time and space. That’s it! Now we can write down the rotation equations that we already found in the new system:

And indeed this is a new symmetry of our universe. It corresponds to putting the observer on a moving train and let him observe from there. According to this observer, the time and space coordinates of events are remixed in such a way, that all objects that were previously standing still are now changing their position with more or less the speed of a train. And these two equations describe exactly that. I know for a non-physicist it seems a bit strange with all the i’s in there, but in the end these equations can be used quite easily.

You may object that this solution seems different from the normal idea that when you get on a train, other objects will change their velocities with exactly the speed of your train, and that’s that (Gallilean transformation). And you’re right: the solution is different. And now comes the beautiful conclusion of this story: the above solution is more elegant, it is different, and, if you do it very precisely, you can measure it is also the only correct solution. It is the basis of the theory of general relativity. I conclude that physics is the search for the simplest solution that fits a problem, but sometimes it is very difficult to be smart enough to understand the simple solution. I hope you now have an idea about how this search works, though I admit this story is much too short for a proper explanation. If you like it, please leave a comment!