Concerning the Fourth Dimension

In tonight's post, I intend to talk about, wait for it, the Fourth Dimension. This is not an easy topic to explore in a straightforward manner. Any discussion of different dimensions tends to evoke scenarios from science fiction, where a protagonist is transported, for instance, to an alternate reality in which cats rule the world and humans are diminutive slaves of their feline overlords. It summons up ideas of alternate Earths and the multiverse. This is not what physicists and mathematicians mean when they talk about multiple dimensions however. For a physicist or mathematician, dimensionality is the least number of coordinates necessary to determine the position of a point within a space, relative to an origin and to a frame of reference. This definition sounds a little dull and obscure when presented this way but what I wish to say about dimensionality, in particular what I wish to say about time, will I think be interesting and thought-provoking even to a mathematically uninclined reader. It sounds lie science fiction but isn't.  I don't believe I have ever read or heard these musings from anyone before. In a nutshell, until last night, I had one view of time and then, lying in bed last night, a completely different way of understanding time occurred to me. It is these two different views I want to talk about.

What is a dimension? Suppose we have a line and on this line a point we call the 'origin'. We can specify the position of any other point on this line with a single number, either positive or negative, its distance from the origin. This line is a space of dimensionality one. Suppose we have instead a plane, a piece of paper. We can specify any point on this plane with two coordinates – if the plane is oriented up-down, left-right, we can specify this point with the coordinates, x and y. The plane is a space of dimensionality two. This is Cartesian geometry, a subject most people learn something about at high school. (Note: the plane doesn't have to flat, it could be the surface of a cylinder or sphere and we could still specify the position of any point with just two coordinates.) We can go further; we can easily extend Cartesian geometry to three dimensions by specifying a point with the coordinates (x,y,z). So far, so good; three dimensional space seems to be the space we actually inhabit. But suppose we could go one step further and talk about a point located in a four dimensional space?

I tend to assume that my readers have some rudimentary knowledge of physics. The concept of dimensionality has been around since at least 1884 and the publication of the satirical novel Flatland, but the idea that space-time could be described as a four-dimensional manifold only first became truly popular as the result of Einstein's discovery of Special Relativity and Minkowski's showing that the interval between two points in space-time could be described as a vector in four-dimensional space, that time could be considered another dimension. Minkowski showed that a Lorentz transformation was equivalent to a rotation through a four-dimensional manifold, a change of the frame of reference. It is because of Einstein and Minkowski that we got the idea of time as a fourth dimension. According to the physics pioneered by Einstein and Minkowski, we can speak of an 'event' as a point in the space-time manifold specified by the four coordinates (x,y,z,t). This notion, that time is another dimension, the fourth dimension, is, I think, widely known. It is in fact so widely known that I wouldn't bother writing about it unless I had something new and interesting to say about the subject. And I believe I do.

One facet of the theory that has always fascinated and mystified me concerns particles. Take an electron. An electron is a point particle – it has no spatial extent. Its width, height, and depth are literally infinitesimal. But it has infinite duration. Its extent along the temporal dimension is infinite – often a typical electron has been around since the Big Bang and will continue to exist forever or until swallowed by a Black Hole. Of course, occasionally an electron is created, at the same time as its sibling particle, the positron, is created, and occasionally an electron disappears as the result of pair annihilation. But, generally speaking, an electron's life-span is vastly greater than its spatial extent. If time is simply another dimension, why this lack of symmetry between a particle's characteristics in the three spatial dimensions and its characteristics in the fourth temporal dimension?

We can conceptualise a particle as a kind of wriggly line through four dimensional space. If we pretend for the moment that there are only two spatial dimensions rather than three, an x dimension and a z dimension, and represent the temporal dimension as lying along the y axis, we can imagine a particle as a line that always goes up. Sometimes it wriggles to the left, sometimes to the right, sometimes forward and sometimes backward, but it is always rising, is even in fact constrained never to make an angle with the y axis greater than a certain amount because it can't go faster than the speed of light.  We can imagine the universe as a vast number of these lines. In the nineteenth century, particles were assumed to be eternal and indestructible – if we pretend this is true for the moment, this is equivalent to saying that the number of lines passing through the x-z plane at some y value is the same as the number of lines passing though the x-z plane at any other y value. Of course, we now know that a particle can turn into another particle, even transform into a couple of particles, but the law that the total mass-energy in the universe remains constant, that the total amount of stuff around always stays the same, can with a little fiddling be found to be compatible with this picture.

This model of the universe, as consisting of squiggly lines passing from the past into the future, is the picture I subscribed to until last night. And then a different way of looking at space-time occurred to me. This inspiration didn't come out of the blue – it was motivated by a clip I saw on Youtube explaining General Relativity. In this clip, the narrator makes a slight offhand comment that particles travel along the temporal dimension at the speed of light. When I heard this it confused me a little but then, after some reflection, it brought me to consider a quite different conception of space-time.

I'll try to explain this new, different model. A particle not only has a position in space at some time, it also has a particular velocity. That is, it has a velocity along the x axis, a velocity along the y axis, and a velocity along the z axis. All three components can have any value less than the speed of light.  It also, and this is key, has a velocity along the t axis, and this velocity is always the speed of light, c. Now, this seems confusing. The velocity along the x axis is the rate of change of x with respect to t. We are now being asked to believe that the rate of change of t with respect to t is c. How can a particle have a velocity along the temporal dimension at all? The solution to this conundrum is that, although space-time has four dimensions, the fourth dimension is not identical to time. I have put this in italics because it strikes me as important, and I'll repeat myself because it is indeed important. We live in a four-dimensional manifold but the fourth dimension isn't time.

Everything in the universe is barrelling along the fourth dimension at the same speed, the speed of light. Permit me to suppose, dear reader, that you are sitting reading this on your computer, that the distance between you and your computer remains constant, that the distance between you and your TV remains constant, that the distance between you and your front door remains constant. The reason all these distances remain constant is because you and the rest of your household accoutrements are moving through the fourth dimension at the same speed, which is very fast indeed. An analogy will make this clear. Suppose you are driving along a multi-lane highway at 100km/hour and the cars ahead of you, behind you, and on either side, are all travelling at the same speed. It will seem to you as if the world is unchanging even though the world around you is moving quite fast, by human standards. If everything is travelling through the fourth dimension at the same speed, we can't tell and live in an eternal 'now'.

This alternate picture can be clarified if we suppose it possible for objects to travel along the fourth dimension at speeds other than c. Suppose we have a four-dimensional sphere of radius r with its centre at t = 0 and its spatial coordinates smack bang in the middle of your living room. It is motionless in all dimensions. At any time before t = – r/c, the sphere doesn't exist. At t= – r/c, a point appears that grows into a sphere of radius r by t=0 and then contracts to a point that disappears at t=r/c. From your perspective, trapped in a permanent now, a point has appeared in your living room, grown into a sphere of radius r, then shrunk back to a point, and then disappeared. If it were possible for objects to travel along the fourth dimension at speeds other than the speed of light, it would seem to us as though things are continually appearing and disappearing. The law that energy is always conserved wouldn't hold true. The reason why this law is indeed valid, why things don't appear and disappear, is because everything is travelling through the fourth dimension at the same speed, the speed of light.

In this post, which I know is less clear than I would like it to have been, I have presented two ways of looking at space-time. There is a lot I haven't talked about. I haven't talked about imaginary numbers at all, even though they are central to Minkowski's theory. Some theoretical models, born from string theory, have it that there are eleven or even fourteen dimensions. I haven't discussed these either. As I said in the previous post, I still have a very shaky understanding of General Relativity. Although I am not a physicist, I hope that you still found this post interesting, and that it will stimulate you to think about these things yourself. After all, even though I am not an expert, I may not be wrong.