The Necessity of Relativity

Relativity holds a rather curious place in physics, as I see it. Other branches – nuclear, mechanics, E&M, so forth, all seem firmly based in the study of physical objects, systems, and principles. Relativity is, in point of fact, much more of a math than a physics. It starts with simple axioms about our understanding of the universe and derives conclusions from them about the way the universe must work. The tools of relativity are not physical laws, but rather logic.

Now, as I said, relativity starts with simple axioms, and works from them to derive its conclusions. The axiom in question in this case is that the speed of light is finite, and constant. No matter who is measuring it, how fast they are going, what they are doing, who else is measuring, all observers always measure the same speed of light. That speed we’ll call c here, though more traditionally it is 299 792 458 meters per second.Now to the experiment.

Alice Cooper is riding in a train which is moving at speed v. At some time, which we’ll call even one, a flash of light is emitted from the floor of the train, straight up towards the ceiling, being absorbed by the ceiling at event two. If we say that the train is of height h, then the entire transit time of the flash, from the floor to the ceiling, as seen by Alice, is h/c. And needless to say, the length of the path of the light flash, from the floor to the ceiling, as seen by Alice, is simply h.

At the same time, a man with no distinguishing characteristics, code named ‘Observer’ is standing at a station watching Alice Cooper’s train pass. He sees the light emitted, watches it travel, and watches it be absorbed by the ceiling of the train. Below is a diagram of what he sees.

The problem should begin to be clear. Once the light is emitted, some time passes before it is absorbed by the ceiling. During this time, the train continues forwards, so the point of absorption is no longer directly above the point the light was emitted from. As such, there is now a horizontal component to the path of the light that was not present in the first case. As such, the light travels further for the station observer than it does for Alice Cooper.And -that- is ridiculous! How could someone measure something as one length, and someone else measure it as another? To make matters worse, since as we said, they both measure the speed of light to be the same thing, c, if they measure the paths as having different lengths, then they must measure the time between the two events as different as well, the station observer measuring a longer time since it takes longer for light to travel along that path.

Clearly a contradiction. Nevertheless – nothing in this thought experiment is strange or alien or impossible. There are no buses going the speed of light, no perfect spheres. If we wanted to, we could put Alice Cooper on a train, we could speed it up to some velocity, and some man could be observing the train. Everything is doable, and the contradiction is simply a logical result of what we know to be true. This is the essence of relativity – it is a mathematical description of how the -must- universe work. It is a necessity.

Let us ask then, exactly what the station observer measures between the two events in terms of distance and time. Let the distance between the emission and the absorption be L, and the time be T. Because the speed of light is constant, we know that the time the station observer measures must be T = L/c.

But what of L? Notice the diagram below, with the conveniently drawn right triangle.

The hypotenuse of the diagrammed triangle is the path the light takes, so it is of length L. The vertical leg of the triangle is simply the height of the train, which is h. The horizontal leg is the distance the point of emission travels during the time interval between events. In other words, v*T. Using the Pythagorean theorem and that T = L/c,

Notice that L, the distance the station observer measures the path to be, is simply h, the distance Alice Cooper measured, scaled by a curious factor. Now look at the time.

Notice that T, the time the station observer measures the path to take, is simply h/c, the time Alice Cooper measured, scaled by the same curious factor. Length and time have been dilated. The conclusion is inescapable, bizarre, completely logical, and utterly fascinating.

So we have two observers, measuring ostensibly the same thing, but each getting markedly different results in their measurements of time and distance. This is a drive by introduction to special relativity.

The important thing to note here is this – many people object to the conclusions of relativity as bizarre, counter intuitive, nonsensical, and downright impossible. However, the problem is not with relativity. Relativity is simply a precise, mathematical description of what -must be true- about the universe based on what we know to be true. There are no special forces at work here – no magic, no hand waving. Each step of the derivation follows logically from the rest. Relativity is simply a mathematical necessity based on our understanding of the universe.

A mathematical necessity. And there’s really little more to say than that.