One consequence of Special Relativity I find particularly interesting is that all particles are always moving at the speed of light. The catch is that this motion is not just through spatial dimensions, but temporal dimensions. That is why a photon, which travels at the speed of light through space, does not experience time (that is to say it is as "young" upon absorption as it was upon emission, even if it has travelled millions of lightyears) - because that photon is using all of it's alloted light speed velocity to travel through spatial dimensions, and therefore has no further velocity to travel through time. This also explains the "twins paradox", where one twin travels through space at near light speed while the other remains on Earth. When the travelling twin returns to Earth he finds his twin has aged many years while he himself has aged little. That is because the travelling twin, who was moving at near the speed of light, had only a little velocity left to travel through time; meanwhile the twin who remained on Earth travelled at far lower speeds and therefore used most of his velocity for travelling through time.

So, every particle is always travelling at light speed through its combined velocity from its motion through the three spatial dimensions and the temporal dimension. In other words, every particle has an associated velocity vector that exists in 4 dimensions and has a fixed length of c (the light speed constant). Changes in speed and motion through the spatial dimensions change the direction this vector points in four dimensional space-time, and this change in direction is responsible for some of the strange consequences of Special Relativity, such as the two examples mentioned.

Now, imagine the set of all possible space-time velocity vectors. This set consists of an infinite number of points in four dimensions, all of equal distance from a center point (because all velocity vectors are of equal length: c). What does such a set of points look like?

In two dimensions, a set of all points equally far away from a center point is the perimeter of a circle. In three dimensions, it is the surface of a sphere. In four dimensions, it is the surface of a four-dimensional sphere, four-sphere, or hypersphere depending on which terminology you prefer. I prefer four-sphere.

However, because particles do not travel backwards in time, what we really have is not a complete four-sphere, but half of a four-sphere - the "time greater than or equal to zero" half. In visualizing this it is easiest to eliminate one spatial dimension, resulting in two spatial dimensions and one temporal dimension, and therefore half of a three-sphere. But this is only to help clarify the concept, as it is rather difficult to visualize a four-dimensional sphere.

Anyway, the space-time velocity vector of every particle is located on the surface of this four-sphere. Because a particle at rest uses all of it's velocity to travel through time, it is travelling at the speed of light through time and its associated velocity vector's location on the surface of this half four-sphere is at the apex or pole. Likewise, because a particle at light speed is using all of its velocity to travel through space, its associated velocity vector is located at the "time equal to zero" equator of our four-sphere, or more accurately the base perimeter of our half four-sphere. Any particle which is in motion, but not at light speed, has an associated velocity vector located somewhere between the equator and the pole of this four-sphere. Because we humans travel nowhere near the speed of light our velocity vectors can never travel far from the pole. Since you're presumably sitting motionless in front of a computer right now, your velocity vector is right at the pole.

One more concept must be understood before I finish up - that of vector projection. Here's a simple example for those unfamiliar with it.

Take a pencil and a piece of paper. Put the eraser of the pencil on the sheet of paper and point the pencil anywhere you like. We can think of this as a three dimensional vector. Now shine a light from directly above the pencil and observe the shadow the pencil makes on the paper. This is the two-dimensional projection of our three-dimensional vector. Move the pencil around (keeping the eraser in contact with the paper) and observe what happens: while the pencil does not change length, its shadow - its projection into two dimensions - does change length. Now take another pencil of the exact same length and place its eraser near that of the original pencil. Orient your two pencils anyway you like and again shine the light down from directly above. While both pencils are always the same length, in many instances their projections are of different lengths.

This is why, even though all particles are always travelling at the speed of light, they appear to have different velocities: their space-time velocity vectors are pointing in different directions and therefore the velocity of the particles project differently into three dimensions.

Let's return to our four-sphere (more accurately, the surface of our half four-sphere), which as you recall is the set of all possible space-time velocity vectors. Well actually lets back up. Lets return to our half three-sphere to visualize this part. Imagine you took a hollow globe and cut it in half. You now half the surface of half of a three-sphere. Place this onto a peice of paper, and shine a light on it from above. Although you can't see it, you know that the shadow being projected onto the paper is a circle - not the perimeter of a circle, but a solid ("filled in") circle. And this circle has the same radius as the globe.

Therefore the surface of a three-sphere projects into two dimensions as a solid circle. But remember a circle is just a sphere that has lost one dimension.

If we were to take the surface of our half four-sphere and project it into three dimensions, what would we get? We are removing one dimension so we get a three-sphere instead of a four-sphere. As with the half-globe projecting to a solid circle, the surface of our half four-sphere projects to a solid sphere. But wait a minute, we projected half of a four-sphere, why should we get a full three-sphere? For the same reason our half-globe didn't project to a half-circle. We are projecting in such a way that we lose a dimension, namely the time dimension. And it was in the time dimension that the four-sphere was halved. Again, the radius of our three-sphere is the same as that of the four-sphere we started with.

So what is the significance of this solid-three sphere resulting from our projection? It is the set of all possible spatial velocity vectors. Because nothing can travel faster than the speed of light, the radius of the sphere is limited to c. Because, in three dimensions only, objects can travel slower than the speed of light, the sphere is solid and velocity vectors are not stuck on the surface as they were in our four-sphere.

And while a motionless body sits at the pole of our four-sphere, it sits at the center of this three-sphere. How about objects travelling at light speed? While they were on the equator of our four-sphere they are on the surface of our three-sphere. This brings us to an interesting conclusion about the geometry of four-spheres: The equator of a four-dimensional sphere, or any cross section of a four-dimensional sphere, is the surface of a three-dimensional sphere. This is analagous to the more intuitive fact that the cross section of a cube is a square.

So where does this all take us? I don't know. I'm sure it's been thought of before, but I found it an interesting study.

Thanks,

Dane

Wow man, that was an interesting read. Just one question, would the rate of change of time be limited to the speed of light? Light appears to always move in straight lines except when effected by gravity, would this suggest that light is differentiated from a higher dimension? I don't fully understand how light is standing still in the 4th dimension because if it was we wouldn't be able to measure its speed, its direction yes if the source kept emitting light but not its speed. Light in different dimensions has always confused the hell out of me.

Nevertheless I enjoyed the read keep up the good work.

Yes, the rate of change of time for any particle is limited to the speed of light. Travelling through time faster than this would amount to increasing the magnitude of a particle's space-time velocity vector, which would violate Special Relativity.

As I understand it (and I don't claim to understand General Relativity very well) when we see light "curve" around a massive object, the light it always travelling in a straight line through space-time; it is space-time itself that is curved. However, nearly every time this concept is explained the "bowling ball on a trampoline" or similar analogy is used to show how a massive object warps the space around it. Except this analogy uses a two dimensional surface to represent four-dimensional space-time, and due to the bowling ball this surface is warped in the direction of another dimension. This suggests four dimensional space-time is warped into a fifth dimension by objects with mass, but because there are only three spatial dimensions and one temporal dimension I don't see how space-time can be warped into a fifth dimension.

As for your last question - we are able to measure the speed of light even though the photons don't experience time because we, as the people doing the measuring, do experience time. When we measure the speed of light and state it in miles per hour, we are talking about what is an hour for us and not for the photons making up that light. Wikipedia's page on Time Dilation can explain this better than I can.

Thanks,

Dane

Ok dude, thanks, you've given me a bit to think about.