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One description of relativistic effects that I've heard/read goes something like this:

Everything moves through spacetime at a constant speed. An object's direction of travel through spacetime can change, though. For instance, if you are at rest, all of your motion through spacetime is through time. If you are walking, though, some of your motion through spacetime will be allocated to traveling through space. Since less of your motion through spacetime will be allocated to traveling through time, your local time will be slower than the local time of your surroundings.

I don't know if this description is a good model or not. It could just be an imperfect analogy as far as I know. If it is a good model, though, it seems to me like light might only be devoting some of its total motion through spacetime to moving through space: it seems that light also travels through time. (This might be incorrect because of the relativity of simultaneity. I don't really know.) If nothing can go through space faster than light but light isn't devoting all of its motion through spacetime to traveling through space, could that mean that spacetime is quantized?

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I more or less understood you, up to the last clause. Could you explain a bit more why you think that has something to do with quantisation? (As for the rest, it's ok as an analogy, except the time dimension sort of works in the opposite sense as the space dimensions: you wind up moving faster through time when you move through space. I would try to explain more but, as I say, you lost me at the end.) –  Retarded Potential Mar 14 '13 at 22:27
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@RetardedPotential: If all of an object's motion through spacetime is devoted to moving through space, it seems like that object wouldn't move at all through time. Light, however, does move through time (maybe?), so it must not be devoting all of it's motion through spacetime to moving through space. Therefore, the "constant speed" at which everything moves through spacetime would seem to be faster than the speed of light in some sense. If there are no possible headings through spacetime that make an object move through space faster than light, it would seem that there is a (continued) –  yakiv Mar 14 '13 at 22:49
    
discontinuity in the range of possible velocities. Would that perhaps mean that spacetime is quantized? (I'm not sure that I'm being any clearer . . .) –  yakiv Mar 14 '13 at 22:51
    
Thanks, I understand where you're coming from now. Time works in the opposite sense as space, as I say. So this does in fact lead to a speed limit. There's no gap. @NathanReed has explained. –  Retarded Potential Mar 15 '13 at 16:50

1 Answer 1

This analogy is based on the 4-velocity, a vector in spacetime that generalizes the 3-dimensional concept of velocity. The 4-velocity of an object can be defined as $dx/d\tau$, where $x$ stands for the 4-dimensional spacetime coordinates of the object, and $\tau$ for its proper time (the time experienced by the object as it moves around). For ordinary objects it always has "length" $c$, the speed of light. By "length" I actually mean the spacetime interval, which is calculated with opposite signs for time and space. Objects that are moving at higher speed have greater space components in this vector, and therefore they have a greater time component as well in order to balance out and maintain a "length" of $c$. The time component is $dt/d\tau$, i.e. the ratio of coordinate time to proper time. When this ratio is larger, it indicates that the object's local time is slowed down relative to the coordinate time, i.e. the object is experiencing time dilation.

Light travels so fast that the proper time it experiences is zero. Unfortunately this makes the definition of 4-velocity break down, since the derivative $dx/d\tau$ would be a division by zero. As an object accelerates toward the speed of light, both the space and time components of its 4-velocity grow toward infinity.

So the speed of light really is the universal speed limit in relativity. To go faster would require a negative proper time, which doesn't physically make sense. While light is traveling through time according to you or me, it has no proper time of its own, so "from its own perspective", so to speak, no time is passing.


As an aside: the title of your question made me think of the following, though your question isn't really about it. There is a theoretical possibility that light could travel at different speeds depending on energy; this is predicted by certain quantum gravity models - which also quantize spacetime. So your question was "accidentally" quite pertinent!

Some observations have been done on gamma-ray bursts to look for this effect, but so far no conclusive evidence of an energy-dependent speed of light has been seen. Here is a fairly recent blog post on the subject, in case you want to read more.

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