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Originally Posted by AndreasJ
From another point of view, everything moves at the speed of light. This would be looking at the four-speed, which always has the magnitude of c - it's just that for ordinary objects the overwhelmingly largest components points in the direction of future time rather than in a spatial direction. In this view time dilatation is simply the fact that more of the constant total velocity points spatially so the particle goes slower in time, and photons (and other massless particles) travel in an exclusively spatial direction, so being in a sense timeless.
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This is an insightful way to look at the question, and it is often expressed this way. I would actually say it slightly differently-- I would say that
any observer always sees the 'rate of flow of time' to be unity-- one second of action elapsed per second on his/her clock (which is really just a statement that all local processes proceed according to a local standard clock, no one knows why, but that is the core concept of time). However, they do not perceive
other observers as having a unity rate of flow of time, when referencing their time to the observers' time (that's time dilation). That's bad, we've lost an invariant. The way to recover that invariant is to look at both space and time, as
AndreasJ says, and include the "rate of flow" of both, together into a
single rate (or "spacetime velocity", if you will). The problem is, space and time are different entities, so are measured in different units, and a conversion is required to be able to unite them into a single rate. The conversion that allows the rate to be invariant is that all distances get multiplied by 1/c, and then we combine the whole business into an effective rate-- an effective one second of action per one second of the observer's clock. So c is handed to us as part of the desire to convert space into a timelike format so that we can get an invariant rate of flow of something. Then massless particles must also move at that speed c, in a vacuum, so that no time can pass for them in "their own frame" (a frame in which they would have no mass so could not exist). What I'm saying is I don't really like the idea that "we are all moving at rate c", I like to say "we are all moving at rate unity", and we think everyone else is also moving through time at rate unity, where we just need to generalize our concept of 'movement through time' to account for an appropriately converted distance contribution.
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From this POV the question of the thread title is trivial - the spatial component of your four-velocity can obviously not be greater than your total four-velocity.
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True, but the question persists, why is that speed limit equal to c? Or as I've put it above, why is 1/c the conversion factor from distance to time that makes the rate invariant? We still have no idea. Could that number ever come from some other number, or would it always just have to be inserted as an observed parameter? Some people might use an anthropic argument, "it had to be c, or very close to it, or we couldn't be here". Although that may be true, I wouldn't call that a scientific explanation (as it relies on untestable assumptions), but it may be the best science can ever do.