# Why is travelling around the speed of light a problem?

I don't fully understand what would happen if we could travel at the speed of light. But I saw somewhere here that it would mean events happen out of order. But why is this a problem. It is said that cause has to happen before the effect, but why does this have to be linear?

And why is it that the speed of light is the maximum? I take it that light (photons?) are massless, so why can't they travel faster?

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I think both aspects of this question have already been covered on this site, but I'm having trouble tracking down the right prior questions... –  David Z May 22 '11 at 23:58
I thought they had been already too, I searched for awhile but couldn't find exactly what I was looking for. –  Jonathan. May 23 '11 at 0:08
Just to be a wiseass, I'll point out it's certainly possible to get light propagating with a phase velocity faster than c (where c=speed of light in vacuum). It's also possible to get light propagating with a group velocity faster than c. Folks do it all the time. Of course, this doesn't answer your question, but I'm not sure anything will. Does Jerry Schirmer's formula help you understand why light travels at the speed of light any more than pointing out "that's the way our universe is"? It's an experimental "fact" that light in vacuum always travels at the same speed. –  Anonymous Coward May 23 '11 at 1:20
Your two questions, one about causality and one about why photons don't go faster than light, are conceptually quite different. They're related in that the same sorts of inquiries led you to both of them, but I don't think their answers are very similar. Could you limit this to just one question you most want answered, then ask the other in a separate thread if you want answers on that, too? –  Mark Eichenlaub May 23 '11 at 2:22

Before we even consider this, consider the kinetic energy of a relativistic particle:

$$E=mc^{2}\left(\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^{2}}}-1\right)$$

This represents the amount of energy required to accelerate the particle from rest (with respect to a given reference frame) to the speed $v$. It should be immediately obvious that this quantity becomes infinite as $v\rightarrow c. {}^{1}$ Since it is impossible to generate an infinite amount of energy, this then means that no particle with $m > 0$ can be accelerated to the speed of light. In fact, since high-energy particles have speeds that are generally extremely close to thespeed of light in a laboratory frame, it has become custom to quote the speed of the particles in terms of their energy--particle physicists routinely talk about "1 GeV" electrons to refer to the kinetic + rest mass energy of the electron in question, rather than bothering to translate this into a translational velocity.

${}^{1}$ This is true unless $m=0$. In the latter case, you can get a finite answer out of the $0\cdot \infty$ in the case of massless particles, but ONLY if they travel at exactly at the speed of light. Were they to travel faster than this, the formula would no longer be the indeterminate form $0 \cdot \infty$, but rather $0\cdot ({\rm complex \,\,number})$, which would be a definite, but nonsensical answer.

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Photons can't travel faster than $c$ and they also cannot travel slower than $c$ because they are massless. The rest mass is defined as $$m_0^2 = E^2 / c^4 - p^2 / c^2$$ and if this is zero, like for photons, you may easily show that $pc^2/E = c$ and this is how the speed is defined for an energy-momentum vector.

More conceptually, nothing can move faster than light in special relativity because according to special relativity, it would be equivalent to traveling backwards in time which would violate causality. A trajectory $$t=\tau, \quad x=V\tau$$ where $V>c$ may be Lorentz-transformed to another trajectory where $$t = a\tau, \quad x = b\tau$$ where $a$ is negative, so the trajectory goes backwards in time. Because the laws of physics must have the same form in this frame, it means that the identity of the cause and the effect may be switched. Because there's no sharp ordering between the cause and the effect, you may kill your grandfather before he met your grandmother, making your own existence inconsistent.

Such things can only be avoided if the cause preceded its effect from the viewpoint of all inertial systems which is equivalent to the effect's belonging to the future light cone of the cause, i.e. to the speed limit $c$ on any velocity that any signal or material object may have.

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