Speed of light - the ultimate speed limit?

One thing that we are taught very early on in physics lectures is that the speed of light in vacuum, c, is the fastest anything can ever move, period. But is that really true? Well, the short answer is yes, and the long answer has some ifs and buts.

Probably the most popular reasoning behind this statement is that Einsteins theory of relativity does not allow for anything to exceed the speed of light. But what it actually says is that information or energy cannot travel faster than light.

Of course, even this is a very strict condition, since every particle, physical system and any radiation can be used to carry information and/or energy, which means that none of those can travel faster than light. To achieve a speed faster than c, you need something that doesn't contain information or energy, but what could that possibly be?

One entity that potentially carries neither, is the group velocity of light. In optical telecommunication, group velocity is the speed of the signal, but by itself it does not contain information. Another velocity associated with waves is the phase velocity, and this too does not necessarily contain energy or information.

Lets say we have light with two different wavelengths propagating through a medium where the refractive index is higher for shorter wavelengths. This causes the shorter wavelength to move more slowly than the longer one, and in this case the group velocity of the light can significantly exceed the speed of light, as is shown in the video below.

What you can also see from the video, is that the beginning of the signal can move at most at the speed of the faster component. Thus, no information or energy is conveyed faster than light, although something does exceed c.

Interestingly enough, if you reverse the effect of the medium, so that the longer wavelength component moves more slowly than the short one, then not only is the group velocity still greater than c, but it also propagates backwards.

In fact, group velocity is just one example of a class of systems, which apparently move faster than light, that is, systems featuring closing speeds. In general, two objects that are moving away or towards each other can produce apparent speeds that are greater than c.

Closing speed is easy to demonstrate with a particle collider as an example. Imagine that there are two streams of particles colliding, and both have a velocity close to c from an outside observers point of view. Then, from the same outside observers point of view, the speed at which the two particle beams meet - the closing speed - will be close to two times the speed of light.

This is completely fine though, because relativity is not concerned with how an outside observer is viewing the system, but how the components of the system see each other. Computing the relative speed from the frame of reference of either of the particle beams, you would always come up with a figure less than c.

More vexingly, it is also possible to have a phase velocity larger than c. This occurs if the material has a refractive index of less than one. In this case, the phase velocity can be arbitrarily high, approaching infinity as the index approaches zero. This is something that actually happens quite often, naturally for x-rays in many materials and with specially engineered meta-materials for visible light.

This may seem problematic, because the phase velocity is equal the velocity of a monochromatic wave and thus the observed speed of that wave is greater than c, right? Actually, it is the absolute value of the phase velocity that determines the actual speed of the monochromatic wave. For a refractive index that is larger than one, it is always real.

When the refractive index is less than one, the phase velocity attains an imaginary part that just shifts the phase of the wave. The absolute value will still be at most equal to c and thus the wave will not break relativity. All is well and good again.

Then again, there are some special cases when the speed of a particle can be higher than the speed of light within a medium (but still less than c). This can happen in nuclear reactors, and the particles that do move faster than light within the medium will emit light in the form of Cherenkov radiation. The first picture in this post shows this type of radiation, produced at the Advanced Test Reactor and below is a similar picture from the Reed Research Reactor.

So no, the speed of light is not a universal top speed for absolutely everything. In some special circumstances it can be exceeded, although you can't really do anything with those effects, other than light up your nuclear reactor in a very cool way.