# What is the relationship between the speed of light and virtual particle production

What is the actual basis of the speed of light? Why is the transfer of information limited to it? Virtual particles that arise and disappear within a certain timeframe, is There a relationship between how long those particles appear before they disappear in some way geared to the speed of light? I've been assuming that the speed of light is the maximum transmission of information from one discrete bit of space time to the next (the Planck limit) which is so much smaller than the virtual particles, the scales are so different that I don't understand the relationships. I apologize for this question not being very well focused

To address your first question about the speed of light, my favourite way of thinking about this is to treat special relativity as a geometrical theory. If you've ever watched a popular science programme about general relativity you'll probably have heard that GR is based on geometry (specifically Riemannian geometry). You can think of SR in the same way.

If you think back to learning about Pythagorus' theorem, this tells you that the distance from the origin to the point in space (x, y, z) is:

$d^2 = x^2 + y^2 + z^2$

Special Relativity extends this idea and defines a quantity called proper time, $\tau$, defined by:

$c^2\tau^2 = c^2t^2 - x^2 - y^2 - z^2$

where $c$ is a constant that will turn out to be the speed of light.

The key thing about Special Relativity is that it states that the proper time is an invariant, that is all observers will calculate it has the same value. All the wierd effects in SR like length contraction and time dilation come from the fact that $\tau$ is a constant.

So what about that constant $c$? Well the quantity $\tau^2$ can't be negative otherwise you can't take the square root - well, you can, but it would give you an imaginary number and this is unphysical. So suppose we let $\tau^2$ get as low as it can i.e. zero, then:

$0 = c^2t^2 - x^2 - y^2 - z^2$

and rearranging this gives:

$c^2 = \frac {x^2 + y^2 + z^2}{t^2}$

but $x^2 + y^2 + z^2$ is just the distance (squared) as calculated by Pythagorus so the right hand side is distance divided by time (squared) so it's a velocity, $v^2$, that is:

$c^2 = v^2$ or obviously $c = v$

So that constant $c$ is actually a velocity, and what's more it's the fastest velocity that anything can travel because if $v > c$ the proper time becomes imaginary. That's why in special relativity there is a maximum velocity for anything to move. Although it's customary to call this the speed of light, in fact it's the speed that any massless particle will move at. It just so happens that light is massless.

• I think I sprained my brain! Feb 2 '12 at 16:12

Let me address the part about virtual particles, since the speed of light has been well covered by John.

Virtual particles that arise and disappear within a certain timeframe, is There a relationship between how long those particles appear before they disappear in some way geared to the speed of light?

Virtual particles are a mathematical formulation arising from a specific method for calculating the probabilities of interaction of elementary particles, using Feynman diagrams.

The diagram on the right following very specific rules (in one to one correspondence with the diagram) allows for writing down an integral which will give the probability of two electrons scattering off each other, an angular distribution and a crossection. The intermediate gamma is called a photon because it carries the quantum numbers of a photon except its mass ( 0 in reality), can have non zero values and therefore it is called virtual ( since imaginary has been used for imaginary numbers). It is a pictorial representation of a function ( called propagator) which will be integrated within the limits of the problem).

Thus one cannot talk of a virtual particle in the same way as with the real particle. The virtual qualifies it as a carrier of the quantum numbers of the synonymous real particle.

Now in general terms, within the Heisenberg Uncertainty Principle, particle pairs can be mathematically represented as virtual particles , but can only become real,i.e. on mass shell, if energy is supplied , as virtual particles are off mass shell. This is not connected to the speed of light, but to the HUP. in the second form

of energy and time.