I finished my post about time travel paradoxes with a promise to write about one very specific example in which faster than light travel (which is equivalent to time travel) is possible. I am going to write it in a slightly weird way - firstly I intend to explain why this particular example is not possible, and then to show that it happens nevertheless.
We all know how an atom looks like - it is just a little ball (nucleus) surrounded by a cloud of electrons. Most of the atom is empty space. The atom has a very interesting property: the electrons that orbit the nucleus can be only on a finite number of orbits = energy levels. I am not going to talk about why it is so, but what is important to note is that the distance between any two energy levels is very large. In fact, the atom can be viewed as a solar system in miniature - the distances between the orbits are comparable to the distance between planets. The electrons decide on which energy level to be according to the energy they have. The more energy an electron has the higher (further from the nucleus) it will be..
Thee is a simple way to cause an electron to jump from orbit to orbit - we just need to hit it with a photon. If the energy the photon has is the energy needed for the electron to move to a higher orbit, it will absorb the photon and move to this higher orbit. After discussing this we can go to the main question I will deal with in this post:
Lets look on a regular piece of wire. If we will connect it to a battery (if you want this battery to be of any use lately don't do it) a current will flow in it. How much current? The usual way to define current is to look on the area of a cut of the wire = A. The current than is the amount of charge that passes in infinitesimal time=dt. To define it precisely we need first to define the density of charge flow. The density of charge flow (J) is the number of free electrons in the wire multiplied by their speed and by their charge: J=nvq. Note - J is a vector. The current is than defined as:
What the formula says exactly? It says that the rate of change of the charge is an integral on the charge flow density. Thus, the charge (electrons) that left this object is exactly the charge that passed the surface of this object - the integral is basically a flow throw the surface area of our object. This means that electrons cannot just jump outside - they must cross the surface somewhere. This conclusion seems obvious, but it isn't. Without this formula, having only the law that the total charge is conserved, we can say that if the electron jumped outside the total charge in the space is the same - so it is possible. But this simple formula says no. However, it doesn't say that it is impossible for the electrons to jump inside our object. To show that this is also impossible we need another formula, so lets develop it.
What is Q? It is the total charge of our 3d object. Therefore it can be expressed as:
In this integral rho is the charge density (it is not J). If we will put this expression into the previous formula, and also use the Gauss theorem to change the right integral in our formula from JdA to divJdV we will get:
The both integrals are dV, so we get that:
It is usually written in this form:
What does this new formula says? It says one important thing - the change in the density of charge in an infinitesimal is equal to minus the divergence of J. This is exactly what we were looking for - now even the electrons inside our 3d object cannot jump. They all must flow passing all the midpoints.
Therefore it is impossible for the electron to jump from one energy level to another in zero time.
But is this correct? No. The math I used is correct, but there is one single thing that is left out - quantum mechanics. The formulas I developed don't take it into account. They are still correct, but they cannot be used on very small scales.
It is now time to explain what happens with out electron. When the electron is being hit by a photon, it consumes the photon and gains its energy. This energy is exactly the energy needed for it to jump to the next energy level (otherwise the photon will not be consumed). The collusion between the photon and the electrons happens in one instance - it is not a process. What this means is that the electron gets energy that corresponds to the next energy level. Therefore it must be in a higher energy level. It cannot just stay. If there would be no photons, than the electron would consume the light energy gradually and would "rise" to the next level. But the energy is transfered in a collusion.
Mathematically, the potential energy is space must be continues. Because of this it is not possible for the electron to have a higher energy while being in a place that corresponds to a lower energy. The reverse is also true - when the electron descends to a lower energy level it releases a photon and jumps down - in zero time.
So, was Einstein wrong in saying that faster than light travel is impossible? Were I wrong when I said in the previous posts that faster than light communication (or travel, it doesn't matter) is equivalent to time travel? This I will answer in the next post.