Faster than light

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:

How much time does it take for the electron to move from one energy level to another?

Answer: Zero.

To explain this, lets firstly look on why this time must be more than zero. In the following paragraph I will develop a mathematical formula - if you don't understand what I am writing just skip it and go to the next paragraph.

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:

I=

dA is an area vector of an infinitesimal part of the cut area A. If, as it is with regular wire J=const we get that I is simply: I=JA. The formula with dA is good for any object in which a current can flow. Because of this we can use it in a slightly different way. Lets think about a general bounded 3d object. This formula is good for it - but instead of saying what the current is, it tells what is the rate of change in the charge of this object.
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:

The plus sign is a convention - it doesn't really matter, because the positive and negative flow direction is defined according to what is easier to work with.
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.

Plans, solutions and Firefox 4

Today I spent almost all the day building a little project of mine. For a long time the wires behind my computer were in a total mess, so I decided to change it. It is very hard to do anything with such mess - I cannot cut the wires (obviously) and I cannot put them inside the wall (although that seems like an interesting idea to try). So I built a nice little wooden box with 8 plugs inside. I connected my computer and all the other equipment I had there to the plugs in the box. The box itself is connected to a power plug in the wall. It also has a button that allows to turn the power off without manually unplugging it. I even put a small CPU fan I had lying around in one of the sides of the box to make sure the inside will remain cool.
While I didn't remove the mess completely in this way, it looks much better now. I don't have any photos - but trust me it is a huge improvement.

So this was a plan and it by completing it I solved a problem. But this was not the solution I thought about when I wrote the post title. After I made sure that I didn't short circuit anything while building the box, I decide to check what Firefox version I have installed. As I suspected I was already automatically updated to FF3 RC2 (I have no idea when this happened) . However, while looking on nearby packages in synaptic I suddenly noticed something called Firefox 4. Since I am a curious being, I installed it. It turned out that this is an alpha version of Firefox 4 - Minefield. It imported all of my extensions and declared all of them to be not compatible except for MR Tech local install. This is probably because when I installed it I edited the .xpi file so that it will claim to be compatible with FF4. I used this extension to disable compatibility checks and all of my extension worked just fine. I usually don't use alpha versions, but to my surprise a bug that annoyed me greatly in FF3 RC1 and RC2 has dissapeared in FF4 - I can download photos without FF crashing when the download finishes. It was probably a conflict between two extensions, but still I am very happy that I can use all of my extension and to be able to download photos without restarting FF all the time.

New semester old semester

As I already wrote there was a long strike in all of the Universities in Israel. This was the longest such strike ever - 89 days. It ended this week. Also this week is the last week of the first semester..
I suppose Israel is the only place where after such strike the semester is not canceled and no compensation is payed to the students. Moreover, we still have to do our exams, on the courses that were taught despite the strike, while studying other courses. The semesters will be shortened also, so we will have to learn in 11 weeks what is normally taught 14 weeks. It was also announced today that due to financial problems there will be no TA classes for one of the courses next week.
For me this is good actually. It looks like I will be able to add a course to this semester, and overall I don't have too much to study. So I am not complaining.
There is no timetable for now but I will probably will have less than 20 hours per week.

For the blog this is bad news unfortunately - I will have much less time to work on it especially during the 2-3 next weeks. I hope that I will manage to use all the time I have wisely.

Calendars and science

This is the post I promised to write about calendars.
We are all familiar with the Gregorian calendar. It is the most used calendar today. However it is not the only one. In Israel for example we have another calendar. And while it is not used as much as the Gregorian (even in Israel), in many places we have dates written according to both Gregorian and Hebrew calendars. The reason why it is still used is mainly religious. The Gregorian calendar is usually regarded as Christian calendar - after all it begins from Jesus birth. Therefore, it is only natural for the orthodox Jews to be against its use.

The Gregorian calendar is an excellent one, in my opinion. However it and all the other calendars suffer from one major problem - they are not universal. I don't mean by this that they cannot by used in other parts of the universe. This is also true but it is only another aspect of the some fundamental problem.

Lets start from the beginning, taking the Gregorian calendar as an example.
First of all, we have the following - a year is 365 or 366 days, divided into twelve months of varying length. The month is usually considered to have 4 weeks and a week is 7 days. You must have noticed that even this description is not perfect - the year can have different number of days and so are months. We have of course rules according to which all these numbers are set in each year. But still the definition is not perfect - while it is not too difficult to make all of the necessary calculations the fact that the devision is not obvious shows that there is something that causes this.

Why we have 365 days in the year? It is the period in which earth orbits the sun. Thus for every planet we have a different duration of the year. Even the duration of one day is different from planet to planet. Therefore, our calendar would be entirely useless on other planets. If lets say we will colonize Mars one day the Gregorian calendar will be of little use there. But if we will make a different calendar for Mars, we will have problems communicating since we will be living according to different calendars. And this is even more important for computers that for humans. So what can be done?

Lets go deeper. What is time? And how can it be measured? It is very difficult to answer this question. And since according to relativity theory, time, gravitation, space and speed are all connected, it is almost impossible to give a really good definition..
In the Gregorian calendar the shortest period of time is one day. This is its basic time unit. But a day is not a good choice for a measure unit because it length changes. Even on earth the length of the day is different at different seasons. In fact all of the problems with the Gregorian calendar and with other calendars come from this simple fact - a day is not a natural measure unit for time.

But then what is? Well one option is the so called Planck time. It is the time needed for light to cross a distance equal to the Planck length. Since the speed of light is invariant the time will also be always equal. The problem with it - it is too small. It is only about 10^-44 sec.
The only other option is to use a system of calendars. This is one calendar for every scale. In this way there can be one calendar for earth and one for mars - and all the communications between the two planets can be done according to the main calendar of the solar system.
While this brings more complexity, this is probably the best choice if we want a calendar that is naturally connected at least to the planet it is used on..