Random Thoughts on the Universe

Sunday, June 25, 2006

Time Travel Troubles

Time travel has always been a popular subject in science fiction, but in the last two decades it has entered the realm of serious science following a number of theoretical physics papers outlining the details of how to construct a time machine. Of course this also requires a serious investigation into several paradoxes inherent in time travel - what happens if you kill your own grandfather before you are born, and therefore make it impossible for you to go back and kill him? If you go back in time and give a copy of Shakespeare's plays to him before he writes them, who actually wrote them? And my favourite, an orphaned boy grows up, goes back in time, fathers himself, then gets a sex change, goes back in time, and is his own mother as well...

So could these happen? There is no answer yet from the scientific community. One solution is to argue that some undiscovered law of physics prevents time machines (or at least hides them). For example, if light falls in a time machine, goes back in time, and then comes out and re-enters the time machine it doubles the energy in the machine. Now that doubled light does the same thing and you have four times as much energy. Obviously with this argument you can quickly argue that as soon as a time machine forms the feedback it generates destroys it! A similiar argument would have the huge energy form an event horizon (black hole) around the time machine so it cannot influence anything else.

It is the other solution which provides an interesting philosophical problem. It has been demonstrated that quantum mechanics can also solve the problems for simple systems, by ALWAYS going to a non-paradoxical solution. The standard example is of a bomb rolled into a time machine in such a way that it goes back in time and destroys itself before being launched. It turns out that the bomb will always come out of the machine slightly off course, and therefore does not destroy its original self but instead knocks it off course, and then the off-course original bomb comes out of the machine off course. No paradox!
So in the grandfather paradox, the solution is that the gun jams, or the grandfather somehow survives the killing attempt, or some other unexpected contigency.

As for the creation paradoxes, such as the plays of Shakespeare, it is possible that the same method prevents the transfer of information. But we also know in quantum mechanics that particles can suddenly appear, travel for a while and then bounce back and travel back in time until they destroy themselves. Perhaps the plays are a large scale version of this effect, where they are spontaneously created? (This actually violates several other laws of physics, and is not much of an answer anyway, but it is interesting...)

There is a third solution which is also quite interesting - multiple universes. There is an idea in quantum mechanics that every time a decision is made, a new universe is created with each universe containing one outcome of the decision. Flip a coin, and in one universe it is heads while another universe is exactly the same except in contains tails. Then time travel is easy! By arriving at a previous time, you have automatically landed in a different copy of the universe (because one copy you arrive, one copy you don't) and if you then kill someone or change something, you go into yet another version of the Universe. And it doesn't matter, because you are born in your original universe, so what you do in the other one has no effect on you or your world!

Tuesday, June 13, 2006

Fractals and the Chaos Game

Today's topic is the Chaos Game (which it should be noted has no relation to chaos) in which a string of random numbers is used to generate very complicated fractal patterns. Two specific examples of this game can be found for Sierpinski's Gasket and Sierpinski's Carpet, as well as the general rules (although it should be noted that the screensavers mentioned on these sites are approximately ten years old, and are no longer maintained)

The general rules of the game are:

  1. The first, and most important step in producing a chaos game fractal is to choose a fractal which will work! The Selection Rules are:
    1. The set of points contained in the fractal MUST be bounded. This means basically the fractal has no points at infinity, or that it can be enclosed in a big enough circle.
    2. It SHOULD have obvious transformations which leave parts of the fractal unchanged. For example, in the Sierpinski Carpet, a large square is composed of eight small squares which are exactly the same as the large square. This the reason why the Mandelbrot set is an unwise choice for the Chaos Game.
  2. Define S to be the set of all points contained in the fractal.
  3. Now create a set, V, containing all of the possible mappings from S into S.
  4. Define T to be a finite subset of V, such that the union of the ranges of the mappings in T is equivalent to S.
  5. Now define the rules for this version of the Chaos Game to be:
    1. Select a point which is known to be in the fractal.(You must know at least one point to determine the location of the fractal anyways). Call this the current point.
    2. Randomly select a mapping in T, and apply that mapping to the current point.
    3. Redefine the image of the current point as the new current point, and repeat the last step and this step.
If these rules appear too complicated, go back to the two specific cases listed above and compare those specific cases to the general rules.

If you follow the rules correctly, you should see a few randomly placed dots slowly turn into a beautiful fractal! So why does it work? Because of the nature of these fractals - they are, by definition, similiar to themselves. Let us start by looking at a very simple shape - the circle. The rules for this game would be choose a point to be the origin and then another random point. Then rotate around the origin by a random amount and draw another point, and then repeat the process. Eventually you will get a circle. Since the circle is invariant under rotations, you can choose ANY point in the circle and any rotation and hit another point on the circle! The random part is simply the order in which the points are drawn, but eventually all points would be drawn in.

The same argument works for the more complicated fractals. Every symmetry of the fractal provides a transformation which leaves the fractal unchanges, so any point you choose can be transformed to another point in the fractal. And again, the random part simply determines the order in which the points are added. But since the rules are simple, and the player has a free choice of the random numbers, it seems magical to see a fractal pattern appear!

Saturday, June 03, 2006

The Many Dimensions of Dimensions III continued

Continued from Previous Post...


Now we move on to the physical effects of extra dimensions and the reasons why they are interesting:

- Kaluza-Klein particles. One of the most universal effects of extra dimensions is the existence of massive particles known as Kaluza-Klein states. If a particle lives in five-dimensions, it will have a component of momentum in the fifth dimensions as well. Since quantum mechanics equates momentum with wavelength, and since the fifth dimension is assumed to be curled up into a ring, it becomes clear that this fifth component of momentum is restricted to have certain values - those values which give a wavelength that is a multiple of the length of the dimension.
And that leads to one of the interesting aspects of Kaluza-Klein states. When viewed in four-dimensions, these discrete values of the fifth component of momentum have the same effect as making the particle heavier. A massless particle in 5D appears in 4D as being an infinite number of particles, with each one having a different mass. One of the main methods of searching for extra dimensions is to look for heavier copies of known particles.
Another effect is the creation of particles. In a higher-dimensional Universe that contain no particles at all, the properties of the higher dimensions can mimic particles. For example the length of a curled up dimension behaves as a massive particle with no spin (called the radion) while other properties of the same curled up dimension can reproduce spin-1 particles such as the photon (which is responsible for electromagnetism) and possibly the particles known as fermions which are responsible for all known forms of matter. This leads to the possibility that there are no such things are particles, only 4D effects of higher dimensional spaces.

- Weakened Gravity : Another interesting effect of higher dimensions is as a solution to the hierarchy problem. It has been known for many years that electromagnetism and the two types of nuclear forces are actually identical at high energies - they are in fact different forms of a single force. But gravity (the only other know force) doesn't fit into this picture. It is far too weak to even be related to the other forces. It is this extreme difference in strengths of forces that is known as the hierarchy problem.
One solution is the presence of extra dimensions. If gravity is travelling in dimensions that the other forces cannot penetrate, it could easily be diluted in strength. Imagine transporting a small amount of water through a hose, and through a wide channel. In the hose the water is kept together and is forced (by water pressure) to travel quickly to its destination. In the channel, the water spreads out and can travel slowly to its destination. The same amount of water is moved, but in the second case it is so spread out that the movement is a lot weaker.
This also leads to another method of detecting extra dimensions. Since they affect only gravity, it is possible to build experiments which measure gravity on tiny distances. Because the extra dimensions dilute it, we expect to see gravity get stronger than expected as we get to smaller distances, where there is less room to dilute it.

- Black Holes: We will probably never have experiments on Earth capable of making black holes in 4D. They require too much energy. But higher dimension black holes can be made with very little energy! So little that the next big experiment - the Large Hadron Collider - could easily produce such things. (And contrary to what some irresponsible journalists have reported, there is no danger involved. These black holes decay too quickly to leave the experiment - if they didn't we would be hit by several each year coming from space. )
The possibility of producing black holes on Earth opens up the possibility of studying gravity in far more detail than we could have ever expected in a four-dimensional Universe.

These are just three of the interesting effects of extra dimensions. The professional journals are filled with articles on all the interesting properties of higher dimensions!

The Many Dimensions of Dimensions III

Today we return to the topic of dimensions. This time we will look at the history and physical effects of having extra dimensions in the Universe....

It seems obvious that the Universe has three dimensions. From the perspective of the reader, there is the forward/backward direction, side-to-side direction, and up-and-down direction. And most of the people reading this site already know that Einstein's theory of relativity makes time into a fourth dimensions like the others. (Actually the time dimension has a few different properties, but those are unimportant for now)

But what if there are more dimensions? Maybe they are too small to see, or maybe light doesn't penetrate them, or perhaps our minds are just not able to comprehend their existence! As strange as this idea sounds, the existence of extra dimensions has become a very hot topic of research in recent years.


As a serious research topic the idea of higher dimensions goes back to the era 1915-1926 with the work of Nordstrom, Kaluza, & Klein. Einstein had already formulated the general theory of relativity in which we live in a four-dimensional universe in which spacetime is distorted by the presence of energy. It reproduced Newton's theory of gravity in calculations where that theory was known to be correct, and improved on it by successfully predicting several other measurements.
Almost immediately people began to wonder what would happen if instead of a four-dimensional Universe we lived in a five-dimensional Universe. The result was amazing - the 5D GR equations reproduced the 4D GR equations, and added four new equations which were identical to Maxwell's equations of electromagnetism! By adding a single dimension, general relativity can predict the existence of electric and magnetic forces!
The only problem was that we do not observe and higher dimensions. The solution given at the time was that they could be tiny. A straight line is one-dimensional, as is a piece of rope when viewed from far away. But if you get closer to the piece of rope, you see that it also has thickness, and in fact has a two-dimensional surface. The same idea work for the Universe - we view it as four-dimensional, but perhaps if you had a very powerfull magnifying glass you would see that each point of space was in fact a very tiny ring or a ball and that there was in fact another unseen dimension. (Of course there are reasonable arguments that our minds could not comprehend these extra dimensions even if visible.)

Unfortunately that was the era in which quantum mechanics was fashionable, and no one bothered to continue with extra-dimensional physics. Then in the 1980's the idea was re-opened. Several new theories, including string theory and supergravity, were found to work only in certain numbers of dimensions (usually 10,11, or 26). Although these theories were later found to be less useful than originally thought, the idea of extra dimensions remained.
Experiments started looking for these extra dimensions and found nothing. It became clearer that if they existed they would be ridiculously small. Then another revolution hit in 1998, with suggestion of a Universe in which all known particles and all electromagnetic and nuclear forces are somehow trapped on a four-dimensional membrane that exists in a five-dimensional (or higher) universe. Until that time all the experiments had focused on searching for the effects of extra dimension on light or on particle properties. The limits on this new model allowed for extra dimensions as large as 1 mm (~1/25 inch).

To Be Continued...

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