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Following on from yesterday's post...

Suppose the hypothesis that the universe is infinite is true, and that any event with a finite probability will happen an infinite number of times. Now, given a set of observations about the universe (such as every observation you have ever made), what is the probability that those observations were made in the first stage of the universe (i.e. the one we have generally assumed that we were in) and what is the probability that those observations were made in one of the macroscopic quantum fluctuations in the time after the heat death of the universe?

The set of observations is finite, so the amount of matter required for them to be made is finite. So quantum fluctuations will cause these observations an infinite number of times for each sufficiently large volume of space. In the first stage, those observations can happen at most once for the same volume of space. So given the set of observations, it's practically certain that they are post-heat-death.

But if the universe is post-heat-death, why can we see so much? A much smaller universe would have allowed for almost as many observations (including the existance of the entire human race) but be much more likely. Perhaps the human race (and hence somebody to observe the universe) cannot exist without all those very distant galaxies.

No-one knows exactly how the universe will end. There are several possibilities. One is that everything will fall into black holes, which will then fall into each other and coalesce until every particle in the universe is in one place just as they are thought to have been at the beginning.

A second possibility is that the expansion of the universe will accelerate faster and faster until all the fundamental particles are ripped apart from each other at speeds exceeding the speed of light so that they can never affect each other again and every little particle effectively ends up alone and impotent in its own universe, allowing nothing more complicated to exist.

A third possibility is that the universe is completely balanced between those two extremes, Goldilocks-style, and will continue to expand but at a decreasing rate so that it would theoretically stop expanding at all, but infinitely far in the future. In this case life within the universe can theoretically go on as normal for a very long time. However, eventually all the stars run out of fuel. The smaller ones evolve into white dwarves, then cool to brown dwarves and black dwarves. The larger ones evolve into black holes and neutron stars. Eventually all the protons in the black dwarves decay into positrons and gamma rays and the black holes evaporate via Hawking radiation until only the neutron stars are left. Much later, the neutron stars quantum tunnel into black holes which then themselves evaporate relativity rapidly. At this point the universe is just a homogeneous sea of electrons, positrons and photons. The electrons and positrons will eventually annihilate leaving only photons. From then on, nothing really changes.

But here's the weird part. We have a universe, empty apart from some weak radio waves, for an infinite period of time. Now, quantum-mechanically it is possible for empty space to just create a small piece of matter and a small piece of antimatter spontaneously, from nothing. In fact, this is happening all the time but normally these annihilate again in a very short time. It's very unlikely, but sometimes these particles will stay around a little bit longer. In some cases they may even be around for long enough to be joined by other fluctuation-generated particles. Very rarely you'll get a whole bunch of such particles together at once. Even more rarely still there will be enough of these particles to form an entire planet or solar system or stellar cluster or galaxy or galactic cluster or even a pile of matter the size of the currently observable universe. These things are all incredibly unlikely but given an infinite amount of time even the most unlikely things will eventually happen so long as they are possible.

So eventually, every sequence of events that has ever been played out will play out again just by random chance. And every possible sequence of events involving a finite amount of matter (including your life, and mine, and all conceivable variations thereupon) will play out just by random chance, an infinite number of times.

When I explained this to my friends, they said "wait a minute, so you believe that (given the universe is flat, the third possibility), at some point in the future the following sequence of events will happen:

1. a perfect replica of the Earth as we know it will spontaneously form from nothing
2. all of the salt dissolved in all of the water in all of the oceans of this replica world will spontaneously leap out of the oceans, hundreds of feet into the air
3. this salt will then spontaneously form itself into a giant peanut orbiting the planet
4. the peanut will spontaneously turn into a small green shining baby and
5. all of this will happen an infinite number of times?

I had to confess that while phrased like that it did seem rather ridiculous, that it what the theory predicted. Some of my friends are rather strange people.

Continuing on yesterday's theme of light being bent by gravity, what happens when gravity is really strong, such as around a block hole? At 1.5 times the Schwarzchild radius (the "point of no return") of a black hole, light is bent so much that it actually orbits around and around the black hole! If you were at that distance from a black hole and looked out at a tangential angle, the surface below you would look perfectly flat and you would see the back of your own head all along the horizon!

But if light going around in circles is weird, time going around in circles is even weirder. According to General Relativity, deep within the bowels of certain rotating black holes it may be possible to move around in a circle and end up not only where you started but also when you started. Gravity is so strong that it bends time into loops and events can occur which are their own cause and their own effect. A time machine. All this is hidden behind the event horizon of the black hole so we could never actually observe this time travel going on but it's pretty weird nonetheless.

This is one of the most mind-boggling things I learnt about while studying physics at university. That and one of the last lectures I ever attended, which was given by Sir Martin Rees and was on the subject of the fate of the universe in the extremely long term. I'll write about that tomorrow.

The first time I realized that grown-ups were not infallible was when I was trying to find out whether beams of light can bend.

Someone (I don't remember exactly who - either a parent or teacher I expect) told me that light always travels in straight lines. Someone else (another parent or teacher) told me that in fact gravitational fields could cause light bend. They couldn't both be right, so one of them had to have been lying to me. This was a very confusing thing for a young child to understand (not the gravity thing, the grown-ups lying thing) and I vaguely remember being rather upset about it.

I got over it, but it wasn't until I was learning about gravity at university that I found out that they were both right, sort of.

Gravity does affect light - that much has been determined experimentally. I have seen Hubble Space Telescope pictures where a single galaxy seems to appear in multiple places (or is even "smeared out" into a circle called an Einstein ring) due to the light being bent by another galaxy.

But gravity works by causing time and space to be curved in the vicinity of massive objects. What does it even mean for a line to be straight if the space it lies in is curved? It's a similar situation to trying to find straight lines on the surface of the Earth. There aren't any very long ones because the surface of the Earth is itself curved (cue comment about Christian fundamentalists). However, we can find the shortest distance between two points. On the surface of the Earth we can do that by finding a "great circle" (i.e. a circle who center whose radius is the radius of the Earth and whose center is at the center of the Earth) that connects the two points, and following it. That's why the shortest route from London to Seattle goes way up North into the Arctic circle (try it with a globe and a piece of string if you don't believe me).

Similarly, we can find the shortest path between two events in spacetime (such as a photon leaving a distant galaxy and that same photon entering your eye or telescope), even if that spacetime is curved by gravity. Some of these shortest paths turn out to be exactly the paths that beams of light follow. So light does actually follow straight lines with the right definition of "straight", even when it is being bent!

What is even weirder is when you look at shortest paths through spacetime when the thing moving through spacetime is not moving at the speed of light. In empty space, these are exactly the straight lines you would expect. But in a gravitational field, they are curves. In fact, if the gravitational field is large enough in spacetime these curves are parabolas - i.e. the sort of curves you get by throwing some object and watching the arc it makes. Now these are obviously not straight paths in space, but it turns out that they are "straight" paths in spacetime. So now when watching a baseball game you can confuse your friends by saying "my, look at that spacetime geodesic" when someone hits a fly ball.

When you sit on a swing and a friend twists the swing up and then lets go, you spin around as the swing untwists. If you are spinning fast enough, you'll notice that your extremities seem to get "pulled away" from the axis of rotation. They don't really, of course, it just seems that way because the force you need to exert on them to keep them moving in a circle is towards you just like the force you'd need to exert to counter an outword pulling force.

According to the theory of General Relativity, if you were standing still and all the matter in the universe were rotating around you (with a speed proportional to distance from you), the motions of all this matter would exert the same forces on you that you feel by spinning around in a non-spinning universe. In other words, there is a sort of symmetry between spinning around in a non-spinning universe and staying still in a spinning universe. This line of thinking suggests that in a universe which was completely empty apart from you (if that were possible) you would be able to spin around without experiencing these centrifugal effects.

But the only reason we feel these effects is because you need to exert a force to accelerate things (like limbs) - inertia. This doesn't seem to have anything to do with the rest of the universe. Or does it? Perhaps inertia only happens because of gravitational interactions with the rest of the universe as a whole, and that whenever we experience the relationship between force and acceleration we can infer the existence of the rest of the universe from that. Perhaps there is something in the mystical idea that "all things are connected" after all.

One of the most beautiful and general principles in physics was discovered by Emmy Noether. Hers is a fascinating story by itself, but I am a physicist not a biographer, so here comes some science-y stuff. Noether's theorem says that for every symmetry in a system, there is an associated quantity that is conserved. To see what that means in practice, it is useful to look at some examples.

Space is symmetrical in that (in the absence of matter like electrons, protons and galaxies), one piece of space looks very much like another piece of space. If I do an experiment in one part of space, then slide it over to another part of space and perform the experiment again, the result will be the same. This symmetry leads to conservation of momentum. If the second piece of space is different from the first piece of space (for example because it has a planet in it) momentum will not be conserved as a particle moves from one piece of space into the second (it will hit the planet and its momentum will change).

Time is also symmetrical in that if I do an experiment at one time and then do it again in the same place at a later time, I'll get the same result. This symmetry leads to the conservation of energy.

Another symmetry that space has is rotational symmetry. If I do an experiment with the apparatus pointing one way, then reorient the apparatus and do the experiment again pointing in a different direction, you'll get the same result. This symmetry leads to the conservation of angular momentum. Near the surface of the earth there is a rotational asymmetry due to gravity (there is a "special" direction - down). This assymmetry causes a pendulum to change angular momentum as it swings backwards and forwards (if you do it in a symmetrical place, such as far away from any sources of gravity, it will go around and around in a circle - it will have constant angular momentum).

Most of the time, our universe acts the same as it would if it was "flipped" the way a mirror-image reflection is flipped. This "mirror image" symmetry leads to the conservation of a property called the "parity" of a fundamental particle. The mirror image version of a particle has the opposite parity. However, it seems that there are some occasions when parity isn't conserved - in these respects our universe acts differently to a hypothetical "mirror universe", identical to ours in every respect except left and right being swapped. The apparent symmetry turned out to be an asymmetry.

I was recently trying to convince someone that that Special Relativity (SR) was more correct (in the situations where it is applicable) than classical (Newtonian) physics. One argument I used is that irrespective of experimental evidence for SR, SR is actually a simpler theory than classical physics, when you write each theory down in their simplest forms.

In this form, physics looks very different from classical "high school" physics. A lot of concepts which are classically very different turn out to be the same thing in relativity. Space turns out to be the same thing as time. Different angles (as in rotation) turn out to be the same thing as motion at different (constant) velocities. Electric fields turn out to be the same thing as magnetic fields.

This theory has a "parameter", a value which isn't predicted by the theory and must be determined experimentally and plugged into the theory to make it complete. This parameter is called "c" and is usually known by its physical meaning "the speed of light in a vacuum".

Now, one could conceivably get classical physics in this same form by plugging in a "c" value of infinity instead of 299,792,458 metres per second. Doing this causes time and space (and angles and velocities, and electricity and magnetism) to separate out like an emulsion of oil and vinegar left to stand for a while. Only the finite value of c causes these concepts to mix (and the smaller the value of c, the more they mix and the more pronounced relativistic effects become).

"Great," you might say, "so Einstein might have been wrong all along and all this weird time dilation/length contraction/mass equals energy stuff could all be bunk." The trouble with that, though, is that with c=infinity, the model corresponds less well to observed experimental results - the time dilation effects that have been measured are not predicted, and the speed of light is predicted to be infinity.

But the place where this model diverges most drastically from reality is magnetism. The c=infinity theory predicts that there should be no magnetism at all. Trying to add magnetism back in to a non-relativistic theory causes all sorts of complexities and irregularities. In fact, it was trying to remove these irregularities that brought about relativity in the first place. Really, the simplest way to have a consistent theory of magnetism is relativity with all the non-intuitive concepts that entails.