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Cartwheel galaxy

Monday, January 30th, 2006

I think what I like most about this image isn't the high technology involved in combining images of the same structure from 4 different telescopes, nor even the deep insights into the physics of galactic collisions that it provides. What I like most about the picture is all the pretty colours.

More specifically, I'm fascinated by what it means that this picture is so colourful. What it does mean is that there within this galaxy there are regions of space where qualitively different things are going on. There's got to be at least 7 or 8 different colours in that picture. Each of these colours represents a region of space with a particular distribution of dust and gas and stars of various different sizes. All the different colours mean that there are lots of different such distributions. In some places, space is mostly full of dust and gas. In others, there are stars which are mostly very old. In others, there are stars which mostly very young. Why different areas of space have such different characteristics is a mystery to me - that's why I find the different colours so fascinating.

Here's another fascinating picture. This isn't a painting, an artist's impression or a computer graphic, it's a photograph (albeit a very high-tech one) taken by the Hubble Space Telescope. If you had a sufficiently powerful telescope and pointed it in the right direction, that's exactly what you'd see. It's really hard to get a sense of the incredible scale of this galaxy from the picture, but it's about 50,000 light years across. That means even the finest details you can see in the dust rings surrounding the galaxy are each hundreds of light years across. This thing is just unimaginably big, yet the size of these fine details compared to the size of the whole thing makes it look like something on a much more familiar scale. One would expect something that big to be quite smooth but that isn't the case.

It seems that the universe has a great deal of structure at all scales, like a fractal.

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My beliefs

Wednesday, January 25th, 2006

There's a fascinating article on The Edge's World Question Center where they ask a big pile of distinguished thinkers about what they believe, but cannot prove. Curiously, right before I came across this article, I was thinking myself about things that I believe but don't have any evidence for - things that I believe simply because believing them makes me happier than not believing them. In the sense that there are some of these things, I suppose I am religious, though not in the traditional "organized religion" sense (I prefer to choose my own beliefs rather than following a pre-packaged set). For your derision I will outline these beliefs here.

I believe that I have free will, that is that there is an irreducible "something" in the universe that is "me" and that is not bound by deterministic laws of physics (i.e. that if it was simulated perfectly on a computer, no matter how much the simulation might act like me it would still not be me). I belive that quantum mechanics may leave room for this sort of dualism.

I believe that we will eventually figure out how to construct a theory of physics that includes both quantum mechanics and gravity as special cases.

I believe that true love is real - that there is more to this feeling than just a chemical reaction in the brain that has evolved to give children a better chance of survival into adulthood by providing them with stable families. (Call me a romantic.)

I also believe that I will live forever. Not in the Christian sense of living forever in heaven - I believe I will live forever in this universe. This is kind of an unusual one, so I feel I should justify it a bit (though I do not pretend for a moment that this constitutes any sort of proof). Suppose for a moment that Everett's many worlds formulation of quantum mechanics is the correct one - i.e. that whenever any fundamental particle could go either one way or another way, the entire universe is effectively "splits" into two universes identical in every respect except that this one particle goes one way in one universe and the other way in the other universe.

Now, suppose one of these choices lead inevitably to my death, while the other choice allowed me to remain alive. From my point of view, the irreducible "me" continues to exist only in one of the universes, so that is the universe that I experience. I cannot experience my own death because (by definition) I am no longer there to experience it once it is complete.

This seems to have worked pretty well so far - there have surely been lots of quantum mechanical events which, if they had turned out in a way differently than the way they did, would have lead to my death. However, that in itself is not very strong evidence since the probability of my death so far has probably been fairly low, quantum-mechanically speaking (if we discount the improbability of my conception in the first place, which I do because I wasn't alive then). Lots of people have survived to my age in my universe, and none of them quantum-mechanically needed to in my universe in the same way that I quantum-mechanically need to in my universe. However, when one day I become the oldest living human being and continue to live for much longer than anyone else has ever done, this line of thought will be more convincing.

That leads to a rather depressing-sounding scenario - I will get to watch everybody that I love die. The only way I get to die is if the universe inevitably gets to a state where no life at all is possible - if it collapses in a "Big Crunch" or expands at an ever accelerating rate leading eventually to all atoms being ripped apart. Which are also rather depressing scenarios in themselves. However, I find these possibilities significantly less depressing than the possibility of dying, so I don't worry about it too much. I'm excited to see what happens!

Believing this behooves me to support certain causes - those which will ensure I continue to be comfortable in the very long term. I want our planet to continue to be a nice place to live. I want the rest of the human race to not become extinct, so that I always have somebody around to talk to. I also want medical technology to continue to improve so that, no matter how many parts of my body start to fall of, I'll always be able to get them stuck back on or replaced so that I can continue to have a good quality of life. With this belief system these worthy causes are also in my personal interest.

Although I can't die I can still suffer a great deal of pain, so this immortality does not excuse me from having to take care not to get into a car accident, and if you say "so you think you're immortal? Prove it by shooting yourself in the head with this gun" I will refuse since the most likely outcome of accepting would be having to live with terrible brain damage.

Despite the depressing sides, this idea holds a certain comfort for me. Not having any limit on my lifespan frees me from thinking "oh I must do this before I die, and this, and this" and getting frustrated that I probably wouldn't be able to achieve them all. There are still lots of things I want to do, but there's no rush as I have plenty of time. It's kind of the opposite of the "live every day as if it was your last" philosophy.

Note that nothing in this theory mentions me by name, so anyone else can apply this theory to themselves just as I can. From your point of view, you will also experience living until the end of the universe, and experience everyone else dying. There is no contradiction here because of the many worlds interpretation of quantum mechanics - you continue to live forever in your universes, I continue to live forever in mine. You die in my universes, and I die in yours. Of course, you might not believe that this is really how things work, but if things do work like this you don't have to believe it for it to happen. Of course, this theory isn't falsifiable - if this theory turns out to be wrong you won't be able to tell me "I told you so", and if it turns out to be right I won't be able to say "I told you so" either (at least to anyone who has been born already).

While checking the links in this post, I discovered that apparently I am not the first person to have thought of this.

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Metascience: the nature of the laws governing the universe

Sunday, January 22nd, 2006

Given what we know about the laws of the universe so far, I suspect that there are not too many of them - i.e. that when we finally figure out how to unify quantum mechanics and general relativity, the resulting "theory of everything" will be conceptually quite simple - perhaps just a few lines of equations when written down in their simplest form (although they might be rather difficult to do actual calculations with).

But what if there are exceptions to these laws of physics? What if there are a finite number of points in spacetime where these equations do not hold, and events happen that are not predicted by these laws? We couldn't do science with these directly - as each of them would only happen once, any experiments around them could not be repeated. There is a great deal of evidence pointing to the existence of one such point - the one the exact moment of the big bang at the beginning of the universe.

I got this idea from thinking about the classification of the finite simple groups. I won't go into great detail about what that actually means, but a very simple introduction follows in the next paragraph for the curious.

A group is just a mathematical object consisting of a set of things and an operation (e.g. addition or multiplication, call it "*") which takes any two of these things (e.g. a and b) and generates a third thing, a*b = c. This operation must also have certain special properties: (a*b)*c = a*(b*c), an "identity" element I such that a*I = I*a = a and an inverse element a-1 for every element a such that a*a-1 = a-1*a = I. The simple groups are just groups with particular properties - kind of like the equivalent of prime numbers for groups, or the chemical elements in chemistry - they can't be broken down into smaller simple groups.

Mathematicians wished to classify the finite simple groups, to find the equivalent of the "periodic table" for them. It turned out to be a rather big job - the result is the biggest theorem in mathematics (so far), consisting of some 15,000 pages in 500 articles by 100 mathematicians over a period of 28 years. It turns out that the simple groups can be classified into 18 different families (each of which is infinitely large). However, strangely there are 26 solitary finite simple groups (called the "sporadic groups") which don't fit into any of these 18 families! The largest of these has 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 elements, which can be thought of as a group of rotations of some object in a space with 196,883 dimensions.

I wonder if the universe works the same way. If it does, perhaps a theory of everything could be made much simpler by including such "sporadic events". By adding a finite number of sporadic events, it might be possible to change the theory of everything from an analog of the "18 families" form to a form analogous to the definition of a finite simple group. In so doing, one could predict when and where these sporadic events occurred (or would occur). We could seek out evidence for the sporadic events predicted to have occurred in the past. For sporadic events in the future, we could go to the place they were predicted to occur at the time that they were predicted to occur and perform experiments to observe them directly and gain evidence for the simplified version of the grand unified theory. Presumably if that were to occur, any alien species who had also achieved our level of scientific knowledge would be there too. I hope that by then we would be mature enough not to go to war with them over who gets to observe it. It would be kind of like the physics version of a pilgrimage to Mecca.

This might make a rather good science fiction short story.

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Time travel

Sunday, September 11th, 2005

Suppose that some time in the future, humankind figures out a way to make possible time travel into the past. Suppose, furthermore, that it turns out that the universe is holonomic - that is, there is only one past and one future and we can't change it (so we can't create an "alternative 1985" like in Back To The Future II). Effectively, the entire history of the universe would be predestined. So the going back in time would also be predestined. Anything that the time-travellers do while on their trip to the past would have already have happened and would always have happened. So we know that the time travellers cannot kill their past selves (or they would not be in there in the future to make the trip). In fact, anything that they do in the past must guarantee that the time travellers are alive and sufficiently healthy in the future to be able to make the trip back (which would not necessarily be the case if the time travellers were not there). So effectively, the time travellers would be their own guardian angels, "protecting" their past selves (deliberately or not) simply by the virtue of existing in the past.

Now, suppose that you are a future human in possession of a time machine. You don't worry about "responsible time travel" since you can't break anything or rewrite history. It's only natural for you to wonder just how far back in time you can go. With sufficient technological advancement, you might be able to go right back to the moment at the very beginning of the universe. And, just maybe by going back to that point in time you actually initiate the creation of the universe - you become God.

Sometimes I wonder if the universe is like that. But I suspect it is not - I find it very difficult to believe that free will is just an illusion.

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Relativity Q&A

Wednesday, July 12th, 2000

It is impossible to travel faster than light, and certainly not desirable, as one's hat keeps blowing off. -- Woody Allen

Q1: Who is this page aimed at?
A1: It's aimed at people who are happy with the basic concepts of classical motion such as speed equals distance divided by time, but who know nothing about relativity (except maybe E=mc^2 and that you can't go faster than the speed of light) and wish to know/understand more.

Q2: What's the deal with relativity, then?
A2: Relativity was invented to account for a peculiar experimental result - that the speed of light is the same no matter how fast you are moving with respect to the source.

Q3: How do you do this experiment?
A3: Suppose you have an accurate timer, which is stopped when a pulse of light goes past it. Then you have another exactly the same. You bring them close together, synchronize them, and then move them slowly apart (Why slowly? You'll find out later). Now, fire a laser beam along the line connecting the two timers. By looking at the difference of the times recorded by the timers and dividing the distance between the two timers by this, you can measure the speed of light, which we'll call c for short from now on. (It's exactly 299,792,458 meters per second if you do the experiment in a perfect vacuum). Now, repeat the experiment but move the laser towards (or away from) the timers at speed v whilst you're firing it. You'll notice that your estimate of the speed of light equals c, not c+v or c-v as you would expect if you know nothing about relativity.

Q4: No, how do you really do this experiment?
A4: Unfortunately it's too difficult to do the experiment so directly in real life, so you have to do it more indirectly. For details, look up "The Michaelson-Morley Experiment" in any elementary textbook about special relativity.

Q5: Isn't this result because c is very large whilst v is very small, so c+v and c-v are roughly the same as c?
A5: Nope, even if v is 99.9999% of c, you'll still get the same result. The speed of light is an absolute constant (that's why it's called c for constant.)

Q6: What does this mean?
A6: It means that almost everything you thought you knew about space, time, speed and motion is wrong - they break down at high speeds (of the order of magnitude of c).

Q7: Why can't you go faster than c?
A7: The kinetic energy of a particle of mass m moving at speed v is not E_c=\frac{1}{2}mv^2 as they tell you in physics lessons in secondary school. The correct formula is \displaystyle E_r=mc^2\left(1-\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\right), which is approximately \frac{1}{2}mv^2 if v is much less than c. Here is a graph of the classical kinetic energy E_c per unit mass and the relativistic kinetic energy E_r per unit mass, plotted against speed:

As you can see from this graph, as the velocity approaches c, the energy approaches infinity, so it requires an infinite amount of energy for any object with non-zero mass to even reach c, let alone go faster. In fact, no information can travel faster than c, even if that information carries no mass.

Q8: But light goes at c. How come?
A8: Particles of light do not have any rest mass - the m in the above equation equals zero.

Q9: So light has no energy?
A9: No. Because it goes at c, you can't use the equation from question 7 to figure out the energy of a photon (a particle of light). The above equation gives zero times infinity, which is undefined. In fact, a photon can have any amount of energy, depending on it's wavelength or frequency. The energy of a photon E equals hf where f is the frequency (oscillations per second) and h is Planck's constant (about 6.626x10-34 Joule-seconds).

Q10: I've heard about "solar sails" - the idea that you can propel a spaceship using the momentum of light. But if the speed of light is finite and the mass of light is zero, then the momentum of light p=mv=0. So how does the solar sail work?
A10: The equation for momentum p=mv is another of those classical equations that are just plain wrong (well, not so much plain wrong as an approximation that only holds for velocities much less than c.) The correct equation is \displaystyle p=\frac{mv}{\sqrt{1-\frac{v^2}{c^2}}} for particles with mass, or p=hcf for photons.

Q11: Okay, I'll accept for a moment that nothing can go faster than the speed of light. Suppose you're on a train that is moving at 0.95c with respect to the ground, and you're skateboarding down the aisle of the train on a jet propelled skateboard at 0.1c. Now I'm going at 1.05c with respect to the ground. What's the explanation of this apparent paradox?
A11: There is no paradox here. There's nothing in physics that says you can't have a train moving at 0.95c. According to relativity the laws of physics are the same in any reference frame so there's nothing special about 0.05c or any other speed (except c) aboard the train. In fact, relativity says that you can't even tell how fast the train is moving by performing any experiment that doesn't rely on the outside of the train.
The problem here is that if A is moving relative to B with speed X and if B is moving relative to C with speed Y, the speed of A relative to C is not X+Y as you think it is. In fact, it is \displaystyle \frac{X+Y}{1+\frac{XY}{c^2}}, which is always less than c as long as X and Y are, and this is approximately X+Y when X and Y are much smaller than c.

Q12: I heard about this thing called time dilation. What's that all about? And why did the timers have to be moved apart slowly in question 3?
A12: Suppose you have a set of twins. One of the twins stays on the Earth, the other goes on a round trip on a spaceship at a speed close to c. Because of the bizarre things that happen at speeds near c, when the travelling twin returns, he will not have aged as much (will have experienced less time than) his brother who stayed on the Earth.

Q13: But surely from the point of view of the twin on the spaceship, it was the earth which went on the relativistic round trip, and to him it should be the Earth-bound brother who should end up younger.
A13: No, the two brothers do not experience the same things. The one on the spaceship experienced an acceleration at the far point of its journey as he stopped moving away from the Earth and started moving back towards it. His reference frame was not "inertial" (did not move at a constant speed) so is not equivalent to the reference frame of his brother.

Q14: So just before the acceleration period, which brother is older?
A14: It may seem strange, but the question is not meaningful. You can't compare the ages of the brothers when they are a long distance apart. You can't compare times over a long distance and you can't compare distances over a long period of time. This is because relativistically, time and distance are two sides of the same coin. When moving at speed, time and distance "change places" to a certain extent - this is the source of time dilation and it's partner, length contraction.

Q15: Length contraction? What's that?
A15: When you're moving relative to something, say a plank of wood, that plank will be shorter (from your point of view) the faster you are moving relative to it, compared to the length it was when you weren't moving.

Q16: Suppose there's a 1 metre wide hole and a 2 metre wide plank. Suppose the plank is moving sufficiently fast that, from the point of view of the hole, the plank is length dilated to 1 metre. Then suppose that at the time the plank is passing over the hole, it goes through the hole. From the point of view of the plank, it's the hole that's length dilated (to 0.5m) so now the plank is too long to go through the hole. What's going on?
A16: The problem here is that the concept of rigidity is a classical one and has no equivalent in relativity. Think of the speed of sound - this is how fast mechanical signals travel through a material. In an ideal rigid body the speed of sound is infinite, but since no information can travel faster than c you cannot have a relativistic rigid body. So the simple answer to this question is that the the plank bends.

Q17: Hang on a sec. The plank bends in the reference frame of the plank, but not in the reference frame of the hole?
A17: Exactly. The fundamental thing here is the relativity of simultaneity. If two events happen simultaneously in one frame of reference, they do not necessarily happen simultaneously in another. This is why you cannot say what the difference in the age of the twins is when they are a long way away - the answer depends on your frame of reference.

Q18: What is the Cherenkov effect?
A18: Cherenkov radiation is a bluish light emitted when a particle moves faster than the speed of light.

Q19: WHAT!?!?!??!!!
A19: Notice that I said "speed of light", not c as I have mostly been using in the rest of this document. c is the speed of light in a vacuum, the speed of light in materials is lower, and depends on the material. The speed of light isn't the absolute speed limit, c is.

Q20: How do you actually do calculations with this stuff? It seems like all the starting points I've been taking for granted - space, time, velocity - aren't really fundamental any more.
A20: You can define a basis for space and time, however it will depend on your velocity, so you'll need a different basis for every reference frame you use. Fortunately, there is a simple formula for converting between reference frames, the Lorentz transform. You can read about this in any elementary special relativity textbook.

Q21: (From Gregg) If an object's mass increases as it's speed increases, where does this mass come from?
A21: From whatever accelerated the object. Mass and energy are the same thing. So when you increase the object's (kinetic) energy by speeding it up, you also increase it's mass. Now, energy can't be created or destroyed, so whatever gave the object this kinetic energy has lost some energy (and, therefore, mass) itself. Note that there isn't any transfer of matter going on in the acceleration process - the accelerated and accelerating objects have the same number of atoms (electrons, quarks...) in them that they started with, but the masses of these particles have changed.

Q22: (From Colin) Mass and energy are interchangeable. Has mankind managed to turn any energy into mass yet?
A22: Oh yes, physicists are doing this every day in particle accelerators. As the particles are accelerated they gain mass, then when they smash into each other they break up into many particles, some of which may well be the same particles that were originally accelerated. The particles that are "created" are effectively the result of turning energy into mass.

Q23: (From Colin) Mass attracts mass (gravity). Mass attracts energy (gravitational lensing etc). Has it been shown practically that energy attracts energy, or energy attracts mass?
A23: Not directly (in a lab) because the amounts of energy we can work with are too small to exert any gravitational attraction. The finest gravitational experients that have been done require masses of the order of a few grams, 1 gram of mass could power 80,000 homes for a year.

However, there are very good reasons to believe that energy does attract mass (and energy). Much of what makes up the "mass" in everyday substances is in fact energy (binding energy holding the protons and neutrons in the nuclei together). So if this energy didn't contribute to the gravitational force, we would expect that different substances (which have different ratios of mass to binding energy) would accelerate differently under gravity (because they would have different ratios of inertial mass to gravitational mass). Accurate experiments (to many significant figures) have been done measuring this ratio for many different substances, and no difference has been found between any of them. So if there is a difference in gravitational attraction between fermions ("matter" particles) and gauge bosons (the virtual "particles" responsible for "energies" of various sorts) it's very small (too small to be detected in any experiments anybody has devised so far).

Q24: (From Rachel) Something about Einstein's theory of relativity bothers me, specifically about the issue of time dilation. According to what I read (pls correct me if I'm wrong), the stronger the gravity the slower the pace of time. This was proven by experiments with clocks that seem to run faster when farther from the Earth, as well as with experiments wherein time delays for radio waves near a sufficiently dense body (such as the Sun) were observed.

Now, I understand that space distortions can be caused by sufficiently dense masses (similar to a rubber sheet weighed down in one part by a small yet heavy stone). But the reasoning regarding time doesn't convince me well. The experiments used to prove time dilation (as far as I know) had to make use of speed (i.e. a relationship between distance and time). This was the case for the experiments using clocks and radio waves.

So I wonder: What if... the apparent (take note: apparent) slowing down of time and the delays were caused, not by the true slowing down of time, but by the "stretching" of distances due to the presence of dense masses in space (much like the stone-on-rubber sheet again)? If so, then aging will occur at the same pace regardless of whether a person experiences high or low gravity.

Are there any experiments that disprove my assumption?

A24: The Pound-Rebka experiment verifies that time passes slower in a stronger gravitational field. By "makes use of speed" do you mean "assumes that the speed of light is the same no matter how strong gravity is"? I don't think any other speeds are involved. The constancy of the speed of light has been verified by other experiments.

Gravity bends space and it bends time, but it bends both in such a way that the speed of light remains constant (if only space were bent and time remained the same, the speed of light would have to change in proportion to the stretching of space).

The stone-on-rubber sheet image is a neat way to visualize matter bending space, but don't confuse the visualization with the physics - that model has some serious oversimplifications, especially where time is concerned.

If you have a question about relativity, email me (or comment below) and I might put it up here. I'm not going to do your homework for you, though.

If you think relativity is strange, just wait until you find out about quantum mechanics.

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