Wednesday, 30 March 2016

How Can Something Finite Not Have An Edge?

It will eventually become clear why there's a picture of a tardis here.

Somebody just wrote that if something is finite it must have an edge and be sitting inside something else.  The purpose of this entry is to show why this need not be so.

I wrote elsewhere about how probability might measure the physical distance between universes or timelines.  It seems to me that if we are living in a world which is constantly branching, the older branches would be in the way of the newer ones off our timeline, so it follows that the newer branches must be closer.  Otherwise they would cross the older ones and for an instant they would be identical, and that wouldn't happen.  It also seems that the less likely events are, the further away they would be.

There is a problem with this though.  Probability is usually between one and zero.  Apparently calculations in quantum physics sometimes come out with the result that something is infinitely probable, and nobody knows quite what to do with this, but since I don't understand what that's about, I'm going to leave it alone for now.

An event which is certain has 100% probability or a probability of one.  A completely impossible event has zero probability or 0%.  Clearly the probability of certain events depends on what's happened before, so if your house is already on fire and the fire brigade has run out of water, the chances of you dying in the fire are higher than if your house wasn't on fire or the fire brigade hadn't run out of water.  Also, not all events with zero probability are impossible.  If an infinite number of dice were rolled, an event which is impossible because they would be frozen facing in particular directions because of their mutual gravitational attraction unless they were infinitely far away from each other, they would have a particular arrangement of numbers on their faces, so looking from one direction, one might have six facing one, another one, another five and so on, and the probability of that arrangement would be zero but it would have happened, assuming them to be truly random.

If probability varies between zero and one, it looks like the multiverse might be very small and crowded and have a limit to it.  This need not be so though.  It could be, for example, that fifty to a hundred percent probability is one unit, then 25% is two, 12.5% is three and so on, meaning that events of zero probability along with impossible events are infinitely far away.

It's been suggested that space could in fact be like this.  In order to imagine that, consider this Escher picture:

(A print of this can be ordered here).  This is a circular picture whose devils and angels shrink the further away they are from the centre, and although this can't be done in reality, in an ideal such world there would be no limit to the shrinkage and to any one of the angels (devils being angels too), the plane would be infinite.  Moving across it would involve shrinking in proportion to how far from the centre one was, and consequently it could have a limit but still be infinite to its denizens.  This can easily be extended to three dimensions.  One could be in the centre of a sphere twenty metres across, but find that once one has swum or flown five metres from that centre, one was half the size one was at the centre and so on, so even though the surface was clearly visible and you might think you could reach it, it would be impossible.

This is known as hyperbolic geometry and it comes down to parallel lines.  We think of parallel lines as staying the same distance apart, but in hyperbolic geometry the distance between them increases.  This is similar to what happens on a pseudosphere, where apparently parallel lines fan out towards the equator from the poles, within limits:

At first, hyperbolic geometry might seem useless, but oddly it's found a home amongst a few Young Earth Creationists (YECs).  If you believe the Universe is only six thousand years old, there is immediately a problem with the sky.  It can be demonstrated that the Universe is more than six thousand years ago as follows:

The moons of Jupiter orbit it at a known rate and can easily be observed through a small telescope.  Jupiter's distance from the Sun can be known because of how long it takes to go round it, since Kepler's Third Law of Planetary Motion states that the square of the time taken to orbit the Sun is proportionate to the cube of the mean distance from the Sun, which in turn follows from the fact that we live in a three-dimensional Universe and the force of gravity diminishes as the square of the distance from a mass.

When Earth is on the opposite side of the Sun to Jupiter, it's 300 million miles further away from it than when it's on the same side because of the width of our orbit.  The movements of the four big Galilean moons are delayed by around a quarter of an hour, a thousand seconds, because of the time taken for light to get here from there.  This means light travels at about 300 000 kilometres a second.  Alternatively, if the speed of light is known but not the diameter of our orbit, it can be worked out from that, the speed of light itself being calculated by shining a light at a rapidly spinning mirror and working out how much the beam reflected back shifts if the rate of rotation is known.  Incidentally I've tried this between Elizabeth House in Leicester and Old John in Bradgate park, and the difficulty is getting a clear enough day for it to work.

Having established that, the next stage is to look at how the stars shift against the background over a period of six months, the background being much more distant objects such as galaxies.  In the above diagram, it can be seen that the star will be in front of the object in the background in December from our perspective, and closer to the next object down in June from our perspective.  Trigonometry makes it easy to work out the angles involved and the base of the triangle is already known to be about 300 million kilometres, so distances to nearby stars can be measured.  This is known as the first rung on the distance ladder.

The next one can be worked out by observing Cepheid variables.  Not all stars shine steadily.  Some of them fluctuate in brightness, and the way in which they fluctuate indicates what kind of variable star they are.  The brightness of a Cepheid variable fluctuates in a curve indicated by the graph above, and is determined by the speed of sound within the star, as it involves a shock wave moving back and forth in the star.  This information is not needed to understand that the brightness of the star is directly linked to how long it takes to flash, but it turns out to be because it pulsates at the speed of sound and similar sizes and temperatures of stars have similar magnitudes of brightness.  The closest Cepheids are near enough to show parallax, and sufficiently bright ones can be seen from far away.  Because their real brightness, or "absolute magnitude", can be known, their distance can be known.  This is slightly complicated by the fact that there is more than one type, but the principle remains the same.  This also enables distances to nearby galaxies to be measured.

The final rung in the distance ladder is the rate at which other galaxies are moving away from us.  For a reason I'll go into in a bit, most other galaxies in the Universe are moving away from Earth, and the further they are away, the faster they are moving away.  This can be measured because the light is stretched to longer wavelengths in a predictable manner, so the further away something is, the redder it is.  This enables the distance of all large visible objects to be measured, until eventually they are moving away faster than light, which is why the night sky is black - the light can never get to us from an object travelling faster than light.

It turns out that the light left distant galaxies billions of years ago and is only now reaching this planet.  Consequently the Universe must be billions of years old.  I made this observation to a fundamentalist Christian postgrad biologist housemate once, and she said she couldn't explain it but there was an answer.  It's no good to say it was created in transit because that would mean God was deceiving us, so Young Earth Creationists need an answer as to why this happens.

They find an answer in hyperbolic geometry.  In my humble opinion, they've spelt the word "hyperbolic" slightly wrong, but they aren't bad people because nobody is, and it could also be said that a detail like the age of the Universe isn't very important in everyday life for most people.  Anyway, their solution to the problem is to suppose that space is warped into a shape rather like the pseudosphere illustrated above, but in four dimensions.  This would mean that we are near the centre of the Universe and that time travels more slowly here because of relativity, which in turn means that you can get away with Earth only being six thousand years old and the rest of the Universe being much older, because time passes more quickly here than further out in the Universe.  This is rather surprising and ingenious, but since it places us near the centre of the Universe, it probably isn't true and it isn't really any better than the idea that the Sun goes round us rather than the other way round.  Nonetheless it does illustrate an application of hyperbolic geometry.  

In such a Universe, a spacecraft travelling away from Earth towards the apparent edge of the Universe would never reach it because as it travelled it would shrink, and as it shrank it would move ever more slowly, meaning that it would never get there.  This is one way in which something finite can get away with not having an edge.

Hyperbolic geometry is what you get if you assume parallel lines move apart.  It applies for real in some situations regardless of what YECs think about the Cosmos.  For instance, inside the event horizon of a black hole it would apply because of the warping of space there, meaning that one would be confronted with relentless movement towards the end with a tantalising prospect of the rest of the Universe being visible but unreachable even if it was possible to travel faster than light because of the shrinkage issue.  If, on the other hand, you assume parallel lines converge, you get Riemann geometry, which surprisingly is probably the way real space is.

Imagine the Earth wrapped in bandages.  If you were to have an infinite supply of bandages stowed somewhere outside the Universe which you could constantly call upon to wrap this planet fairly evenly, as you wrapped it, it would become an ever-larger sphere with an ever-flatter surface due to its size.  However, once your layer of bandages was more than a few billion light years thick, you would in fact find it was no longer convex or even flat, but concave. You would end up inside the bandages surrounded by them in all directions, with the ground in all directions away from you as if it was hollow and you were inside it, until eventually you would be trapped inside a cosmic layer of bandages many billions of light years thick.

In fact this would never happen for various reasons.  One of these is that the Universe is constantly expanding in all directions.  Getting back to the idea of galaxies moving away from us in all directions, this sounds at first like cosmic egocentrism, but the real reason this happens is that space is getting bigger, and this is where things get misleading.

Here's a picture of the Hubble Deep Field on which I've doodled a few arrows:

Just to say, the Hubble Deep Field is a photograph taken by the Hubble Space Telescope of an area of the constellation of the Plough less than a hundredth the apparent size of the Sun in which three thousand galaxies can be seen.  It's not unusual except that there are unusually few stars in the way, so that suggests that over the whole sky the same project, if there weren't any stars nearby, would show more than 70 billion galaxies.

The arrows show the directions the galaxies are likely to be moving in, that is, away from each other and away from the Milky Way too.  The distances, on the whole, are getting bigger.  They are sometimes getting smaller because even galaxies orbit something and if the orbit is edge on they will be moving towards us if they're close enough, but on the whole everything is moving away from everything else because space constantly gets bigger.

This is usually when a very misleading image is introduced of a balloon inflating and people are asked to imagine being an ant on that balloon with spots moving away in all directions from her.  The reason this is misleading is that it makes it seem like the Universe is expanding into something.  Whereas this might be the case, it doesn't follow from the description just made that it is, and this is one reason Riemann geometry is important.

What is really going on is that large distances constantly increase.  If that's so, it would be fair to ask why the stars in galaxies themselves don't get more scattered, why planets and suns don't get farther apart and ultimately why atoms don't just get ripped apart by this expansion, which is also known as the Cosmological Constant or Dark Energy.  The answer is that it isn't expanding enough to overcome the attractive forces which pull everything together, although it is expanding enough to make sure that everything won't be pulled together into a black hole.  The Universe will go on expanding forever.  As it expands, from any point in the Universe it's only possible to see as far as the speed of light will allow, because great distances expand faster than the speed of light.  Since nothing can travel faster than light, that means that two bits of matter separated by that distance can't interact in any way.  This distance may be getting constantly smaller.  Some cosmologists believe that it will eventually get smaller than a galaxy, then a solar system, then a planet and ultimately a person, meaning that objects of smaller and smaller size will get ripped apart, including in the end any people who might be left alive at that point.  This is called the Big Rip, but it might not happen.

The question of what the Universe is expanding into may be an improper question because it isn't certain that space is a "thing" at all so much as a relationship between objects which describes how far away they are and what direction they're in from each other.  This graph shows income distribution in the UK from 1979 to 2009:

It would only help to think of this graph as a street of stripy skyscrapers if you were distressed by being in one of the purple ones and preferred not to think about it, or if you were in one of the beige ones and didn't want to think about the people in the purple bits.  They aren't real objects.  Likewise, temperature is a scale but not a "thing" as such, and so on.  It could be that space is merely a combination of angles, i.e. directions, and distances, each expressible as a scale, rather than a container for objects, as it were.  If this is so, the idea of Riemann geometry is roughly this:  There is a maximum distance between two points and once this distance is reached, directions reverse.  You can think of this as a four-dimensional sphere, but ultimately that's misleading because then you might start asking yourself what it's expanding into, and the answer is "nothing".  The expansion merely represents the idea that distances tend to increase, including the maximum possible distance between two points.

One odd consequence of this is that it means that on the whole any sufficiently large object is bigger on the inside than the outside.  Go back to the idea of wrapping Earth in bandages and instead imagine it being contained in ever-larger hollow spheres.  In that situation, each sphere has a smaller surface than its volume would lead you to expect.  If this planet is at the centre of a sphere half the size of the Universe, it would take one and a half times longer for light to get to the surface of that sphere than it "should".  This means that the volume of the sphere concerned is getting on for four times its expected size.  Also, over large enough distances parallel lines converge, meaning that a square is kind of this shape:

and a cube is like this:

There would be some variation and some squares and cubes would be more symmetrical, but all of them would be wonky.  Many of them couldn't be reflected onto themselves.  Also, this applies to all squares and cubes, and in fact all shapes which look regular on a small scale such as spheres, circles, squares and cubes.

The reason space doesn't have an edge, therefore, is that it isn't a "thing", or it may not be.  Temperature and velocity both have limits, but those limits recede in such a way that infinite energy would be needed to reach them.  The same is true of space.  This is confusing but it does at least mean that tardises are kind of real, except for the time travel bit.