Pythonism

code and the oracular

Journeys In Spacetime

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When they removed Einstein’s brain (don’t worry it was after his death) they found the region that usually takes the role of spatial reasoning was enlarged, actually it had crowded out part of the language region.

If you want to be like Einstein I suggest you get into visualising geometry and topology of all sorts. The easiest is to visualise a perfectly flat plane, with straight and curved journeys, triangles, squares and other shapes, but once you’ve done that you can move on up to subtler and more complex manifolds, like the sphere and then the torus, which is the shape of a ring doughnut. It’s got a hole.

Let’s allow the proportions of our torus to be about the same as the doughnut. Imagine you were an ant confined to crawl on the surface of this torus. In how many directions could you set out (following a straight line), such that you would eventually return to where you started ?

There are two immediately obvious ways to achieve a closed loop. Let’s call North a circle resting on the topmost bit, as if you had placed a doughnut on a flat table and rested a piece of paper on top. the bit where the paper contacts the torus is the circle that represents north, journeys north have to intersect this circle at a right angle. If you start on an outermost point of the torus and head due north then you will circumnavigate the body by travelling along the shorter circumference which means you will be going through the hole. If you move due east, which must be a line at right angles to north you will circumnavigate the torus in the longest circumference, and this means you will be circumnavigating the hole.

Setting out at an angle to the meridians is a little subtler. lets denote the angle from due north you set out in as arctan a. as long as a is a rational number then you will trace a curved helix and come back to your start eventually, although in many cases this will take several orbits around. If however a is irrational then you will never reach start exactly. Of course that means you will never return to the exact zero dimensional point of your initial position, although you may come arbitrarily near.

Contrast this with a sphere… if you are on a sphere it is apparent that every straight line journey will return you to start eventually.

But now lets move to an analogous case. We do the clever trick of trying to consider what a toroidal space of 4 dimensions might be like. In the same way as the ant is only able to move on the surface of the 3 torus, we will now be constrained to the 3 dimensional space that is the surface of the 4 torus. Although its hard or even impossible to visualise 4D objects, by using analogies we can still make some deductions.

I make it that journeys north will be journeys in a specific plane. in this case by analogy a direction that specified one of our neverending jourmeys would be given by two angles, one relative to the north plane and one to the east west. Journeys like this would be circumnavigating the hole. Here’s a problem for you, do the angles given by arctan a and arctan b need to have irrational a and irrational b or just one of the two for the neverending journey ? Its a good one for using a mixture of visualisation and reasoning by analogy.

Another intersting phenomenon is that if you pointed a sufficiently powerful telescope in one of these directions you might see the back of your own head ! When the apparent size of the back of your head is a maximum, you are looking north. Move away from north just a fraction and the size gets a lot smaller. An infinitesimal movement again and it might disappear, you’ve hit another irrational value of a. Of course if toroidal spacetime is like ours then infinitesimals are pretty hard to find, since even the planck length is infinitely huger. Math spaces are not exactly like physics spaces…

Are there possible directions in a closed 4d manifold such as our own spacetime may be where you will never return ? if the manifold is a 4 torus then by analogy this should be the case. If our spacetime is a 4 sphere then it seems this would not happen. every direction eventually returns you to start. Of course if the space is not topologicaly closed then you can go off forever, just as you could on a flat and open plane. I think closed universes are much more fun, and also they may avoid endless heat death and expansion where the average separation of all particles tends to infinity as time goes on to infinity. To me this is a depressing thought, I want a big crunch and a second chance at everything !

I am fascinated by the idea that a journey in a specific direction in a 4D toroidal universe would go on forever without ever returning to start, even though the actual size of the universe is finite. What an everlasting voyage of discovery this would be ! You would be circumnavigating your homespace endlessly on a different path each circuit, and this could go on forever. You would also be going round the hole forever too. It’s hard to see what this hole would be like. Maybe superadvanced aliens in a universe like that would learn to take hyperspatial shortcuts by jumping out of their 3d volume, crossing the hole in a straight line, and popping back in on the other side.

It’s also worth adding that if distances in any of these spaces have a minimum value, i.e. if space itself is quantised then there can be no endless journeys, since there are only a finite range of locations you can be in. Here’s another thought – what must the angles be to ensure you visit every possible location. This would be useful if you ran a tourist agency promising a “see absolutely everything” grand tour of your universe ! In our universe the planck length is the smallest distance we can make observations about, I think that means that space is effectively quantized. But doesn’t that mean that there can be no straight line journeys.? only zig zags as we pop into the nearest cell of space that touches our ideal straight line ? This reminds me of trying to make neat diagonals in a paint program, which is limited by the square grid of pixels on the screen !

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Written by Luke Dunn

August 19, 2009 at 4:17 pm

Posted in Math

Tagged with , , , ,

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