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Musings On Relativity and Accuracy

OK. Just been watching this

http://www.youtube.com/watch?v=tflf05x-WVI

It’s made me think. The race time measured by the runner is less than the race time measured by the observer. As the runner speeds up, the difference between those two times increases. So are there times that cannot be registered on the runner’s clock due to the jumps between steps in time of the observer? Just as with x=y^2. y goes 1, 2, 3, 4 and x goes 1, 4, 9, 16.

Of course, the answer is no because the universe is analogue, not digital. But there’s something about that that messes with my head a little. I blame modern computer science and it’s finite space view on things. Lets say x goes 1, 2, 3, 4 then y goes 1, 1.41421…, 1.73205…, 2.

Right. Lets take this one step further. In a computer, with a number like π you can’t store the whole number as it goes on forever. We have to store it in the number of 1s and 0s allocated so we have to approximate it. This is the reason (1/3)*3 often doesn’t come out as 1 on a calculator. But what if we had a computer that wasn’t bound by the number of 1s and 0s allocated to the number? Well then you’d be looking at an analogue computer. Now I’m going to take an analogue computer I know. This is called the Hartree Differential Analyser. This computer was mechanical. It used gears, rods and various other connections to calculate continuous equations. You didn’t just put in 5 * 5 and get 25. You would insert a graph of a line representing values of x for x * 5 and get a graph of the answers. Awesome! What’s the catch? Well you’re now limited by the mechanical accuracy of your machine, not it’s design. While this machine could theoretically give you a 100% accurate of any equation inputted, in practice it can’t. You can’t machine those gears to touch each other perfectly. There will always be a small gap between them where they can wriggle. Those jumps between what each 64bit number of then computer can represent has now become jumps based upon that wiggle in the gears.

So why am I talking about this? Every analogue system on this earth created by man has inaccuracies. They can provide constant calculation within a tolerance. They are, in some senses, still digital. This is at the man made macro level though. Lets say you could get past that. You now have inaccuracies at the molecular level in the movement of molecules in a substance (assuming greater than absolute zero temperatures). OK. Lets assume you could get past that. You now have inaccuracies at the atomic level. The vibration of the atoms. Depending on your measurements, the movement of the electrons around the nucleus may even have an effect. Lets assume you could overcome that. Lets assume an absolute zero system. I’m also going to assume that the fundamental particles don’t change but there may be another inaccuracy. I just don’t have the knowledge of what goes on there. Now your only possible inaccuracy is that within time itself and space itself. Of course, inaccuracies in space may arise through its bending caused by the mass of all the bodies within having an effect on each other but lets say we’ve taken those into account.

So here’s the final thing. The last thing that may cause jumps in the possible time registered on each device now we have them analogue and all the possible inaccuracies sorted. Are time and space truly analogue or do they have fixed values? That’s like asking can I throw a ball from here to there without it passing the space in the middle. If not, then we have infinite accuracies and that means the universe itself does not behave like any mechanical system I know. The universe is behaving more like a mechanical construct. The universe itself is not something that exists physically but merely as a continuous mathematical construct. Those, of course, I may be getting my ideas of what can be physical wrong and likely am. But lets say the universe is digital. A body can move from a to b without passing through the space in between. Now either the possible speeds something can move at, the distance between a and b and the time between two consecutive points in time are such that they always align up nicely and a body will merely jump between well defined speeds that are more accurate than the universe needs, or there are inaccuracies as to the body appearing in the wrong place between two times as it should be in the space in between. Now this can’t happen as physics would drift. Or more it can but is unlikely as the universe would probably collapse. Well. Maybe it will collapse but it just hasn’t happened yet. That’s a fun thought. Maybe the accuracies of space only have to be such that that collapse will take at least slightly more than the life of the universe?

I guess both of these possibilities give rise to a universe based purely on maths. Universe that are physical entities floating in a sea of conception. It bares the thought that the maths has to be at least accurate enough and nothing more. Just like the universe bashed rocks together until the atoms arrange such that they were accurate enough to sustain life for a period of time that is long enough and nothing more. Maybe before the universe, this conceptual aether bashed maths together until it were at a point where it is accurate enough to sustain the universe for long enough and nothing more. Maybe the universe will eventually bash those rocks together enough to sustain life permanently? Though the second law of thermodynamics would then pull some pretty glum faces and throw the decay of the universe at that god-like fish in the sea that is here. Then maybe that aether will eventually create a set of maths whereby the universe can exist indefinitely in steady state. Where the second law of thermodynamics has a sweat spot. 

Ok. So I think I’ve tortured math and physics enough now. I better get to bed soon. Oh. And the answer is it doesn’t matter if Usain Bolt jumps in well defined points in time because those points of time are more accurate than the physical universe needs them to be. Like your hard drive being bigger than it needs to be to hold Portal 2. Now have a mug of hot chocolate and get some rest.

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