Registered: Dec 2005
I too am a big fan of the "Problem" you purpose. In fact, I too am interested in the solving the same problem: To write a computer simulation of th universe.
(In fact, on a more philosophical note, I think that deep down everyone is at least a little interested in making this happen. :)) It's one of the main reasons that I majored in physics in college.
Let me offer you a suggestion, hopefully it will help. I'm not a complete expert, so some things might not be exactly accurate. But here's what I can offer:
So it maybe that in this discussion of special relativity you are not thinking quite generally enough. Often the simply mathematical structure of special relativity gets disguised in all sorts of "flowery language".
Here's all that's really going on:
In Newtonian mechanics we are concerned with a continuous 3-dimensional set called R^3. We define a metric, or a "measure of distance" on the set R^3 called the Euclidean metric, defined as Sqrt[x^2 + y^2 +z^2], that we must have before we can even start talking about distances, etc. The metric is not a mathematical axiom. It is in fact an empirically determined postulate. Now, any particle's position in R^3 can be expressed as a parameterized curve from R -> R^3. This curve is x(t), called the world line of the particle. The parameter t is called "time".
Now in Special Relativity, here's all we do:
1. Change from R^3 to R^4.
2. Define a new metric: Sqrt[x^2 + y^2 + z^2 - (c^2)t^2]. c is the "speed of light".
3. Now every particle (ie pong ball) has a position function which is a map R -> R^4, still a parameterized curve, but this time in R^4 rather than R^3. The parameter to this curve is now called L (lambda, actually), giving us a function u(L) = (x(L), y(L), z(L), t(L)), a point in R^4 instead of R^3.
All we have done, really, is just add another dimension, to make time just like another component of space. However, we change the metric a little so that time is still different. All the stuff about light cones, simultaneity, and observers observing different things all comes from this simple model of world lines in R^4 with the "Lorentz metric" instead of the "Euclidean metric". An "observer" treats the fourth coordinate t as if in the Newtonian world. This is what makes observations different for each observer. But overall the universe is simply treating t as another space coordinate (with different metric component). So we can still in fact talk of "one universe".
The statement that a particle cannot travel out of it's light cone is simply the statement that the "magnitude" of du/dL must be negative.
Hope this helps. Best,
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