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Relativistic Calculator

One program I wrote to support my writing activity is a Relativistic Calculator. This was written originally for a QT user interface, but then rewritten for Internet. These were not a complete rewrite, but rather a rewriting of the user interface only.

This calculator is set up to calculate how fast a particular frame of reference would be going after a period of time if it were subjected to a constant acceleration for a long period of time. I have not been able to find a closed form solution to this problem, so I wrote an approximation. Whatever the time being calculated is, it is divided into short 15 minute segments. If something were accelerating for 15 minutes at a reasonable acceleration it would not be going fast enough for any relativistic effect to occur. These short intervals are then added together using Relativistic addition to come up with the final speed and location.

For the Internet version of the program I added a capability of calculating a time require to go a given distance. This is done using a binary search on the time. That is, I take a time and calculate how far the acceleration will take the frame of reference. The interval is cut in half, and the calculation made on the middle. These are compared against the distance. This approach is continued until the distance is located to within the tolerance.

This approach is somewhat flawed in that the computing time is order n, where n is the time interval. I could rewrite this using a binary approach, and cut the size to order ln(n), and at some time in the future I intend to. Also, this assumes the travel is in a straight line, that the acceleration is constant, and the acceleration and velocity are in the same direction, none of which is certain. Each of these issues need to be address.

Part of the problem is that I have not found any books with a statement of the Lorentz equations in more than one dimension. To express these in three dimensions (or four dimensions) would be much more difficult, and I have not taken the time to calculate them. I think a more reasonable approach would be to use transformations which would transform the frame of reference from its current frame to a "stopped" frame, then do the calculations, and transform it back to the current frame. I haven't done this yet, and am not sure exactly how the transformations would take place.

One concern that I have is about the accuracy of the Lorentz equations. These equations were all developed in an environment where there was not acceleration. There was an assumption made that the distances vertically from the direction of travel did not change, which I don't think is justified (although it may be that it doesn't make any difference). Also, if there were an acceleration argument to the "proper" transformations, the Lorentz transformations ignores this because the Michelson-Morley experiment the transformations were based on didn't have any acceleration involved. These are theoretical issues I need to contemplate.

Another issue which comes up is the issue of General Relativity, which is to say, gravity. The only books I've seen on General Relativity tend to get their results from hand waving and general statements rather than a strict mathematical approach. I do have a book written by Einstein on the Special and the General theory of Relativity. The problem is that the book contains at least two errors of a conceptual fashion in the description of General Relativity. Whether this is because the theory itself is flawed (and it would be at a very basic level) or whether Einstein was simplifying the explanation of his results is something I have not been able to determine.

I have yet to be convinced that General Relativity is accurate.

My concern with General Relativity was that I needed this for a novel I was writing. I did find a statement, however, which indicates that regardless of the accuracy of General Relativity, the accelerations, time, and speeds I was handling were small enough that General Relativity didn't make any difference.

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