Accelerating Bodies in General Relativity: An Investigation
By Rick Boozer
I am currently thinking about writing a computer program that would predict the effects of time dilation (i.e., the slowing down of time) on a constantly accelerating spacecraft that departs Earth and reaches relativistic speeds. I think the acceleration should default to be one earth gravity as it was with the “torch ships” depicted in the old Heinlein science fiction novels. Granted the technology does not currently exist to produce such spacecraft (and might not ever exist), but I thought it could be an interesting exercise to give both myself and others a better insight on how certain aspects of Einstein’s General Theory of Relativity function.
Once I began this endeavor, I found one particular equation in a myriad of physics websites and textbooks. That equation being the one for describing the instantaneous velocity of an accelerating body at any dilated time t as measured on the spaceship:
where acceleration as viewed from the spacecraft is represented by a and c denotes the speed of light in a vacuum.
But what I really wanted was an equation to give me the distance traveled from the starting point when the shipboard time was plugged into it. I found that equation at one place and nowhere else; that is, the University of Cincinnati’s website. Specifically it was on a web page authored by Michael L. Sitko. Here is that equation:
I couldn’t help but wonder, “Why is Sitko’s page the only place that I found that had the distance equation? Is it because it is extraordinarily difficult to derive?” Well, my curiosity got the better of me. I had to find out just how hard it was to come up with the distance equation from the velocity equation. All it took was a little integral and differential calculus and a spare hour or two. I thought those readers of this website with enough mathematical background might be interested in seeing my derivation. Here is a scan of the page of calculations that I did.
CRC Standard Mathematical Tables: 21st Edition. CRC Press, 1973, page 421
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