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I'm having some issues with a problem that I is related, though not identical, to Bell's spaceship paradox (this is not for a homework or anything, it just sort of occurred to me when I was thinking about something else).

Consider two particles that are initially at rest in some frame [itex]O[/itex], that are separated by a distance [itex]L[/itex]. We will say that particle 1 is initially at [itex]x=0[/itex] and particle 2 is initially at [itex]x=L[/itex]. At time [itex]t=0[/itex], they both start accelerating in the same direction, which is parallel to their separation, in such a way as to always maintain a constant separation, as viewed in their own constantly changing MCRF, which I will refer to as [itex]O^{\prime}[/itex]. (This is the reason I am saying it is related to Bell's spaceship paradox - their separation is constant as viewed in in [itex]O^{\prime}[/itex], rather than as viewed in [itex]O[/itex]). My goal is to get equations for the movement of each particle as viewed in [itex]O[/itex].

For the simpler problem of one particle accelerating at a constant acceleration [itex]a^{\prime}[/itex], I have the following equations:

(1): [itex]\beta =\beta (a^{\prime},t^{\prime})=\tanh (\frac{a^\prime}{c}t^{\prime})[/itex]

(2): [itex]x=x(a^{\prime},t^{\prime})=\frac{c^{2}}{a^{\prime}}[\cosh (\frac{a^\prime}{c}t^{\prime})-1][/itex]

(3): [itex]t=t(a^{\prime},t^{\prime})=\frac{c}{a^{\prime}}\sinh (\frac{a^{\prime}}{c}t^{\prime})[/itex]

Now, one way I could solve this is to simply apply equation (2) to either the front particle or the back particle, and then use my knowledge that Lorentz contraction will occur to determine the equation for the other particle. But, I want to solve this without directly applying Lorentz contraction (although admittedly I used Lorentz contraction to derive equations (1)-(3)).

The issue I am having is that both particles by assumption have constant acceleration [itex]a^{\prime}[/itex] as viewed in [itex]O^{\prime}[/itex], so as far as I can tell, equation (1) should apply to both particles. Then, inverting equation (3) to find [itex]t^{\prime}[/itex] as a function of [itex]a^{\prime}[/itex] and [itex]t[/itex] and substituting into equation (1), we have [itex]\beta = \beta (a^{\prime},t)[/itex] is the same for both particles, and differentiating with respect to [itex]t[/itex], [itex]a=a(a^{\prime},t)[/itex] is the same for both particles. If this is the case, how could Lorentz contraction ever occur? I.e., for [itex]O[/itex] to see the distance between the two particles contracting as the particles speed up, it must see that the speed of the back particle is increasing faster than that of the front particle, but my analysis here has that the acceleration of both particles is always the same as viewed in [itex]O[/itex] (which is clearly wrong, since they are the same as viewed in [itex]O^{\prime}[/itex]). So, I have stumbled onto a contradiction.

Does anyone see my mistake? If so, what is it, and how should the analysis in the above paragraph proceed?

Again, I don't want to just apply Lorentz contraction to solve the above problem - instead, I want to analyze it using equations (1)-(3), and hopefully see Lorentz contraction re-emerge as a result. Basically, this is one of those issues where I know two ways to solve a problem that both seem correct to me, but that contradict each other. It is not enough for me to know that one is right - I want to also know why the other is wrong.

Thanks very much for any help that you can give.

-HJ Farnsworth

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# Two particles, constant acceleration in their own frame

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