Rindler Motion in Special Relativity: Hyperbolic Trajectories - Comments

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  • #2
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Some historical remarks:
This sort of motion is also known as hyperbolic motion, which was implicitly used by Minkowski in his famous talk on space and time (1908), search for "acceleration-vector" and "hyperbola of curvature" in
http://web.mit.edu/redingtn/www/netadv/SP20130320.html

The name was given by Max Born 1909, see
Born's remarks on hyperbolic motion
where he used the formula
$$\begin{cases}
x=-q\xi,\\
t=\frac{p}{c^{2}}\xi.\end{cases}$$
where ##q=\sqrt{1+p^{2}/c^{2}}##. The modern formula follows with ##\xi=c^{2}/g## and ##p=c\sinh(g\tau/c)##.

A nice summary was given by Sommerfeld in 1910, see
Sommerfeld's remarks on hyperbolic motion
where he used an imaginary time coordinate and imaginary rapidity,
$$x=r\cos\varphi,\ y=y,\ z=z,\ l=r\sin\varphi$$
 
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  • #3
Orodruin
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Great Insight, but a few things I would like to underline.

The relations ##V\cdot V = 1## (in units where ##c = 1##) and ##V\cdot A = 0## are generally true, not only for hyperbolic motion. The first by definition and the second as a direct consequence of that definition (just differentiating 1 wrt the proper time). This also gives a very straight-forward way of integrating the equation of motion for constant proper acceleration in terms of the proper time. We would have ##A\cdot A = -g^2## and ##V\cdot V = 1## generally gives the possibility of parametrising ##V## as ##V^0 = \cosh(\theta)## and ##V^1 = \sinh(\theta)## for motion in one spatial dimension. Differentiating wrt proper time then directly leads to
$$
-\dot\theta^2 = -g^2 \quad \Longrightarrow \quad \dot \theta = \pm g \quad \Longrightarrow \quad \theta = \pm g\tau \mp \theta_0,
$$
where ##\theta_0## is an integration constant. Directly integrating the 4-velocity then leads to
$$
t = \frac{1}{g} \sinh(g\tau - \theta_0) + t_0, \quad x = \pm \frac{1}{g} [\cosh(g\tau - \theta_0) - 1] + x_0,
$$
where ##t_0## and ##x_0## are integration constants chosen such that ##t(\theta_0/g) = t_0## and ##x(\theta_0/g) = x_0##. Clearly, ##t_0## and ##x_0## are just translations of the solution in time and space, whereas ##\theta_0## represents a shift in the proper time. I have always found this direct integration a more direct way of deriving the hyperbolic motion than that you would typically find in textbooks, which typically is based on using coordinate time, solving coupled differential equations, and/or reference to the instantaneous rest frame.

Edit: A small inconsistency, you also introduce the 4-velocity as ##(V^0, V^1)##, but later you place the spatial component first.
 
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  • #5
ZapperZ
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Coincidentally, the Rindler motion is covered in this just-released video when discussing the Unruh effect in accelerating frames:


Zz.
 
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