A Einstein's Elevator Trajectories: Desloge & Philpott 1987, Hamilton 1978

Mathieu Rouaud
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Newton's theory predicts parabolic trajectories. But what kind of trajectories within the framework of Special Relativity?
Hello,
Some papers describe the vertical motion of a ray of light or a non-zero mass particle in a uniformly accelerated reference frame in special relativity:
  • Desloge, E. A., & Philpott, R. J. (1987). Uniformly accelerated reference frames in special relativity. American Journal of Physics, 55(3), 252–261. https://doi.org/10.1119/1.15197 (world lines on page 258)
  • Hamilton, J. D. (1978). The uniformly accelerated reference frame. American Journal of Physics, 46(1), 83–89. https://doi.org/10.1119/1.11169 (world lines for a ray of light on page 85, for a massive particle on page 86)
But in the case of a non-vertical initial velocity what is the trajectory? What kind of curve does a particle draw on a vertical wall of the elevator? Do you know reference papers or books on this subject?
Thank you for your answers.
 
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A point particle in a homogeneous electric field, neglecting radiation reaction, realizes a particle with constant proper acceleration. The trajectories are hyperbolae.
 
Wikipedia (Rindler coordinates): "we obtain a picture which looks suspiciously like the family of all semicircles through a point and orthogonal to the Rindler horizon"
Thus, the trajectories of photons in the accelerated elevator seem to be circular!
 
Mathieu Rouaud said:
Wikipedia (Rindler coordinates): "we obtain a picture which looks suspiciously like the family of all semicircles through a point and orthogonal to the Rindler horizon"
Thus, the trajectories of photons in the accelerated elevator seem to be circular!
Semi-circular, yes. Given of course a very large elevator where spacetime is still flat.
 
Well, but circles in a Lorentzian plane are in fact hyperbolae (or light cones), namely (in "Minkoski-Cartesian coordinates")
$$\eta_{\mu \nu} x^{\mu} x^{\nu}=\text{const}.$$
See the picture in Wikipedia just close the quoted passage.
 
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