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

[Mathieu Rouaud]

TL;DR

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.

 

[Dale]

The Wikipedia entry for Rindler coordinates has a section on geodesics:

https://en.m.wikipedia.org/wiki/Rindler_coordinates

 

[vanhees71]

A point particle in a homogeneous electric field, neglecting radiation reaction, realizes a particle with constant proper acceleration. The trajectories are hyperbolae.

 

[Mathieu Rouaud]

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!

 

[Dale]

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.

 

[vanhees71]

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.