Relativity visualized

# Simulating the relativistic bike ride

To simulate the relativistic bike ride, the first step is to portray the surroundings from the point of view of an observer at rest. This observer is located at the same position as the biker. To represent his point of view a so-called cubemap is created: six images on the sides of a cube that give the view into the six directions in space. In a second step the view of the moving biker is computed from the cubemap. For each pixel in the image, the direction of the light ray hitting that pixel is transformed from the frame of the biker into the frame of the imaginary observer at rest. This transformation is done with the formula for relativistic aberration. Subsequently, the colour of the pixel is determined from the cubemap.

There is also an interactive version of the relativistic bike ride. Sitting on an exercise bike, you steer and speed up and down. In front of you on the screen, the view that you should have at nearly the speed of light is projected in real time. For the real time simulation, the cubemap must be created anew for each momentary position of the biker. To produce one image, the surrounding scene of approximately 20000 polygons and 200 Megabyte of data for the so-called texture is drawn six times. A single high-end computer can do this in less than 10 ms creating a cubemap with 6 million pixels. The relativistic transformation takes less than 2 ms so that image rates of more than 60 frames per second are possible.

## Riding the relativistic bike yourself

Naturally, a relativistic bike ride is most interesting when riding yourself. You can interactively ride through Tübingen in the special exhibition “Sonderausstellung Albert Einstein” in the Deutsches Museum in Munich between May 7 and December 31, 2005. In the special exhibition “Sonderausstellung Albert Einstein” in the Historisches Museum Bern, visitors can follow Einstein's daily route from his apartment to his place of work in the patent office of Bern, interactively and at nearly the speed of light, June 16, 2005 to April 17, 2006.

The relativistic bike ride illustrates the view that one should have of one's surroundings when moving at nearly the speed of light. Now, what would the opposite situation be like, when the camera is at rest, but an object moves by at nearly the speed of light? The answer follows from the principle of relativity: It makes no difference whether we regard the object or the camera as moving; the resulting movie will be the same in both cases. The explanation, however, is different for the two different points of view. For the case of the moving object, [10] describes how light travel times make a fast moving object appear distorted and rotated. Movies of such scenes can be found on the web ([8]).

## Acknowledgement

These simulations use a detailed three-dimensional model of the city centre of Tübingen that has been constructed at the Max Planck Institute for Biological Cybernetics, Tübingen. We thank Prof. H. Bülthoff for his kind permission to use this model. More information on the Virtual Tübingen project: Virtual Tübingen.

This project was supported by the DFG (SFB 382, SFB TR7).

## Summary

A computer simulation makes it possible for us to move through the old city centre of Tübingen at nearly the speed of light. On that trip we see amazing things: The houses in front of us recede into the distance when we accelerate; at the same time the edges of the houses close to us are bent, the more strongly the faster we move.

The reason for these strange pictures is the effect called aberration: One and the same light ray has different directions for a moving observer and for an observer at rest on the roadside. In everyday life, this effect is small. But in the simulation, when we rush through Tübingen at nearly the speed of light, it is dramatically large.

## References

 [1] A. Einstein, Annalen der Physik 1905, 17, 891 [2] G. Gamov, Mr. Tompkins' seltsame Reisen durch Kosmos und Mikrokosmos, Vieweg Verlag, Wiesbaden 1984 [3] G. Gamow, Mr. Tompkins in Paperback, Cambridge University Press, Cambridge 1967 [4] A. Lampa, Z. Physik 1924, 72, 138 [5] R. Penrose, Proc. Cambr. Phil. Soc. 1959, 55, 137 [6] J. Terrell, Phys. Rev. 1959, 116, 1041 [7] U. Kraus, Am. J. Phys. 2000, 68, 56, online version: http://www.spacetimetravel.org/ , section Brightness and color of rapidly moving objects [8] C. Zahn, U. Kraus, http://www.spacetimetravel.org/, fast moving objects: e. g. sections Motion near the cosmic speed limit, Rolling Wheels [9] M. L. Boas, Am. J. Phys. 1961, 29, 283 [10] U. Kraus et al., Physik Journal 1 (2002) Nr. 7/8, 77, online version: http://www.spacetimetravel.org/ , section Through the city at nearly the speed of light