Visible shape of moving bodies

Abstract

We show that if an extended object moves with not only a relativistic but even a nonrelativistic speed, an observer at rest sees the shape of this object distorted, and the distortion depends on the way the object is observed. This phenomenon is due to different retardation times of light emitted by various parts of the object. Moreover, the observer at rest sees the spatial position and speed of objects in an incorrect way. If an extended object moves with a relativistic speed, the relativistic aberration phenomenon occurs, which was analyzed by Einstein. The essence of the effect is that the observer at rest sees the image of a moving small body rotated by some angle. The analysis of these phenomena reported in well-known papers by Terrell and Penrose fails to correctly address the effects related to different retardation times of light emitted by various parts of the extended object but coming to the observer at rest at the same time. In particular, it follows from their studies that the observer at rest sees the image of a moving extended object, for example, a cube or a sphere, not flattened in the direction of motion (as follows from the Lorentz transformation) but only ‘rotated’ by the relativistic aberration angle. We report correct expressions for the images of rods parallel and perpendicular to the velocity of motion as seen by an observer at rest. In particular, if a cube moving sufficiently fast passes by a remote observer at rest, the image of the cube face turned to the observer is contracted in the direction of motion in accordance with the Lorentz transformations, but is not ‘rotated’, while the image of its rear face (with respect to direction of motion) ‘rotates’ by some angle. The image of the cube is therefore distorted. A history of theoretical predictions and experimental observations of this phenomenon is presented. We discuss Gamow’s relativistic street car paradox, which shows that Terrell’s and Penrose’s results are incorrect in the general case of motion of objects. Results of our study explain the Gamow street car paradox in an easily comprehensible way. Physical problems are presented that can be solved significantly more easily if the formulas for the relativistic aberration and light retardation effects are used. We show that assertions made by some astronomers regarding the observation of superluminal motion of some galaxies and supernova jets are incorrect because the effects discussed here were ignored in their calculations.