Two-dimensional appearance of a relativistic cube

Abstract

The two-dimensional or ’’photographic’’ appearance of a cube moving at relativistic speeds is investigated. It is shown that the appearance of the cube depends upon whether the camera is held fixed with the optic axis perpendicular to the direction of motion or panned to follow the cube as it moves. The Lorentz contraction, as it is usually defined, is not visible in either case. However, if the camera is rotated, the photographic appearance has some of the characteristics of a picture of a stationary rotated cube.

Projection of relativistically moving objects on a two-dimensional plane, the `train' paradox and the visibility of the Lorentz contraction

“…These transformations are appropriately referred to as nonlinear Terrell transformations. The closest investigation to these transformations was given by Hickey [9], who, however, applied methods of mapping out the tangents to points on an object. The latter also provides no room for resolving the 'train' paradox.…”

“…Since the early classic work of Penrose [1], Terrell [2] and Weisskopf [3] on the appearance of relativistically moving objects several refinements and extensions of the above work have been carried out (see, e.g., [4,5,6,7,8,9,10,11]). Three problems seem, however, to remain outstanding.…”

The appearance, apparent speed, and removal of optical effects for relativistically moving objects

“…That is, for a given value of y, points higher and lower on the object will appear behind points closer to the center. 5,8 To quantify this curvature we calculate d c ‫ץ(‬ 2 x a /‫ץ‬z 2 ) from Eq. ͑15͒ and evaluate it at (xЈ,y,z)ϭ(…”

“…8 As the object appears to be approaching, the length appears longer than this quantity and when the object appears to be receding, the length appears shorter than this quantity. 5 To quantify this length change, we calculate ‫ץ‬x a /‫ץ‬xЈ from Eq. ͑15͒ and evaluate the derivative at (xЈ,y,z)ϭ(…”

“…Although the two-dimensional projection of a relativistically moving object onto a two-dimensional plane has been derived, 5 equations have not been derived that directly transform a two-dimensional projection of an object in its rest frame to the two-dimensional projection of the apparent coordinates of the object. In this paper we derive this transformation and apply it to some photographic images.…”