Abstract
This thought experiment addresses loss of volume that occurs when a three-dimensional object is
contracted due to relativistic processes. Specifically we investigate the dependence of contraction
due to orientation relative to direction of motion. Length contraction states that relative constant
velocity will lead to a shortened distance in the direction of motion, when viewed from a
separate inertial reference frame. When applied to two dimensional shapes, this results in a
reduced area. The problem initially focuses on a square pyramid with the height perpendicular to
the direction of motion. Using geometry, it is found that the area of the base is invariant under
rotation. The problem is then expanded to general shapes. Any shape can be approximated by
inscribing circles within the perimeter, coming arbitrarily close to the shape’s actual area. Since
circles are fundamentally invariant under rotation, the change in area of a circle is independent of
orientation. This argument is used to postulate that any two dimensional shape, when contracted
in one direction, will have a reduced area that is invariant under rotation. Therefore the
contracted area and subsequent reduced volume depend only on the Lorentz factor γ, and not on
relative orientation of the object. A key next step is to determine if this can be extrapolated to
three dimensions, using spheres instead of circles.
Morgan Andrew Davis
Hanjoon Kim
John Patrick McCulloch
Mason Duran Waaler
Click here for PDF file: 2012[4]