Astronomy Research:
Dr. G.H. George

  Most Recent Work:

Resolution of the “Bug-Rivet paradox” (similar to the “pole in the barn paradox” in special relativity), 2012 May, updated 2015 July.

Presentation on the topic of black holes to Shad Valley students at Memorial University, 2005 July 25.

  An investigation of the visual appearance of rapidly moving objects.

The finite speed of light will cause visual distortions in the appearance of objects moving at a significant fraction of the speed of light c relative to the observer.   If, in the standard cartesian (x, y, z) coordinate system or cylindrical polar (r, θ, z) coordinate system, an object has a velocity k parallel to the z-axis and if the observer sees that a reference point on that object achieves its closest approach at a distance d from the observer in the z = 0 plane at observer’s time t = 0, then the apparent position of the reference point at any time t will be given by

z = c*v*{(ct+d) - sqrt[v^2 (ct+d)^2 + (c^2 - v^2) d^2] / c} / (c^2 - v^2)
    r = d ;   theta = theta_o

An application of this result to the visual appearance of a superluminal [faster than light] circle is illustrated in a QBASIC program.

The apparent speed u is given by

u = c*v*{c - v^2 (ct+d)/sqrt[v^2 (ct+d)^2 + (c^2 - v^2) d^2]}
       / (c^2 - v^2)

These results can be extended to the case of objects moving at or beyond light speed (such as the images of spotlights sweeping out rapidly over a distant surface).   The effects of Lorentz contractions on real extended objects can also be incorporated.

In the special case of an object moving directly toward the observer, the expression for the apparent speed u simplifies to

u = c*v/(c-v) [approaching];
u = c*v/(c+v) [receding]

and the ratio of apparent length to true length follows the related law

L(app)/L(true) = c/(c-v) [approaching];
                 = c*/(c+v) [receding]

For a real object, the true length also changes with speed, so that the ratio of apparent length to true rest length is

L(app)/L(rest) = ... = sqrt{(c+v)/(c-v)} [approaching]
L(app)/L(rest) = ... = sqrt{(c-v)/(c+v)} [receding]

Publications include:

Dr. G.H. George
Faculty of Engineering and Applied Science
S.J. Carew Building
Memorial University of Newfoundland
St. John’s, Newfoundland, Canada
A1B 3X5

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Created 1997 09 12 and most recently modified 2017 10 31.