On the Perception of Moving Contours by a Single Eye
In this article we compute what a single "eye" would hypothetically see when certain lines/contours move. It is not a trivial matter, since at time t=0 the eye is acted upon by photons that depart from points on the line/contour at different moments in the past (and not by photons that leave these points at a common instant). Four lines/contours are investigated: (1) a horizontal line moving in the positive x-direction; (2) a vertical line moving in the positive x-direction; (3) a circular contour moving in the positive x-direction; and (4) a clockwise-rotating straight line which, at t=0, coincides with the x-axis.
The observing eye is permanently at rest at point z on the positive z-axis, where z is comparable to a given lineís/contourís rest dimensions. All of the lines/contours are assumed to lie in the xy-plane. In the cases of the translating lines/contours, the lineís/contourís center moves along the x-axis at the constant, relativistic speed of .95c. At time t=0 these linesí/contoursí centers are at the origin. In the case of the rotating line, the line center is permanently at rest at the origin and its ends move at a speed of .95c.
The general approach is to break each line/contour up into a large number of segments, with each segment approximating a point source of photons. Photons are presumably emitted constantly and in all directions from each such point. However only one photon, emitted by a given point at the retarded time, can reach the eye at time t=0. This photon will presumably determine where the eye perceives that particular point to be at time t=0. (It will, of course, not be where the point actually is at time t=0.) Collectively all of the photons, reaching the eye at time t=0, determine the lineís/contourís perceived shape.
2. Horizontal Line Segment Translating Along the x-axis.
Let us assume that a straight line segment, of rest length L = 2 meters, moves in the positive x-direction at constant speed .95c. The segment is coincident with the x-axis. At time t=0 the segmentís center is at the origin. Theoretically its measured length would be length-contracted to g-1L. The question is, what would the segment look like to the eye at z=L? Fig. 2_1 depicts the retarded points (i.e. the apparent segment). At t=0 the entire segment appears not yet to have reached the origin. Furthermore, the apparent length is more than 3 times the rest length (and many more times the contracted length)!
Apparent Horizontal Line Segment
3. Vertical Line Segment Translating Along the x-axis.
Fig. 3_1 depicts the line segment of Sect. 2, rotated 90 degrees (all other things being equal). At time t=0 the middle of the segment is perceived to be well to the left of the origin, and points above/below the middle are perceived to be even further to the left.
Apparent Vertical Line Segment
4. Contour is a Circle (when resting).
Fig. 4_1 depicts the contour that is a circle of radius R = 1 meter in its rest frame. When moving at a speed of .95c the circle would theoretically be measured to be length-contracted into an oval. However, this is not what is perceived. Note that the observing eye in this case has been located further out on the z-axis, at z = 10R.
Apparent Circular Contour
5. Rotating Line Segment.
Figure 5_1 depicts the eyeís perception of a straight line, of length L=2 meters, that rotates clockwise. At time t=0 the line coincides with the x-axis. The observing eye is at z=L/2.
Apparent Rotating Line Segment