On the Perception of Moving Contours by a Single Eye
Abstract. The appearance of moving contours to a single eye are computed.
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. 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 is measured to coincide with the x-axis. The observing eye is at z=L/2.
Apparent Rotating Line Segment
6. Physical Reality.
The lesson of this article is that measured reality doesn't always agree with what we see from afar. Virtually every physicist agrees that what we measure, and not necessarily what we see, constitutes physical reality. When the objects of our perception move with relativistic speeds, then we must not believe everything we see. Are the arms of spiral galaxies really curved as much as they appear to be when viewed from afar? Would they be straighter if we could measure them with clocks and reference systems at rest relative to their centers? And then there's the problem that we can't even measure things up close when the things are very small; we can only compute the probabilities that a tiny system's parts are here or there. That is of course another very challenging story.