The effect of the relativity correction on the law of reflection from a moving mirror
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Abstract
This thesis utilizes the Lorentz transformation formulas from the special theory of relativity in dealing with the law of reflection from a moving mirror. It was found necessary in the proof to consider the source of the light ray and its final destination. For this reason the mirror was taken to be one side of a square. The light source was at one corner of the opposite side of the square and the light ray was so directed that it would be reflected to the other corner. For a physical treatment of the problem a cubical box would be necessary, but for the mathematical treatment the square was sufficient.
In the first part of the thesis the square was moving away from the earth at a fixed angle and a constant velocity. The earth was considered to be the stationary system. If an observer was placed on the square he would note that the light would be reflected to the corner of the square. The problem was to investigate the effect of the relativity correction on the law of reflection from a moving mirror and as a subsidiary problem the effect of a moving or stationary ether upon reflection from a mirror at rest. A medium, called the ether, for the light to pass through was brought into the problem. It was assumed stationary with respect to the earth.
The first formula derived was the law of reflection from a moving mirror using the methods of classical physics. This law was general in that it did not involve the source of light. It dealt entirely with the angles of incidence and reflection, and the velocity of the mirror with respect to the earth.
Next, a classical treatment of the moving square was made. This was treated as a geometrical problem. It was assumed for this case and the next that the quantities that could be measured physically were the velocity of the square, the angle at which it was moving, and the angle of incidence from the point of view of an observer on the mirror. It was assumed in the construction that the reflected ray would hit the corner of the square. Thus the formula for the angle of reflection was derived. For this angle to be correct it would have to agree with the angle given by the law of reflection from a moving mirror. To show that the two angles were not identical a numerical example was worked out. The angles were widely divergent.
Then the Lorentz transformation formulas were applied. Actually, the only formula needed was the one which showed how much a length would apparently be shortened in its direction of motion. This shortening is from the stationary observer's point of view. The effect of this apparent shortening was to alter the angles in the geometric construction. The equations for the angles of incidence and reflection were then formulated in terms of the quantities assumed to be known.
This angle of reflection to be correct must agree with the angle from the law of reflection from a moving mirror for the same angle of incidence. The relativity correction was applied to the law of reflection. It was assumed that the two angles of reflection were identically equal. It was then proven that this assumption was correct.
It should be noted that the normal that the stationary observer uses for calculating the angles of incidence and reflection is the normal from his point of view and not that of the observer on the square. The stationary ether had no effect upon the outcome of the problem.
In the second part of the thesis the square was considered to be stationary and the ether moving with a constant velocity. The problem was to find out what effect this moving ether has upon the light being reflected to the corner of the square. This problem was divided into two cases which were assumed to be independent. They were (1) the effect of the ether upon the light and (2) the effect of the ether upon the mirror.
In the first case the ether can change the velocity and the direction of the light ray. By taking a particular orientation of the square it was shown that if a transformation was applied that would lead to the correct results the velocity of the ether would have to reduce to zero. However, this gives a result that the observer has already obtained without considering a moving ether and transformations.
In the second case the ether can distort the square. To get the correct results the transformation would have to bring it back to a square. Again the velocity of the ether would have to reduce to zero. The observer would have measured his apparatus as a square without taking into account any distortions.
Thus in both cases a moving ether is artificial and meaningless. Therefore in both parts of the problem the ether whether moving or stationary adds nothing to the solution.
It is well known that light reflects from a mirror in a square according to the formula i = r and the light is reflected to the corner of the square if the mirror is at rest relative to an observer. If this is true then light reflected from a mirror in a square in motion relative to an observer should lead the observer from his calculations to predict that the light will be reflected to the corner of the square. This paper shows that it is necessary to apply the relativity correction to the law of reflection from a moving mirror to have theory agree with facts.
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Thesis (M.A.)--Boston University
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