We derive a formula for the law of reflection of a plane-polarized light beam from an inclined flat mirror in uniform rectilinear motion by applying the Huygens–Fresnel principle. We then use this formula and the postulates of special relativity to show that the moving mirror is contracted along the direction of its motion by the usual Lorentz factor. The result emphasizes the reality of Lorentz contraction by showing that the contraction is a direct consequence of the first and second postulates of special relativity, and is not a consequence of the relativistic measurement of the length.
REFERENCES
1.
For a beautiful historical overview and a good introductory account of the problem of reflection and refraction of electromagnetic waves from moving boundaries, see
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3.
See, for example, M. Born and E. Wolf, Principles of Optics (Cambridge U.P., Cambridge, 1999), 7th ed.
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6.
See, for example, E. Hecht, Optics (Addison–Wesley, Reading, MA, 1987), 2nd ed.
7.
A word of caution! The visual appearance of a high-velocity moving object (that is, the process of seeing) and the measurement of its shape (that is, the process of relativistic measurement) are completely different. See, for example,
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An illustrative example of this misconception is the discussion associated with Fig. 15-3 in R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison–Wesley, Reading, MA, 1989), Vol. 1, p. 15-6. See in particular, part (c) of the caption, which states “Illustration of the diagonal path taken by the light beam in a moving ‘light clock.’ ”
9.
The result follows from the requirement that the phase of a plane-polarized electromagnetic wave must be an invariant quantity under a Lorentz transformation. See, for example, Sec. 2.9 in Ref. 5 and J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), 2nd ed., Sec. 11.3(d).
10.
It is worth noticing that Eq. (14) also can be obtained by a direct application of the addition law for relativistic velocities. See, for example, L. Landau and E. Lifshitz, The Classical Theory of Fields (Addison–Wesley, Cambridge, 1951). Still another way to derive the aberration formula is discussed in Ref. 9.
11.
The statement that the lateral dimensions of a uniformly moving object remain unchanged with respect to the lateral dimensions of the same object at rest can be easily proved by using only Einstein’s first postulate. See, for example, Problem 6.1 in A. A. Pinsky, Problems in Physics (MIR, Moscow, 1988).
12.
For a more detailed discussion on the length of a moving object, the process of measurement, and Lorentz contraction in special relativity, see T. A. Moore, Six Ideas That Shaped Physics, Unit R: The Laws of Physics Are Frame-Independent (WCB/McGraw–Hill, Boston, 1998).
13.
The physical length of a uniformly moving object would equal the object’s proper length, that is, the object’s length measured in the reference frame in which the object is at rest.
14.
The statement that Lorentz contraction of a plane mirror in uniform rectilinear motion implies Lorentz contraction of any uniformly moving object can be clarified by imagining the object’s surface to consist of an infinite number of infinitely small plane mirrors having different inclination angles with respect to the object’s motion.
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