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Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 9, Iss. 3 — May. 1, 1924
  • pp: 223–233

ABERRATION OF LIGHT IN TERMS OF THE THEORY OF RELATIVITY AS ILLUSTRATED ON A CONE AND A PYRAMID

VLADIMIR KARAPETOFF

JOSA, Vol. 9, Issue 3, pp. 223-233 (1924)
http://dx.doi.org/10.1364/JOSA.9.000223


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Abstract

A ray of light is propagated in the <i>XY</i> plane and its angle with the <i>X</i>-axis is determined by two observers, <i>S</i> and <i>S</i>′, who are in relative motion to each other along the <i>X</i>-axis at a velocity <i>q</i> (expressed as a fraction of the velocity of light). If the angles measured by them are δ and δ′, then it is proved that tan½δ/tan½δ′=cos(45°′½α)/cos(45°-½α), where Sin α=<i>q</i>. This relationship is interpreted geometrically by means of a cone in oblique coordinates. The axes of coordinates are <i>X, Y</i>, and time <i>T</i>. The <i>T</i> and <i>X</i> axes are different for the two observers, and the four axes <i>T, X, T</i>′, <i>X</i>′, form a Lorentzian Plane to which Y is perpendicular. The properties of the cone are considered in detail and a mechanical model of the cone, built of wood and iron, is described. At a desired velocity <i>q</i>, one can compare directly the angles δ and δ′. It is also shown that the relationship between the angles δ, δ′, and α can be represented on a pyramid. References are made to other articles by the same writer, in which oblique axes are also used for the solution of problems in the theory of relativity.

Citation
VLADIMIR KARAPETOFF, "ABERRATION OF LIGHT IN TERMS OF THE THEORY OF RELATIVITY AS ILLUSTRATED ON A CONE AND A PYRAMID," J. Opt. Soc. Am. 9, 223-233 (1924)
http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-9-3-223


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References

  1. For a popular description of this model see Science and Invention, 11, p. 442; 1923.
  2. The Physical Review, 23, p. 239; 1924.
  3. See, for example, O. W. Richardson, The Electron Theory of Matter, p. 300; 1916.
  4. See, for example, Snyder and Sisam, Analytic Geometry of Space, index under "Cone."
  5. The foregoing proof is, of course, but a particular case of the invariancy of the expression x2+y2+z2-c2t2, when z=0.
  6. See, for example, Seaver's Mathematical Handbook, p. 143, formulae 1342.
  7. See, for example, Richardson, loc. cit., p. 306, eq. (14).

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