Abstract

The modified holodiagram is used to solve and visualize in a graphical way a number of problems that are important for the evaluation of ultrahigh-speed recordings. A simplified diagram is introduced to explain the focusing effect of fast-moving light sources or observers. The diagram is used to show the distortion of an orthogonal coordinated system to simplify the study of apparent deformations of arbitrary shaped rigid objects. These distortions are compared with those of pulse fronts or wave fronts of light observed with holographic light-in-flight recordings.

© 1985 Optical Society of America

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References

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  1. N. Abramson, “Light-in-Flight Recording. 3: Compensation for Optical Relativistic Effects,” Appl. Opt. 23, 4007 (1984).
    [CrossRef] [PubMed]
  2. V. F. Weisskopf, “Do Rapidly Moving Objects Appear Lorentz-Contracted?” presented at Theoretical Physics Institute, U. Colorado (Summer 1960).
  3. N. Abramson, The Making and Evaluation of Holograms (Academic, London, 1981).
  4. J. Terrell, “Invisibility of the Lorentz Contraction,” Phys. Rev. 116, 1041 (1959).
    [CrossRef]
  5. R. Bhandari, “Visual Appearance of a Moving Vertical Line,” Am. J. Phys. 38, 1200 (1970).
    [CrossRef]
  6. P. M. Mathews, M. Lakshmanan, “On the Apparent Visual Forms of Relativistically Moving Objects,” Il Nuovo Cimento 12, 168 (1972).
  7. G. D. Scott, H. J. van Driel, “Geometrical Appearances at Relativistic Speeds,” Am. J. Phys. 38, 971 (1970).
    [CrossRef]
  8. N. Abramson, “Light-in-Flight Recording. 2: Compensation for the Limited Speed of the Light Used for Observation,” Appl. Opt. 23, 1481 (1984).
    [CrossRef] [PubMed]

1984

1972

P. M. Mathews, M. Lakshmanan, “On the Apparent Visual Forms of Relativistically Moving Objects,” Il Nuovo Cimento 12, 168 (1972).

1970

G. D. Scott, H. J. van Driel, “Geometrical Appearances at Relativistic Speeds,” Am. J. Phys. 38, 971 (1970).
[CrossRef]

R. Bhandari, “Visual Appearance of a Moving Vertical Line,” Am. J. Phys. 38, 1200 (1970).
[CrossRef]

1959

J. Terrell, “Invisibility of the Lorentz Contraction,” Phys. Rev. 116, 1041 (1959).
[CrossRef]

Abramson, N.

Bhandari, R.

R. Bhandari, “Visual Appearance of a Moving Vertical Line,” Am. J. Phys. 38, 1200 (1970).
[CrossRef]

Lakshmanan, M.

P. M. Mathews, M. Lakshmanan, “On the Apparent Visual Forms of Relativistically Moving Objects,” Il Nuovo Cimento 12, 168 (1972).

Mathews, P. M.

P. M. Mathews, M. Lakshmanan, “On the Apparent Visual Forms of Relativistically Moving Objects,” Il Nuovo Cimento 12, 168 (1972).

Scott, G. D.

G. D. Scott, H. J. van Driel, “Geometrical Appearances at Relativistic Speeds,” Am. J. Phys. 38, 971 (1970).
[CrossRef]

Terrell, J.

J. Terrell, “Invisibility of the Lorentz Contraction,” Phys. Rev. 116, 1041 (1959).
[CrossRef]

van Driel, H. J.

G. D. Scott, H. J. van Driel, “Geometrical Appearances at Relativistic Speeds,” Am. J. Phys. 38, 971 (1970).
[CrossRef]

Weisskopf, V. F.

V. F. Weisskopf, “Do Rapidly Moving Objects Appear Lorentz-Contracted?” presented at Theoretical Physics Institute, U. Colorado (Summer 1960).

Am. J. Phys.

R. Bhandari, “Visual Appearance of a Moving Vertical Line,” Am. J. Phys. 38, 1200 (1970).
[CrossRef]

G. D. Scott, H. J. van Driel, “Geometrical Appearances at Relativistic Speeds,” Am. J. Phys. 38, 971 (1970).
[CrossRef]

Appl. Opt.

Il Nuovo Cimento

P. M. Mathews, M. Lakshmanan, “On the Apparent Visual Forms of Relativistically Moving Objects,” Il Nuovo Cimento 12, 168 (1972).

Phys. Rev.

J. Terrell, “Invisibility of the Lorentz Contraction,” Phys. Rev. 116, 1041 (1959).
[CrossRef]

Other

V. F. Weisskopf, “Do Rapidly Moving Objects Appear Lorentz-Contracted?” presented at Theoretical Physics Institute, U. Colorado (Summer 1960).

N. Abramson, The Making and Evaluation of Holograms (Academic, London, 1981).

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Figures (8)

Fig. 1
Fig. 1

A traveler moves at constant speed (0.6c) to the right. His lines of sight are aberrated from angle γ to the angles drawn in the diagram (the q lines). Along each q line the separation of the intersections by the ellipsoids of observation have a constant value, the q value. Doppler ratio, apparent speed of time, and apparent angular and longitudinal magnification are all functions of q, while transversal Doppler shift, time dilation, and Lorentz contraction depend only on the q line representing γ = 90°.

Fig. 2
Fig. 2

To the traveler an arbitrary point G of the stationary world appears to exist at K which is found by drawing a line of constant Y value from G to the sphere. Light rays emitted by the traveler at angle γ are aberrated by his velocity to angle α, while his lines of sight are aberrated from angle γ to angle β.

Fig. 3
Fig. 3

The traveler in the center of the circle is moving to the right. The radii inside the circle represent directions of the traveler emitted light rays as seen by himself, while the lines leaving A represent those outgoing rays after they have been aberrated (as seen by the rester). Thus, light thrown in the forward direction appears focused as if by a positive lens, while light thrown backward appears diverged as if by a negative lens.

Fig. 4
Fig. 4

Same situation as in Fig. 3. The radii inside the circle, however, now represent the traveler’s line of sight (direction of his telescope) instead of outgoing light rays. The lines leaving B represent the traveler’s aberrated lines of sight. Thus, to the traveler the stationary world appears in the forward direction demagnified as if by a negative lens while in the backward direction it appears magnified as if by a positive lens.

Fig. 5
Fig. 5

The traveler looks in the direction (γ) perpendicular to his direction of travel which is to the right. His line of sight is aberrated backward to angle β. He sees a stationary object at C as if it existed at R, while the viewing angle in the image is unchanged. The object therefore appears rotated at angle γ. When it approaches, its back side is seen even before the object seems to be at the closest range.

Fig. 6
Fig. 6

A flat surface (ss) perpendicular to the direction of travel will appear distorted to the traveler (B) into the curved surface s′–s′, which is constructed in the following way: Draw lines parallel to the x axis from the points where ss intersects the ellipsoids of observation until they reach corresponding points on the spheres of observation. The same can be done going from the intersections with the aberrated lines of sight (q lines) to the original lines of sight (at angle γ). Connecting these points results in the traveler’s impression s′–s′ of the stationary surface ss. (a) Ellipsoids of observation and aberrated lines of sight, (b) Spheres of observation and original lines of sight.

Fig. 7
Fig. 7

Orthogonal coordinate system of the stationary world (a) appears to the traveler transformed into that of (b). The traveler exists at the small circle (i,O) and is moving to the right at a speed of 0.6c. This situation is identical to that when the observer is stationary while the coordinate system is moving to the left. From the diagram we find that flat surfaces are transformed into hyperboloids that point like arrows in the direction of travel. The plane (ii) through the observer is transformed into a cone. The back side can be seen on all objects that have passed this cone. The separation of advancing hyperboloids is increased, while that of those moving away is decreased.

Fig. 8
Fig. 8

Orthogonal coordinate system identical to that in Fig. 7(a) is made up of light using horizontal light rays and vertical pulse fronts (wave fronts) that pass from right to left. An observer makes a light-in-flight recording and finds that the flat pulse fronts appear transformed into a set of paraboloids with himself at the common focal point. The back side is seen on all pulse fronts.

Equations (4)

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d = t · c 1 ( υ c ) 2 .
l T l R = l ( υ / c ) 2 = l / k ,
l T l R = cos ξ,
M a · M l = 1 ,

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