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Project PHYSNET Physics Bldg. Michigan State University East Lansing, MI

MISN-0-44

APPEARANCES AT

RELATIVISTIC SPEEDS

observer

v

light rays

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APPEARANCES AT RELATIVISTIC SPEEDS

byPeter Signell, Michigan State University

1. Replacing an Erroneous View . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. Simple Object: a Cube. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

3. No Contraction, No Retardationa. View From Side of Slowly Moving Cube ................. 2

b. View of a Slowly Moving Rotated Cube . . . . . . . . . . . . . . . . . 24. Retardation, No Contraction

a. Photon Departure Positions and Times ..................3b. A Photons Lead Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3c. Distance Traveled by Cube During Lead Time . . . . . . . . . . . 4

5. Retardation and Contractiona. Modification Due to Lorentz Contraction . . . . . . . . . . . . . . . . 4b. Apparent Angle of Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

A. Additional References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

B. Journal Excerpts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7


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APPEARANCES AT RELATIVISTIC SPEEDS

by

Peter Signell, Michigan State University

1. Replacing an Erroneous View

This topic, the appearance of objects moving at relativistic speeds,should be viewed as a recreation inasmuch as its final result is not usedelsewhere in the physical sciences. Nevertheless, the derivation and expo-sition in 1959 led to widespread discussion; in one stroke it demolished apicture which physicists had believed in and taught for fifty years. More-

over, it replaced a complicated erroneous picture with a simple correctone. We also recommend the derivation to you because the techniquesand principles used are useful in other areas.

2. Simple Object: a Cube

In order to simplify the derivation, we will deal solely with an objectwhich, in its rest frame (when at rest), has the appearance of a cube withsides of length . The cube is assumed to be so far away from you, theobserver, that the light rays coming to you from its various parts can beconsidered to be parallel. Also, we will only consider its appearance asthe object passes directly in front of you as shown in Figref1. Note also

top

E

F

A

B C

D

v

light rays

observer, assumed

far away compared

to size of cube

Figure 1. View from above. The cube is drawn as it would

appear in its rest frame.

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MISN-0-44 2

front

A

B C

D

l

l

Figure 2. As in Fig. 1, view from the front.What corner is hidden behind corner B?

the labels of the outside corners of this solid cube.

3. No Contraction, No Retardation

3a. View From Side of Slowly Moving Cube. This is the ordinaryapproximate appearance one obtains for an object moving at speeds thatare very small compared to the speed of light. For a cube of side weexpect the appearance to be as shown in Figure 2.

3b. View of a Slowly Moving Rotated Cube. Now suppose thatthe cube has been rotated about an axis perpendicular to both the lineto the eye and the direction of motion, as shown in Figure 3a. Then the

observer is predicted to see the appearance as shown in Figure 3b.

frontleft

B

C

l

l

E

top

ls in

ls in

lcos

lcos

F A D

E B C(a) (b)

Figure 3. A rotated cube: (a) top view; (b) side view.

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top

E1 E2

B1 B2 C1 C2

E B C

photons traveling

together toward

the observer

v

Figure 4. View of the cube from above. The dashed linesshow the position of the cube at some particular time whilethe solid lines show its position at a later time.

4. Retardation, No Contraction

4a. Photon Departure Positions and Times. We go back to theunrotated cube and start from the appearance at any one instant as re-sulting from all of the photons1 arriving at the observer at that instant.Photons do not travel instantaneously from the cube to the observer. The

speed of light is finite. Due to the varying distances of the parts of thecube from the observer, and our requirement that the photons from themmust all arrive at the eye at the same time, the photons must have lefttheir points of origin on the cube at various times in the past. This re-tardation effect means that a photon of light must have left corner Eearlier than one from cornerB to arrive at the observer at the same time.To see this, compare Figure 1 and Figure 4. In Figure 4, photon Emusthave left cornerEat timetE, whereas photonsB and Cleft at timetB,C.

4b. A Photons Lead Time. The extra distance theEphoton musttravel is, since that is the length of side BE(remember that our photonsare supposed to be traveling parallel paths due to the large distance tothe observer). Thus the Ephoton must start out earlier than the B andCphotons by a time given by:

E-photons lead time = extra distance

photons speed=

c=tB,C tE.(see Fig. 4)

1(fotan, a particle of light).

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4c. Distance Traveled by Cube During Lead Time. During thistime the cube travels a distance given by:

cubes distance = cubes speed time =v

c=

v

c =tB,C tE.

Considering only the retardation effect, the cube would appear as in Fig. 5.

5. Retardation and Contraction

5a. Modification Due to Lorentz Contraction. Using relativity,2

one finds that a moving length in the direction of motion becomes con-tracted from its rest-frame or proper valueL0 to L(v) = L0

1 v2/c2.

SideAD of the cube in Fig. 1 has its entire length in the direction of mo-tion. Therefore the Lorentz contraction modifies Fig. 5 in the mannershown in Fig. 6.

5b. Apparent Angle of Rotation. This is the final, correct appear-ance. It is astonishing that it is exactly the rest-frame appearance of thecube but rotated through an angle of:

= sin1 (v/c) = cos1

1 v2/c2 ,

as shown in Fig. 3b.

Show that Figs. 5 and 6 reduce to Fig.2 asv/c 0.

Show that, for ordinary everyday speeds, is too small to be seen.

Describe the appearance of the cube as v/c 1.

Prior to Terrells inclusion of the effects of retardation, physicists as-sumed that the appearance would be that produced by the Lorentz con-

2

See The Length Contraction and Time Dilation Effects of Special Relativity(MISN-0-13).

frontleftl

F A D

E B C

v

lv_c

l Figure 5. From the side, noLorentz contraction but with retar-dation.

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l

F A D

E B C

vl 1- v

c

2

2

frontleft

lv_c

Figure 6. Appearance from theside, with Lorentz contraction andretardation.

traction alone. Show, under the assumption of only Lorentz contrac-tion, that the front side of the cube in Fig.2 would have the sameappearance as a planar square that had bee rotated through an angle

= cos1

1 v2

/c2

(Note that the sides AD and BC in Fig. 1 are inthe direction of motion and hence are Lorentz-contracted).

Suppose a distant piece of straight line has its length in the directionof its motion. Show that it will have the appearance of a rest-frame linerotated through an angle = cos1

1 v2/c2.

Acknowledgments

Steve Smith, Mark McChesney, Douglas Ullmer, and Ray Van Aus-dal gave valuable feedback on an earlier version of this module. JimLinnemann has made very useful suggestions that improved the modulesignificantly. Preparation of this module was supported in part by theNational Science Foundation, Division of Science Education Developmentand Research, through Grant #SED 74-20088 to Michigan State Univer-sity.

A. Additional References

Invisibility of the Lorentz Contraction, James Terrell, PhysicalReview 116, 1041 (1959). See the first page of his article, reproducedby permission in Appendix B.

Observation of Length by a Single Observer, Roy Weinstein,American Journal of Physics 28, 607 (1960). He considers the prob-lem of a line segment oriented along the direction of motion, withLorentz contraction and retardation, and without the restriction ofparallel rays (large distance from object to observer). See his ab-

stract, reproduced by permission in Appendix B.

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The Visual Appearance of Rapidly Moving Objects, V. F. Weis-skopf, Physics Today, Vol. 13, No. 9, 24 (Sept., 1960). The de-scription in this module is based mainly on Professor Weisskopfs

exposition of Terrells paper. Weisskopf shows the appearance of thecube along its entire trajectory and makes interesting comments.

Scientific American, Vol. 203, No. 1, 74 (1960).

Apparent Shape of Large Objects at Relativistic Speeds, Mary L.Boas, American Journal of Physics 29, 283 (1961). She considersobjects of finite shape and distance. Her abstract is reproduced bypermission in Appendix B.

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B. Journal Excerpts

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MISN-0-44 ME-1

MODEL EXAM

1. See Output Skills K1-K5 in this modulesID Sheet. The actual exammay contain any number of these skills.

Brief Answers:

1. See this modulestext.

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