24

THE

VISUAL

APPEARANCE

OF RAPIDLY MOFING OBJECTS

By V. F. Weisskopf

I WOULD like to draw the attention of physiciststo a recent paper by James Terrell1 in which hedoes away with an old prejudice held by practically

all of us. We all believed that, according to specialrelativity, an object in motion appears to be con-tracted in the direction of motion by a factor 11 —(v/c)-]1/-. A passenger in a fast space ship, lookingout of the window, so it seemed to us, would seespherical objects contracted to ellipsoids. This is defi-nitely not so according to Terrell's considerations,which for the special case of a sphere were also carriedout by R. Penrose.2 The reason is quite simple. Whenwe see or photograph an object, we record light quantaemitted by the object when they arrive simultaneouslyat the retina or at the photographic film. This impliesthat these light quanta have not been emitted simul-taneously by all points of the object. The points fur-ther away from the observer have emitted their partof the picture earlier than the closer points. Hence,if the object is in motion, the eye or the photographgets a "distorted" picture of the object, since the ob-ject has been at different locations when different partsof it have emitted the light seen in the picture.

In special relativity, this distortion has the remark-able effect of canceling the Lorentz contraction so thatobjects appear undistorted but only rotated. This isexactly true only for objects which subtend a smallsolid angle.

In order to understand the situation thoroughly letus consider the distortion of the picture we see of amoving object under nonrelativistic conditions, wherelight moves with light velocity c only in the stationaryframe of reference of the observer, and a moving ob-ject does not suffer a Lorentz contraction. In the frame

i j . Terrell, Phys. Rev. 116, 1041 (1959).2 R Penrose, Proc. Cambridge Phil. Soc. 55, 137 (1959); see also

H Salecker and E. Wigner, Phys. Rev. 109, 571 (1958).

of the object moving with the velocity v the lightvelocity would be c — v in the direction of motion andc + v in the opposite direction.

We first consider the case of a cube of dimension Imoving parallel to an edge and observed from a di-rection perpendicular to the motion. The observationis made at great distance in order to keep the sub-tended angle small (see Fig. 1). The square ABCDfacing the observer will be seen undistorted since allpoints have the same distance from the observer.The square ABEF facing in the opposite direction ofthe motion (the rear side in regard to the motion, notin regard to the observer's position) is invisible whenthe cube is not in motion. However when it moves itbecomes visible since the light from E and F is emit-ted l/c seconds earlier when the points E and F were(v/c)l further behind at E' and F'. Hence the faceABEF will be seen as a rectangle with a height / anda width (v/c)l. The picture of the cube, therefore, isa distorted one. In an undistorted picture of a rotated

E_'_ E

~ F

B

V

To Observer

Picture ( v - Vz c)

E B c

Non Relat

F A

E B CD

Relativistk

F A

Fig. 1. A cube moving with velocity vseen by an observer at an angle of 90°.

PHYSICS TODAY

v = C

M

Motion3'

25

.in

r

r

P cture

I 1,3

Relativistic

5 " \ ^

Non Relot./ . Observer

Fig. 2. A cube moving with light velocityviewed at an angle 0=180° —«. PointsI, II, III are light pulses coming from ob-ject points 1, 2, 3 and arriving simulta-neously at the observer. (Nonrelativistic.)

cube both faces should be foreshortened; if the faceABEF is shortened by the factor v/c, the other faceABCD should be foreshortened by (1 — fl2/c2)1/2,whereas here ABCD appears as a square. Hence thepicture of the cube appears dilated in the directionof motion. A similar consideration for a movingsphere shows that it would appear as an ellipse elon-gated in the direction of motion by a factor (1 +v-/c-y/2.

We get even more paradoxical results by consid-ering the picture of a moving cube in a nonrelativisticworld, seen not at 90° to the direction of motion butat 180 — a degrees where a is a small angle. Wenow look at the object to the left when it is comingtowards us from the left. We will assume now thatv/c — 1 in order to simplify our considerations. Whatis the picture then? Fig. 2 illustrates the situation.The edges AB, CD, EF are denoted by the numbers1, 2, 3. We assume that the edge 1 emits its lightquanta at the time t = 0. Where must the edges 2and 3 emit their light such that it travels in a com-mon front with the light from 1, in order to arrivesimultaneously at the observer? It is easily seen that2 must emit its light much earlier; in fact it musthappen when it is at the point marked V which isdetermined by the equality of the distances (2'2) and(2'M). The interval (2'2) is the distance which theedge 2 travels between the emission of light by 2and 1. The length (2'M) is the distance which thelight travels from 2' in order to be "in line" withthe light emitted by 1. Both light and edge travelwith the speed c. We can see that the distance (\M)is equal to (1 2) which is the size I of the cube. Thelight seen from edge 3 is emitted much later, when

Victor F. Weisskopf is a member of the Laboratory for Nuclear Sci-ence and professor of physics at the Massachusetts Institute of Tech-nology, Cambridge, Mass.

the edge is at 3'. The point 3' is determined by theequality of the distances (3 3') and (1AO. A simplecalculation shows that (3'N) = / sin a (1 — cos a) \

What then is the picture we see of the cube? It isindicated in the figure by the points I, II, III whichrepresent the positions of the light quanta comingfrom the object and form the picture. We will see astrongly deformed cube with the edge 1 in the middle,the edge 2 on the left of 1 as if we were looking frombehind (from the left to the right) and the edge 3quite far to the right of 1. Again we see a pictureelongated in the direction of flight. The face betweenthe edges 1 and 2 appears as a true square.

We now will show that relativity theory simplifies thesituation. It removes the distortion of the picture andwhat remains is an undistorted but rotated aspect ofthe object. We can see this directly with the examplesquoted. Consider the cube when looked at perpendicu-lar to its motion; the Lorentz contraction reduces thedistance between the edges AB and CD by the factor(I — v2/c2)1/2 and leaves the distance between ABand EF unchanged. Therefore the picture of the faceABCD is foreshortened precisely by the amount nec-essary to represent an undistorted view of a cubeturned by an angle whose sine is v/c. In the case ofthe cube moving with light velocity towards us theLorentz contraction reduces the distance between theedges 1 and 3 to zero. The picture one sees then is aregular square corresponding to the rear face andnothing else, since edge 3 coincides with edge 1. Hencewe see an undistorted picture directly from behind.The object is undistorted but turned by an angle of(180 — a) degrees.

We can show by means of the following considera-tion that this result is quite generally true for anyobject. Let us consider an assembly of light pulsesoriginating from N points of the object, traveling all

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26

Fig. 3. A "picture". A, B, C,D are points of the object.The four crosses are the lightpulses making up the "pic-ture".

in the same direction given by a vector k, and suchthat the light pulses are all in one plane perpendicu-lar to k (see Fig. 3). Then they will arrive simul-taneously at the eye of the observer and produce theshape which is seen. We will call such an assembly oflight pulses a "picture" of the object. Under nonrela-tivistic conditions a "picture" does not remain a pic-ture when seen from a moving frame of reference. Thereason is that, in the moving frame, the plane of thelight pulses is no longer perpendicular to the directionof the propagation. In a relativistic world a "picture"remains a "picture" in any frame of reference. Thelight pulses would arrive simultaneously at a camerain every system of reference.

This fact can be proven immediately in the fol-lowing way: The light pulses form a wave front orcan be imagined as moving embedded in an electro-magnetic wave exactly where this wave has a crest. Itis known that electromagnetic waves are transverse inall frames of reference. That means that a wave frontor the plane of the wrave crest is perpendicular to thedirection of propagation in any system. We can alsoshow that the distance between the light pulses is aninvariant magnitude. Here we only need to introducea coordinate system where the x-axis is parallel to thepropagation. Then for two light pulses of the picturethe invariant (x1 — %„)'- + (y1 — y.,)2 + (zx — z2)2 —c2(t1 — t2)

2 is equal to the square of the distance dbetween the two pulses, since d2 = (yx — y2)

2 + (zx -22)2 and xx — x.-. when tx =tn. The latter relationexpresses the fact that the pulses are in a plane per-pendicular to the propagation.

The only thing that is not invariant is the directionof propagation, the vector k. The transformation ofthis direction is given by the well-known aberrationformula. A light beam whose direction includes theangle 0 with the x-axis is seen including an angle ffwith the x-axis in a system moving with the velocity valong the x-axis: *

L — ?'2/c2)2 sin d

1 + (v/c) cos 6 '

We can conclude the following result from the invari-ance of the "picture": The picture seen from a movingobject observed at the angle 0 is the same as one wouldsee in the system where the object is at rest, but ob-

* The angles refer to the direction in which the light beam is seen;that means a direction opposite to the motion of the light pulses.

served at the angle (/. Hence we see an undistortedpicture of a moving object, but a picture in which theobject is seemingly rotated by the angle ff — 0. Aspherical object still appears as a sphere.

This must not by any means be interpreted as in-dicating that there is no Lorentz contraction. Of course,there is Lorentz contraction, but it just compensatesfor the elongation of the picture caused by the finitepropagation of light.

It is instructive to plot the angle tf as a function0. Fig. 4 shows this relation for v = 0, for a small

Fig. 4. The angle of observation 9' of a light beamrelative to the direction of v, seen by an observerin the moving system, versus the same angle 0 asseen in the rest system. Curve 1 is for v = 0, curve 2is for v — c/2, curve 3 is for v = c.

value of v/c and also for the case v/c « 1. We seethat the apparent rotation is always negative, whichmeans that the object is turned such that it revealsmore of its trailing side to the observer. In the ex-treme case of v « c, ff is extremely small for allvalues of Q except when 180 — 6* is of the order[1 - ( D / C ) 2 ] 1 / 2 . Since 0 goes from 180° to 0° whenan object moves by, we find for the case v a c thatwe see the front side of the object only at the verybeginning; it turns around facing its trailing side atus quite early when we still see it coming at us andremains doing so until it leaves us and naturally isseen from behind. This paradoxical situation is per-haps not so surprising when one is reminded of thefact that the aberration angle is almost 180° whenv « c. Hence the light which we see coming fromthe object when it is moving towards us, has leftthe object backwards when observed from the ob-ject itself.

The situation becomes clearer when we look closerat the distribution of the emitted light as seen from

PHYSICS TODAY

11

the observer. Let us assume that the moving objectemits radiation which is isotropic in its own rest sys-tem, i.e., its intensity is independent of the emissionangle ff. This radiation does not at all appear iso-tropic in the nonmoving system; there it seems con-centrated in the forward direction. If v « c, most ofthe light appears to be emitted such that it includes avery small angle 9 with v. This is a well-known ef-fect which causes an isotropic emission to look as ifalmost all radiation is emitted in the form of a fo-cused headlight beam. One example of this effect isthe radiation of electrons running along a circle witha velocity near c as one finds it in synchrotrons. Inthis case the radiation in the rest system is not com-pletely isotropic; it is essentially a dipole radiation.Still it appears as an emission sharply peaked in thedirection of flight.

The apparent angular distribution 1(6) of the radi-ated intensity in the system at rest is connected withthe angular distribution 1^(9') in the system of themoving object by an expression which is related tothe aberration formula:

1(6) _ sin 6'dd' _

h{B')

K(6) =

s in

[1 + (v/c) cos 8J

where 9 is the angle of observation, which means thatthe forward direction is near 9 — T:. K(9) is plotted inFig. 5 as a function of 9 and we see that the width ofthe "headlight" beam is of the order [1 — (v/c)2]1/2.

The factor K(9) also determines the Doppler shiftof the light. If the emitted light has the frequency m0

in the system moving with the object, the observerat rest sees a frequency o> = -K1/2w0. The frequencyis increased or decreased by the square root of thefactor by which the intensity is enhanced or reduced.

We note that, for 0 — 90° there is a reduction offrequency due to the relativistic Doppler shift.

We now describe what is seen when an object ismoving by with a velocity near that of light. First,when the angle of vision is still near 180°, we see thefront face of the object, strongly Doppler-shifted tovery high frequencies and with high intensity. Weare looking into the "headlight" beam of the radia-tion. When the angle of vision becomes of the order7T — [1 — (v/c)-]1/2 the color shifts towards lower fre-quencies, the intensity drops, and the object seems toturn. When 9 « 7 r - 2 1 / 2 [ l - (i>A)2]1/4, still an an-gle close to ISO0, the intensity becomes very low, weare out of the "headlight" beam, the color is now ofmuch lower frequencies than it would be in the sys-tem moving with the object; the object has turned allaround and we are looking at its trailing face. Thefront is invisible because the beams emitted forwardin the moving system are concentrated into the smallangle of the "headlight". The picture seen at anglessmaller than TT — 21/2[1 — (vA) 2 ] 1 / 4 remains essen-tially the same until the object disappears. It is thepicture expected when the object is receding. How-ever it appears already when the object is movingtoward us.

We would like to emphasize that all these consid-erations are exact only for objects which subtend avery small solid angle. Only then the picture consistsessentially of parallel light pulses. If the angle sub-tended is finite we must expect different rotation an-gles for different parts of the picture and this wouldlead to some distortions. It has been shown by Pen-rose,2 however, that the picture of a sphere retains acircular circumference even for large angles of vision.

It is most remarkable that these simple and impor-tant facts of the relativistic appearance of objectshave not been noticed for 55 years until J. Terrelldiscovered and fully recognized them in his recentpublication.

"7c = 0.9

Fig. 5. The ratio K of the emit-ted intensity per solid anglemeasured in the observer's sys-tem to the intensity per solidangle measured in the systemmoving with the object, as func-tion of the angle 6 of observa-tion. K also determines the Dop-pler shift. The observed frequencyoi is related to the emitted fre-quency by w = K 1^0.

SEPTEMBER 1960