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Spatial Distortion Help

By — McGraw-Hill Professional
Updated on Sep 12, 2011

Introduction

Relativistic speeds—that is, speeds high enough to cause significant time dilation—cause objects to appear foreshortened in the direction of their motion. As with time dilation, relativistic spatial distortion occurs only from the point of view of an observer watching an object speed by at a sizable fraction of the speed of light.

Point Of View: Length

If we travel inside a space ship, regardless of its speed, everything appears normal as long as our ship is not accelerating. We can cruise along at 99.9 percent of the speed of light relative to the Earth, but if we are inside a space ship, the ship is always stationary relative to us. Time, space, and mass appear normal from the point of view of passengers on a relativistic space journey. However, as we watch the space ship sail by from the vantage point of the Earth, its length decreases as its speed increases. Its diameter is not affected. The extent to which this happens is the same as the extent to which time slows down.

Let L be the apparent length of the moving ship as a fraction of its length when it is standing still relative to an observer. Let u be the speed of the ship as a fraction of the speed of light. Then

L = (1 − u 2 ) 1/2

This effect is shown in Fig. 20-4 for various relative forward speeds. The foreshortening takes place entirely in the direction of motion. This produces apparent physical distortion of the ship and everything inside, including the passengers. It’s sort of like those mirrors in fun houses that are concave in only one dimension and reflect your image “all scrunched up.” As the speed of the ship approaches the speed of light, its observed length approaches zero.

Relativity Theory Spatial Distortion Suppositions And Cautions

Fig. 20-4 . As an object moves faster and faster, it grows shorter and shorter along the axis of its motion.

Suppositions And Cautions

This is a curious phenomenon. You might wonder, based on this result, about the shapes of photons, the particles of which visible light and all other EM radiation are comprised. Photons travel at the speed of light. Does this mean that they are infinitely thin, flat disks or squares or triangles hurtling sidelong through space? No one has ever seen a photon, so no one knows how they are shaped. It is interesting to suppose that they are two-dimensional things and as such have zero volume. However, if they have zero volume, how can we say they exist?

Scientists know a lot about what happens to objects as they approach the speed of light, but we must not extrapolate and try to say what would happen if the speed of light could be attained by a material thing. We will see shortly that no physical object (such as a space ship) can reach the speed of light, so the notion of a real object being squeezed down to zero thickness is nothing more than an academic fantasy. As for photons, comparing them with material particles such as bullets or baseballs is an unjustified intuitive leap. We cannot bring a photon to rest, nor can we shoot a bullet or throw a baseball at the speed of light. As they might say in certain parts of the country, “Baseballs and photons ain’t the same animals.”

Spatial Distortion Practice Problem

Problem 

Suppose that a space ship measures 19.5 m long at rest. How long will it look if it zooms by at a speed of 2.40 × 10 8 m/s?

Solution 

First, convert the speed to a fraction of the speed of light and call this fraction u:

u = (2.40 × 10 8 )/(3.00 × 10 8 )

= 0.800

Then use the formula for spatial distortion to find L , the fraction of its at-rest length:

L = (1 − u 2 ) 1/2

= (1 − 0.800 2 ) 1/2

= (1 − 0.640) 1/2

= 0.360 1/2

= 0.600

Finally, multiply the at-rest length of the vessel, 19.5 m, by 0.600 to get 11.7 m. This is how long the vessel will appear to be as it whizzes by at 2.40 × 10 8 m/s.

Practice problems of these concepts can be found at: Relativity Theory Practice Test

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