Relativistic Effects Help (page 2)
Point Of View: Length
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.
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, it 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 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. 16-4 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.
Suppositions And Cautions
Spatial distortion is a curious phenomenon. You might wonder, based on this result, about the shapes of photons. They are the particles of which visible light and all other EM radiation are comprised. Photons travel at the speed of light. Does that mean 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 that they exist?
Scientists know a lot about what happens to objects as they approach the speed of light, but it’s intellectually dangerous to extrapolate and claim to know 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 places, “Baseballs and photons ain’t the same animals.”
Another interesting effect of relativistic speeds is an increase in the masses of objects as they move faster and faster. This increase occurs to the same extent as the decrease in length and the slowing down of time.
Point Of View: Mass
If we travel inside a space ship, regardless of its speed, the masses of all the objects in the ship with us appear normal as long as our ship is not accelerating. However, from the vantage point of Earth, the mass of the ship and the masses of all the atoms inside it increase as its speed increases.
Let m be the mass of the moving ship as a multiple of its mass when it is stationary relative to an observer. Let u be the speed of the ship as a fraction of the speed of light. Then
m = 1/(1 – u 2)1/2 = (1 – u 2)– 1/2
This is the same as the factor k that we defined a little while ago. It is always greater than or equal to 1.
Look again at Fig. 16-4. As the space ship moves faster, it scrunches up. Imagine now that it also becomes more massive. The combination of smaller size and greater mass produces a dramatic increase in density at relativistic speeds.
Suppose that the rest mass (the mass when stationary) of our ship is 10 metric tons. When it speeds by at half the speed of light, its mass increases to a little more than 11 metric tons. At 80 percent of the speed of light, its mass is roughly 17 metric tons. At 95 percent of the speed of light, the ship’s mass is about 32 metric tons. At 99.9 percent of the speed of light, the ship’s mass is more than 220 metric tons. And so it can go indefinitely. As the speed of the ship approaches the speed of light, its mass grows larger and larger without limit.
Speed Is Self-limiting
It’s tempting to suppose that the mass of an object, if it could be accelerated all the way up to the speed of light, would become infinite. After all, as u approaches 1 (or 100 percent), the value of m in the preceding formula increases without limit. However, it’s one thing to talk about what happens as a measured phenomenon or property approaches some limit; it is another thing entirely to talk about what happens when that limit is actually reached, assuming that it can be reached.
No one has ever seen a photon at rest. No one has ever seen a space ship moving at the speed of light, nor will they ever. No finite amount of energy can accelerate any real object to the speed of light. This is so because of the way in which the mass increases as the speed of an object approaches the speed of light. Even if it were possible to move a real object at the speed of light relative to some point of observation, the mass-increase factor, as determined by the preceding formula, would be meaningless. To calculate it, we would have to divide by zero, and division by zero is not defined. (If you tell a theoretical mathematician that “one over zero equals infinity,” you will get, at the very least, a raised eyebrow.)
The more massive a speeding space ship becomes, the more powerful is the rocket thrust necessary to get it moving faster. As a space ship approaches the speed of light, its mass becomes arbitrarily great. This makes it harder and harder to give it any more speed. Using a mathematical technique called integral calculus , astronomers and physicists have proven that no finite amount of energy can propel a space ship to the speed of light. The mass increases too fast. The function “blows up.”
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