Relativistic Effects Help (page 3)
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.”
You’ve heard expressions such as electron rest mass , which refers to the theoretical mass of an electron when it is not moving relative to an observer. If an electron is observed whizzing by at relativistic speed, it has a mass greater than its rest mass and thus will have momentum and kinetic energy greater than is implied by the formulas used in classical physics. This, unlike spatial distortion, is more than a mere “mind experiment.” There is little practical concern about spatial distortion in most situations, at least nowadays. (A thousand years from now, when we are roaming among the stars, we should expect that things will be different!) When electrons move at high enough speed, they attain properties of much more massive particles and acquire some of the properties of x-rays or gamma rays such as are emitted by radioactive substances. There is a name for high-speed electrons that act this way: beta particles .
Physicists take advantage of the relativistic effects on the masses of protons, helium nuclei, neutrons, and other subatomic particles. When these particles are subjected to powerful electrical and magnetic fields in a device called a particle accelerator , they get moving so fast that their mass increases because of relativistic effects. When the particles strike atoms of matter, the nuclei of those target atoms are fractured. It takes quite a wallop to break up the nucleus of an atom! When this happens, energy can be released in the form of infrared, visible light, ultraviolet, x-rays, and gamma rays, as well as a potpourri of exotic particles.
If astronauts ever travel long distances through space in ships moving at speeds near the speed of light, relativistic mass increase will be a dire concern. While the astronaut’s own bodies won’t seem unnaturally massive, and the objects inside the ship will appear to have normal mass too, the particles whizzing by outside will gain real mass. It is scary enough to think about what will happen when a 1-kg meteoroid strikes a space ship traveling at 99.9 percent of the speed of light. But that 1-kg stone will mass more than 22 kg when u = 0.999, that is, at 99.9 percent of the speed of light. As if this is not bad enough, every atomic nucleus outside the ship will strike the vessel’s shell at relativistic speed, producing deadly radiation.
Relativistic time dilation and mass increase have been measured under controlled conditions, and the results concur with Einstein’s formulas stated earlier. Thus these effects are more than mere tricks of the imagination.
To measure time dilation, a superaccurate atomic clock was placed on board an aircraft, and the aircraft was sent up in flight to cruise around for a while at several hundred kilometers per hour. Another atomic clock was kept at the place where the aircraft took off and landed. Although the aircraft’s speed was only a tiny fraction of the speed of light and the resulting time dilation was exceedingly small, the accumulated discrepancy was large enough to measure. When the aircraft arrived back at the terminal, the clocks, which had been synchronized (when placed right next to each other, of course!) before the trip began, were compared. The clock that had been on the aircraft was a little behind the clock that had been resting comfortably on Earth.
To measure mass increase, particle accelerators are used. It is possible to determine the mass of a moving particle based on its known rest mass and the kinetic energy it possesses as it moves. When the mathematics is done, Einstein’s formula is always shown to be correct.
Practice problems of this concept can be found at: Special and General Relativity Practice Problems
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