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Special Relativity Length, Momentum, Ami Energy Special Relativity Length, Momentum, ami Energy he speed of light is the speed limit for all matter. Suppose that two spaceships are T both traveling at nearly the speed of light and they are moving directly toward each other. The realms of space-time for each spaceship differ in such a way that the relative speed of approach Mass converts to energy and is still less than the speed of light! For example, if both spaceships vice versa. are traveling toward each other at 80% the speed of light with respect to Earth, an observer on each spaceship would measure the speed of approach of the other spaceship as 98% the speed of light. There are no circumstances where the relative speeds of any material objects surpass the speed of light. Why is the speed of light the universal speed limit? To under- stand this, we must know how motion through space affects the length, momentum, and energy of moving objects. 16.1 Length Contraction For moving objects, space as well as time undergoes changes. When viewed by an outside observer, moving objects appear to contract along the direction of motion. The amount of contraction is related to the amount of time dilation. For everyday speeds, the amount of contraction is much too small to be measured. For relativistic speeds, the contraction would be noticeable. A meter stick aboard a Figure 16.1 1 spaceship whizzing past you at 87% the speed of light, for example, A meter stick traveling at 87% would appear to you to be only 0.5 meter long. If it whizzed past at the speed of light relative to an 99.5% the speed of light, it would appear to you to be contracted to observer would be measured as one tenth its original length. The width of the stick, perpendicular to only half as long as normal. the direction of travel (Figure 16.1), doesn't change. As relative speed gets closer and closer to the speed of light, the measured lengths of objects contract closer and closer to zero. 232 Chapter 16 Special Relativity—Length, Momentum, and Energy Do people aboard the spaceship also see their meter sticks—and everything else in their environment—contracted? The answer is no. People in the spaceship see nothing at all unusual about the lengths of things in their own reference frame. If they did, it would violate the first postulate of relativity Recall that all the laws of physics are the same in all uniformly moving reference frames. Besides, there is no relative speed between the people on the spaceship and the things they observe in their own reference frame. However, there is a relative speed between themselves and our frame of reference, so they will see our meter sticks contracted and us as well. A rule of relativity is that changes due to alterations of space-time are always en in the frame of reference of the "other guy." Figure 16.2 A to the frame of reference of the meter stick, its length is 1 meter. Observers m this frame see our meter sticks contracted. The effects of relativity are always attributed to "the other guy." The contraction of speeding objects is the contraction of space tself. Space contracts in only one direction, the direction of motion. Lengths along the direction perpendicular to this motion are the same in the two frames of reference. So if an object is moving hori- zontally, no contraction takes place vertically (Figure 16.3). 0 is. 0.87c u-.0.995c =0.999c o=C (1) gure 16.3 A relative speed increases, contraction in the direction of motion increases. gths in the perpendicular direction do not change. 233 Relativistic length contraction is stated mathematically: L= Lol /1— (v2 I c2) In this equation, v is the speed of the object relative to the observer, „ LINK TO BIOLOGY c is the speed of light, L is the length of the moving object as mea- r sured by the observer, and Lo is the measured length of the object at rest.* Suppose that an object is at rest, so that v = 0. When 0 is substi- tuted for v in the equation, we find L= Lo, as we would expect. It was stated earlier that if an object were moving at 87% the speed of light, it would contract to half its length. When 0.87c is substituted for v in the equation, we find L = 0.5L0. Or when 0.995c is substituted for v, we find L= 0.1L0, as stated earlier. If the object could reach the speed c, its length would contract to zero. This is one of the reasons that the .c e st speed of light is the upper limit for the speed of any material object. ztor Eart,, s s . U rise eramlifeti .e secoridel-seemingly;too4',„: III Question .f.brietto.reach the groun A spacewoman travels by a spherical planet so fast that it appears below before decaying.;,, to her to be an ellipsoid (egg shaped). If she sees the short diameter But because niuons Move'. as half the long diameter, what is her speed relative to the planet? at nearly the speed of light, length contraction dramatically shortens their distance to Earth. You are hit by hundreds of muons every second! ?Awn impact, like that of 16.2 Momentum and Inertia all high-speed elementary in Relativity 1particles, causes biologi- cal mutations. So we see If we push an object that is free to move, it will accelerate. If we a link between the effects maintain a steady push, it will accelerate to higher and higher of relativity and the evo- speeds. If we push with a greater and greater force, we expect the lution of living creatures acceleration in turn to increase. It might seem that the speed should on Earth. increase without limit, but there is a speed limit in the universe— the speed of light. In fact, we cannot accelerate any material object enough to reach the speed of light, let alone surpass it. We can understand this from Newton's second law, which Newton originally expressed in terms of momentum: F= Amy/At (which reduces to the familiar F = ma, or a = Fl m). The momentum form, interestingly, remains valid in relativity theory. Recall from Chapter 7 that the change of momentum of an object is equal to the III Answer The spacewoman passes the spherical planet at 87% the speed of light This equation land those that follow) is simply stated as a "guide to thinking" about the ideas of special relativity. The equations are given here without any explanation as to how they are derived. 234 Chapter 16 Special Relativity—Length, Momentum, and Energy fl impulse applied to it. Apply more impulse and the object acquires more momentum. Double the impulse and the momentum doubles. Apply ten times as much impulse and the object gains ten times as ver, much momentum. Does this mean that momentum can increase without any limit, even though speed cannot? Yes, it does. We learned that momentum equals mass times velocity In equa- tion form, p = my (we use p for momentum). To Newton, infinite ti- momentum would mean infinite speed. Not so in relativity. Einstein was showed that a new definition of momentum is required. It is ght, I) in mu •o, peed it the where v is the speed of an object and cis the speed of light. Notice ect. that the square root in the denominator looks just like the one in the formula for time dilation in the previous chapter. It tells us that the relativistic momentum of an object of mass m and speed v is larger than mu by a factor of 1/V1— (v2/c2). At relativistic speed, momentum increases dramatically. As v approaches c, the denominator of the equation approaches zero. el This means that the momentum approaches infinity! An object pushed to the speed of light would have infinite momentum and would require an infinite impulse, which is clearly impossible. So nothing that has mass can be pushed to the speed of light, much less beyond it. Here is another reason that c is the speed limit in the universe. What if v is much less than c? Then the denominator of the equation is nearly equal to I and pis nearly equal to mu. Newton's definition of momentum is valid at low speed. We often say that a particle pushed close to the speed of light acts as if its mass were increasing, because its momentum—its "iner- tia in motion"—increases more than its speed increases. The quan- tity m in the equation above is called the rest mass of the object. It is the a true constant, a property of the object no matter what speed it has. iould Subatomic particles are routinely pushed to nearly the speed of light. The momenta of such particles may be thousands of times Oct more than the Newton expression mu predicts. One way to look at the momentum of a high-speed particle is in terms of the "stiffness" of its trajectory The more momentum it has, the harder it is to At deflect it—the "stiffer" is its trajectory. If it has a lot of momentum, it gum more greatly resists changing course. om to the ELECTROMAGNETS Figure 16.4 If the momentum of the elec- trons were equal to the Newtonian value my, the beam would follow the dashed line. But because the relativistic momentum, or inertia in motion, about is greater, the beam follows the iation as SCREEN "stiffer" trajectory shown by the ELECTRON BEAM) solid line.
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