Chapter 2: Q3CQ (page 61)
Question: The Lorentz transformation equations have x and t and x' and t'. Why no v and v' ?
Answer
The event does not occur at a particular velocity: therefore, Lorentz transformation doesn’t have v and v' .
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A projectile is a distance r from the center of a heavenly body and is heading directly away. Classically, if the sum of its kinetic and potential energies is positive, it will escape the gravitational pull of the body, but if negative, it cannot escape. Now imagine that the projectile is a pulse of light energy E. Since light has no internal energy ,E is also the kinetic energy of the light pulse. Suppose that the gravitational potential energy of the light pulse is given by Newton’s classical formula U=-(GMm/r), where M is the mass of the heavenly body and m is an “effective mass” of the light pulse. Assume that this effective mass is given by .
Show that the critical radius for which light could not escape the gravitational pull of a heavenly body is within a factor of 2 of the Schwarzschild radius given in the chapter. (This kind of “semiclassical” approach to general relativity is sometimes useful but always vague. To be reliable, predictions must be based from beginning to end on the logical, but unfortunately complex, fundamental equations of general relativity.)
Question: You are gliding over Earth's surface at a high speed, carrying your high-precision clock. At points and on the ground are similar clocks, synchronized in the ground frame of reference. As you pass overclock. it and your clock both read . (a) According to you, do clocksand advance slower or faster than yours? (b) When you pass overclock , does it read the same time. an earlier time, or later time than yours? (Make sure your answer agrees with what ground observers should sec.) (c) Reconcile any seeming contradictions between your answers to parts (a) and (b).
Question: Two objects isolated from the rest of the universe collide and stick together. Does the system’s final kinetic energy depend on the frame of reference in which it is viewed? Does the system’s change in kinetic energy depend on the frame in which it is viewed? Explain your answer.
In Example 2.5, we noted that Anna could go wherever she wished in as little time as desired by going fast enough to length-contract the distance to an arbitrarily small value. This overlooks a physiological limitation. Accelerations greater than about are fatal, and there are serious concerns about the effects of prolonged accelerations greater than Here we see how far a person could go under a constant acceleration of 1g, producing a comfortable artificial gravity.
(a) Though traveller Anna accelerates, Bob, being on near-inertial Earth, is a reliable observer and will see less time go by on Anna's clock than on his own Thus, , where u is Anna's instantaneous speed relative to Bob. Using the result of Exercise with replacing F/m, substitute for then integrate to show that
(b) How much time goes by for observers on Earth as they “see” Anna age 20 years?
(c) Using the result of Exercise 119, show that when Anna has aged a time t’, she is a distance from Earth (according to Earth observers) of
(d) If Anna accelerates away from Earth while aging 20 years and then slows to a stop while aging another 20. How far away from Earth will she end up and how much time will have passed on Earth?
(a) Determine the Lorentz transformation matrix giving position and time in framefromthose in framein the classical limitlocalid="1657533931071" . (b) Show that it yields equations (2-1).
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