Elementary General Relativity v3.2--Macdonald.pdf: eKnigu

Elementary General Relativity v3.2--Macdonald.pdf

Size 2.6Mb
Date Mar 20, 2005

to obtain the Schwartzschild metric. The geodesic equations are then solved and applied to the classical solar system tests of general relativity. There is a discussion of the Kerr metric, including gravitomagnetism and its observation by the LAGEOS satellites. The chapter closes with short sections on the binary pulsar and black holes. In this chapter, as elsewhere, I have tried to provide the cleanest possible calculations. Chapter 4 applies general relativity to cosmology. We obtain the RobertsonWalker metric in an elementary manner without using the ﬁeld equation. We then solve the ﬁeld equation with a nonzero cosmological constant for a ﬂat Robertson-Walker spacetime. WMAP data allows us to determine all unknown parameters in the solution, giving the new “standard model” of the universe with dark matter and dark energy. There have been many spectacular astronomical discoveries and observations since 1960 which are relevant to general relativity. We describe them at appropriate places in the book. Some 50 exercises are scattered throughout. They often serve as examples of concepts introduced in the text. If they are not done, they should be read. Some tedious (but always straightforward) calculations have been omitted. They are best carried out with a computer algebra system. Some material has been placed in 14 appendices to keep the main line of development visible. The appendices occasionally require more background of the reader than the text, but they may be omitted without loss of continuity. Appendix 1 gives the values of various physical constants. Appendix 2 contains several approximation formulas used in the text....

The inertial frame postulate for a ﬂat spacetime
An inertial frame can be constructed with any event E as origin, with any orientation, and with any inertial ob ject at E at rest in it. 14...

where ∆so = tS − tS is the time between the observation of the pulses at S and ∆se = tR − tR is the time between the emission of the pulses at R. (We use ∆s rather than ∆t to conform to notation used later in more general situations.) If a clock at R emits pulses of light at regular intervals to S , then Eq. (1.5) states that an observer at S sees (actually sees ) the clock at R going at the same rate as his clock. Of course, the observer at S will see all physical processes at R proceed at the same rate they do at S . We will encounter situations in which ∆so = ∆se . Deﬁne the redshift z= ∆so − 1. ∆se (1.6)...

1.3 The Metric Postulate The spacetime interval ∆s is deﬁned physical ly, independently of any inertial frame. But there is a simple formula for ∆s in terms of the coordinate diﬀerences of two events in an inertial frame:...

Lightlike separated events. In this case E and F can be on the worldline of a pulse of light. With c = 1, a universal light speed means c = |∆x/∆t| = 1 , i.e., |∆x| = |∆t|. Since by deﬁnition ∆s = 0 for lightlike separated events, Eq. (1.11) is satisﬁed....

Fig. 1.10: ∆s2 = ∆t2 − ∆x2 for spacelike separated events E and F . See the text....

Suppose that ∆t = 40 hours and the speed of the airplane with respect to the ground is 1000 km/hr. Substitute values to obtain ∆sa − ∆sg = 1.4 × 10−7 s....

1.3 The Metric Postulate In all of the above experiments, the source of the light is at rest in the inertial frame in which the light speed is measured. If light were like baseballs, then the speed of a moving source would be imparted to the speed of light it emits. Strong evidence that this is not so comes from observations of certain neutron stars which are members of a binary star system and which emit X-ray pulses at regular intervals. These systems are described in Sec. 3.1. If the speed of the star were imparted to the speed of the X-rays, then various strange eﬀects would be observed. For example, X-rays emitted when the neutron star is moving toward the Earth could catch up with those emitted earlier when it was moving away from the Earth, and the neutron star could be seen coming and going at the same time! See Fig. 1.11. This does not happen; an analysis of the arrival times of the pulses at Earth made in 1977 by K. Brecher shows that no more than two parts in 109 of the speed of the source is added to the speed of the X-rays. (It is not possible to “see” the neutron star in orbit around its com- Fig. 1.11: The speed of light is independent of the panion directly. The speed speed of its source. of the neutron star toward or away from the Earth can be determined from the Doppler redshift of the time between pulses. See Exercise 1.6.) Finally, recall from above that the universal light speed statement of the metric postulate implies the statements about timelike and spacelike separated events. Thus the evidence for a universal light speed is also evidence for the other two statements. The universal nature of the speed of light makes possible the modern deﬁnition of the unit of length: “The meter is the length of the path traveled by light during the time interval of 1/299,792,458 of a second.” Thus, by deﬁnition, the speed of light is 299,792,458 m/sec....

in an inertial frame. Eq. (1.17), unlike Eq. (1.16), is symmetric in all four coordinates of the inertial frame. Also, Eq. (1.17) shows that the worldline is a straight line in the spacetime. Thus “straight in spacetime” includes both “straight in space” and “straight in time” (constant speed). See Exercise 1.1. The worldlines are called geodesics . Exercise 1.10. Eq. (1.17) parameterizes the worldline of an inertial particle with x0 . Show the worldline can also parameterized with s, the proper time along the worldline....

2.1 History of Theories of Gravity moon. In 1846, U. LeVerrier, a French mathematician, calculated that a new planet, beyond Uranus, could account for the discrepancy. He wrote J. Galle, an astronomer at the Berlin observatory, telling him where the new planet should be – and Neptune was discovered! It was within 1 arcdegree of LeVerrier’s prediction. Even today, calculations of spacecraft tra jectories are made using Newton’s theory. The incredible accuracy of his theory will be examined further in Sec. 3.3. Nevertheless, Einstein rejected Newton’s theory because it is based on prerelativity ideas about time and space which, as we have seen, are not correct. For example, the acceleration in Eq. (2.1) is instantaneous with respect to a universal time....

2.2 The Key to General Relativity and Moon causes a diﬀerence in their acceleration toward the Sun. The lunar laser experiment shows that this does not happen. This is something that the Dicke and Braginsky experiments cannot test. The last experiment we shall consider as evidence for the three postulates is the terrestrial redshift experiment . It was ﬁrst performed by R. V. Pound and G. A. Rebka in 1960 and then more accurately by Pound and J. L. Snider in 1964. The experimenters put a source of gamma radiation at the bottom of a tower. Radiation received at the top of the tower was redshifted: z = 2.5 × 10−15 , within an experimental error of about 1%. This is a gravitational redshift . According to the discussion following Eq. (1.6), an observer at the top of the tower would see a clock at the bottom run slowly. Clocks at rest at diﬀerent heights in the Earth’s gravity run at diﬀerent rates! Part of the result of the Hafele-Keating experiment is due to this. See Exercise 2.1. We showed in Sec. 1.3 that the assumption Eq. (1.4), necessary for synchronizing clocks at rest in the coordinate lattice of an inertial frame, is equivalent to a zero redshift between the clocks. This assumption fails for clocks at the top and bottom of the tower. Thus clocks at rest in a small coordinate lattice on the ground cannot be (exactly) synchronized. We now show that the experiment provides evidence that clocks at rest in a small inertial lattice can be synchronized. In the experiment, the tower has (upward) acceleration g , the acceleration of Earth’s gravity, in a small inertial lattice falling radially toward Earth. We will show shortly that the same redshift would be observed with a tower having acceleration g in an inertial frame in a ﬂat spacetime. This is another example of small regions of ﬂat and curved spacetimes being alike. Thus it is reasonable to assume that there would be no redshift with a tower at rest in a small inertial lattice in gravity, just as with a tower at rest in an inertial frame. (It is desirable to test this directly by performing the experiment in orbit.) In this way, the experiment provides evidence that the condition Eq. (1.4), necessary for clock synchronization, is valid for clocks at rest in a small inertial lattice. Loosely speaking, we may say that since light behaves “properly” in a small inertial lattice, light accelerates the same as matter in gravity. We now calculate the Doppler redshift for a tower with acceleration g in an inertial frame. Suppose the tower is momentarily at rest when gamma radiation is emitted. The radiation travels a distance h, the height of the tower, in the inertial frame. (We ignore the small distance the tower moves during the ﬂight of the radiation. We shall also ignore the time dilation of clocks in the moving tower and the length contraction – see Appendix 4 – of the tower. These eﬀects are far too small to be detected by the experiment.) Thus the radiation takes time t = h/c to reach the top of the tower. (For clarity we do not take c = 1.) In this time the tower acquires a speed v = g t = g h/c in the inertial frame. From Exercise 1.6, this speed causes a Doppler redshift z= gh v = 2. c c (2.2)...