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Elementary General Relativity v3.2--Macdonald.pdf |
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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...
i.e., if the times in the two directions are equal. A reformulation of the definition will make it more transparent. Suppose the pulse from Q to P is the reflection of the pulse from P to Q. Then tQ = tQ 15...
Thus, subtracting the equations Eq. (1.8) gives tQ − t P = t P − tQ , i.e., the clocks at P and Q are synchronized. 17...
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....
where κ is the Newtonian gravitational constant, M is the mass of the central body, and r is the distance to the center of the central body. Eq. (2.1) implies that the planets orbit the Sun in ellipses, in accord with Kepler’s findings. See Appendix 8. By taking the distance r to the Earth’s center to be sensibly constant near the Earth’s surface, we see that Eq. (2.1) is also in accord with Galileo’s findings: a is constant in time and is independent of the mass and composition of the falling ob ject. Newton’s theory has enjoyed enormous success. A spectacular example occurred in the nineteenth century. Observations of the position of the planet Uranus disagreed with the predictions of Newton’s theory of gravity, even after taking into account the gravitational effects of the other known planets. The discrepancy was about 4 arcminutes – 1/8th of the angular diameter of the 29...
As (y 1, y 2 ) varies, (x, y , z ) varies on the surface. Assign coordinates (y 1, y 2 ) to the point (x, y , z ) on the surface given by Eq. (2.3). For example, Fig. 2.6 shows spherical coordinates (y 1, y 2 ) = (φ, θ) on a sphere of radius R. The coordinates are assigned by the usual parameterization x = R sin φ cos θ, y = R sin φ sin θ, z = R cos φ. (2.4) 35...
The global coordinate postulate for a curved spacetime
The events of a curved spacetime can be labeled with coordinates (y 0 , y 1 , y 2 , y 3 ). In the next two sections we give the metric and geodesic postulates of general relativity. We first express the postulates in local inertial frames. This local form of the postulates gives them the same physical meaning as in special relativity. We then translate the postulates to global coordinates. This global form of the postulates is unintuitive and complicated but is necessary to carry out calculations in the theory. We can use arbitrary global coordinates in flat as well as curved spacetimes. We can then put the metric and geodesic postulates of special relativity in the same global form that we shall obtain for these postulates for curved spacetimes. We do not usually use arbitrary coordinates in flat spacetimes because inertial frames are so much easier to use. We do not have this luxury in curved spacetimes. It is remarkable that we shall be able to describe curved spacetimes intrinsical ly , i.e., without describing it as curved in a higher dimensional flat space. Gauss created the mathematics necessary to describe curved surfaces intrinsically in 1827. G. B. Riemann generalized Gauss’ mathematics to curved spaces of higher dimension in 1854. His work was extended by several mathematicians. Thus the mathematics necessary to describe curved spacetimes intrinsically was waiting for Einstein when he needed it....
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