Elementary общая Относительность v3.2 Macdonald, Elementary General Relativity v3.2--Macdonald.pdf:

Elementary общая Относительность v3.2 Macdonald

Elementary General Relativity v3.2--Macdonald.pdf

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Date Mar 20, 2005

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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...

Equations (1.4) and (1.5) correspond to z = 0. If, for example, z = 1 (∆so /∆se = 2), then the observer at S would see clocks at R, and all other physical processes at R, proceed at half the rate they do at S . If the two “pulses” of light in Eq. (1.6) are successive wavecrests of light emitted at frequency fe = (∆se )−1 and observed at frequency fo = (∆so )−1 , then Eq. (1.6) can be written z= fe − 1. fo (1.7)...

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....

The local planar frame postulate for a curved surface
A local planar frame can be constructed at any point P of a curved surface with any orientation. We shall see that local planar frames at P provide surface dwellers with an intuitive description of properties of a curved surface at P . However, in order to study the surface as a whole, they need global coordinates, deﬁned over the entire surface. There are, in general, no natural global coordinate systems to single out in a curved surface as planar frames are singled out in a ﬂat surface. Thus they attach global coordi1 2 Fig. 2.6: Spherical coordinates (φ, θ) nates (y , y ) in an arbitrary manner. The only restrictions are that diﬀerent points on a sphere. must have diﬀerent coordinates and nearby points must receive nearby coordinates. In general, the coordinates will have no geometric meaning; they merely serve to label the points of the surface. One common way for us (but not surface dwellers) to attach global coordinates to a curved surface is to parameterize it in three dimensional space: x = x(y 1, y 2 ), y = y (y 1, y 2 ), z = z (y 1, y 2 ) . (2.3)...

The geodesic postulate for a curved surface, global form
Parameterize a geodesic with areclength s . Then in every global coordinate system y i + Γi k y j y k = 0, ¨ j˙˙ i = 1, 2. (2.18) 41...

2.5 The Geodesic Postulate We now assume that the metric of a local inertial frame at E satisﬁes Eq. (2.19) for the same reasons as given above for local planar frames. Then Appendix 9 translates the local form of the geodesic postulate for curved spacetimes to the global form:...

2.6 The Field Equation We can roll a ﬂat piece of paper into a cylinder without distorting distances along curves in the paper and thus without changing K . Thus K = 0 for the cylinder. Viewed from the outside, the cylinder is curved, and so K = 0 seems “wrong”. However, viewed from within (and remember, we are describing curved surfaces and spacetimes without reference to a higher dimensional space), the rolling does not distort distances in the paper. Thus surface dwellers could not detect the curvature seen from the outside. Thus K “should” be zero for the cylinder. The formula expressing K in terms of distances was given by Gauss. If g12 = 0, then + g1 ∂ ∂ g1 1 1 . − −2 −1 i 2 2 ∂2 222 ∂2 g11 ∂1 g22 (2.23) K = − (g11 g22 ) 2 1 i ≡ ∂ /∂ y 11 Exercise 2.12. Show that Eq. (2.23) gives K = 1/R2 for a sphere of radius R. Use Eq. (2.11). Exercise 2.13. Generate a surface of revolution by rotating the parameterized curve y = f (u), z = h(u) about the z -axis. Let (r, θ, z ) be cylindrical coordinates and parameterize the surface with coordinates (u, θ). Use ds2 = dr2 + r2 dθ2 + dz 2 to show that the metric is f2 . +h2 0 0 f2 u 1 Exercise 2.14. If y = Re−u and z = R 0 (1 − e−2t ) 2 dt, then the surface of revolution in Exercise 2.13 is the pseudosphere of Fig. 2.11. Show that K = −1/R2 for the pseudosphere. Exercise 2.15. If y = 1 and z = u, then the surface of revolution in Exercise 2.13 is a cylinder. Show that K = 0 for a cylinder using Eq. (2.23). The local forms of our curved spacetime postulates show that in many respects a small region of a curved spacetime is like a small region of a ﬂat spacetime. We might suppose that all diﬀerences between the regions vanish as the regions become smaller. This is not so. To see this, refer to Fig. 2.7. Let ∆ r be the small distance between ob jects at the top and bottom of the cabin and let ∆ a be the small tidal acceleration between them. In a ﬂat spacetime ∆ a = 0 for inertial particles. In the cabin ∆ a = 0 , but ∆ a → 0 as ∆ r → 0 . However, ∆ a/∆ r → 0, as ∆ r → 0 : from / Eq. (2.1), da/dr = 2κM /r3 . In a ﬂat spacetime, da/dr = 0. Here is a diﬀerence between regions of the spacetime in the cabin and a ﬂat spacetime which does not vanish as the regions become smaller. The metric and geodesic postulates describe the behavior of clocks, light, and inertial particles in a curved (or ﬂat) spacetime. But to apply these postulates, we must know the metric of the spacetime. Our ﬁnal postulate for general relativity, the ﬁeld equation , determines the metric. Loosely speaking, the equation determines the “shape” of a spacetime, how it is “curved”. 45...

To specify the two sides of this equation, we need several deﬁnitions. Deﬁne the Ricci tensor Rj k = Γpk Γt p − Γpp Γt k + ∂k Γpp − ∂p Γpk . t j j j t j (2.25)...