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CALTECH PH 136A - Relativistic Stars and Black Holes

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Chapter 25Relativistic Stars and Black HolesVersion 0425.1.K.pdf, 11 May 2005.Please send comments, suggestions, and errata via email to [email protected] or on paperto Kip Thorne, 130-33 Caltech, Pasadena CA 9112525.1 IntroductionHaving sketched the fundamentals of Einstein’s theory of gravity, general relativity, we shallnow illustrate his theory by means of several concrete applications: stars and black holes inthis chapter, gravitational waves in Chap. 26, and the large-scale structure and evolution ofthe universe in Chap. 27.While stars and black holes are the central thread of this chapter, we study them less fortheir own intrinsic interest than for their roles as vehicles by which to understand generalrelativity: Using them we shall elucidate a number of issues that we have already met:the physical and geometric interpretations of spacetime metrics and of coordinate systems,the Newtonian limit of general relativity, the geodesic motion of freely falling particles andphotons, local Lorentz frames and the tidal forces measured therein, proper reference frames,the Einstein field equations, the local law of conservation of 4-momentum, and the asymptoticstructure of spacetime far from gravitating sources. Stars and black holes will also serveto introduce several new physical phenomena that did not show up in our study of thefoundations of general relativity: the gravitational redshift, the “many-fingered” nature oftime, event horizons, and spacetime singularities.We begin this chapter, in Sec. 25.2, by studying the geometry of the curved spacetimeoutside any static star, as predicted by the Einstein field equation. In Sec. 25.3 we studygeneral relativity’s description of the interiors of static stars. In Sec. 25.4 we turn attention tothe spherically symmetric gravitational implosion by which a nonrotating star is transformedinto a black hole, and to the “Schwarzschild” spacetime geometry outside and inside theresulting static, spherical hole. In Sec. 25.5 we study the “Kerr” spacetime geometry of aspinnning black hole. Finally, in Sec. 25.6 we elucidate the nature of “time” in the curvedspacetimes of general relativity.1225.2 Schwarzschild’s Spacetime GeometryOn January 13, 1916, just seven weeks after formulating the final version of his field equation,G = 8πT, Albert Einstein read to a meeting of the Prussian Academy of Sciences in Berlina letter from the eminent German astrophysicist Karl Schwarzschild. Schwarzschild, as amember of the German army, had written from the World-War-One Russian front to tellEinstein of a mathematical discovery he had made: he had found the world’s first exactsolution to the Einstein field equation.Written as a line element in a special coordinate system (coordinates named t, r, θ,φ) that Schwarzschild invented for the purpose, Schwarzschild’s solution takes the form(Schwarzschild 1916a)ds2= −(1 − 2M/r)dt2+dr2(1 − 2M/r)+ r2(dθ2+ sin2θdφ2) , (25.1)where M is a constant of integration. The connection coefficients, Riemann tensor, and Ricciand Einstein tensors for this metric can be computed by the methods of Chaps. 23 and 24;see Ex. 25.1. The results are tabulated in Box 25.1. The key bottom line is that the Einsteintensor vanishes. Therefore, the Schwarzschild metric (25.1) is a solution of the Einstein fieldequations with vanishing stress-energy tensor.Many readers know already the lore of this subject: The Schwarzschild spacetime isreputed to represent the vacuum exterior of a nonrotating, spherical star; and also theexterior of a spherical star as it implodes to form a black hole; and also the exterior andinterior of a nonrotating, spherical black hole; and also a wormhole that connects two differentuniverses or two widely separated regions of our own universe.How does one discover these physical interpretations of the Schwarzschild metric (25.1)?The tools for discovering them—and, more generally, the tools for interpreting physicallyany spacetime metric that one encounters—are a central concern of this chapter.When presented with a line element such as (25.1), one of the first questions one istempted to ask is “What is the nature of the coordinate system?” Since the metric coefficientswill be different in some other coordinate system, surely one must know something aboutthe coordinates in order to interpret the line element.Remarkably, one need not go to the inventor of the coordinates to find out their nature.Instead one can turn to the line element itself: the line element (or metric coefficients) containfull information not only about the details of the spacetime geometry, but also about thenature of the coordinates. The line element (25.1) is a good example:Look first at the 2-dimensional surfaces in spacetime that have constant values of t andr. We can regard {θ, φ} as a coordinate system on each such 2-surface; and the spacetimeline element (25.1) tells us that the geometry of the 2-surface is given in terms of thosecoordinates by(2)ds2= r2(dθ2+ sin2θdφ2) (25.2)(where the prefix(2)refers to the dimensionality of the surface). This is the line element(metric) of an ordinary, everyday 2-dimensional sphere expressed in standard spherical polarcoordinates. Thus, we have learned that the Schwarzschild spacetime is spherically symmet-ric, and moreover that θ and φ are standard spherical polar coordinates. Here is an example3of extracting from a metric information about both the coordinate-independent spacetimegeometry and the coordinate system being used.Note, further, from Eq. (25.2) that the circumferences and surface areas of the spheres(t, r) = const in Schwarzschild spacetime are given bycircumference = 2πr , Area = 4πr2. (25.3)This tells us one aspect of the geometric interpretation of the r coordinate: r is a radial coor-dinate in the sense that the circumferences and surface areas of the spheres in Schwarzschildspacetime are expressed in terms of r in the standard manner (25.3). We must not go further,however, and assert that r is radius in the sense of being the distance from the center of oneof the spheres to its surface. The center, and the line from center to surface, do not lie on thesphere itself and they thus are not described by the spherical line element (25.2). Moreover,since we know that spacetime is curved, we have no right to expect that the distance fromthe center of a sphere to its surface will be given by distance = circumference/2π = r as inflat


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CALTECH PH 136A - Relativistic Stars and Black Holes

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