Contents25 Relativistic Stars and Black Holes 125.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.2 Schwarzschild’s Spacetime Geometry . . . . . . . . . . . . . . . . . . . . . . 225.3 Static Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.3.1 Birkhoff’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.3.2 Stellar Interior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.3.3 Local Energy and Momentum Conservation . . . . . . . . . . . . . . 1225.3.4 Einstein Field Equation . . . . . . . . . . . . . . . . . . . . . . . . . 1425.3.5 Stellar Mo dels and Their Properties . . . . . . . . . . . . . . . . . . . 1525.4 Gravitational Implosion of a Star to Form a Black Hole . . . . . . . . . . . . 2025.5 Spinning Black Holes: The Kerr Spacetime . . . . . . . . . . . . . . . . . . . 3125.5.1 The Kerr Metric for a Spinning Black Hole . . . . . . . . . . . . . . . 3125.5.2 Dragging of Inertial Frames . . ..................... 3225.5.3 The Light-Cone Structure, and the Horizon . . . . . . . . . . . . . . 3325.5.4 Evolution of Black Holes: Rotational Energy and Its Extraction . . . 3525.6 The Many-Fingered Nature of Time . . . . . . . . . . . . . . . . . . . . . . . 410Chapter 25Relativistic Stars and Black HolesVersion 1025.1.K.p df, 6 May 2009Please send comments, suggestions, and errata via email to [email protected] or on paper toKip Thorne, 350-17 Caltech, Pasadena CA 91125Box 25.1Reader’s Guide• This chapter relies significantly on– The special relativity portions of Chap. 1.– Chapter 22, on the transition from special relativity to general relativity.– Chapter 23, on the fundamental concepts of general relativity.• Portions of this chapter are a foundation for the applications of general relativitytheory to gravitational waves (Chap. 25) and to cosmology (Chap. 26).25.1 OverviewHaving 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. 25, and the large-scale structure and evolution ofthe universe in Chap. 26.While stars and black holes are the central thread of this chapter, we study them less fortheir o wn 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 r elativity, the geodesic motion of freely falling particles andphotons, local Lorentz frames and the tidal forces measured therein, proper reference frames,the E instein field equations, the local law of conservation of 4-momentum, and the asymptotic12structure of spacetime far from gravitating sources. Stars and blac k 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, eve nt 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.25.2 Schwarzschild’s Spacetime GeometryOn January 13, 1916, just seven weeks after formulating the final version of his field equation,G =8πT,AlbertEinsteinreadtoameetingofthePrussianAcademyofSciencesinBerlinaletterfromtheeminentGermanastrophysicistKarlSchwarzschild. Schwarzschild,asamember of the German army, had written from the World-War-One Russian front to tellEinstein of a mathematical discovery he hadmade: hehadfoundtheworld’sfirstexactsolution to the Einstein field equation.Written as a line element in a sp ecial coordinate system (coordinates named t, r, θ,φ)thatSchwarzschildinventedforthepurpose,Schwarzschild’ssolutiontakestheform(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. 22 and 23;see Ex. 25.1. The results are tabulated in Box 25.2. 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 Sc hwarzschild 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;andalsoawormholethatconnectstwodifferentuniverses 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 coefficients3will 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.Wecanregard{θ, φ} as a
View Full Document
Unlocking...