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CALTECH PH 236A - General Relativity

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Ph 236a: General Relativity 31 October 2007WEEK 6: Schwarzschild Black Holes, and WormholesRecommended Reading:1. Blandford and Thorne, Secs. 25.2, 25.3.1, 25.4, 25.6 of version 0625.1.p df, availableon the website http://www.pma.caltech.edu/Courses/ph136/yr2006/text.html . Thisdiscusses Schwarzschild black holes a nd gravitational implosion to form a black hole,but not wormholes.2. MTW Chapter 31 ( Schwarzschild Geometry), Chapter 32 (Gravitational Collapse),and Section 34.2 (“Infinity in Asymptotically Flat Spacetime”).Note: Section 32.4 provides an exact solution for the interior of an imploding star. Itrelies on something we have not yet studied: a Friedmann cosmological model withhyperspherical geometry for its homogeneous space slices. I do not expect you tocomprehend fully the details of this section, though if you choose to study Boxes 27.1and 27.2 of MTW you can then understand fully.]3. S.W. Hawking and G.F.R. Ellis, The Large Scale Structure of Space-Time, last partof Sec. 5.5 on the Reissner-Nordstr¨om solution. [A copy wi l l be on our web site byThursday evening.] B riefer and less technical treat ments of this material a re containedin MTW E xercises 3 1.8, 32.1 and 34.3 and in Carroll (Ref. 6 below). A more completetreatment is in Ref. 8 below.Note: In the copy on our course website, I have inserted abol ute value signs in variousformulae on page 157 t o correct typos.Possible Supplementary Reading:4. Hartle Gravity: Chapter 12. This is an especially good treatment of Schwarzschildblack holes and wormholes — significantly more lucid than MTW, I think.5. Schutz (A First Course in General Relativity): pp. 288–297 on Schwarzschild solutionand black holes.6. Carroll Spacetime and Geometry: Secs. 5.1, 5.2, 5.3, 5.6, 5.7, 5.8 on Schwarzschildwormholes and black holes; and Sec. 6.5 on t he Reissner-Nordstrom Solution.7. Wald (General Relativity): Secs. 6.1 and 6.4.8. J. C. Graves and D. R. Brill, “Oscillatory Character of Reissner-Nordstrom Metricfor an Ideal Charged Wormhole”, Physical Review, 120, 1507–1513 (1960). Thiswas the first paper to deduce the causal structure of Reissner-Nordstr¨om. It carriesthe analysis up to the analo g of Kruskal-Szekeres coordinates, but does not do thefinal step of bringi ng infinity in to a finite locat ion via a conformal transformation.Penrose had not yet invented that idea when this paper was written. N evertheless, thispaper might be more accessible and understandable than the much terser discussionin Hawking and Ellis (Ref. 1 above). Historically, t his work was the undergraduate,Senior thesis of John Graves at Princeton in 1959; Dieter Brill, then a grad studentjust finishing his PhD, was Graves’ advi sor.************************************************************1ProblemsNote: Each problem is worth 10 points unless otherwise indicated. As usual, themaximum number of points that will be given for this set is 50.1. Interpretation of a Metric and Coordinate SystemA spacetime has the following metricds2= −dt2to/t − 1+ (1 − to/t)dz2+ t2(dθ2+ sin2θdφ2) . (1)Here tois a constant, and t increases upward from 0.a. What are t he coordinates being used?b. What symmetries does this spacetime have, and how are the coordinates relatedto those symmetries?c. Without solving the geodesic equation, explain on the basis of symmetries whythe curves {z, θ, φ} = constant are geodesics, and why these geodesics are theworld lines of freely falling observers.d. Plot, a s a function of their proper time, the proper distances between two freelyfalling observers with the same {θ, φ} but slig htly different z. Do the same forobservers with the same z but slightly different {θ, φ}. Discuss the physicalmeaning of these plots.e. At what values of the coordinates does the metric (1) become singular?f. What i s t he relationship of this spacetime to the Schwarzschild spacetime?g. Based on that relation, what is the physical nature of each of the singularitiesyou identified in part e?2. The Bertotti-Robinson Solution of the Einstein Field EquationBlandford and Thorne Exercise 25.2 .3. Gore at the SingularityBlandford and Thorne Exercise 25.8 .4. Slices of Simultaneity in Schwarzschild SpacetimeBlandford and Thorne, Exercise 25.12.5. Nonradial Light Cones [5 Points]MTW Exercise 31. 2.6. Eddington-Finkelstein and Kruskal-Szekeres ComparedMTW Exercise 31. 5.7. Rindler and Kruskal-Szekeres ComparedDivide Minkowsk i spacetime up into four regions bounded by leftward a nd right-ward traveling light rays, like the four regi ons of Kruskal-Szekeres. In region I intro-duce Rindler coordinates, defined as follows: If the Lorentz coordinates are denoted2(T, X, Y, Z) and the Rindler coordinates are (t, x, Y, Z), then the transformation t oRindler in region I isX = x cosh(gt) , T = x sinh(gt) , (1)where g is a constant acceleration. Notice that these Rindler coordinates are not hingbut the proper reference frame of a uniformly accelerated observer [Eq. (6.17) ofMTW], with the spatial origin moved: t = ξ0′, x = ξ1′+ g−1. Notice also thesimilarity to Eq. (31.17a) for Kruskal-Szekeres coordinates.a. Show that the spacetime metric in Rindler coordinates isds2= −g2x2dt2+ dx2+ dY2+ dZ2. (2)b. Construct Rindler coordinates for regions II, III, and IV, by analogy with Eqs.(31.17b,c,d).c. Derive the Minkowsk i spacetime metric in Rindler coo rdinates for all four regionsfrom these transformations.d. Draw coordinate diagrams using the Lorentz coordinates and using t he Rindlercoordinates, and explore the rela tionships between them in t he same manner asMTW explo res the relationship between Kruskal-Szekeres and Schwarzschild.8. Rindler Approximation to the Schwarzschild metric near the horizon of ablack hole[Note: Again, I did this in my Monday lecture, but you may find it useful to do thedetails yourself.]Build the proper reference frame of a static observer who is at rest at location(ro, θo, φo) very slightly above the horizon of a Schwarzschild black hole. Introduceslightly different normalizations for the coo rdinates than usual. In particular, use asthe time coordinate Schwarzschild time t rather than the observer’s proper time τ,and use for the vertical spatial coordinate the proper distance x measured from thehole’s horizon rather than from the observer’s location. More specifically, use as coor-dinates t, x =Rr2Mdr/p1 − 2M/r ≃ 4Mp1 − 2M/r, Y = ro(θ − θo) ≃ 2M(θ − θo),Z = rosin θo(φ − φo) ≃ 2M sin


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