# CALTECH PH 236A - The Schwarzschild Solution (3 pages)

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## The Schwarzschild Solution

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## The Schwarzschild Solution

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3
School:
California Institute of Technology
Course:
Ph 236a - Relativity
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Week 6 The Schwarzschild Solution Yanbei Chen Dated November 9 2009 Due on Tuesday November 17 2009 I READING This week we discussed the Schwarzschild Geometry which we now understand is the exterior solution of a spherical star the Birkhoff Theorem states that a spherically symmetric vacuum space time must be a Schwarzschild spacetime We discussed the structure of a spherical star and started discussing the more interesting features of the Schwarzschild space time near the r 2M surface Here are possible reading materials My lectures were roughly following Chapter 5 of Carroll except Section 5 7 which I did not have time to discuss Secs 24 1 24 3 of Blandford and Thorne covers similar material but in a somewhat different style Sec 9 5 of Shapiro and Teukolsky Black Holes White Dwarfs and Neutron Stars for a discussion on maximum mass of stars II PROBLEMS Each problem is worth 10 points unless otherwise noted The maximum number of points you can get for this assignment is 50 although you could choose to do problems that worth more than 50 points You should choose to do those problems that you will learn the most from 1 Schwarzschild Geometry in Isotropic Coordinates Show either by hand or using a computer that if one changes radial coordinates from r to r via the transformation r r 1 M 2r 2 the Schwarzschild metric takes on the new form 2 4 M 1 M 2r 2 2 2 2 dt 1 dr r d 2 sin2 d 2 ds 1 M 2r 2r 1 2 This new t r coordinate system is called isotropic because in it the spatial part of the metric is written as a function of r times the flat 3 dimensional Euclidean metric in spherical coordinates and this Euclidean metric does not pick out any direction as special 2 Embedding Diagram for a Wormhole Embedding Diagrams are sometimes useful in depicting space times usually a particular slicing of some spacetime is isometrically embedded into a 3 Euclidean space See Blandford and Thorne 24 3 5 for detailed discussions a Show that in the isotropic coordinates of Eq 2 the radial

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