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The de ection of a light ray in this case is proportional to (a) m 1 m 2 (b) m 1 m 2 (c) m 2 + m 2 1 2 (d) m 1 + m 2 14. The G del metric, which is a physically unrealistic model that allows for o the possibility of time travel, is given by ds 2 = (dt + e x dy)2 dx 2 1 2 x 2 e dy dz 2 2
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Suppose that the stress-energy tensor be given by 1 0 = x e 0 0 e x 0 0 2 x 0 e 0 0 0 0 0 0
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If the cosmological constant is not zero, the Einstein eld equations in this case require that (a) = 2 /2 (b) = 4 /2
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(c) (d) 15. = 2 = 4 /2
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Rotating black holes are described by (a) the Reissner-Nordstr m solution (b) the Kerr solution (c) the Schwarzschild metric Consider two xed observers near a Schwarzschild black hole. One observer at r = 3m emits a pulse of ultraviolet light (wavelength about 400 nm) to a second xed observer at r = 5m. The second observer nds that the light is redshifted, with a wavelength such that the light is (a) green (b) yellow (c) blue (d) orange The best description of the coordinate singularity in the Schwarzschild metric is that (a) it has no effect whatsoever; it is just an artifact of our coordinate choice. Nothing happens here and particles that cross the coordinate singularity can return to the outside, given enough energy. (b) it is an artifact of our choice of coordinates; however, it pinpoints the surface of the event horizon. Once you pass this point you cannot return. (c) it is the location where the geometry blows up. A space with k = 1 can be said to have (a) zero curvature (b) positive curvature (c) negative curvature (d) embedded curvature In a perfect uid, the diagonal components of the stress-energy tensor are given by (a) T a b = diag( , P, P, P) (b) They all vanish. (c) T a b = diag( , P, P, P) (d) T a b = diag( , 0, 0, 0) For a type N spacetime (a) the only nonzero Weyl scalar is principal null direction
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(b) the Weyl scalars all vanish (c) the only nonzero Weyl scalar is 0 or 4 , and there are two principal null directions (d) there is one doubly repeated null direction and the nonzero Weyl scalars are 0 , 2 , and 4
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Quiz and Exam Solutions
1. c 2. a 3. c 4. a 5. b
1. c 2. a 3. b 4. a 5. c
1. b 2. c 3. b 4. c 5. d
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Quiz and Exam Solutions
1. c 6. a 2. a 7. d 3. c 8. b 4. d 9. c 5. b 10. b
1. a 6. a 2. c 7. d 3. a 8. a 4. d 5. c
1. c 6. c 2. b 7. a 3. a 8. a 4. d 5. a
1. a 2. b 3. a 4. a 5. a
1. d 2. b 3. a
1. c 2. a 3. a 4. d 5. b
1. a 2. b 3. b 4. a 5. d
1. b 2. b 3. a 4. c 5. a
Quiz and Exam Solutions
1. c 6. a 2. a 3. b 4. a 5. c
1. a 2. b 3. b
FINAL EXAM
1. 6. 11. 16. c d b a 2. 7. 12. 17. a c a b 3. 8. 13. 18. c d d b 4. 9. 14. 19. a a a c 5. 10. 15. 20. b a b a
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References and Bibliography
Books
Of course it is impossible to write a math or science book without relying on previous publications. Below is a list of the materials used in the preparation of the manuscript. Hopefully, we have not forgotten to list anyone. Since those studying the subject will likely have the desire to purchase additional books, I have included my comments. Adler, R., Bazin, M., and Schiffer, M., Introduction to General Relativity, 2d ed., McGraw-Hill, New York, 1975. We relied heavily on this book for the development of the manuscript. Unfortunately it is out of print, which is really too bad. Although the presentation is a bit old fashioned, it includes some nice discussions on the Schwarzschild
Copyright 2006 by The McGraw-Hill Companies, Inc. Click here for terms of use.
References and Bibliography
solution and the Petrov classi cation. I also enjoy their use of Lagrangian methods throughout the text. Carlip, S., Quantum Gravity in 2 + 1 Dimensions, Cambridge University Press, Cambridge, 1998. Carroll, S.M., An Introduction to General Relativity Spacetime and Geometry, Addison-Wesley, San Francisco, 2004. This is a nice edition to the books in the eld that came out recently. Clearly written. Chandrasekhar, S., The Mathematical Theory of Black Holes, Clarendon Press, Oxford, 1992. A good reference book that includes description of the Newman-Penrose formalism. Trying to read it cover to cover might induce a headache. D Inverno, R., Introducing Einstein s Relativity, Oxford University Press, Oxford, 1992. A nice concise presentation that s a good place to start building mathematical background for the subject. s are relatively short. Grif ths, J.B., Colliding Plane Waves in General Relativity, Clarendon Press, Oxford, 1991. A specialized book that we referenced for discussion of Petrov classi cation and the meaning of the Weyl scalars. More on the advanced side, but good if you are interested in gravitational waves. Hartle, J.B., Gravity: An Introduction to Einstein s General Relativity, AddisonWesley, San Francisco, 2002. Good for discussions on the physics with a slow and gentle introduction to the math. Also, some discussion of experimental/observational research like LIGO. Hartle would be a good compliment to this book since we took a more mathematical approach, and so you can nd what s missing from our book here. Hawking, S.W. and Ellis, G.F.R., The Large-Scale Structure of Spacetime, Cambridge University Press, Cambridge, 1973. Great read if you are more mathematically inclined. This is a personal favorite. Hughston, L.P. and Tod, K.P., An Introduction to General Relativity, Cambridge University Press, Cambridge, 1990. Very slim volume, covers topics quickly in a few pages. It kind of has a mathematically formal prose.
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