Black Holes

by Raine, Derek; Thomas, Edwin

• ISBN13: 9781860945861
• ISBN10: 1860945864
• Format: Hardcover
• Copyright: 2005-12-30
• Publisher: Imperial College Pr

Summary

This introduction to the fascinating subject of black holes fills a significant gap in the literature which exists between popular, non-mathematical expositions and advanced textbooks at the research level. It is designed for advanced undergraduates and first year postgraduates as a useful stepping-stone to the advanced literature. The book provides an accessible introduction to the exact solutions of Einstein's vacuum field equations describing spherical and axisymmetric (rotating) black holes. The geometry and physical properties of these spacetimes are explored through the motion of particles and light. The use of different coordinate systems, maximal extensions and Penrose diagrams is explained. The association of the surface area of a black hole with its entropy is discussed and it is shown that with the introduction of quantum mechanics black holes cease to be black and can radiate. This result allows black holes to satisfy the laws of thermodynamics and thus be consistent with

Table of Contents

Preface
1 Relativistic Gravity	1	(12)
1.1 What is a black hole?	1	(2)
1.2 Why study black holes?	3	(1)
1.3 Elements of general relativity	3	(10)
1.3.1 The principle of equivalence	3	(1)
1.3.2 The Newtonian affine connection	4	(1)
1.3.3 Newtonian gravity	5	(1)
1.3.4 Metrics in relativity	6	(1)
1.3.5 The velocity and momentum 4-vector	7	(1)
1.3.6 General vectors and tensors	8	(1)
1.3.7 Locally measured physical properties	9	(1)
1.3.8 Derivatives in relativity	9	(2)
1.3.9 Acceleration 4-vector	11	(1)
1.3.10 Paths of light	11	(1)
1.3.11 Einstein's field equations	11	(1)
1.3.12 Symmetry and Killing's equation	11	(2)
2 Spherical Black Holes	13	(44)
2.1 The Schwarzschild metric	13	(5)
2.1.1 Coordinates	14	(1)
2.1.2 Proper distance	15	(1)
2.1.3 Proper time	15	(1)
2.1.4 Redshift	15	(1)
2.1.5 Interpretation of M and geometric units	16	(1)
2.1.6 The Schwarzschild radius	16	(1)
2.1.7 The event horizon	17	(1)
2.1.8 Birkoff's theorem	17	(1)
2.1.9 Israel's theorem	17	(1)
2.2 Orbits in Newtonian gravity	18	(1)
2.2.1 Energy	18	(1)
2.2.2 Angular momentum	18	(1)
2.2.3 The Newtonian effective potential	18	(1)
2.2.4 Classification of Newtonian orbits	18	(1)
2.3 Particle orbits in the Schwarzschild metric	19	(9)
2.3.1 Constants of the motion	19	(1)
2.3.2 Energy	20	(1)
2.3.3 Angular momentum	20	(1)
2.3.4 The effective potential	21	(1)
2.3.5 Newtonian approximation to the metric	22	(1)
2.3.6 Classification of orbits	22	(1)
2.3.7 Radial infall	23	(1)
2.3.8 The locally measured energy of a particle	24	(1)
2.3.9 Circular orbits	25	(1)
2.3.10 Comparison with Newtonian orbits	26	(1)
2.3.11 Orbital velocity in the frame of a hovering observer	27	(1)
2.3.12 Energy in the last stable orbit	27	(1)
2.4 Orbits of light rays	28	(3)
2.4.1 Radial propagation of light	29	(1)
2.4.2 Capture cross-section for light	29	(1)
2.4.3 The view of the sky for a stationary observer	30	(1)
2.5 Classical tests	31	(1)
2.6 Falling into a black hole	32	(4)
2.6.1 Free-fall time for a distant oserver	32	(2)
2.6.2 Light-travel time	34	(1)
2.6.3 What the external observer sees	34	(1)
2.6.4 An infalling observer's time	35	(1)
2.6.5 What the infalling observer feels	35	(1)
2.7 Capture by a black hole	36	(2)
2.7.1 Case I: Capture of high angular momentum particles	37	(1)
2.7.2 Case II: Capture of low energy particles	37	(1)
2.8 Surface gravity of a black hole	38	(3)
2.8.1 The proper acceleration of a hovering observer	38	(1)
2.8.2 Surface gravity	39	(1)
2.8.3 Rindler coordinates	39	(2)
2.9 Other coordinates	41	(2)
2.9.1 Null coordinates	42	(1)
2.9.2 Eddington Finkelstein coordinates	42	(1)
2.10 Inside the black hole	43	(2)
2.10.1 The infalling observer	44	(1)
2.11 White holes	45	(1)
2.12 Kruskal coordinates	46	(3)
2.12.1 The singularities at r = 0 and cosmic censorship	47	(1)
2.12.2 The spacetime of a collapsing star	48	(1)
2.13 Embedding diagrams	49	(2)
2.14 Asymptotic flatness	51	(2)
2.14.1 The Penrose–Carter diagram for the Schwarzschild metric	52	(1)
2.14.2 The Penrose–Carter diagram for the Newtonian metric	52	(1)
2.15 Non-isolated black holes	53	(2)
2.15.1 The infinite redshift surface	53	(1)
2.15.2 Trapped surfaces	54	(1)
2.15.3 Apparent horizon	54	(1)
2.16 The membrane paradigm	55	(2)
3 Rotating Black Holes	57	(46)
3.1 The Kerr metric	58	(1)
3.2 The event horizon	58	(2)
3.2.1 The circumference of the event horizon	59	(1)
3.2.2 The area of the event horizon	59	(1)
3.3 Properties of the Kerr metric coefficients	60	(1)
3.3.1 Identities	60	(1)
3.3.2 Contravariant components	60	(1)
3.4 Interpretation of m, a and geometric units	61	(1)
3.5 Extreme Kerr black hole	62	(1)
3.6 Robinson's theorem	62	(1)
3.7 Particle orbits in the Kerr geometry	62	(4)
3.7.1 Constants of the motion	63	(1)
3.7.2 Energy	64	(1)
3.7.3 Angular momentum	64	(1)
3.7.4 The Carter integral	64	(1)
3.7.5 The radial equation	65	(1)
3.7.6 The effective potential	65	(1)
3.8 Frame-dragging	66	(3)
3.8.1 Orbits of zero angular momentum particles	68	(1)
3.8.2 Orbits with non-zero angular momentum	68	(1)
3.9 Zero angular momentum observers (ZAMOs)	69	(3)
3.9.1 Some applications of ZAMOs	70	(2)
3.10 Photon orbits	72	(2)
3.10.1 The photon effective potential	72	(1)
3.10.2 Azimuthal motion	73	(1)
3.10.3 Photon capture (Toss-section	73	(1)
3.11 The static limit surface	74	(2)
3.12 The infinite redshift surface	76	(1)
3.13 Circular orbits in the equatorial plane	76	(6)
3.13.1 Innermost (marginally) stable circular orbit	77	(2)
3.13.2 Period of a circular orbit	79	(1)
3.13.3 Energy of the innermost stable orbit	80	(1)
3.13.4 Angular momentum of the innermost stable orbit	81	(1)
3.13.5 Marginally bound orbits	81	(1)
3.13.6 Unbound orbits	81	(1)
3.14 Polar orbits	82	(3)
3.14.1 Orbital period	84	(1)
3.15 The ergosphere	85	(4)
3.15.1 Negative energy orbits	85	(1)
3.15.2 Angular momentum of a negative energy particle	86	(1)
3.15.3 The Penrose process	87	(1)
3.15.4 Realising the Penrose process	87	(2)
3.16 Spinning up a black hole	89	(3)
3.16.1 From Schwarzschild to extreme Kerr black hole	91	(1)
3.17 Other coordinates	92	(1)
3.18 Penrose-Carter diagram	93	(4)
3.18.1 Interior solutions and collapsing stars	97	(1)
3.19 Closed timelike lines	97	(1)
3.20 Charged black holes	98	(5)
4 Black Hole Thermodynamics	103	(34)
4.1 Black hole mechanics	103	(4)
4.1.1 Surface gravity	103	(2)
4.1.2 Redshift	105	(1)
4.1.3 Conservation of energy	106	(1)
4.2 The area, of a Kerr black hole horizon cannot decrease	107	(3)
4.2.1 Area change by accretion	107	(1)
4.2.2 Area change produced by the Penrose process	108	(1)
4.2.3 The area theorem	109	(1)
4.2.4 Irreducible mass	109	(1)
4.2.5 Maximum energy extraction	109	(1)
4.2.6 Naked singularities	110	(1)
4.3 Scattering of waves	110	(6)
4.3.1 Superradiance	110	(6)
4.4 Thermodynamics	116	(5)
4.4.1 Horizon temperature	118	(2)
4.4.2 The four laws of black hole thermodynamics	120	(1)
4.5 Hawking radiation	121	(10)
4.5.1 Introduction	121	(1)
4.5.2 Casimir effect	122	(1)
4.5.3 Thermal vacua in accelerated frames	122	(5)
4.5.4 Hawking radiation	127	(4)
4.6 Properties of radiating black holes	131	(3)
4.6.1 Entropy and temperature	131	(1)
4.6.2 Radiating black holes	132	(1)
4.6.3 Black hole in a box	133	(1)
4.7 Entropy and microstates	134	(3)
5 Astrophysical Black Holes	137	(10)
5.1 Introduction	137	(1)
5.2 Stellar mass black holes	138	(4)
5.2.1 Formation	138	(2)
5.2.2 Finding stellar mass black holes	140	(1)
5.2.3 The black hole at the centre of the Galaxy	141	(1)
5.3 Supermassive black holes in other galaxies	142	(3)
5.3.1 Intermediate mass black holes	143	(1)
5.3.2 Mini black holes	144	(1)
5.4 Further evidence for black hole spin	145	(1)
5.5 Conclusions	146	(1)
References	147	(4)
Bibliography	151	(2)
Index	153