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El. knyga: Exact Solutions in Three-Dimensional Gravity

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"Due to the big amount of existing nowadays exact solutions in 2+1 Einstein gravity the purpose of the present book is to bring to the audience of experts and young researchers a complete and concise list of exacts solutions with emphasis on their physical and geometrical properties in (2+1) gravity from its beginning in 1963 to present. Emphasis is addressed to solutions to the Einstein equations in the presence of matter and fields, for instance, point particle solutions, perfect fluids, cosmological spacetimes, dilatons, inflatons, stringy solutions. The second part of this book deals with solutions to vacuum topologically massive gravity with a cosmological constant; there exist two big families of spacetimes: the inhomogeneous Bianchi class of solutions, the Kundt spacetimes and the Cotton type N wave fields"--

A self-contained text, systematically presenting the determination and classification of exact solutions in three-dimensional Einstein gravity. This book explores the theoretical framework and general physical and geometrical characteristics of each class of solutions, and includes information on the researchers responsible for their discovery. Beginning with the physical character of the solutions, these are identified and ordered on the basis of their geometrical invariant properties, symmetries, and algebraic classifications, or from the standpoint of their physical nature, for example electrodynamic fields, fluid, scalar field, or dilaton. Consequently, this text serves as a thorough catalogue on 2+1 exact solutions to the Einstein equations coupled to matter and fields. The solutions are also examined from different perspectives, enabling a conceptual bridge between exact solutions of three- and four-dimensional gravities, and therefore providing graduates and researchers with an invaluable resource on this important topic in gravitational physics. Including contributions by David Chow, Christopher N. Pope and Ergin Sezgin (chapters 16-19).

A self-contained and unique text, for graduate students and researchers, systematically presenting the determination and classification of exact solutions in three-dimensional Einstein gravity. This book explores the theoretical framework and general physical and geometrical characteristics of each class of solutions, and includes information on the researchers responsible for their discovery. Including contributions by David Chow, Christopher N. Pope and Ergin Sezgin (chapters 16-19).

Recenzijos

'2+1 dimensions is the lowest dimension in which Einstein gravity works, and it exhibits some interesting features, such as space-time being flat where there is no matter. The topic has been studied for about 50 years, and there are, remarkably, large numbers of exact solutions that have been found. This book presents a comprehensive account of them It is obviously a labour of love, which runs to over 400 pages, and which will be of interest to relativists, providing links between the solutions and those for the more familiar 3+1 space- time.' Alan Heavens, The Observatory

Daugiau informacijos

A self-contained and unique text systematically presenting the determination and classification of exact solutions in three-dimensional Einstein gravity. Including contributions by David Chow, Christopher N. Pope and Ergin Sezgin (chapters 16-19).
Preface xvii
1 Introduction 1(18)
1.1 Main Features of (2 + 1) Gravity
1(6)
1.1.1 Field Equations and Curvature Tensors
2(1)
1.1.2 Matter Distribution Locally Curves the Spacetime
2(1)
1.1.3 Point Particles Produce Global Effects on the Spacetime
3(1)
1.1.4 Newtonian Limits
3(2)
1.1.5 No Geodesic Deviation for Dust
5(1)
1.1.6 No Dynamic Degrees of Freedom
6(1)
1.1.7 Black Holes in (2 + 1) Gravity
7(1)
1.1.8 Gravity in the Presence of Other Fields and Matter
7(1)
1.2 Algebraic Classification
7(4)
1.2.1 Classification of the Cotton-York Tensor
7(2)
1.2.2 Classification of the Energy-Momentum Tensor
9(2)
1.2.3 Classification of the Traceless Ricci Tensor
11(1)
1.3 Brown-York Energy, Mass, and Momentum for Stationary Metrics
11(4)
1.3.1 Summary of Quasilocal Mass, Energy, and Angular Momentum
14(1)
1.4 Decomposition with Respect to a Frame of Reference
15(4)
1.4.1 Kinematics of the Frame
15(1)
1.4.2 Perfect Fluid Referred to a Frame of Reference
16(3)
2 Point Particle Solutions 19(8)
2.1 Staruszkiewicz Point Source Solutions
19(2)
2.1.1 Relationship Between the Deficit Angle and Mass
20(1)
2.2 Staruszkiewicz Single Point Source Solution
21(1)
2.2.1 No Parallelism With the (3 + 1) Schwarzschild Solution
22(1)
2.3 Staruszkiewicz Two Point Sources Solution
22(1)
2.4 Deser-Jakiw-'t Hooft Static N Point Sources Solution
23(1)
2.4.1 Energy and Euler Invariant
23(1)
2.4.2 Energy-Momentum Tensor for N Point Particles
24(1)
2.5 Clement Rotating Point-Particles Solution
24(3)
3 Dust Solutions 27(17)
3.1 Cornish-Frankel Dust Heaviside Function Solution
27(1)
3.2 Giddings-Abott-Kuchar Dust Solutions
28(2)
3.2.1 Time-Dependent Class of Dust Solutions Omega = ln (t f (x, y))
29(1)
3.2.2 Static Class of Dust Solutions Omega = ln g(x, y)
30(1)
3.3 Barrow-Shaw-Tsagas Anisotropic Dust Solution; Lambda = 0
30(3)
3.4 BST Diagonal Anisotropic Dust Solutions with Lambda
33(1)
3.5 BST (t, x, y)-Dependent Cosmological Solutions with Comoving Dust
34(6)
3.5.1 BST Class 2 of Solutions
37(1)
3.5.2 BST Class 1 Spacetime
38(1)
3.5.3 BST Class 3 of Dust Solutions
39(1)
3.6 Rooman-Spindel Dust Godel Non-Diagonal Model
40(4)
4 A Shortcut to (2+1) Cyclic Symmetric Stationary Solutions 44(14)
4.1 Cyclic Symmetric Stationary Solutions in Canonical Coordinates
44(4)
4.1.1 Banados-Teitelboim-Zanelli Solution in Canonical Polar rho Coordinate
45(1)
4.1.2 BTZ Solution Counterpart
46(1)
4.1.3 Coussaert-Henneaux Metrics
47(1)
4.2 Static AdS Black Hole
48(4)
4.2.1 Static BTZ Solution
49(2)
4.2.2 Static AdS Solution Counterpart
51(1)
4.3 Symmetries of the Stationary and Static Cyclic Symmetric BTZ Metrics
52(6)
4.3.1 Symmetries of the AdS Metric for Negative M, M = -alpha2
56(2)
5 Perfect Fluid Static Stars; Cosmological Solutions 58(8)
5.1 Static Circularly Symmetric Fluid Solutions
58(1)
5.1.1 Cotton Tensor Types
59(1)
5.2 Incompressible Static Star
59(3)
5.2.1 Collas Static Star with Constant Density mu0
60(1)
5.2.2 Giddings-Abott-Kuchat Static Star with mu0
60(1)
5.2.3 Cornish-Frankel Static Star with mu0
61(1)
5.3 Cornish-Frankel Static Polytropic Solutions
62(4)
5.3.1 Static Star with a Stiff Matter p(r) = mu(r)
64(1)
5.3.2 Static Star with Pure Radiation p = mu(r)/2
65(1)
6 Static Perfect Fluid Stars with A 66(14)
6.1 Equations for a (2+1) Static Perfect Fluid Metric
67(2)
6.1.1 General Perfect Fluid Solution with Variable rho(r)
68(1)
6.2 Canonical Coordinate System {t, N, 0}
69(1)
6.3 Perfect Fluid Solutions for a Barotropic Law p = gamma rho
70(1)
6.4 Perfect Fluid Solutions for a Polytropic Law p = Cpgamma
71(2)
6.5 Oppenheimer-Volkoff Equation
73(1)
6.6 Perfect Fluid Solution with Constant Density
74(6)
6.6.1 (3+1) Static Spherically Symmetric Perfect Fluid Solution
76(2)
6.6.2 Comparison Table
78(2)
7 Hydrodynamic Equilibrium 80(12)
7.1 Generalized Buchdahl's Theorem
80(1)
7.2 Stellar Equilibrium in (2 + 1) Dimensions with Λ
81(4)
7.2.1 Cruz-Zanelli Existence of Hydrostatic Equilibrium for Lambda < or = to 0
83(1)
7.2.2 No Buchdahl's Inequality in (2 + 1) Hydrostatics
83(1)
7.2.3 Static Star with Constant Density mu0 and Lambda = -1/l2 < or = to 0
84(1)
7.3 Buchdahl Theorem in d Dimensions
85(7)
7.3.1 Buchdahl's Inequalities
86(3)
7.3.2 Constant Density Solution
89(3)
8 Stationary Circularly Symmetric Perfect Fluids with A 92(16)
8.1 Stationary Differentially Rotating Perfect Fluids
93(1)
8.2 Garcia Stationary Rigidly Rotating Perfect Fluids
94(8)
8.2.1 Rigidly Rotating Perfect Fluid Solution with W(r) = J/(2r2)
96(1)
8.2.2 Garcia Interior Solution with Constant Energy Density
97(3)
8.2.3 Interior Perfect Fluid Solution to the BTZ Black Hole
100(1)
8.2.4 Alternative Parametrization
100(2)
8.2.5 Barotropic Rotating Perfect Fluids Without Lambda
102(1)
8.3 Lubo-Rooman-Spindel Rotating Perfect Fluids
102(6)
8.3.1 Equations for Rigidly Rotating Fluids
104(1)
8.3.2 Garcia Representation of Stationary Perfect Fluid Solutions
105(1)
8.3.3 Barotropic Class of Solutions p = gamma mu it
105(1)
8.3.4 Constant Density Stationary Solution; p = p(r), mu = mu0
105(1)
8.3.5 Lubo-Rooman-Spindel Perfect Fluids u = theta° and grr = 1
106(1)
8.3.6 LBR Rotating Perfect Fluid with mu0
106(1)
8.3.7 Rooman-Spindel Rotating Fluid Model; gtt = -1 = -grr
107(1)
9 Friedmann-Robertson-Walker Cosmologies 108(13)
9.1 Einstein Equations for FRW Cosmologies
108(2)
9.1.1 Einstein Equations for (3+1) FRW Cosmology
108(1)
9.1.2 Einstein Equations for (2+1) FRW Cosmology
109(1)
9.2 Barotropic Perfect Fluid FRW Solutions
110(3)
9.2.1 Barotropic Perfect Fluid (3 + 1) Solutions
110(1)
9.2.2 Barotropic Perfect Fluid (2 + 1) Solutions
111(1)
9.2.3 Comparison Between (3+1) and (2+1) Barotropic Solutions
112(1)
9.3 Polytropic Perfect Fluid FRW Solutions
113(1)
9.3.1 Polytropic Perfect Fluid (3 + 1) Solutions
113(1)
9.3.2 Polytropic Perfect Fluid (2 + 1) Solutions
114(1)
9.3.3 Comparison Between (3+1) and (2+1) Polytropic Solutions
114(1)
9.4 Mann-Ross Collapsing Dust FRW Solutions with Lambda
114(7)
9.4.1 Cosmological dS-FRW Solution
115(1)
9.4.2 Asymptotically AdS-FRW Dust Solution
115(1)
9.4.3 Matching the AdS-FRW Dust to the Static BTZ
116(1)
9.4.4 Determination of Kij
117(2)
9.4.5 Gidding-Abbott-Kuchar Dust FRW Solution
119(2)
10 Dilaton-Inflaton Friedmann-Robertson-Walker Cosmologies 121(21)
10.1 Equations for a FRW Cosmology with a Perfect Fluid and a Scalar Field
122(5)
10.1.1 Einstein Equations for (3+1) FRW Dilaton Cosmology
122(1)
10.1.2 Einstein Equations for (2+1) FRW Cosmology
123(2)
10.1.3 Correspondence Between (3+1) and (2+1) Solutions
125(2)
10.2 Single Scalar Field to Linear State Equations; Lambda = 0
127(4)
10.2.1 (2+1) Solutions for a Scalar Field
127(1)
10.2.2 (3+1) Solutions for a Scalar Field
128(1)
10.2.3 Slow Roll Spatially Flat FRW Solutions
129(2)
10.3 Spatially Flat FRW Solutions for Barotropic Perfect Fluid and Scalar Field
131(4)
10.3.1 Spatially Flat FRW (3+1) Solutions; gamma4 not = to 2Gamma4
131(2)
10.3.2 Spatially Flat FRW (2+1) Solutions; gamma3 not = 2Gamma3
133(1)
10.3.3 Barrow-Saich Solution; gamma = 2Gamma
134(1)
10.4 Single Scalar Field Spatially Flat FRW Solutions to pφ + rhoφ=Gamma&rho&phiBeta
135(4)
10.4.1 Spatially Flat (3+1) Solutions with V(phi) = A(alphaphi2/1=beta)-phi2beta/(1-beta))
135(1)
10.4.2 Spatially Flat (2+1) Solutions with V(phi) = A(alphaphi2/1=beta)-phi2beta/(1-beta))
136(1)
10.4.3 Barrow-Burd-Lancaster (2+1) and Madsen (3+1) Solutions
137(2)
10.5 Scalar Field Solutions for a Given Scale Factor
139(3)
10.5.1 Second (2+1) BBL Solution
139(2)
10.5.2 (3+1) Generalization of the Second (2+1) BBL Solution
141(1)
11 Einstein-Maxwell Solutions 142(98)
11.1 Stationary Cyclic Symmetric Einstein-Maxwell Fields
143(10)
11.1.1 Stationary Cyclic Symmetric Maxwell Fields
143(2)
11.1.2 General Stationary Metric and Einstein Equations
145(3)
11.1.3 Complex Extension and Real Cuts
148(1)
11.1.4 Positive A Solutions
149(1)
11.1.5 Characterizations of Einstein-Maxwell Solutions
149(3)
11.1.6 Static Cyclic Symmetric Equations for Maxwell Fields
152(1)
11.2 Electrostatic Solutions; b not = to 0, a = 0
153(6)
11.2.1 General Electrostatic Solutions
154(1)
11.2.2 Gott-Simon-Alpern, Deser-Mazur, and Melvin Electrostatic Solution
155(1)
11.2.3 Charged Static Peldan Solution with Lambda
156(3)
11.3 Magnetostatic Solutions; a not = to 0, b = 0
159(9)
11.3.1 General Magnetostatic Solutions
160(1)
11.3.2 Melvin, and Barrow-Burd-Lancaster Magnetostatic Solution
161(1)
11.3.3 Peldan Magnetostatic Solution with Lambda
162(3)
11.3.4 Hirschmann-Welch Solution with Lambda
165(3)
11.4 Cataldo Static Hybrid Solution
168(5)
11.4.1 Mass and Energy
170(1)
11.4.2 Field, Energy-Momentum, and Cotton Tensors
171(2)
11.5 Uniform Electromagnetic Solutions Fmunu;sigma = 0
173(7)
11.5.1 General Uniform Electromagnetic Solution for a not = to 0, not = to b
173(2)
11.5.2 Uniform "Stationary" Electromagnetic A = r/(bl2)(dt - wodphi) Solutions
175(1)
11.5.3 Matyjasek-Zaslayskii Uniform Electrostatic A = r/(bl2) dt Solution
176(3)
11.5.4 Uniform "Stationary" Electromagnetic A = r/(al2)(dphi+W0dt) Solutions
179(1)
11.5.5 No Uniform Stationary Magnetostatic Solution for Lambda = -1/l2
180(1)
11.6 Constant Electromagnetic Invariants' Solutions
180(11)
11.6.1 General Constant Invariant FmunuFmunu=2gamma for a not = to 0 not = to b
181(1)
11.6.2 Constant Electromagnetic Invariant FF = 2/l2 Solution
182(1)
11.6.3 Constant Electromagnetic Invariant FF = -2/l2 Solution for b not - to 0
182(1)
11.6.4 Constant Electromagnetic Invariant FF = 2/l2 Stationary Solution for a not = to 0
183(1)
11.6.5 Vanishing Electromagnetic Invariant FF = 0 Solution
184(1)
11.6.6 Kamata-Koikawa Solution
185(4)
11.6.7 Proper Kamata-Koikawa Solution, ro0 = ±Q/square root of Lambda
189(2)
11.7 Ayon-Cataldo-Garcia Stationary Hybrid Solution
191(8)
11.7.1 ACG Hybrid Solution Allowing for BTZ Limit
192(2)
11.7.2 Mass, Energy, and Momentum
194(4)
11.7.3 Constant Electromagnetic Invariants' Hybrid Solution for Lambda = 0
198(1)
11.8 Stationary Solutions for a not = to 0 or b not = to 0
199(5)
11.8.1 Stationary Magneto-Electric Solution for a not = to 0 = b
199(3)
11.8.2 Stationary Electromagnetic Solution for b not = to 0 = a
202(2)
11.9 Garcia Stationary Solutions for a = 0 and b no = to 0
204(11)
11.9.1 Alternative Representation of the Einstein Equations
205(1)
11.9.2 Garcia Stationary Electromagnetic Solution with BTZ limit
206(7)
11.9.3 Garcia Stationary Solution with BTZ-Counterpart Limit
213(2)
11.10 Generating Solutions via SL(2, R)-Transformations
215(2)
11.11 Transformed Electrostatic b not = to 0 Solutions
217(11)
11.11.1 Stationary Electromagnetic Solution
218(1)
11.11.2 Clement Spinning Solution
219(5)
11.11.3 Martinez-Teitelboim-Zanelli Solution
224(4)
11.12 Transformed Magnetostatic a not = to 0 Solutions
228(7)
11.12.1 Stationary Magneto-Electric Solution
229(1)
11.12.2 Dias-Lemos Magnetic BTZ-Solution Counterpart
229(6)
11.13 Transformed Cataldo Hybrid Static Solution
235(3)
11.13.1 Mass, Energy and Momentum
237(1)
11.14 Summary on Electromagnetic Maxwell Solutions
238(2)
12 Black Holes Coupled To Nonlinear Electrodynamics 240(17)
12.1 Nonlinear Electrodynamics in (2+1) Dimensions
241(1)
12.2 General Nonlinear Electrostatic Solution
242(2)
12.2.1 Static Charged Peldan Solution
244(1)
12.3 Cataldo-Garcia Nonlinear EBI Charged Black Hole
244(5)
12.3.1 Static Cyclic Symmetric EBI Solution
246(1)
12.3.2 Cataldo-Garcia Black Hole to EBI
247(2)
12.4 Regular Black Hole Solution
249(3)
12.4.1 Regularity
250(1)
12.4.2 Horizons
251(1)
12.4.3 Thermodynamics
252(1)
12.5 Coulomb-Like Black Hole Solution
252(4)
12.5.1 Horizons for the Coulomb-Like Solution
254(2)
12.6 Stationary Nonlinear Electrodynamics Black Holes
256(1)
13 Dilaton Field Minimally Coupled to (2 + 1) Gravity 257(29)
13.1 Scalar Field Minimally Coupled to Einstein Gravity
257(1)
13.2 Static Black Hole Coupled to a Scalar Psi(r) = k ln(r)
258(5)
13.2.1 Quasi Local Momentum, Energy, and Mass
261(1)
13.2.2 Classification of the Energy-Momentum and Cotton Tensors
262(1)
13.3 General Static Chan-Mann Solution
263(3)
13.3.1 Regular F(r)+ Function for the Metric g+
264(1)
13.3.2 Chan-Mann Solution
265(1)
13.4 Stationary Solution Coupled to Psi(r) = k ln(r)
266(4)
13.4.1 Momentum, Energy, and Maas for a Rotating Dilaton
268(1)
13.4.2 Classification of the Energy-Momentum and Cotton Tensors
269(1)
13.5 Stationary Dilaton Solutions Generated via SL(2, R) Transformations
270(4)
13.5.1 Sub-Class of Rotating Dilaton Black Holes
272(1)
13.5.2 Rotating Chan-Mann Dilaton Black Hole
273(1)
13.6 Dilaton Coupled to Einstein-Maxwell Fields
274(1)
13.6.1 Einstein-Maxwell-Scalar Field Equations
274(1)
13.7 Static Charged Solution Coupled to Psi(r) = k ln(r)
275(4)
13.7.1 Quasi-Local Mass, Momentum, Energy for Charged Dilaton
277(1)
13.7.2 Algebraic Classification of the Field, Energy-Momentum, and Cotton Tensors
277(2)
13.8 Stationary Charged Dilaton Generated via SL(2, R)
279(5)
13.8.1 Quasi-Local Mass and Momentum
280(2)
13.8.2 Algebraic Classification of the Field, Energy-Momentum, and Cotton Tensors
282(2)
13.8.3 Particular Stationary Charged Dilaton via SL(2, R) Transformation
284(1)
13.9 Summary of Dilaton Minimally Coupled to Gravity
284(2)
14 Scalar Field Non-Minimally Coupled to (2+1) Gravity 286(6)
14.1 Einstein Equations for Non-Minimally Coupled Scalar Field
286(1)
14.1.1 Martinez-Zanelli Black Hole Solution with Tmuto the mu = 0
287(1)
14.2 Stationary Black Hole for a Non-Minimally Coupled Scalar Field
287(5)
14.2.1 Quasi-Local Momentum, Energy, and Mass
288(1)
14.2.2 Algebraic Classification of the Ricci, Energy-Momentum, and Cotton Tensors
289(3)
15 Low-Energy (2+1) String Gravity 292(11)
15.1 n-Dimensional Heterotic String Dynamical Equations
292(3)
15.1.1 String Frame
292(1)
15.1.2 Einstein Frame
293(2)
15.2 Dynamical Equations in (2+1) String Gravity
295(1)
15.3 Horne-Horowitz Black String
296(2)
15.4 Horowitz-Welch Black String
298(2)
15.5 Chan-Mann String Solution
300(3)
15.5.1 Einstein-Maxwell-Scalar Field Equations
300(1)
15.5.2 Static and Stationary Black String Solutions
301(2)
16 Topologically Massive Gravity 303(4)
16.1 Chern-Simons Action and Field Equations of TMG
304(1)
16.2 Exact Vacuum Solutions of TMG with Lambda
305(2)
17 Bianchi-Type (BT) Spacetimes in TMG; Petrov Type D 307(26)
17.1 Generalities on Bianchi-Type (BT) 3D Spaces
307(2)
17.2 Nutku-Baekler-Ortiz "Timelike" BT VIII Spacetime
309(2)
17.2.1 Nutku Timelike Biaxially Squashed Metric
310(1)
17.3 Nutku-Baekler-Ortiz "Spacelike" Squashed BT VIII Spacetime
311(3)
17.3.1 Spacelike Biaxially Squashed Metric; Nutku Solution Counterpart
313(1)
17.4 Nutku-Baekler-Ortiz Solutions of Bianchi Type III
314(3)
17.4.1 Nutku-Baekler-Ortiz BT III Timelike Solution with Lambda = 0
315(1)
17.4.2 Nutku-Baekler-Ortiz BT III Spacelike Solution with Lambda = 0
316(1)
17.5 Timelike Biaxially Squashed Metrics
317(10)
17.5.1 Representation of the Vacuum Biaxially Squashed Solutions
317(2)
17.5.2 Eigenvectors of the Cotton Tensor; Triad Formulation
319(1)
17.5.3 Complex Extension Toward the Spacelike Squashed Metric
320(1)
17.5.4 Alternative Metric Representation of ds2sl
321(6)
17.6 Spacelike Biaxially Squashed Metrics
327(6)
17.6.1 Eigenvectors of the Cotton Tensor; Triad Formulation
328(1)
17.6.2 Alternative Metric Representation of ds2sl
329(4)
18 Petrov Type N Wave Metrics 333(12)
18.1 Brinkmann-Like 3D Metric
333(1)
18.2 AdS3 Non-Covariantly Constant TN-Waves
334(6)
18.2.1 AdS3 TN-Waves with Lambda not = to 0
336(1)
18.2.2 Nutku TN-Wave Solution
336(1)
18.2.3 Clement TN-Wave Solution
337(1)
18.2.4 Ayon-HassaIne TN-Wave Solution
337(1)
18.2.5 Olmez-Sarioglu-Tekin TN-Wave Solution
338(1)
18.2.6 Dereli-Sarioglu TN-Wave Solution
338(1)
18.2.7 Carlip-Deser-Waldron-Wise TN-Wave Solution
338(1)
18.2.8 Gibbons-Pope-Sezgin TN-Wave Solution
338(1)
18.2.9 Anninos-Li-Padi-Song-Strominger TN-Wave Solution
338(1)
18.2.10 Garbarz-Giribet-Vasquez TN-Wave Solution
339(1)
18.3 pp-Wave Solutions; Lambda = 0
340(5)
18.3.1 Martinez-Shepley pp-Wave Solution; Lambda = 0
340(1)
18.3.2 Aragon pp-Wave Solution; Lambda = 0
341(1)
18.3.3 Percacci-Sodano-Vuorio pp-Wave Solution; Lambda = 0
341(1)
18.3.4 Hall-Morgan-Perjes pp-Wave Solution; Lambda = 0
341(1)
18.3.5 Dereli-Tucker pp-Wave Solution; Lambda = 0
342(1)
18.3.6 Deser-Steif pp-Wave Solution; Lambda = 0
342(1)
18.3.7 Clement pp-Wave Solution; Lambda = 0
342(1)
18.3.8 Cavaglia pp-Wave Solution; Lambda = 0
343(1)
18.3.9 Dereli-Sarioglu pp-Wave Solution; Lambda = 0
343(1)
18.3.10 Garcia-Hehl-Heinicke-Macias pp-Wave Solution; Lambda = 0
343(1)
18.3.11 Macias-Camacho pp-Wave Solution; Lambda = 0
344(1)
19 Kundt Spacetimes in TMG 345(59)
19.1 Null Geodesic Vector Field
345(2)
19.2 General Kundt Metrics
347(3)
19.2.1 3D Kundt Metric
348(2)
19.3 3D Canonical Kundt Metric
350(7)
19.3.1 Petrov Classification of the Cotton and Traceless Ricci Tensors
352(3)
19.3.2 Sub-Branch W1(r) of the General Kundt Metric in TMG
355(1)
19.3.3 Kundt Metric Structure for W1(r)
356(1)
19.4 Type II CSI Kundt Metric; W1 = 2mu/3
357(2)
19.4.1 Negative Cosmological Constant; Lambda = -m2
358(1)
19.4.2 Positive Cosmological Constant; Lambda = m2
359(1)
19.4.3 Zero Cosmological Constant; Lambda = 0
359(1)
19.5 Type D CSI Kundt Solutions; W1 = 2mu/3, F0 = 0
359(1)
19.6 Petrov Type III Kundt Metrics
360(1)
19.7 Type III Kundt Solution; Lambda = 0, W1 = 0
361(1)
19.7.1 Type N pp-Wave Limit
362(1)
19.8 Type III Kundt Solution; Lambda = 0, W1 = -2/r
362(4)
19.8.1 Type N Limit
365(1)
19.9 Type III Kundt Solution; Lambda = -m2, W1 = -2m
366(3)
19.9.1 Type III Kundt Solution; Lambda = -m2, W1 = -2m, mu ±m
366(1)
19.9.2 Type III Kundt Solution; Lambda = -m2, W1 = -2m, mu = m
367(1)
19.9.3 Type III Kundt Solution; Lambda = -m2, W1 = -2m, mu = -m
368(1)
19.10 Type III Kundt Metric; Lambda = -m2, W1 = -2m coth(m r)
369(15)
19.10.1 Type III Kundt Solution; Lambda = -m2, W1 = -2m phi
375(2)
19.10.3 Type III Kundt Solution; Lambda = -m2, W1 = -2m coth(m r), mu = -m
377(4)
19.10.4 Type III Kundt Solution; Lambda = -m2, W1 = -2m coth(mr), mu = m
381(3)
19.11 Type III Kundt Metric; Lambda = -m2; W1 = -2 m tanh(m r)
384(14)
19.11.1 Type III Kundt Solution; Lambda = -m2, W1 = -2m tanh(mr), mu # ±m
386(5)
19.11.2 Type III Kundt Solution; Lambda = -m2, W1 = -2m tanh(mr), mu = -m
391(3)
19.11.3 Type III Kundt Solution; Lambda = -m2, W1 = -2m tanh(mr), mu = m
394(4)
19.12 Type III Kundt Metric; Lambda = m2, W1 = -2m cot(m r), mu
398(6)
19.12.1 Solution Phi Through VP
399(2)
19.12.2 Multi Exponent-Integral Representation of Phi
401(3)
20 Cotton Tensor in Riemannian Spacetimes 404(17)
20.1 Bianchi Identities and the Irreducible Decomposition of the Curvature
405(4)
20.2 Cotton 2-Form
409(6)
20.3 Conformal Correspondence
415(1)
20.4 Criteria for Conformal Flatness
416(1)
20.5 Classification of the Cotton 2-Form in 3D
417(4)
20.5.1 Euclidean Signature
418(1)
20.5.2 Lorentzian Signature
419(2)
References 421(9)
Index 430
Alberto A. Garcķa-Dķaz is Emeritus Professor at the Center for Research and Advanced Studies of the National Polytechnic Institute, Mexico (CINVESTAV-IPN). His research throughout his career has focused on algebraic classification in four dimensional gravity, nonlinear electrodynamics and dilaton fields.
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