Exact Solutions (Existence and Uniqueness) / Symmetries / Classifications

  1. Schwarzschild K: Über das Gravitationsfeld einer Kugel aus inkompressibler Flüssigkeit nach der Einsteinschen Theorie, Sitz.-Ber. Preuß. Akad. Wiss., Berlin (1916), 424 - 434 [in German]
    English translation: Preprint physics/9905030
  2. Curzon H E J, Proc. Lond. Math. Soc. 23 (1924), 477
  3. Wahlquist H D: Phys. Rev. 172 (1968), 1291
  4. Boyer R H: Geodesic Killing Orbits and Bifurcate Killing Horizons, Proc. R. Soc. Lond. A 311 (1969), 245
    NB: Written posthumously by J Ehlers and J L Stachel.
  5. Wainwright J: A Class of Algebraically Special Perfect Fluid Space-Times, Commun. Math. Phys. 17 (1970), 42
  6. Ernst F J: J. Math. Phys. 17 (1976), 54
  7. Stephani H: A Note on Killing Tensors, Gen. Rel. Grav. 9 (1978), 789
  8. Allnut J A: A Petrov Type-III Perfect Fluid Solution of Einstein's Equations, Gen. Rel. Grav. 13 (1981), 1017
    NB: $(\mu+3p) < 0$; one spacelike KVF; shearing, expanding, twist-free repeated PND; non-geodesic, shearing, expanding, irrotational $u^{\mu}$; non-conformally flat spacelike 3-surfaces orthogonal to $u^{\mu}$.
  9. Friedrich H: On the Regular and the Asymptotic Characteristic Initial Value Problem for Einstein's Vacuum Field Equations, Proc. R. Soc. Lond. A 375 (1981), 169
    NB: Employs NP spin-frame formalism. Very technical. (+ - - -). Communicated by S W Hawking.
  10. Friedrich H: The Asymptotic Characteristic Initial Value Problem for Einstein's Vacuum Field Equations as an Initial Value Problem for a First-Order Quasilinear Symmetric Hyperbolic System, Proc. R. Soc. Lond. A 378 (1981), 401
    NB: Employs NP spin-frame formalism. Very technical. (+ - - -). Communicated by S W Hawking.
  11. Karlhede A, M A H MacCallum: On Determining the Isometry Group of a Riemann Space, Gen. Rel. Grav. 14 (1982), 673
  12. Herlt E, H Stephani: Algebraically Special, Shearfree, Diverging, and Twisting Einstein-Maxwell Fields, Class. Quantum Grav. 1 (1984), 95
  13. Kramer D: A New Solution for a Rotating Perfect Fluid In General Relativity, Class. Quantum Grav. 1 (1984), L3
    NB: Abelian $G_{2}$, stationary and axisymmetric, rigidly rotating, $(\rho+3p)=const$, Petrov type D.
  14. Neugebauer G, E Herlt: Einstein-Maxwell Fields Inside and Outside Rotating Sources as Minimal Surfaces, Class. Quantum Grav. 1 (1984), 695
  15. McIntosh C B G: Symmetries of Vacuum Type-N Metrics, Class. Quantum Grav. 2 (1985), 87
  16. Bonnor W B, W Davidson: Petrov Type II Perfect Fluid Spacetimes with Vorticity, Class. Quantum Grav. 2 (1985), 775
  17. Stephani H, E Herlt: Twisting Type-N Vacuum Solutions with Two Non-Commuting Killing Vectors Do Exist, Class. Quantum Grav. 2 (1985), L63
    NB: Comments on McIntosh.
  18. Kitchingham D W: The Application of the Homogeneous Hilbert Problem of Hauser and Ernst to Cosmological Models with Spatial Axes of Symmetry, Class. Quantum Grav. 3 (1986), 133
  19. Garfinkle D, Q Tian: Spacetimes With Cosmological Constant And A Conformal Killing Field Have Constant Curvature, Class. Quantum Grav. 4 (1987), 137
    NB: Results in either de Sitter or anti-de Sitter, depending on sign of $\Lambda$.
  20. McIntosh C B G et al: Type II and III Twisting Vacuum Metrics and Symmetries, Class. Quantum Grav. 4 (1987), 117
  21. Patra A C, D Roy: A New Solution For A Rotating Perfect Fluid In General Relativity, Class. Quantum Grav. 4 (1987), 195
    NB: Refers to Kramer (1984).
  22. Senovilla J M M: On Petrov Type-D Stationary Axisymmetric Rigidly Rotating Perfect-Fluid Metrics, Class. Quantum Grav. 4 (1987), L115
  23. Stephani H: Some Perfect Fluid Solutions of Einstein's Field Equations without Symmetries, Class. Quantum Grav. 4 (1987), 125
  24. Kramer D: , Class. Quantum Grav. 5 (1988), 393
  25. Kramer D: Two Charged Masses in Equilibrium, Class. Quantum Grav. 5 (1988), 1435
    NB: Electrostatic Einstein-Maxwell field.
  26. Castejón-Amenedo J, M A H MacCallum, V S Manko: On an Axisymmetric Solution of the Vacuum Einstein Equations for a Stationary Rotating Mass, Class. Quantum Grav. 6 (1989), L211
  27. Hall G S: The Global Extension of Local Symmetries in General Relativity, Class. Quantum Grav. 6 (1989), 157
  28. Hoenselaers C, B G Schmidt: Exact Solutions for a Simple Model of Radiation Damping, Class. Quantum Grav. 6 (1989), 867
  29. Stephani H, R Grosso: Perfect Fluids with 4-Velocity Spanned by Two Commuting Killing Vectors, Class. Quantum Grav. 6 (1989), 1673
    NB: $\vec{u}=\left(\,\partial_{t}+\Omega\,\partial_{\varphi}\,\right)/ \sqrt{-H}$, differential rotation $\Omega$, relates to solutions that are stationary and axisymmetric, required: the metric be regular on the symmetry axis and there exists a closed surface with $p=0$.
  30. Bohr H, K Buchner: Computer Simulations of Stationary Cylindrical Solutions to Einstein's Equations for Perfect Fluids, Class. Quantum Grav. 7 (1990), 771
    NB: Metric regular on the $z$-axis; by varying initial conditions, NO asymptotically flat spacetimes were obtained.
  31. Barnes A, R R Rowlingson: Killing Vectors in Conformally Flat Perfect Fluid Spacetimes, Class. Quantum Grav. 7 (1990), 1721
  32. Bonnor W B: The C-Metric in Bondi's Coordinates, Class. Quantum Grav. 7 (1990), L229
  33. Carminati J: Type D Perfect-Fluid Spacetimes with a Non-Null Electromagnetic Field, Class. Quantum Grav. 7 (1990), 1543
  34. Chinea F J, L M González-Romero: Interior Gravitational Field of a Stationary, Axially Symmetric Perfect Fluid in Irrotational Motion, Class. Quantum Grav. 7 (1990), L99
  35. Coley A A, B O J Tupper: Spacetimes Admitting Inherting Conformal Killing Vector Fields, Class. Quantum Grav. 7 (1990), 1961
  36. Garc\'{\i}a A D: Type D Divergenceless Charged Perfect Fluid Solutions, Class. Quantum Grav. 7 (1990), 1299
  37. Gaffet B: The Einstein Equations with Two Commuting Killing Vectors, Class. Quantum Grav. 7 (1990), 2017
  38. Hoenselaers C, Z Perjés: Multipole Moments Of Axisymmetric Electrovacuum Spacetimes, Class. Quantum Grav. 7 (1990), 1819
  39. Horský J, N V Mitskievitch: Generalised Kramer's Metric, Class. Quantum Grav. 7 (1990), 1523
    NB: Refers to Kramer (1988).
  40. Uggla C, K Rosquist, R T Jantzen: Geometrizing the Dynamics of Bianchi Cosmology, Phys. Rev. D 42 (1990), 404
  41. Arianrhod R et al: Principal Null Directions of the Curzon Metric, Class. Quantum Grav. 8 (1991), 1519
    NB: See Curzon (1924) and end of last page.
  42. Coley A A: Fluid Spacetimes Admitting a Conformal Killing Vector Parallel To The Velocity Vector, Class. Quantum Grav. 8 (1991), 955
  43. Coley A A, B O J Tupper: Fluid Spacetimes Admitting Covariantly Constant Vectors and Tensors, Gen. Rel. Grav. 23 (1991), 1113
  44. Fackerell E D, R P Kerr: Einstein Vacuum Field Equations with a Simple Non-Null Killing Vector, Gen. Rel. Grav. 23 (1991), 861
    NB: One spacelike KVF.
  45. Hewitt C G: Algebraic Invariant Curves in Cosmological Dynamical Systems and Exact Solutions, Gen. Rel. Grav. 23 (1991), 1363
  46. Bonnor W B: Physical Interpretation of Vacuum Solutions of Einstein's Equations. Part I. Time-Independent Solutions, Gen. Rel. Grav. 24 (1992), 551
  47. Carminati J, F I Cooperstock: Herlt Metrics and Gravitational-Electrostatic Balance in General Relativity, Gen. Rel. Grav. 24 (1992), 881
  48. Castejón-Amenedo J, A A Coley: Exact Solutions with Conformal Killing Vector Fields, Class. Quantum Grav. 9 (1992), 2203
    NB: Purpose is to formalise the role of the conformal group $C_{r}$ in general relativity analoguous to the isometry group $G_{r}$.
  49. Chinea F J et al: Singularity-Free Space-Times, Phys. Rev. D 45 (1992), 481
    NB: Refers to Senovilla J M M, Phys. Rev. Lett. 64 (1990), 2219.
  50. Koutras A: Killing Tensors from Conformal Killing Vectors, Class. Quantum Grav. 9 (1992), 1573
  51. Kyriakopoulos E: Interior Axisymmetric Stationary Perfect Fluid Solution of Einstein's Equations, Class. Quantum Grav. 9 (1992), 217
    NB: $\gamma=\frac{2}{3}$.
  52. Ludwig G, Y B Yu: Type N Twisting Vacuum Gravitational Fields, Gen. Rel. Grav. 24 (1992), 93
    NB: Relates to the Hauser metric.
  53. Manko V S: The Exterior Gravitational Field of a Static and Stationary Mass with an Arbitrary Set of Multipole Moments, Gen. Rel. Grav. 24 (1992), 35
    NB: Exact asymptotically flat vacuum solution.
  54. Manko V S, N R Sibgatullin: Kerr Metric Endowed with Magnetic Dipole Moment, Class. Quantum Grav. 9 (1992), L87
  55. Senovilla J M M: New Family of Stationary and Axisymmetric Perfect-Fluid Solutions, Class. Quantum Grav. 9 (1992), L167
    NB: Model with differential rotation.
  56. Sklavenites D: Stationary and Static Axisymmetric Perfect Fluid Solutions, Gen. Rel. Grav. 24 (1992), 935
  57. Wolf T, G Neugebauer: About the Non-Existence of Perfect Fluid Bodies with the Kerr Metric Outside, Class. Quantum Grav. 9 (1992), L37
  58. Aguirregabiria J M et al: Exterior Gravitational Field of a Magnetized Spinning Source Possessing an Arbitrary Mass-Quadrupole Moment, Phys. Rev. D 48 (1993), 622
    NB: Exact asymptotically flat solution of Einstein-Maxwell equations, model for an axisymmetric neutron star (Manko/Sibgatullin involved).
  59. Ellis G F R: Exact and Inexact Solutions of the Einstein Field Equations, in The Renaissance of General Relativity and Cosmology, Eds. G Ellis, A Lanza, J Miller, (Cambridge: Cambridge University Press, 1993)
  60. Kind S, J Ehlers: Initial-Boundary Value Problem for the Spherically Symmetric Einstein Equations for a Perfect Fluid, Class. Quantum Grav. 10 (1993), 2123
    NB: Quite technical.
  61. Kind S, J Ehlers, B G Schmidt: Relativistic Stellar Oscillations Treated as an Initial Value Problem, Class. Quantum Grav. 10 (1993), 2137
    NB: Linearised perturbations of a star in hydrostatic equilibrium. Quite technical.
  62. Mars M, J M M Senovilla: Axial Symmetry and Conformal Killing Vectors, Class. Quantum Grav. 10 (1993), 1633
  63. McLenaghan R G, N Van den Bergh: Spacetimes Admitting Killing 2-Spinors, Class. Quantum Grav. 10 (1993), 2179
  64. Rácz I: Maxwell Fields in Spacetimes Admitting Non-Null Killing Vectors, Class. Quantum Grav. 10 (1993), L167
    NB: $\mbox{\pounds}_{\vec{\xi}}\, F_{\mu\nu} = 0$.
  65. Stephani H: A Note on the Solutions of the Diverging, Twisting Type N Vacuum Field Equations, Class. Quantum Grav. 10 (1993), 2187
    NB: ONE reference: KSMH `80.
  66. Sussman R A: New Solutions for Heat Conducting Fluids with a Normal Shear-Free Flow, Class. Quantum Grav. 10 (1993), 2675
    NB: $0 = \sigma = \omega$, non-zero bulk viscous pressure and energy current density; solutions of Petrov type D and $0$. "...reducing to the `Stephani Universe' as heat conduction vanishes."
  67. Arianrhod R, A W-C Lun, C B G McIntosh, Z Perjés: Magnetic Curvatures, Class. Quantum Grav. 11 (1994), 2331
  68. Bonnor W B, J B Griffiths, M A H MacCallum: Physical Interpretation of Einstein's Equations. Part II. Time-Dependent Solutions, Gen. Rel. Grav. 26 (1994), 687
  69. Coley A A, D McManus: On Spacetimes Admitting Shear-Free, Irrotational, Geodesic Time-like Congruences, Class. Quantum Grav. 11 (1994), 1261
    NB: General $T_{ab}$.
  70. Finley III J D, J F Plebański, M Przanowski: Third-Order ODEs for Twisting Type-N Vacuum Solutions, Class. Quantum Grav. 11 (1994), 157
  71. Garc\'{\i}a A: A New Stationary Axisymmetric Perfect Fluid Type D Solution With Differential Rotation, Class. Quantum Grav. 11 (1994), L45
    NB: Abelian $G_{2}$, $\partial_{t}$, $\partial_{\phi}$, refers to Senovilla ibid 1992, $u^{\mu}=A(x)\,(\,\delta^{\mu}\!_{t} +\omega(x)\,\delta^{\mu}\!_{\phi}\,)$.
  72. McIntosh C B G, A Arianrhod, S Wade, C A Honselaers: Electric and Magnetic Weyl Tensors: Classification and Analysis, Class. Quantum Grav. 11 (1994), 1555
  73. Neugebauer G, R Meinel: General Relativistic Gravitational Field of a Rigidly Rotating Disk of Dust: Axis Potential, Disk Metric, and Surface Mass Density, Phys. Rev. Lett. 73 (1994), 2166
  74. Neugebauer G, R Meinel: General Relativistic Gravitational Field of a Rigidly Rotating Disk of Dust: Solutions in Terms of Ultraelliptic Functions, Phys. Rev. Lett. 75 (1995), 3046
  75. Meinel R, G Neugebauer: Asymptotically Flat Solutions to the Ernst Equation with Reflection Symmetry, Class. Quantum Grav. 12 (1995), 2045
    NB: Overlaps with P Kordas, Class. Quantum Grav. 12 (1995), 2037.
  76. Uggla C, R T Jantzen, K Rosquist: Exact Hypersurface-Homogeneous Solutions in Cosmology and Astrophysics, Phys. Rev. D 51 (1995), 5522. Also: Preprint gr-qc/9503061.
  77. Finley III J D, J F Plebański, M Przanowski: An Iterative Approach to Twisting and Diverging, Type-N, Vacuum Einstein Equations: A (Third-Order) Resolution of Stephani's `Paradox', Class. Quantum Grav. 14 (1997), 489
    NB: Ref. to Stephani H, Class. Quantum Grav. 10 (1993), 2187.
  78. Claudel C M, K P Newman: The Cauchy Problem for Quasi-Linear Hyperbolic Evolution Problems with a Singularity in the Time, Proc. R. Soc. Lond. A 454 (1998), 1073
    NB: Very technical.
  79. Christodoulou D: On the Global Initial Value Problem and the Issue of Singularities (Review), Class. Quantum Grav. 16 (1999), A23
    NB: Einstein field equations in vacuum (asymptoticaly flat cases) and with massless scalar field.
  80. Friedrich H, G Nagy: The Initial Boundary Value Problem for Einstein's Vacuum Field Equation, Commun. Math. Phys. 201 (1999), 619
    NB: Very technical. (+ - - -). Communicated by H Nicolai.
  81. Klainerman S, F Nicolò: On Local and Global Aspects of the Cauchy Problem in General Relativity (Topical Review), Class. Quantum Grav. 16 (1999), R73
    NB: Einstein field equations in vacuum (asymptoticaly flat cases). Very technical.
  82. Friedrich H, A D Rendall: The Cauchy Problem for the Einstein Equations, in Einstein's Field Equations and their Physical Interpretation, ed. B G Schmidt, (Berlin: Springer, 2000), 127. Also: Preprint gr-qc/0002074, AEI-2000-012.
  83. Rendall A D: Local and Global Existence Theorems for the Einstein Equations, Max-Planck-Gesellschaft Living Reviews Series, No. 2000-1

Selected References
Last revision: Tue, 15-8-2000 (This page is under construction)