In metric theories of gravitation, particularly general relativity, a static spherically symmetric perfect fluid solution (a term which is often abbreviated as ssspf) is a spacetime equipped with suitable tensor fields which models a static round ball of a fluid with isotropicpressure.
Such solutions are often used as idealized models of stars, especially compact objects such as white dwarfs and especially neutron stars. In general relativity, a model of an isolated star (or other fluid ball) generally consists of a fluid-filled interior region, which is technically a perfect fluid solution of the Einstein field equation, and an exterior region, which is an asymptotically flatvacuum solution. These two pieces must be carefully matched across the world sheet of a spherical surface, the surface of zero pressure. (There are various mathematical criteria called matching conditions for checking that the required matching has been successfully achieved.) Similar statements hold for other metric theories of gravitation, such as the Brans–Dicke theory.
In this article, we will focus on the construction of exact ssspf solutions in our current Gold Standard theory of gravitation, the theory of general relativity. To anticipate, the figure at right depicts (by means of an embedding diagram) the spatial geometry of a simple example of a stellar model in general relativity. The euclidean space in which this two-dimensional Riemannian manifold (standing in for a three-dimensional Riemannian manifold) is embedded has no physical significance, it is merely a visual aid to help convey a quick impression of the kind of geometrical features we will encounter.
Short historyEdit
We list here a few milestones in the history of exact ssspf solutions in general relativity:
1939: Tolman gives seven ssspf solutions, two of which are suitable for stellar models,
1949: Wyman ssspf and first generating function method,
1958: Buchdahl ssspf, a relativistic generalization of a Newtonian polytrope,
1967: Kuchowicz ssspf,
1969: Heintzmann ssspf,
1978: Goldman ssspf,
1982: Stewart ssspf,
1998: major reviews by Finch & Skea and by Delgaty & Lake,
2000: Fodor shows how to generate ssspf solutions using one generating function and differentiation and algebraic operations, but no integrations,
2001: Nilsson & Ugla reduce the definition of ssspf solutions with either linear or polytropicequations of state to a system of regular ODEs suitable for stability analysis,
2002: Rahman & Visser give a generating function method using one differentiation, one square root, and one definite integral, in isotropic coordinates, with various physical requirements satisfied automatically, and show that every ssspf can be put in Rahman-Visser form,
2003: Lake extends the long-neglected generating function method of Wyman, for either Schwarzschild coordinates or isotropic coordinates,
2004: Martin & Visser algorithm, another generating function method which uses Schwarzschild coordinates,
2004: Martin gives three simple new solutions, one of which is suitable for stellar models,
2005: BVW algorithm, apparently the simplest variant now known
ReferencesEdit
Oppenheimer, J. R. & Volkov, G. B. (1939). "On massive neutron cores". Phys. Rev. 55 (4): 374–381. Bibcode:1939PhRv...55..374O. doi:10.1103/PhysRev.55.374. The original paper presenting the Oppenheimer-Volkov equation.
Oppenheimer, J. R. & Snyder, H.. (1939). "On continued gravitational collapse". Phys. Rev. 56 (5): 455–459. Bibcode:1939PhRv...56..455O. doi:10.1103/PhysRev.56.455.
Misner, Charles; Thorne, Kip S. & Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. ISBN0-7167-0344-0. See section 23.2 and box 24.1 for the Oppenheimer-Volkov equation.
Schutz, Bernard F. (1985). A First Course in General Relativity. Cambridge: Cambridge University Press. ISBN0-521-27703-5. See chapter 10 for the Buchdahl theorem and other topics.
Bose, S. K. (1980). An introduction to General Relativity. New York: Wiley. ISBN0-470-27054-3. See chapter 6 for a more detailed exposition of white dwarf and neutron star models than can be found in other gtr textbooks.
Lake, Kayll (1998). "Physical Acceptability of Isolated, Static, Spherically Symmetric, Perfect Fluid Solutions of Einstein's Equations". Comput. Phys. Commun. 115 (2–3): 395–415. arXiv:gr-qc/9809013. Bibcode:1998CoPhC.115..395D. doi:10.1016/S0010-4655(98)00130-1. S2CID 17957408. eprint version An excellent review stressing problems with the traditional approach which are neatly avoided by the Rahman-Visser algorithm.
Nilsson, U. S. & Uggla, C. (2001). "General Relativistic Stars: Linear Equations of State". Annals of Physics. 286 (2): 278–291. arXiv:gr-qc/0002021. Bibcode:2000AnPhy.286..278N. doi:10.1006/aphy.2000.6089. S2CID 6381238. eprint version
Nilsson, U. S. & Uggla, C. (2001). "General Relativistic Stars: Polytropic Equations of State". Annals of Physics. 286 (2): 292–319. arXiv:gr-qc/0002022. Bibcode:2000AnPhy.286..292N. doi:10.1006/aphy.2000.6090. S2CID 17998564. eprint version The Nilsson-Uggla dynamical systems.
Lake, Kayll (2003). "All static spherically symmetric perfect fluid solutions of Einstein's Equations". Phys. Rev. D. 67 (10): 104015. arXiv:gr-qc/0209104. Bibcode:2003PhRvD..67j4015L. doi:10.1103/PhysRevD.67.104015. S2CID 119447644. eprint version Lake's algorithms.
Martin, Damien & Visser, Matt (2004). "Algorithmic construction of static perfect fluid spheres". Phys. Rev. D. 69 (10): 104028. arXiv:gr-qc/0306109. Bibcode:2004PhRvD..69j4028M. doi:10.1103/PhysRevD.69.104028. S2CID 119397218. eprint version The Rahman-Visser algorithm.
Boonserm, Petarpa; Visser, Matt & Weinfurtner, Silke (2005). "Generating perfect fluid spheres in general relativity". Phys. Rev. D. 71 (12): 124037. arXiv:gr-qc/0503007. Bibcode:2005PhRvD..71l4037B. doi:10.1103/PhysRevD.71.124037. S2CID 10332787. eprint version The BVW solution generating method.
October 19, 2023
static, spherically, symmetric, perfect, fluid, metric, theories, gravitation, particularly, general, relativity, static, spherically, symmetric, perfect, fluid, solution, term, which, often, abbreviated, ssspf, spacetime, equipped, with, suitable, tensor, fie. In metric theories of gravitation particularly general relativity a static spherically symmetric perfect fluid solution a term which is often abbreviated as ssspf is a spacetime equipped with suitable tensor fields which models a static round ball of a fluid with isotropic pressure Such solutions are often used as idealized models of stars especially compact objects such as white dwarfs and especially neutron stars In general relativity a model of an isolated star or other fluid ball generally consists of a fluid filled interior region which is technically a perfect fluid solution of the Einstein field equation and an exterior region which is an asymptotically flat vacuum solution These two pieces must be carefully matched across the world sheet of a spherical surface the surface of zero pressure There are various mathematical criteria called matching conditions for checking that the required matching has been successfully achieved Similar statements hold for other metric theories of gravitation such as the Brans Dicke theory In this article we will focus on the construction of exact ssspf solutions in our current Gold Standard theory of gravitation the theory of general relativity To anticipate the figure at right depicts by means of an embedding diagram the spatial geometry of a simple example of a stellar model in general relativity The euclidean space in which this two dimensional Riemannian manifold standing in for a three dimensional Riemannian manifold is embedded has no physical significance it is merely a visual aid to help convey a quick impression of the kind of geometrical features we will encounter Short history EditWe list here a few milestones in the history of exact ssspf solutions in general relativity 1916 Schwarzschild fluid solution 1939 The relativistic equation of hydrostatic equilibrium the Oppenheimer Volkov equation is introduced 1939 Tolman gives seven ssspf solutions two of which are suitable for stellar models 1949 Wyman ssspf and first generating function method 1958 Buchdahl ssspf a relativistic generalization of a Newtonian polytrope 1967 Kuchowicz ssspf 1969 Heintzmann ssspf 1978 Goldman ssspf 1982 Stewart ssspf 1998 major reviews by Finch amp Skea and by Delgaty amp Lake 2000 Fodor shows how to generate ssspf solutions using one generating function and differentiation and algebraic operations but no integrations 2001 Nilsson amp Ugla reduce the definition of ssspf solutions with either linear or polytropic equations of state to a system of regular ODEs suitable for stability analysis 2002 Rahman amp Visser give a generating function method using one differentiation one square root and one definite integral in isotropic coordinates with various physical requirements satisfied automatically and show that every ssspf can be put in Rahman Visser form 2003 Lake extends the long neglected generating function method of Wyman for either Schwarzschild coordinates or isotropic coordinates 2004 Martin amp Visser algorithm another generating function method which uses Schwarzschild coordinates 2004 Martin gives three simple new solutions one of which is suitable for stellar models 2005 BVW algorithm apparently the simplest variant now knownReferences EditOppenheimer J R amp Volkov G B 1939 On massive neutron cores Phys Rev 55 4 374 381 Bibcode 1939PhRv 55 374O doi 10 1103 PhysRev 55 374 The original paper presenting the Oppenheimer Volkov equation Oppenheimer J R amp Snyder H 1939 On continued gravitational collapse Phys Rev 56 5 455 459 Bibcode 1939PhRv 56 455O doi 10 1103 PhysRev 56 455 Misner Charles Thorne Kip S amp Wheeler John Archibald 1973 Gravitation San Francisco W H Freeman ISBN 0 7167 0344 0 See section 23 2 and box 24 1 for the Oppenheimer Volkov equation Schutz Bernard F 1985 A First Course in General Relativity Cambridge Cambridge University Press ISBN 0 521 27703 5 See chapter 10 for the Buchdahl theorem and other topics Bose S K 1980 An introduction to General Relativity New York Wiley ISBN 0 470 27054 3 See chapter 6 for a more detailed exposition of white dwarf and neutron star models than can be found in other gtr textbooks Lake Kayll 1998 Physical Acceptability of Isolated Static Spherically Symmetric Perfect Fluid Solutions of Einstein s Equations Comput Phys Commun 115 2 3 395 415 arXiv gr qc 9809013 Bibcode 1998CoPhC 115 395D doi 10 1016 S0010 4655 98 00130 1 S2CID 17957408 eprint version An excellent review stressing problems with the traditional approach which are neatly avoided by the Rahman Visser algorithm Fodor Gyula Generating spherically symmetric static perfect fluid solutions 2000 Fodor s algorithm Nilsson U S amp Uggla C 2001 General Relativistic Stars Linear Equations of State Annals of Physics 286 2 278 291 arXiv gr qc 0002021 Bibcode 2000AnPhy 286 278N doi 10 1006 aphy 2000 6089 S2CID 6381238 eprint version Nilsson U S amp Uggla C 2001 General Relativistic Stars Polytropic Equations of State Annals of Physics 286 2 292 319 arXiv gr qc 0002022 Bibcode 2000AnPhy 286 292N doi 10 1006 aphy 2000 6090 S2CID 17998564 eprint version The Nilsson Uggla dynamical systems Lake Kayll 2003 All static spherically symmetric perfect fluid solutions of Einstein s Equations Phys Rev D 67 10 104015 arXiv gr qc 0209104 Bibcode 2003PhRvD 67j4015L doi 10 1103 PhysRevD 67 104015 S2CID 119447644 eprint version Lake s algorithms Martin Damien amp Visser Matt 2004 Algorithmic construction of static perfect fluid spheres Phys Rev D 69 10 104028 arXiv gr qc 0306109 Bibcode 2004PhRvD 69j4028M doi 10 1103 PhysRevD 69 104028 S2CID 119397218 eprint version The Rahman Visser algorithm Boonserm Petarpa Visser Matt amp Weinfurtner Silke 2005 Generating perfect fluid spheres in general relativity Phys Rev D 71 12 124037 arXiv gr qc 0503007 Bibcode 2005PhRvD 71l4037B doi 10 1103 PhysRevD 71 124037 S2CID 10332787 eprint version The BVW solution generating method Retrieved from https en wikipedia org w index php title Static spherically symmetric perfect fluid 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