In metric theories of gravitation, particularly

general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...

, a static spherically symmetric perfect fluid solution (a term which is often abbreviated as ssspf) is a

spacetime In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...

equipped with suitable

tensor field In mathematics and physics, a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space or manifold) or of the physical space. Tensor fields are used in differential geometry, ...

s which models a static round ball of a fluid with

isotropic In physics and geometry, isotropy () is uniformity in all orientations. Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also ...

pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and eve ...

. Such solutions are often used as idealized models of

star A star is a luminous spheroid of plasma (physics), plasma held together by Self-gravitation, self-gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night sk ...

s, especially compact objects such as

white dwarf A white dwarf is a Compact star, stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very density, dense: in an Earth sized volume, it packs a mass that is comparable to the Sun. No nuclear fusion takes place i ...

s and especially

neutron star A neutron star is the gravitationally collapsed Stellar core, core of a massive supergiant star. It results from the supernova explosion of a stellar evolution#Massive star, massive star—combined with gravitational collapse—that compresses ...

s. 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 In physics, a perfect fluid or ideal fluid is a fluid that can be completely characterized by its rest frame mass density \rho_m and ''isotropic'' pressure . Real fluids are viscous ("sticky") and contain (and conduct) heat. Perfect fluids are id ...

solution of the

Einstein field equation In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Albert Einstein in 1915 in the ...

, and an exterior region, which is an asymptotically flat

vacuum solution In general relativity, a vacuum solution is a Lorentzian manifold whose Einstein tensor vanishes identically. According to the Einstein field equation, this means that the stress–energy tensor also vanishes identically, so that no matter or n ...

. 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 physics, the Brans–Dicke theory of gravitation (sometimes called the Jordan–Brans–Dicke theory) is a competitor to Einstein's general theory of relativity. It is an example of a scalar–tensor theory, a gravitational theory in which the ...

. 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 In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

(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

We 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 In fluid mechanics, hydrostatic equilibrium, also called hydrostatic balance and hydrostasy, is the condition of a fluid or plastic solid at rest, which occurs when external forces, such as gravity, are balanced by a pressure-gradient force. I ...

, 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 In astrophysics, a polytrope refers to a solution of the Lane–Emden equation in which the pressure depends upon the density in the form P = K \rho^ = K \rho^, where is pressure, is density and is a Constant (mathematics), constant of Propo ...

, *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 In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a '' function'' (or '' mapping''); * linearity of a '' polynomial''. An example of a linear function is the function defined by f(x) ...

polytropic A polytropic process is a thermodynamic process that obeys the relation: p V^ = C where ''p'' is the pressure, ''V'' is volume, ''n'' is the polytropic index, and ''C'' is a constant. The polytropic process equation describes expansion and com ...

equations of state In physics and chemistry, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or internal energy. Most mod ...

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 In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of ''nested round spheres''. There are several different types of coordinate chart which are ''adapted'' to this family of nested spheres; the best known is t ...

, 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

References

* The original paper presenting the Oppenheimer-Volkov equation. * * See ''section 23.2'' and ''box 24.1'' for the Oppenheimer-Volkov equation. * See ''chapter 10'' for the Buchdahl theorem and other topics. * See ''chapter 6'' for a more detailed exposition of white dwarf and neutron star models than can be found in other gtr textbooks.
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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.
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The Nilsson-Uggla dynamical systems.
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Lake's algorithms.
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The Rahman-Visser algorithm. *{{cite journal , author1=Boonserm, Petarpa , author2=Visser, Matt , author3=Weinfurtner, Silke , name-list-style=amp , title=Generating perfect fluid spheres in general relativity , journal=Phys. Rev. D , year=2005 , volume=71 , issue= 12 , pages=124037 , doi=10.1103/PhysRevD.71.124037, arxiv = gr-qc/0503007 , bibcode = 2005PhRvD..71l4037B , s2cid=10332787 }
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The BVW solution generating method. Exact solutions in general relativity