Talk:Static spherically symmetric perfect fluid/to do

Tasks not requiring expert attention:

• correct typos

Tasks requiring expert attention:

• gotta start with section helping out laypeople,
• make sure to explain why we say fluid ball rather than gas ball,
• physics of nuclear fusion, neutron drip, whatever, irrelevant for our limited purposes,
• exterior region of any static spherically symmetric stellar model is always part of the Schwarzschild vacuum, which gives us the Kepler mass of our massive object,
• link to hydrostatic equilibtrium, equation of state, barotropic, polytropic, etc., in Newtonian theory, and adumbrate relativistic generalization,
• OV equation or link to separate article; can avoid standard discussion by citing textbooks and referring to discussion in terms of say the BVW model, where OV equation naturally will fall out in slightly disguised form (the idea is to write out the Einstein equations with pressure and density on the RHS, then to make either these two functions of radius or else the two metric functions of radius the variables; this gives the ODE defining the ssspf class, e.g. the BVW equation, or else gives the metric in terms of pressure and density, respectively),
• desiderata include
  • eigenvalue conditions (isotropic pressure, no heat flow),
  • energy conditions,
  • regularity (no conical singularities at origin, finite pressure and density there),
  • accleration of static fluid elements is everywhere radial and outward pointing,
  • causality (appropriate speed of sound; a bit tricky!),
  • stability (decreasing pressure and density),
  • equation of state: actually, not much is known for sure for neutron stars, etc., so theorists are in habit of allowing equation of state to be whatever falls out from their solution; very few known examples actually admin a simple equation EOS (Schwarzschild fluid, Tolman fluid, Buchdahl fluid),
• biggest problem is not getting stress-energy tensor in diagonal form, but getting the diagonal entries positive,
• at a minimum, state Rahman-Visser method and how far it goes toward guaranteeing all these desiderata,
• explain quasilocal mass $m(r)=4\pi \,\int _{0}^{r}\xi ^{2}\,\rho (\xi )\,d\xi$ and how this cannot be interpreted in gtr as saying that the mass is the 'sum' of the density over the volume of the fluid ball, but m(R) is nonetheless the Kepler mass,
• at a minumum, explain the BVW method and how Lie point symmetries of the master ODE (the BVW equation), which is first order linear, give a notion of gauge transformations and solution generation,
• list some examples of famous solutions with fluid type eigenvalues in their BVW form (Schwarzchild vacuum, de Sitter lambdavacuum, Schwarzschild fluid, Tolman IV, Buchdahl, Martin III),
• link to background on Lie point symmetries of an ODE,
• Lie point symmetries in context of special classes of exact solutions are generally a mixture of two types of transformations: first, change coordinate representation without changing physics, and second, perturb physical quantities without changing the geometric meaning of coordinates to obtain a distinct solution in the same class,
• example of a guage transformation is $\zeta \rightarrow k\,\zeta$,
• example of a perturbation is BVW's Theorem II, which changes the pressure, acceleration, and tidal tensor, but leaves the hyperslices, density, and radial coordinate invariant (thus, changes EOS)
• eventually, will compare this simple observation with more challenging case of Weyl vacuums and so forth,
• link to background on extrinsic curvature tensor and junction conditions,
• use either BVW form or Lake forms to explain why all examples regular at origin resemble Schwarzschild fluid near the center,
• link to article on Schwarzschild fluid and its matching in detail,
• as main example, use Tolman IV fluid (which admits exact barotropic equation of state, has easily located surface r = R, easily computed mass, etc.,
• typical features: accelleration maximal just beneath surface, density positive but pressure zero at surface, density and pressure have inflection points, three-spherical near center, but tidal tensor and hyperslice curvature tensor splits into radial/transverse components as approach surface, unusually, has explicit EOS,
• (possibly) link to Misner-Sharp mass, mention Misner-Hernandez mass formula,
• Stewart ssspf is a simple example of an exact fluid solution which is unstable (since density actually increases with radius, while pressure decreases as it should),
• McVittie ssspf is a simple example of a fluid ball obtained in isotropic chart which looks compact in the chart, but density and pressure both vanish at the 'surface' and this actually lies at spatial infinity, so this fluid has no surface,
• Martin III ssspf is a simple example of a fluid ball which is diffuse (maximal acceleration near center, density and pressure fall rapidly, but has surface at finite radius,
• many possible variants of BVW form, Martin-Visser form, etc., including other first order linear master equations, but BVW equation is appparently the simplest of all.

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