Abstract
We consider the spherically symmetric metric with a comoving perfect fluid and non-zero pressure—the Lemaître metric—and present it in the form of a calculational algorithm. We use it to review the definition of mass, and to look at the apparent horizon relations on the observer’s past null cone. We show that the introduction of pressure makes it difficult to separate the mass from other physical parameters in an invariant way. Under the usual mass definition, the apparent horizon relation, that relates the diameter distance to the cosmic mass, remains the same as in the Lemaître–Tolman case.
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Albani V.V., Iribarrem A.S., Ribeiro M.B., Stoeger W.R.: Differential density statistics of the galaxy distribution and the luminosity function. Astrophys. J. 657, 760–772 (2007)
Araújo M.E., Arcuri R.C., Bedran J.L., de Freitas L.R., Stoeger W.R.: Integrating Einstein field equations in observational coordinates with cosmological data functions: nonflat Friedmann–Lemaitre–Robsertson–Walker cases. Astrophys. J. 549, 716–720 (2001)
Araújo M.E., Roveda S.R.M.M., Stoeger W.R.: Spherically symmetric dust solution of the field equations in observational coordinates with cosmological data functions. Astrophys. J. 560, 7–14 (2001)
Araújo, M.E., Stoeger, W.R.: Exact spherically symmetric dust solution of the field equations in observational coordinates with cosmological data functions. Phys. Rev. D 60(104020), 1–7 [Errata in Phys. Rev. D 64(049901), 1 (2001)]
Araújo M.E., Stoeger W.R., Arcuri R.C., Bedran J.L.: Solving Einstein field equations in observational coordinates with cosmological data functions: spherically symmetric universes with cosmological constant. Phys. Rev. D 78, 063513 (2008)
Bishop N., Haines P.: Observational cosmology and numerical relativity. Q. Math. 19, 259 (1996)
Cahill M.E., McVittie G.C.: Spherical symmetry and mass–energy in general relativity. I. General theory. J. Math. Phys. 11, 1382–1391 (1970)
Ellis G.F.R.: Relativistic cosmology. In: Sachs, R.K. (eds) General relativity and cosmology. Proceedings of International School of Physics “Enrico Fermi” (Varenna), Course XLVII, pp. 104–179. Academic Press, London (1971)
Ellis G.F.R., Nel S.D., Maartens R., Stoeger W.R., Whitman A.P.: Ideal observational cosmology. Phys. Rep. 124, 315–417 (1985)
Etherington, I.M.H.: On the definition of distance in general relativity. Philos. Mag. VII 15, 761–773 (1933) [Errata in Gen. Relativ. Gravit. 39, 1055 (2007)]
Hellaby C.: Multicolour observations, inhomogeneity & evolution. Astron. Astrophys. 372, 357–363 (2001)
Hellaby C.: The mass of the cosmos. Mon. Not. R. Astron. Soc. 370, 239–244 (2006)
Hellaby C., Alfedeel A.H.A.: Solving the observer metric. Phys. Rev. D 79(043501), 1–10 (2009)
Ishak M.: On perfect fluid models in non-comoving observational spherical coordinates. Phys. Rev. D 69, 124027 (2004)
Krasiński, A.: Inhomogeneous Cosmological Models. Cambridge University Press, London (1997). ISBN 0 521 48180 5
Krasiński A.: Editor’s note: the expanding universe. Gen. Relativ. Gravit. 29, 637–639 (1997)
Kristian J., Sachs R.K.: Observations in cosmology. Astrophys. J. 143, 379 (1966)
Lemaître, G.: L’Univers en expansion. Ann. Soc. Sci. Bruxelles A 53, 51–85 (1993) [reprinted in Gen. Relativ. Gravit. 29, 641–680 (1997)]
Lu T.H.-C., Hellaby C.: Obtaining the spacetime metric from cosmological observations. Class. Quantum Gravit. 24, 4107–4131 (2007)
Maartens R., Matravers D.R.: Isotropic and semi-isotropic observation in cosmology. Class. Quantum Gravit. 11, 2693–2704 (1994)
Maartens, R., Humphreys, N.P., Matravers, D.R., Stoeger, W.R.: Inhomogeneous universes in observational coordinates. Class. Quantum Gravit. 13, 253 (1996) [Errata in Class. Quantum Gravit. 13, 1689 (1996)]
McClure M.L., Hellaby C.: Determining the metric of the cosmos: stability, accuracy, and consistency. Phy. Rev. D 78(044005), 1–17 (2008)
Misner C.W., Sharp D.H.: Relativistic equations for adiabatic, spherically symmetric gravitational collapse. Phys. Rev. 136, B571–B576 (1964)
Mustapha N., Hellaby C., Ellis G.F.R.: Large scale inhomogeneity versus source evolution—can we distinguish them observationally?. Mon. Not. R. Astron. Soc. 292, 817–830 (1997)
Penrose R.: General relativistic energy flux and elementary optics. In: Hoffman, B. (eds) Perspectives in Geometry and Relativity: Essays in Honour of Vaclav Hlavaty, pp. 259–274. Indiana University Press, Bloomington (1966)
Podurets, M.A.: On one form of Einstein’s equations for a spherically symmetrical motion of a continuous medium. Astron. Z. 41, 28–32 (1964) [Sov. Astron. AJ 8, 19–22 (1964)]
Ribeiro M.B., Stoeger W.R.: Relativistic cosmology number counts and the luminosity function. Astrophys. J. 592, 1–16 (2003)
Stoeger W.R., Ellis G.F.R., Nel S.D.: Observational cosmology: III. Exact spherically symmetric dust solutions. Class. Q. Gravit. 9, 509–526 (1992)
Stoeger W.R., Nel S.D., Ellis G.F.R.: Observational cosmology: IV. Perturbed spherically symmetric dust solutions. Class. Q. Gravit. 9, 1711–1723 (1992)
Stoeger W.R., Nel S.D., Ellis G.F.R.: Observational cosmology: V. solutions of the first order general perturbation equations. Class. Q. Gravit. 9, 1725–1751 (1992)
Stoeger W.R., Nel S.D., Maartens R., Ellis G.F.R.: The fluid-ray tetrad formulation of Einstein’s field equations. Class. Quantum Gravit. 9, 493–507 (1992)
Temple G.: New systems of normal coordinates for relativistic optics. Proc. R. Soc. Lond. A 168, 122 (1938)
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Alfedeel, A.H.A., Hellaby, C. The Lemaître model and the generalisation of the cosmic mass. Gen Relativ Gravit 42, 1935–1952 (2010). https://doi.org/10.1007/s10714-010-0971-y
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DOI: https://doi.org/10.1007/s10714-010-0971-y