# On shear free normal flows of a perfect fluid

01 Jan 1973-General Relativity and Gravitation (Kluwer Academic Publishers-Plenum Publishers)-Vol. 4, Iss: 2, pp 105-129

TL;DR: In this article, the authors consider the case where the flow lines of a perfect fluid form a time-like shear-free normal congruence and show that all the degenerate fields admit at least a one-parameter group of local isometries with space-like trajectories.

Abstract: Flows of a perfect fluid in which the flow-lines form a time-like shear-free normal congruence are investigated. The space-time is quite severely restricted by this condition on the flow: it must be of Petrov Type I and is either static or degenerate. All the degenerate fields are classified and the field equations solved completely, except in one class where one ordinary differential equation remains to be solved. This class contains the spherically symmetric non-uniform density fields and their analogues with planar or hyperbolic symmetry. The type D fields admit at least a one-parameter group of local isometries with space-like trajectories. All vacuum fields which admit a time-like shear-free normal congruence are shown to be static. Finally, shear-free perfect fluid flows which possess spherical or a related symmetry are considered, and all uniform density solutions and a few non-uniform density solutions are found. The exact solutions are tabulated in section 7.

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TL;DR: In this article, the importance of an inhomogeneous framework in the analysis of cosmological observations is discussed and a review of the recent developments in the field is presented, which shows that inhomogeneities are not an alternative to the FLRW models, but an exact perturbation of the latter.

Abstract: Recently, inhomogeneous generalizations of the Friedmann?Lema?tre?Robertson?Walker (FLRW) cosmological models have gained interest in the astrophysical community and are more often employed to study cosmological phenomena. However, in many papers the inhomogeneous cosmological models are treated as an alternative to the FLRW models. In fact, they are not an alternative, but an exact perturbation of the latter, and are gradually becoming a necessity in modern cosmology. The assumption of homogeneity is just a first approximation introduced to simplify equations. So far this assumption is commonly believed to have worked well, but future and more precise observations will not be properly analysed unless inhomogeneities are taken into account. This paper reviews recent developments in the field and shows the importance of an inhomogeneous framework in the analysis of cosmological observations.

225 citations

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TL;DR: In this article, it was shown that any shear-free perfect fluid with the acceleration proportional to the vorticity vector must be either nonexpanding or nonrotating.

Abstract: In this paper we provide fully covariant proofs of some theorems on shear-free perfect fluids. In particular, we explicitly show that any shear-free perfect fluid with the acceleration proportional to the vorticity vector (including the simpler case of vanishing acceleration) must be either non-expanding or non-rotating. We also show that these results are not necessarily true in the Newtonian case, and present an explicit comparison of shear-free dust in Newtonian and relativistic theories in order to see where and why the differences appear.

71 citations

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19 Sep 2013TL;DR: In this article, the authors discuss the known phenomenology of apparent and trapping horizons for analytical solutions of General Relativity and alternative theories of gravity, focusing on spherically symmetric inhomogeneities in a background cosmological spacetime.

Abstract: From the microscopic point of view, realistic black holes are time-dependent and the teleological concept of the event horizon fails. At present, the apparent or trapping horizon seem to be its best replacements in various areas of black hole physics. We discuss the known phenomenology of apparent and trapping horizons for analytical solutions of General Relativity and alternative theories of gravity. These specific examples (we focus on spherically symmetric inhomogeneities in a background cosmological spacetime) are useful as toy models for research on various aspects of black hole physics.

70 citations

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TL;DR: In this article, all perfect fluid spacetimes with a purely electric Weyl tensor have been shown to have an alignment between the fluid 4-velocity and a canonical null tetrad determined by the tensor.

Abstract: All perfect fluid spacetimes with a purely electric Weyl tensor are shown to have an alignment between the fluid 4-velocity and a canonical null tetrad determined by the Weyl tensor. If, in addition, it is assumed that the flow is irrotational, the eigenframes of the shear and Weyl tensors coincide. In all but two rather special cases, it is proved that the vectors of this eigenframe are hypersurface orthogonal and consequently that a coordinate system exists in which the metric, shear and Eab (the electric part of the Weyl tensor) are all diagonal. Geodesic Petrov type D spacetimes are shown to be either Bianchi type 1 or to belong to the class of solutions considered by Szekeres (1975) and Szafron (1977). The Allnutt solutions (1982) are shown to be the only purely electric type D fields in which the shear is non-degenerate and in which the acceleration vector lies in the plane spanned by the principal null vectors. The field equations are partially integrated in two classes where no solutions are yet known.

68 citations

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TL;DR: In this article, a method is developed for treating Einstein's field equations, applied to static spheres of fluid, in such a manner as to provide explicit solutions in terms of known analytic functions.

Abstract: A method is developed for treating Einstein's field equations, applied to static spheres of fluid, in such a manner as to provide explicit solutions in terms of known analytic functions. A number of new solutions are thus obtained, and the properties of three of the new solutions are examined in detail. It is hoped that the investigation may be of some help in connection with studies of stellar structure. (See the accompanying article by Professor Oppenheimer and Mr. Volkoff.)

2,264 citations

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01 Oct 1968

TL;DR: In this paper, a presentation of the fundamentals of special relativity, general relativity, and cosmology with particular emphasis and development of the electromagnetic stress-energy tensor in special relativity is presented.

Abstract: : Relativity and Cosmology is a presentation of the fundamentals of special relativity, general relativity, and cosmology- with particular emphasis and development of (1) the electromagnetic stress-energy tensor in special relativity, (2) the poynting-Robertson effect, (3) the experimental basis for special-relativistic gravitational theories, (4) the geometric foundation for Reimannian geometry, (5) the derivation of the cosmological metric from group theory, (6) a discussion of cosmological models, (7) rotation of coordinate systems in general relativity, (8) electromagnetism in media in general relativity, (9) rotationally symmetric general-relativistic solutions, (10) Godel's cosmological model, and (11) geodesic precession.

315 citations