Singularities in Einstein–conformally coupled Higgs cosmological models

The dynamics of Einstein–conformally coupled Higgs field (EccH) system is investigated near the initial singularities in the presence of Friedman–Robertson–Walker symmetries. We solve the field equations asymptotically up to fourth order near the singularities analytically, and determine the solutions numerically as well. We found all the asymptotic, power series singular solutions, which are (1) solutions with a scalar polynomial curvature singularity but the Higgs field is bounded (‘Small Bang’), or (2) solutions with a Milne type singularity with bounded spacetime curvature and Higgs field, or (3) solutions with a scalar polynomial curvature singularity and diverging Higgs field (‘Big Bang’). Thus, in the present EccH model there is a new kind of physical spacetime singularity (‘Small Bang’). We also show that, in a neighbourhood of the singularity in these solutions, the Higgs sector does not have any symmetry breaking instantaneous vacuum state, and hence then the Brout–Englert–Higgs mechanism does not work. The large scale behaviour of the solutions is investigated numerically as well. In particular, the numerical calculations indicate that there are singular solutions that cannot be approximated by power series.     

Further remark concerning spherically symmetric nonstatic separable solutions to the Einstein equations in the comoving frame

We consider nonstatic spherically symmetric fluid solutions to the Einstein equations which, in the comoving frame, have metric coefficients that are separable functions of their arguments and that have an origin. Subject to the vanishing of the heat flux, we show that all such solutions with shear and non-vanishing shear viscosity have a scalar polynomial singularity at the origin if the fluid satisfies both the weak and strong energy conditions. When combined with previous results [1] we conclude that for the metric forms under consideration, the only fluid solutions to the Einstein equations with vanishing heat flux which satisfy the energy conditions and are free of singularities at the origin are the Robertson-Walker solutions.     

The Initial Singularity of Ultrastiff Perfect Fluid Spacetimes Without Symmetries

We consider the Einstein equations coupled to an ultrastiff perfect fluid and prove the existence of a family of solutions with an initial singularity whose structure is that of explicit isotropic models. This family of solutions is ‘generic’ in the sense that it depends on as many free functions as a general solution, i.e., without imposing any symmetry assumptions, of the Einstein-Euler equations. The method we use is a that of a Fuchsian reduction.     

A New Solution to Einstein's Field Equations

We construct a new exact solution to the vacuum Einstein field equations. This solution possesses a naked physical singularity. The norm of the Riemann curvature tensor of the solution takes infinity at some points and the solution does not have any event horizon around the singularity. A detailed analysis of this new singularity is also presented.     

Singularities in general relativity

The problem of singularities is examined from the stand-point of a local observer. A singularity is defined as a state with an infinite proper rest mass density. The approach consists of three steps: (i) The complete system of equations describing a non-symmetric motion of a perfect fluid under assumption of adiabatic thermodynamic processes and of no release of nuclear energy is reduced to six Einstein field equations and their four first integrals for six remaining unknown componentsgik. (ii) A differential relation for the behavior of the rest mass density is deduced. It shows that any inhomogeneity and anisotropy in the distribution and motion of a non-rotating ideal fluid accelerates collapse to a singularity which will be reached in a finite proper time. Collapse is also inevitable in a rotating fluid in the case of extremely high pressure when the relativistic limit of the equation of state must be applied. In the case of a lower or zero pressure the relation does not give an unambiguous answer if the matter is rotating. (iii) The influence of rotation on the motion of an incoherent matter is investigated. Some qualitative arguments are given for a possible existence of a narrow class of singularity-free solutions of Einstein equations. Assuming rotational symmetry the Einstein partial differential equations together with their first integrals are reduced to a system of simultaneous ordinary differential equations suitable for numerical integration. Without integrating this system the existence of the class of singularity-free solutions is confirmed and exactly delimited. These solutions, representing a new general relativistic effect, are, however, of no importance for the application in cosmology or astrophysics. It is proved that in all the other cases interesting from the point of view of application the occurrence of a point singularity in incoherent matter with a rotational symmetry is inevitable even if the rotation is present.Read on 15 May 1970 at the Gwatt Seminar on the Bearings of Topology upon General Relativity     

Analysis of an FRW cosmological model

The Einstein field equations for the Friedmann universe reduce to a system of three first-order equations for the space-like components and a constraint from the temporal component. We analyse the system from the viewpoints of symmetry and singularity analyses. The solutions of particular relevance to Cosmology are highlighted.      

Eleven Spherically Symmetric Constant Density Solutions with Cosmological Constant

Einstein's field equations with cosmological constant are analysed for a static, spherically symmetric perfect fluid having constant density. Five new global solutions are described. One of these solutions has the Nariai solution joined on as an exterior field. Another solution describes a decreasing pressure model with exterior Schwarzschild-de Sitter spacetime having decreasing group orbits at the boundary. Two further types generalise the Einstein static universe. The other new solution is unphysical, it is an increasing pressure model with a geometric singularity.     

Spherical Gravitational Collapse: Tangential Pressure and Related Equations of State

We derive an equation for the acceleration of a fluid element in the spherical gravitational collapse of a bounded compact object made up of an imperfect fluid. We show that non-singular as well as singular solutions arise in the collapse of a fluid initially at rest and having only a tangential pressure. We obtain an exact solution of the Einstein equations, in the form of an infinite series, for collapse under tangential pressure with a linear equation of state. We show that if a singularity forms in the tangential pressure model, the conditions for the singularity to be naked are exactly the same as in the model of dust collapse.     

A Note on the Relationship Between Solutions of Einstein, Ramanujan and Chazy Equations

The Einstein equation for the Friedmann-Robertson-Walker metric plays a fundamental role in cosmology. The direct search of the exact solutions of the Einstein equation even in this simple metric case is sometime a hard job. Therefore, it is useful to construct solutions of the Einstein equation using a known solutions of some other equations which are equivalent or related to the Einstein equation. In this work, we establish the relationship the Einstein equation with two other famous equations namely the Ramanujan equation and the Chazy equation. Both these two equations play an important role in the number theory. Using the known solutions of the Ramanujan and Chazy equations, we find the corresponding solutions of the Einstein equation.     

A Non-geometric Relativistic Theory of Gravitation

A non-geometric relativistic theory of gravitation is developed by defining a semi-metric to replace the metric tensor as gravitational vector potential. The theory show that the energy-momentum tensor of the gravitational field belong to the gravitational source, gravitational radiation is contained in Einstein’s field equations that including the contribution of gravitational field, the real physical singularity in the gravitational field can be eliminated, and the dark matter in the universe is interpreted as the matter of pure gravitational field.     

Gravitational Collapse of Self-Similar Perfect Fluid in 2 + 1 Gravity

Perfect fluid with kinematic self-similarity is studied in 2+1 dimensional spacetimes with circular symmetry, and various exact solutions to the Einstein field equations are given. These include all the solutions of dust and stiff perfect fluid with self-similarity of the first kind, and all the solutions of perfect fluid with a linear equation of state and self-similarity of the zeroth and second kinds. It is found that some of these solutions represent gravitational collapse, and the final state of the collapse can be either a black hole or a null singularity. It is also shown that one solution can have two different kinds of kinematic self-similarity.     

Matching Spherical Dust Solutions to Construct Cosmological Models

Conditions for smooth cosmological models are set out and applied to inhomogeneous spherically symmetric models constructed by matching together different Lemaître–Tolman–Bondi solutions to the Einstein field equations. As an illustration the methods are applied to a collapsing dust sphere in a curved background. This describes a region which expands and then collapses to form a black hole in an Einstein de Sitter background. We show that in all such models if there is no vacuum region then the singularity must go on accreting matter for an infinite LTB time.     

Spherically symmetric solutions of Einstein + non-polynomial gravities

We obtain the static spherically symmetric solutions of a class of gravitational models whose additions to the General Relativity (GR) action forbid Ricci-flat, in particular, Schwarzschild geometries. These theories are selected to maintain the (first) derivative order of the Einstein equations in Schwarzschild gauge. Generically, the solutions exhibit both horizons and a singularity at the origin, except for one model that forbids spherical symmetry altogether. Extensions to arbitrary dimension with a cosmological constant, Maxwell source and Gauss-Bonnet terms are also considered.     

Conformal Continuations in Gravitation Theory with Lagrangian <Emphasis Type="Italic">F</Emphasis>(<Emphasis Type="Italic">R</Emphasis>)

Static spherically-symmetric vacuum solutions of gravitation theory equations with Lagrangian f(R) are examined, where R is a scalar curvature and f is an arbitrary function. Equations of f(R)-theories are reduced to the Einstein scenario — general relativity theory (GRT) equations with a source in the form of a scalar field with potential — with the use of the well-known conformal transformation. The necessary and sufficient conditions of existence of solutions admitting conformal continuations are formulated. This means that the central singularity of the Einstein scenario is mapped into a regular sphere S_trans of the Jordan scenario (that is, into the manifold corresponding to the initial formulation of the theory), and a solution of the field equations can be smoothly continued through it. The value of curvature R on the sphere S_trans corresponds to an extremum of the function f(R). Concrete examples are considered. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 46–51, September, 2005.     

A pure geometric approach to stellar structure

The present work represents a step in dealing with stellar structure using a pure geometric approach. Geometric field theory is used to construct a model for a spherically symmetric configuration. In this case, two solutions have been obtained for the field equations. The first represents an interior solution which may be considered as a pure geometric one in the sense that the tensor describing the material distributions is not a phenomenological object, but a part of the geometric structure used. A general equation of state for a perfect fluid, is obtained from, and not imposed on, the model. The second solution gives rise to Schwarzschild exterior field in its isotropic form. The two solutions are matched, at a certain boundary, to evaluate the constants of integration. The interior solution obtained shows that there are different zones characterizing the configuration: a central radiation dominant zone, a probable convection zone as a physical interpretation of the singularity of the model, and a corona like zone. The model may represent a type of main sequence stars. The present work shows that Einstein’s geometerization scheme can be extended to gain more physical information within material distribution, with some advantages.     

Incompressible Einstein–Maxwell fluids with specified electric fields

The Einstein–Maxwell equations describing static charged spheres with uniform density and variable electric field intensity are studied. The special case of constant electric field is also studied. The evolution of the model is governed by a hypergeometric differential equation which has a general solution in terms of special functions. Several classes of exact solutions are identified which may be considered as charged generalizations of the incompressible Schwarzschild interior model. An analysis of the physical features is undertaken for the uniform case. It is demonstrated that uniform density spheres with constant electric field intensity are not realizable with isotropic pressures. This highlights the necessity of studying the criteria for physical admissability of gravitating spheres in general relativity which are solutions to the Einstein–Maxwell equations.     

Gravitational fields near space-like and null infinity

Near space-like infinity an initial value problem for the conformal Einstein equations is formulated such that: (i) the data and equations are regular, (ii) space-like and null infinity have a finite representation, with their structure and location known a priori, and (iii) the setting relies entirely on general properties of conformal structures.A first analysis of this problem shows that the solutions develop in general a certain type of logarithmic singularity at the set where null infinity touches space-like infinity. These singularities form an intrinsic part of the solutions' conformal structure. Conditions on the free initial data near space-like infinity are derived which ensure that for solutions developing from these data singularities of this type cannot occur.     

Quiescent Cosmological Singularities

The most detailed existing proposal for the structure of spacetime singularities originates in the work of Belinskii, Khalatnikov and Lifshitz. We show rigorously the correctness of this proposal in the case of analytic solutions of the Einstein equations coupled to a scalar field or stiff fluid. More specifically, we prove the existence of a family of spacetimes depending on the same number of free functions as the general solution which have the asymptotics suggested by the Belinskii–Khalatnikov–Lifshitz proposal near their singularities. In these spacetimes a neighbourhood of the singularity can be covered by a Gaussian coordinate system in which the singularity is simultaneous and the evolution at different spatial points decouples. Received: 17 January 2000 / Accepted: 27 November 2000     

Generation of static solutions of the self-consistent system of Einstein-Maxwell equations

A theorem is proved, according to which to each solution of the Einstein equations with an arbitrary momentum-energy tensor in the right hand side there corresponds a static solution of the self-consistent system of Einstein-Maxwell equations. As a consequence of this theorem, a method is established of generating static solutions of the self-consistent system of Einstein-Maxwell equations with a charged grain as a source of vacuum solutions of the Einstein equations.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 98–102, February, 1988.