On a Framework for Generating Nonstatic Solutions of the Einstein Field Equations

In this paper we present a simple generalisation of all spherically symmetric static solutions. We present a framework to obtain both anisotropic and isotropic models with and without a barotropic equation of state of the form p = . The nonstaticity in some models necessarily requires a nonzero heat flux which dictates the thermodynamics of our models.     

Scalar-tensor cosmologies with a potential in the general relativity limit: Time evolution

We consider Friedmann–Lemaître–Robertson–Walker flat cosmological models in the framework of general Jordan frame scalar-tensor theories of gravity with arbitrary coupling function and potential. For the era when the cosmological energy density of the scalar potential dominates over the energy density of ordinary matter, we use a nonlinear approximation of the decoupled scalar field equation for the regime close to the so-called limit of general relativity where the local weak field constraints are satisfied. We give the solutions in cosmological time with a particular attention to the classes of models asymptotically approaching general relativity. The latter can be subsumed under two types: (i) exponential convergence, and (ii) damped oscillations around general relativity. As an illustration we present an example of oscillating dark energy.     

Astrophysically significant solutions of the Einstein and Einstein-Maxwell equations from the Laplace’s seed

The stationary solutions given by Amenedo and Manko generated from known solutions of Laplace’s equation as seed have been generalised to include the electromagnetic field. Further, the exterior solution of an axially symmetric rotating body with higher multipole moments and a solution corresponding to a Kerr object embedded in a gravitational field are given. We also give a method for constructing stationary vacuum solutions from static magnetovac solutions and vice versa and discuss a specific application of this method.     

One-soliton solutions from Laplace’s seed

One-soliton solutions of axially symmetric vacuum Einstein field equations are presented in this paper. Two sets of Laplace’s solutions are used as seed and it is shown that the derived solutions reduce to some already known solutions when the constants are properly adjusted. An analysis of the solutions in terms of the Ernst potential is also presented. It is found that the solutions do not reduce to the Euclidean form at spatial infinity. However, in the static limit, Weyl solutions are obtained for half integral ∂-values.     

On singularity-free spacetimes

We consider here the metric for the singularity-free family of fluid models. The metric is unique for cylindrically symmetric space-time with metric potentials being separable functions of radial and time coordinates in the comoving coordinates. It turns out that fluid models separate out into two classes, withρ ≠μp in general butρ = 3p in particular andp =ρ. It is shown that in both the cases radial heat flow can be incorporated without disturbing the singularity-free character of the spacetime. The geodesics of the singularity-free metric are studied and the geodesic completeness is established. Several previously known solutions are derived as particular cases.     

A class of stationary rotating string cosmological models

We obtain a one parameter class of stationary rotating string cosmological models of which the well-known Gödel universe is a particular case. By suitably choosing the free parameter function, it is always possible to satisfy the energy conditions. The rotation of the model hinges on the cosmological constant which turns out to be negative. String-dust distribution in Gödel-type universes is also briefly discussed.     

Remarks on exact solutions

This paper first discusses the historical context of the influential “Exact Solutions” book, which was co-authored by Malcolm MacCallum. It then makes various technical points about such solutions. It is useful to characterize solutions in terms of the properties of matter they contain or the kinds of test particle motions which are possible. It is also interesting to consider which isometries are implied by particular kinds of matter behaviour. Judging the validity of an approximation by the smallness of the tensor components can be misleading. Finally, an example is given of a result which is obvious in Newtonian theory but little understood in General Relativity.     

The nearly flat universe

We study here what it means for the Universe to be nearly flat, as opposed to exactly flat. We give three definitions of nearly flat, based on density, geometry and dynamics; all three definitions are equivalent and depend on a single constant flatness parameter ɛ that quantifies the notion of nearly flat. Observations can only place an upper limit on ɛ, and always allow the possibility that the Universe is infinite with k = −1 or finite with k = 1. We use current observational data to obtain a numerical upper limit on the flatness parameter and discuss its implications, in particular the “naturalness” of the nearly flat Universe.     

On the structure and multipole moments of axially symmetric stationary metrics

The structure of the stationary metrics [1], generated from Laplace’s solutions as seed, is investigated. The expressions for the equatorial and polar circumferences, the surface area of the event horizon, location of singular points and the Gaussian curvatures of the metrics [1] are derived and their variations with the field parameter α₀ are studied. The multipole moments are calculated with the help of coordinate invariant Geroch-Hansen technique. These investigations expose some interesting properties of the metrics, some of which are known in the literature and some deserve a new interpretation.     

The International Atomic Time and the Velocity of Light

Atomic clocks distributed around the world communicate with one another by means of radio signals. The synchronization signals sent by a transmitting station always reach the receiving station on time, at any hour of the day and in any season, despite the motion of the Earth. For some authors this means that these signals propagate isotropically (with one way velocity c), even with respect to the Earth's surface. In fact this may not be so; we show that the proper working of the network says nothing about the one-way velocity, as it is consistent with another theory, empirically (almost) equivalent to special relativity, in which the one-way speed of light has a directional dependence in moving frames.     

Nonlinear gravito-electromagnetism

There is a non-linear and covariant electromagnetic analogy for gravity, in which the full Bianchi identities are Maxwell-type equations for the free gravitational field, encoded in the Weyl tensor. This tensor gravito-electromagnetism is based on a covariant generalization of spatial vector algebra and calculus to spatial tensor fields, and includes all non-linear effects from the gravitational field and matter sources. The non-linear vacuum Bianchi equations are invariant under spatial duality rotation of the gravito-electric and gravito-magnetic tensor fields. The super-energy density and super-Poynting vector of the gravitational field are natural duality invariants, and satisfy a super-energy conservation equation.     

On the Meaning of Lorentz Covariance

In classical mechanics, Galilean covariance and the principle of relativity are completely equivalent and hold for all possible dynamical processes. In contrast, in relativistic physics the situation is much more complex. It will be shown that Lorentz covariance and the principle of relativity are not completely equivalent. The reason is that the principle of relativity actually only holds for the equilibrium quantities that characterize the equilibrium state of dissipative systems. In the light of this fact it will be argued that Lorentz covariance should not be regarded as a fundamental symmetry of the laws of physics.     

The Equivalence Principle as a Symmetry

It is shown that the extra coordinate of 5D induced-matter and membrane theory is related in certain gauges to the inertial rest mass of a test particle. This implies that the Weak Equivalence Principle is a geometric symmetry, valid only in the limit in which the test mass is negligible compared to the source mass. Exact solutions illustrate this, and show the way to possible resolutions of the cosmological-constant and hierarchy problems.     

The Weyl Tensor and Equilibrium Configurations of Self-Gravitating Fluids

It is shown that (except for two well defined cases), the necessary and sufficient condition for any spherically symmetric distribution of fluid to leave the state of equilibrium (or quasi-equilibrium), is that the Weyl tensor changes with respect to its value in the state of equilibrium (or quasi-equilibrium).     

Smart Geodesic Tracing in GRworkbench

We have developed a new tool for numerical work in General Relativity: GRworkbench. We discuss how GRworkbench's implementation of a numerically-amenable analogue to Differential Geometry facilitates the development of robust and chart-independent numerical algorithms. We consider, as an example, geodesic tracing on two charts covering the exterior Schwarzschild space-time.     

Gravitational collapse and naked singularities

Gravitational collapse is one of the most striking phenomena in gravitational physics. The cosmic censorship conjecture has provided strong motivation for research in this field. In the absence of a general proof for censorship, many examples have been proposed, in which naked singularity is the outcome of gravitational collapse. Recent developments have revealed that there are examples of naked singularity formation in the collapse of physically reasonable matter fields, although the stability of these examples is still uncertain. We propose the concept of ‘effective naked singularities’, which will be quite helpful because general relativity has limitation in its application at the high-energy end. The appearance of naked singularities is not detestable but can open a window for the new physics of strongly curved space-times.     

Special Stäckel electrovac spacetimes

Classification of all electrovac specetimes permitting the separation of variables in the Hamilton-Jacobi equation for a charged test particle is carried out. This separation requires the existence of a complete set consisting of Killing’s vectors and tensors of a special kind. Every complete set defines its own type of metric and electromagnetic potential in the separable coordinate system. There exist seven types of separation of variables for electromagnetic spaces. For every type an additional classification is carried out by transformation of coordinates without any disturbance of the separation conditions, the gradient transformation of electromagnetic potential and the conformal-constant transformation of metric. The key step in solving the problem is the extraction of an autonomous subsystem which determines the metric from only the Einstein-Maxwell equations for every type of separation of variables. Representatives of all classes of metrics and electromagnetic potential are given for every type of separation of variables with the exception of the spaces found in the well-known work by Carter. The problem is solved in terms of metric formalism. The classes of electrovac spacetimes obtained are found to be related to Petrov’s classification.