Kerr-Taub-NUT spacetime with Maxwell and dilaton fields

In the present work, a theoretical framework focussing on local geometric deformations is introduced in order to cope with the problem of how to join spacetimes with different geometries and physical properties. This framework is used to show that two Lorentzian manifolds can be matched by considering local deformations of the associated spacetime metrics. Based on the fact that metrics can be suitably matched in this way, it is shown that the underlying geometric approach allows the characterization of local spacetimes in general relativity. Furthermore, it is shown that said approach not only extends the conventional thin shell formalism, but also allows the treatment of geometric problems that cannot be treated with standard gluing techniques.

Transversality theory is used to study the curvature of generic space-times in general relativity and some applications to unique determination of the metric by the curvature are given.

This chapter begins consideration of foundations for general relativity. This is both a relativistic theory of gravitation and a freeing from absolute or background structures (as Chaps. 8 to 10 explain further). The latter perspective emphasizes Background Independence, a notion widely used in Quantum Gravity programs as well. Given the substantial conceptual and technical differences incurred by Background Independence, we further motivate the first—and only observationally established—seat of this—general relativity—by firstly briefly surveying observational evidence for general relativity. Secondly, by outlining useful features of some of its simpler cosmological and black hole solutions.

We describe the ideas behind the package `isometry', implemented in Maple to calculate isometry groups of dimensions 2, 3 and 4 in General Relativity. The package extends the functionality of previous programs written to perform invariant classification of space-times in General Relativity. Programming solutions used to surmount problems encountered with the calculation of eigenvectors and the determination of the signs of expressions are described. We also show how the package can be used to find the Killing vectors of a space-time.

We discuss the generalization of the NUT spacetime in General Relativity (GR) within the framework of the (dynamical) Einstein--Chern-Simons (ECS) theory with a massless scalar field. These configurations approach asymptotically the NUT spacetime and are characterized by the `electric' and `magnetic' mass parameters and a scalar `charge'. The solutions are found both analytically and numerically. The analytical approach is perturbative around the Einstein gravity background. Our results indicate that the ECS configurations share all basic properties of the NUT spacetime in GR. However, when considering the solutions inside the event horizon, we find that in contrast to the GR case, the spacetime curvature grows (apparently) without bound.

We construct new classes of cosmological solution to the five dimensional Einstein–Maxwell-dilaton theory, that are non-stationary and almost conformally regular everywhere. The base geometry for the solutions is the four-dimensional Bianchi type IX geometry. In the theory, the dilaton field is coupled to the electromagnetic field and the cosmological constant term, with two different coupling constants. We consider all possible solutions with different values of the coupling constants, where the cosmological constant takes any positive, negative or zero values. In the ansatzes for the metric, dilaton and electromagnetic fields, we consider dependence on time and two spatial directions. We also consider a special case of the Bianchi type IX geometry, in which the geometry reduces to that of Eguchi–Hanson type II geometry and find a more general solution to the theory.

The theory of nonlinear electrodynamics has got a lot of attentions in recent years. It was shown that Born-Infeld nonlinear electrodynamics is not the only modification of the linear Maxwell's field which keeps the electric field of a charged point particle finite at the origin, and other type of nonlinear Lagrangian such as exponential and logarithmic nonlinear electrodynamics can play the same role. In this paper, we generalize the study on the exponential nonlinear electrodynamics by adding a scalar dilaton field to the action. By suitably choosing the coupling of the matter field to the dilaton field, we vary the action and obtain the corresponding field equations. Then, by making a proper ansatz, we construct a new class of charged dilaton black hole solutions coupled to the exponential nonlinear electrodynamics field in the presence of two Liouville-type potentials for the dilaton field. Due to the presence of the dilaton field, the asymptotic behavior of these solutions are neither flat nor (A)dS. In the limiting case where the nonlinear parameter $\beta^2$ goes to infinity, our solution reduces to the Einstein-Maxwell dilaton black holes. We obtain the mass, temperature, entropy and electric potential of these solutions. We also study the behaviour of the electric field as well as the electric potential of these black holes near the origin. We find that the electric field has a finite value \textit{near} the origin, which is the same as the electric field of Born-Infeld nonlinear electrodynamics, but it can diverge exactly at $r=0$ depending on the model parameters. We also investigate the effects of the dilaton field on the behaviour of the electric field and electric potential. Finally, we check the validity of the first law of black hole thermodynamics on the black hole horizon.

Recent theoretical work determines the correct coupling constant of a scalar field to the Ricci curvature of spacetime in general relativity. The periodicity in the redshift distribution of galaxies observed by Broadhurst {\em et al.}, if genuine, determines the coupling constant in the proposed scalar field models. As a result, these observations contain important information on the problem whether general relativity is the correct theory of gravity in the region of the universe at redshifts z<0.5.

For static, spherically symmetric space-times in general relativity (GR), a no-go theorem is proved: it excludes the existence of wormholes with flat and/or AdS asymptotic regions on both sides of the throat if the source matter is isotropic, i.e., the radial and tangential pressures coincide. It explains why in all previous attempts to build such solutions it was necessary to introduce boundaries with thin shells that manifestly violate the isotropy of matter. Under a simple assumption on the behavior of the spherical radius $r(x)$, we obtain a number of examples of wormholes with isotropic matter and one or both de Sitter asymptotic regions, allowed by the no-go theorem. We also obtain twice asymptotically flat wormholes with anisotropic matter, both symmetric and asymmetric with respect to the throat, under the assumption that the scalar curvature is zero. These solutions may be on equal grounds interpreted as those of GR with a traceless stress-energy tensor and as vacuum solutions in a brane world. For such wormholes, the traversability conditions and gravitational lensing properties are briefly discussed. As a by-product, we obtain twice asymptotically flat regular black hole solutions with up to four Killing horizons. As another by-product, we point out intersection points in families of integral curves for the function $A(x) = g_{tt}$, parametrized by its values on the throat.

A certain class of exact solutions of Einstein Maxwell spacetime in general relativity is discussed to demonstrate that at the level of theory, when certain parametric resonance condition is met, the interaction of electromagnetic field with a gravitational wave will display certain Lyapunov instability and lead to exponential amplification of a gravitational wave train described by certain Newman–Penrose component of the Weyl curvature. In some way akin to a free electron laser in electromagnetic theory, by the conversion of electromagnetic energy into gravitational energy in a coherent way, the feasibility of generating a pulsed-laser-like intense beam of gravitational wave is displayed.

We characterize the spherical topology of black holes in a special case of spherical symmetry. For this purpose, we carefully consider the metrics space of topological black holes B4. By adjusting some of the parameters, we infer the appropriate components and the equatorial geodesics on the line element of the topological black holes. We also construct the minimum retraction ρ2:B4→ C 3 2. As a result, the minimum deformation retract has also been inferred to explain the effects on their center of mass in binary system. This would help us to better understand some of possible applications in astrophysics and the Cosmology, such as the natural phenomena of merging two black holes and curved the space-time in general relativity. In addition, we find the relation between limit folding and limit minimum retraction on the n-dimension. The folding on connected sum of essential sub-manifolds of the topological black hole have constructed for different types and characteristics.

The Noether-charge realization and the Hamiltonian realization for the $\diff({\cal M})$ algebra in diffeomorphism invariant gravitational theories are studied in a covariant formalism. For the Killing vector fields, the Nother-charge realization leads to the mass formula as an entire vanishing Noether charge for the vacuum black hole spacetimes in general relativity and the corresponding first law of the black hole mechanics. It is analyzed in which sense the Hamiltonian functionals form the $\diff({\cal M})$ algebra under the Poisson bracket and shown how the Noether charges with respect to the diffeomorphism generated by vector fields and their variations in general relativity form this algebra. The asymptotic behaviors of vector fields generating diffeomorphism of the manifold with boundaries are discussed. In order to get more precise estimation for the "central extension" of the algebra, it is analyzed in the Newman-Penrose formalism and shown that the "central extension" for a large class of vector fields is always zero on the Killing horizon. It is also checked whether the Virasoro algebra may be picked up by choosing the vector fields near the horizon. The conclusion is unfortunately negative.

Many singular space-times are described locally in geodesic normal coordinates with the singularity described by the time coordinate equal to zero. Under appropriate conditions the time coordinate in this description is uniquely determined. Furthermore, every timelike curve in the coordinate patch is inextendible and has the same limit point as exactly one time coordinate line. The proof of uniqueness of the time coordinate employs methods from the theory of first order partial differential equations in a new way.

In an attempt to understand the notions of ‘invariance’ and ‘covariance’ of a differential operator, which are of great importance in mathematical physics, we were led to study the actions of a group of transformations of a manifold M (the space-time in general relativity) on the sections of a vector bundle over M (the tensor or spinor fields of a given type). Several equivalent characterizations of these actions are given. A similar study is made for a Lie algebra of vector fields.

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