Remarks on the Taub-NUT solution in Chern–Simons modified gravity

The problem of having an accurate description of the spacetime around neutron stars is of great astrophysical interest. For astrophysical applications, one needs to have a metric that captures all the properties of the spacetime around a neutron star. Furthermore, an accurate appropriately parameterised metric, i.e., a metric that is given in terms of parameters that are directly related to the physical structure of the neutron star, could be used to solve the inverse problem, which is to infer the properties of the structure of a neutron star from astrophysical observations. In this work we present such an approximate stationary and axisymmetric metric for the exterior of neutron stars, which is constructed using the Ernst formalism and is parameterised by the relativistic multipole moments of the central object. This metric is given in terms of an expansion on the Weyl-Papapetrou coordinates with the multipole moments as free parameters and is shown to be extremely accurate in capturing the physical properties of a neutron star spacetime as they are calculated numerically in general relativity. Because the metric is given in terms of an expansion, the expressions are much simpler and easier to implement, in contrast to previous approaches. For the parameterisation of the metric in general relativity, the recently discovered universal 3-hair relations are used to produce a 3-parameter metric. Finally, a straightforward extension of this metric is given for scalar-tensor theories with a massless scalar field, which also admit a formulation in terms of an Ernst potential.

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.

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.

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.

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.

In this research paper, we determine the nature of conformal [Formula: see text]-Ricci–Bourguignon soliton on a general relativistic spacetime with torse forming potential vector field. Besides this, we evaluate a specific situation of the soliton when the spacetime admitting semi-symmetric energy–momentum tensor with respect to conformal [Formula: see text]-Ricci–Bourguignon soliton, whose potential vector field is torse-forming. Next, we explore some characteristics of curvature on a spacetime that admits conformal [Formula: see text]-Ricci–Bourguignon soliton. In addition, we turn up some physical perception of dust fluid, dark fluid and radiation era in a general relativistic spacetime in terms of conformal [Formula: see text]-Ricci–Bourguignon soliton. Finally, we examine necessary and sufficient conditions for a 1-form [Formula: see text], which is the [Formula: see text]-dual of the vector field [Formula: see text] on general relativistic spacetime to be a solution of the Schrödinger–Ricci equation.

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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.

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.

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.

We present an exact solution describing a stationary and axisymmetric object with electromagnetic and dilaton fields. The solution generalizes the usual Kerr-Taub-NUT (Newman-Unti-Tamburino) spacetime in general relativity and is obtained by boosting this spacetime in the fifth dimension and performing a Kaluza-Klein reduction to four dimensions. We also discuss the physical parameters of this solution and calculate its gyromagnetic ratio.