In the numerical investigation of the physical merger of two black holes, it is crucial to locate a black hole locally. This is usually done utilizing an apparent horizon. An alternative proposal is to identify a geometric horizon (GH), which is char
acterized by a surface in the spacetime on which the curvature tensor or its covariant derivatives are algebraically special. This necessitates the choice of a special null frame, which we shall refer to as an algebraically preferred null frame (APNF). The GH is then identified by surfaces of vanishing scalar curvature invariants but, unfortunately, these are difficult to compute. However, the algebraic nature of a GH means that the APNF plays a central role and suggests a null frame approach to characterizing the GH. Indeed, if we employ the Cartan-Karlhede algorithm to completely fix the null frame invariantly, then all of the remaining non vanishing components of the curvature tensor and its covariant derivatives are Cartan scalars. Hence the GH is characterized by the vanishing of certain Cartan scalars. A null frame approach is useful in the numerical investigation of the merger of two black holes in general, but we will focus on the application to identifying a GH. We begin with a review of the use of APNF and GH in previous work. The APNF is then defined and chosen so that the Weyl tensor is algebraically special, and we must examine the covariant derivatives of the Weyl tensor in this frame. We show how to invariantly fix the null frame, and hence characterize the APNF, and describe how to then identify the GH using the zero-set of certain Cartan scalars. Our ultimate aim is to apply this frame formalism to the numerical collapse of two black holes. As an example, we investigate the axisymmetric evolution of a two black hole Kastor-Traschen spacetime.
A teleparallel geometry is an n-dimensional manifold equipped with a frame basis and an independent spin connection. For such a geometry, the curvature tensor vanishes and the torsion tensor is non-zero. A straightforward approach to characterizing t
eleparallel geometries is to compute scalar polynomial invariants constructed from the torsion tensor and its covariant derivatives. An open question has been whether the set of all scalar polynomial torsion invariants, $mathcal{I}_T$ uniquely characterize a given teleparallel geometry. In this paper we show that the answer is no and construct the most general class of teleparallel geometries in four dimensions which cannot be characterized by $mathcal{I}_T$. As a corollary we determine all teleparallel geometries which have vanishing scalar polynomial torsion invariants.
The quasi-spherical Szekeres dust solutions are a generalization of the spherically symmetric Lemaitre-Tolman-Bondi dust models where the spherical shells of constant mass are non-concentric. The quasi-spherical Szekeres dust solutions can be conside
red as cosmological models and are potentially models for the formation of primordial black holes in the early universe. Any collapsing quasi-spherical Szekeres dust solution where an apparent horizon covers all shell-crossings that will occur can be considered as a model for the formation of a black hole. In this paper we will show that the apparent horizon can be detected by a Cartan invariant. We will show that particular Cartan invariants characterize properties of these solutions which have a physical interpretation such as: the expansion or contraction of spacetime itself, the relative movement of matter shells, shell-crossings and the appearance of necks and bellies.
Dynamical solutions for an evolving multiple network of black holes near a cosmological bounce dominated by a scalar field are investigated. In particular, we consider the class of black hole lattice models in a hyperspherical cosmology, and we focus
on the special case of eight regularly-spaced black holes with equal masses when the model parameter $kappa > 1$. We first derive exact time evolving solutions of instantaneously-static models, by utilizing perturbative solutions of the constraint equations that can then be used to develop exact 4D dynamical solutions of the Einstein field equations. We use the notion of a geometric horizon, which can be characterized by curvature invariants, to determine the black hole horizon. We explicitly compute the invariants for the exact dynamical models obtained. As an application, we discuss whether black holes can persist in such a universe that collapses and then subsequently bounces into a new expansionary phase. We find evidence that in the physical models under investigation (and particularly for $kappa > 1$) the individual black holes do not merge before nor at the bounce, so that consequently black holes can indeed persist through the bounce.
In theories such as teleparallel gravity and its extensions, the frame basis replaces the metric tensor as the primary object of study. A choice of coordinate system, frame basis and spin-connection must be made to obtain a solution from the field eq
uations of a given teleparallel gravity theory. It is worthwhile to express solutions in an invariant manner in terms of torsion invariants to distinguish between different solutions. In this paper we discuss the symmetries of teleparallel gravity theories, describe the classification of the torsion tensor and its covariant derivative and define scalar invariants in terms of the torsion. In particular, we propose a modification of the Cartan-Karlhede algorithm for geometries with torsion (and no curvature or nonmetricity). The algorithm determines the dimension of the symmetry group for a solution and suggests an alternative frame-based approach to calculating symmetries. We prove that the only maximally symmetric solution to any theory of gravitation admitting a non-zero torsion tensor is Minkowski space. As an illustration we apply the algorithm to six particular exact teleparallel geometries. From these examples we notice that the symmetry group of the solutions of a teleparallel gravity theory is potentially smaller than their metric-based analogues in General Relativity.
We review current theoretical cosmology, including fundamental and mathematical cosmology and physical cosmology (as well as cosmology in the quantum realm), with an emphasis on open questions.
It is necessary to make assumptions in order to derive models to be used for cosmological predictions and comparison with observational data. In particular, in standard cosmology the spatial curvature is assumed to be constant and zero (or at least v
ery small). But there is, as yet, no fully independent constraint with an appropriate accuracy that gaurentees a value for the magnitude of the effective normalized spatial curvature $Omega_{k}$ of less than approximately $0.01$. Moreover, a small non-zero measurement of $Omega_{k}$ at such a level perhaps indicates that the assumptions in the standard model are not satisfied. It has also been increasingly emphasised that spatial curvature is, in general, evolving in relativistic cosmological models. We review the current situation, and conclude that the possibility of such a non-zero value of $Omega_k$ should be taken seriously.
We investigate the existence of invariantly defined quasi-local hypersurfaces in the Kastor-Traschen solution containing $N$ charge-equal-to-mass black holes. These hypersurfaces are characterized by the vanishing of particular curvature invariants,
known as Cartan invariants, which are generated using the frame approach. The Cartan invariants of interest describe the expansion of the outgoing and ingoing null vectors belonging to the invariant null frame arising from the Cartan-Karlhede algorithm. We show that the evolution of the hypersurfaces surrounding the black holes depends on an upper-bound on the total mass for the case of two and three equal mass black holes. We discuss the results in the context of the geometric horizon conjectures.
A class of positive curvature spatially homogeneous but anisotropic cosmological models within an Einstein-aether gravitational framework are investigated. The matter source is assumed to be a scalar field which is coupled to the expansion of the aet
her field through a generalized exponential potential. The evolution equations are expressed in terms of expansion-normalized variables to produce an autonomous system of ordinary differential equations suitable for a numerical and qualitative analysis. An analysis of the local stability of the equilibrium points indicates that there exists a range of values of the parameters in which there exists an accelerating expansionary future attractor. In general relativity, scalar field models with an exponential potential $V=V_0e^{-2kphi}$ have a late-time inflationary attractor for $k^2<frac{1}{2}$; however, it is found that the existence of the coupling between the aether and scalar fields allows for arbitrarily large values of the parameter $k$.
Inflationary spatially homogeneous cosmological models within an Einstein-Aether gravitational framework are investigated. The matter source is assumed to be a scalar field which is coupled to the aether field expansion and shear scalars through the
generalized harmonic scalar field potential. The evolution equations are expressed in terms of expansion-normalized variables to produce an autonomous system of ordinary differential equations suitable for numerical and local stability analysis. An analysis of the local stability of the equilibrium points indicate that there exists a range of values of the parameters in which there exists an accelerating expansionary future attractor.
