arXiv subjects

Alan Coley

Publications and source records attributed to Alan Coley.

At least 19 recordsLinked to original sources

arXiv2026

A Complete Invariant Analysis of the Kerr Spacetime and its Photon Region

We present an invariant characterization of the Kerr spacetime, and utilize the invariant structure of the spacetime to derive a function whose zeros identify a special family of null geodesics. Each member of this family is tangent to every photon surface in the Kerr photon region, offering a method of invariantly characterizing photon surfaces in axially symmetric spacetimes and thereby a providing a computational tool for efficiently computing the geodesic equations for any part of the photon region. The invariant that identifies all of the spherical photon orbits is parameterized by a Lorentz parameter, where the parameter is effectively an inclination angle of the spherical photon orbits through the equatorial plane. We also show how the invariant determines the constants of motion for all spherical orbits in the photon region. Finally, we briefly derive invariants which identify the other geometrically important surfaces such as the ergosurfaces and local horizons.

gr-qc↗

arXiv2023

Periodic boundary conditions and $G_2$ cosmology

In the standard concordance cosmology the spatial curvature is assumed to be constant and zero (or at least very small). In particular, in numerical computations of the structure of the universe using N-body simulations, exact periodic boundary conditions are assumed which constrains the spatial curvature. In order to confirm this qualitatively, we numerically evolve a special class of spatially inhomogeneous $G_2$ models with both periodic initial data and non periodic initial data using zooming techniques. We consequently demonstrate that in these models periodic initial conditions do indeed suppress the growth of the spatial curvature as the models evolve away from their initial isotropic and spatially homogeneous state, thereby verifying that the spatial curvature is necessarily very small in standard cosmology.

gr-qc↗

arXiv2023

Invariant description of static and dynamical Brans-Dicke spherically symmetric models

We investigate spherically symmetric static and dynamical Brans-Dicke theory exact solutions using invariants and, in particular, the Newman Penrose formalism utilizing Cartan scalars. The GR limit of these solutions is examined through the use of Cartan invariants via the Cartan-Karlhede algorithm and is additionally supported by analysis of scalar polynomial invariants. It is determined that the appearance of horizons in these spacetimes depends primarily on one of the parameters, $n$, of the family of solutions. In particular, expansion-free surfaces appear which, for a subset of parameter values, define additional surfaces distinct from the standard surfaces (e.g., apparent horizons) identified in previous work. These surfaces in static spherically symmetric Brans-Dicke solutions was previously shown to correspond to the Schwarzschild horizon in general relativity when an appropriate limit exists between the two theories. We show additionally that other geometrically defined horizons exist for these cases, and identify all solutions for which the corresponding general relativity limit is not a Schwarzschild one, yet still contains horizons. The identification of some of these other surfaces was noted in previous work and is characterized invariantly in this work. In the case of the family of dynamical Brans-Dicke solutions, we identify similar invariantly defined surfaces as in the static case and present an invariant characterization of their geometries. Through the analysis of the Cartan invariants, we determine which members of these families of solutions are locally equivalent, through the use of the Cartan-Karlhede algorithm. In addition, we identify black hole surfaces, naked singularities, and wormholes with the Cartan invariants. The aim of this work is to demonstrate the usefulness of Cartan invariants for describing properties of exact solutions.

gr-qc↗

arXiv2023

Global dynamics of two models for Quintom Friedman-Lema\^itre-Robertson-Walker Universes

We comprehensively analyze the dynamics for the gravitational field equations for the Chiral-Quintom theory in a Friedman-Lema\^itre-Robertson-Walker cosmology with an additional matter source. We consider a new set of dimensionless variables and write the field equations in the equivalent form of an algebraic-differential system. Specifically, we consider two families of quintom models where the two scalar fields interact in the kinetic sector. We mathematically focus on the dynamical effect of spatial curvature. Physically, we find two periods of inflation related to the Universe's early and late-time acceleration phases.

gr-qc↗

arXiv2022

Teleparallel Geometry with a Single Affine Symmetry

In teleparallel geometries, symmetries are represented by affine frame symmetries which constrain both the (co)frame basis and the spin-connection (which are the primary geometric objects). In this paper we shall study teleparallel geometries with a single affine symmetry, utilizing the locally Lorentz covariant approach and adopting a complex null gauge. We first introduce an algorithm to study geometries with an affine frame symmetry, which consists of choosing coordinates adapted to the symmetry, constructing a canonical frame, and solving the equations describing the symmetry. %for the spin connection (to determine the six real Lorentz functions. All of the constraints on the geometry are determined in the case of a single affine symmetry, but there are additional constraints arising from the field equations for a given theory of teleparallel gravity. In particular, we find that in $f(T)$ teleparallel gravity there will be severe constraints on the geometry arising from the antisymmetric part of the field equations.

gr-qc↗

arXiv2022

The dynamics of scalar-field Quintom cosmological models

We shall present a complete (compactified) dynamical systems analysis of the Quintom model comprised of an interacting quintessence scalar field and a phantom. We find a range for the model parameters $\kappa, \lambda$ such that there are expanding Quintom cosmologies that undergo two inflationary periods, and this behaviour is not destabilized by spatial curvature. We also discuss a class of bouncing cosmologies. Finally, the linear cosmological perturbations are studied.

gr-qc↗

arXiv2021

Geometric horizons in binary black hole mergers

We numerically study the algebraic properties of the Weyl tensor through the merger of two non-spinning black holes (BHs). We are particularly interested in the conjecture that for such a vacuum spacetime, which is zeroth-order algebraically general, a geometric horizon (GH), on which the spacetime is algebraically special and which is identified by the vanishing of a complex scalar invariant (${\mathcal{D}}$), characterizes a smooth foliation independent surface (horizon) associated with the BH. In the first simulation we investigate the level-$0$ sets of $\text{Re}({\mathcal{D}})$ (since $\text{Im}({\mathcal{D}})= 0$) in the head-on collision of two unequal mass BHs. In the second simulation we shall investigate the level-$\varepsilon$ sets of $|{\mathcal{D}}|$ through a quasi-circular merger of two non-spinning, equal mass BHs. The numerical results, as displayed in the figures presented, provide evidence that a (unique) smooth GH can be identified throughout all stages of the binary BH merger.

gr-qc↗

arXiv2021

Curvature Invariants and the Geometric Horizon Conjecture in a Binary Black Hole Merger

We study curvature invariants in a binary black hole merger. It has been conjectured that one could define a quasi-local and foliation independent black hole horizon by finding the level--$0$ set of a suitable curvature invariant of the Riemann tensor. The conjecture is the geometric horizon conjecture and the associated horizon is the geometric horizon. We study this conjecture by tracing the level--$0$ set of the complex scalar polynomial invariant, $\mathcal{D}$, through a quasi-circular binary black hole merger. We approximate these level--$0$ sets of $\mathcal{D}$ with level--$\varepsilon$ sets of $|\mathcal{D}|$ for small $\varepsilon$. We locate the local minima of $|\mathcal{D}|$ and find that the positions of these local minima correspond closely to the level--$\varepsilon$ sets of $|\mathcal{D}|$ and we also compare with the level--$0$ sets of $\text{Re}(\mathcal{D})$. The analysis provides evidence that the level--$\varepsilon$ sets track a unique geometric horizon. By studying the behaviour of the zero sets of $\text{Re}(\mathcal{D})$ and $\text{Im}(\mathcal{D})$ and also by studying the MOTSs and apparent horizons of the initial black holes, we observe that the level--$\varepsilon$ set that best approximates the geometric horizon is given by $\varepsilon = 10^{-3}$.

gr-qc↗

arXiv2020

Observer-based invariants for cosmological models

We consider the equivalence problem for cosmological models in four-dimensional gravity theories. A cosmological model is considered as a triple $(M, {\bf g},{\bf u})$ consisting of a spacetime $(M, {\bf g})$ and a preferred normalized time-like vector field ${\bf u}$ tangent to a congruence of fundamental observers. We introduce a modification of the Cartan-Karlhede algorithm by restricting to frames adapted to ${\bf u}$ and including the covariant derivatives of ${\bf u}$ along with the Riemann tensor and its covariant derivatives. To fix the frame we make use of quantities relative to the fundamental observers, such as the anisotropic pressure tensor, energy flux vector, electric and magnetic parts of the Weyl tensor and the kinematical quantities of ${\bf u}$. This provides a simpler way to construct a list of invariants relative to the fundamental observers that completely characterizes the model, independent of coordinates. As an illustration of the algorithm, we consider several well-known cosmological models from General Relativity.

gr-qc↗

arXiv2019

Static Spherically Symmetric Einstein-aether models I: Perfect fluids with a linear equation of state and scalar fields with an exponential self-interacting potential

We investigate the field equations in the Einstein-aether theory for static spherically symmetric spacetimes and a perfect fluid source and subsequently with the addition of a scalar field (with an exponential self-interacting potential). We introduce more appropriate dynamical variables that facilitate the study of the equilibrium points of the resulting dynamical system and, in addition, we discuss the dynamics at infinity. We study the qualitative properties of the models with a particular interest in their asymptotic behaviour and whether they admit singularities. We also present a number of new solutions.

gr-qc↗

arXiv2017

Identification of black hole horizons using scalar curvature invariants

We introduce the concept of a geometric horizon, which is a surface distinguished by the vanishing of certain curvature invariants which characterize its special algebraic character. We motivate its use for the detection of the event horizon of a stationary black hole by providing a set of appropriate scalar polynomial curvature invariants that vanish on this surface. We extend this result by proving that a non-expanding horizon, which generalizes a Killing horizon, coincides with the geometric horizon. Finally, we consider the imploding spherically symmetric metrics and show that the geometric horizon identifies a unique quasi-local surface corresponding to the unique spherically symmetric marginally trapped tube, implying that the spherically symmetric dynamical black holes admit a geometric horizon. Based on these results, we propose a suite of conjectures concerning the application of geometric horizons to more general dynamical black hole scenarios.

gr-qc↗

arXiv2017

Stiff fluid spike solutions from Bianchi type V seed solutions

In this paper we expand upon our previous work [1] by using the entire family of Bianchi type V stiff fluid solutions as seed solutions of the Stephani transformation. Among the new exact solutions generated, we observe a number of important physical phenomena. The most interesting phenomenon is exact solutions with intersecting spikes. Other interesting phenomena are solutions with saddle states and a close-to-FL epoch.

gr-qc↗

arXiv2017

Persistent Black Holes in Bouncing Cosmologies

In this paper we explore the idea that black holes can persist in a universe that collapses to a big crunch and then bounces into a new phase of expansion. We use a scalar field to model the matter content of such a universe {near the time} of the bounce, and look for solutions that represent a network of black holes within a dynamical cosmology. We find exact solutions to Einstein's constraint equations that provide the geometry of space at the minimum of expansion and that can be used as initial data for the evolution of hyperspherical cosmologies. These solutions illustrate that there exist models in which multiple distinct black holes can persist through a bounce, and allow for concrete computations of quantities such as the black hole filling factor. We then consider solutions in flat cosmologies, as well as in higher-dimensional spaces (with up to nine spatial dimensions). We derive conditions for the black holes to remain distinct (i.e. avoid merging) and hence persist into the new expansion phase. Some potentially interesting consequences of these models are also discussed.

gr-qc↗

arXiv2017

Black holes as echoes of previous cosmic cycles

The existence of exact solutions which represent a lattice of black holes at a scalar-field-dominated cosmological bounce suggests that black holes could persist through successive eras of a cyclic cosmology. Here we explore some remarkable cosmological consequences of this proposal. In different mass ranges pre-big-bang black holes could explain the dark matter, provide seeds for galaxies, generate entropy and even drive the bounce itself. The cycles end naturally when the filling factor of the black holes reaches unity and this could entail a dimensional transition.

gr-qc↗

arXiv2017

Spikes and matter inhomogeneities in massless scalar field models

We shall discuss the general relativistic generation of spikes in a massless scalar field or stiff perfect fluid model. We first investigate orthogonally transitive (OT) $G_2$ stiff fluid spike models both heuristically and numerically, and give a new exact OT $G_2$ stiff fluid spike solution. We then present a new two-parameter family of non-OT $G_2$ stiff fluid spike solutions, obtained by the generalization of non-OT $G_2$ vacuum spike solutions to the stiff fluid case by applying Geroch's transformation on a Jacobs seed. The dynamics of these new stiff fluid spike solutions is qualitatively different from that of the vacuum spike solutions, in that the matter (stiff fluid) feels the spike directly and the stiff fluid spike solution can end up with a permanent spike. We then derive the evolution equations of non-OT $G_2$ stiff fluid models, including a second perfect fluid, in full generality, and briefly discuss some of their qualitative properties and their potential numerical analysis. Finally, we discuss how a fluid, and especially a stiff fluid or massless scalar field, affects the physics of the generation of spikes.

gr-qc↗

arXiv2016

On the first $G_1$ stiff fluid spike solution in General Relativity

Using the Geroch transformation we obtain the first example of an exact stiff fluid spike solution to the Einstein field equations in a closed form exhibiting a spacelike $G_1$ group of symmetries (i.e., with a single isometry). This new solution is of Petrov type I and exhibits a spike crossing which persists to the past, which allows us to better understand spike crossings in the context of structure formation.

gr-qc↗

arXiv2014

Observational Constraints on the Averaged Universe

Averaging in general relativity is a complicated operation, due to the general covariance of the theory and the non-linearity of Einstein's equations. The latter of these ensures that smoothing spacetime over cosmological scales does not yield the same result as solving Einstein's equations with a smooth matter distribution, and that the smooth models we fit to observations need not be simply related to the actual geometry of spacetime. One specific consequence of this is a decoupling of the geometrical spatial curvature term in the metric from the dynamical spatial curvature in the Friedmann equation. Here we investigate the consequences of this decoupling by fitting to a combination of HST, CMB, SNIa and BAO data sets. We find that only the geometrical spatial curvature is tightly constrained, and that our ability to constrain dark energy dynamics will be severely impaired until we gain a thorough understanding of the averaging problem in cosmology.

astro-ph.CO↗

arXiv2014

Demonstration of the spike phenomenon using the LTB models

We demonstrate the occurrence of permanent spikes using the Lemaitre-Tolman-Bondi models, chosen because the solutions are exact and can be analyzed by qualitative dynamical systems methods. Three examples are given and illustrated numerically. The third example demonstrates that spikes can form directly in the matter density, as opposed to indirectly in previous studies of spikes in the Kasner regime. Spikes provide an alternative general relativistic mechanism for generating exceptionally large structures observed in the Universe.

gr-qc↗