arXiv subjects

Charles Hellaby

Publications and source records attributed to Charles Hellaby.

At least 19 recordsLinked to original sources

arXiv2026

Mirror Symmetry and Double Signature Change

The black mirror spacetime proposed by Tzanavaris, Boyle, and Turok [1] connects the two exterior regions of the extended Schwarzschild black hole directly to each other, with no intervening interior region. Using techniques adapted from previous work on signature change, we reexamine the black mirror spacetime as a model of double signature change, and investigate whether there is a surface layer at the horizon, that is, a distributional curvature singularity corresponding to an impulsive gravitational wave. We confirm that the black mirror spacetime does not contain any such singularity, and compare our result with previous claims that the curvature components are analytic. We also discuss the global structure of the black mirror spacetime, and examine what happens to worldlines and curves passing through.

gr-qc↗

arXiv2021

Rotation, Embedding and Topology for the Szekeres Geometry

Recent work on the Szekeres inhomogeneous cosmological models uncovered a surprising rotation effect. Hellaby showed that the angular $(θ, ϕ)$ coordinates do not have a constant orientation, while Buckley and Schlegel provided explicit expressions for the rate of rotation from shell to shell, as well as the rate of tilt when the 3-space is embedded in a flat 4-d Euclidean space. We here investigate some properties of this embedding, for the quasi-spherical recollapsing case, and use it to show that the two sets of results are in complete agreement. We also show how to construct Szekeres models that are closed in the 'radial' direction, and hence have a 'natural' embedded torus topology. Several explicit models illustrate the embedding as well as the shell rotation and tilt effects.

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arXiv2020

Reversing the Null Limit of the Szekeres Metric

The null limits of the Lemaitre-Tolman and Szekeres spacetimes are known to be the Vaidya and news-free Robinson-Trautman metrics. We generalise this result to the case of non-zero $Λ$, and then ask whether the reverse process is possible -- is there a systematic procedure to retrieve the timelike-dust metric from the null-dust case? We present such an algorithm for re-constructing both the metric and matter tensor components of the timelike-dust manifold. This undertaking has elucidated the null limit process, highlighted which quantities approach unity or zero, and necessitated a careful discussion of how the functional dependencies are managed by the transformations and substitutions used.

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arXiv2017

Calculating Observables in Inhomogeneous Cosmologies I: General Framework

We lay out a general framework for calculating the variation of a set of cosmological observables, down the past null cone of an arbitrarily placed observer, in a given arbitrary inhomogeneous metric. The observables include redshift, proper motions, area distance and redshift-space density. Of particular interest are observables that are zero in the spherically symmetric case, such as proper motions. The algorithm is based on the null geodesic equation and the geodesic deviation equation, and it is tailored to creating a practical numerical implementation. The algorithm provides a method for tracking which light rays connect moving objects to the observer at successive times. Our algorithm is applied to the particular case of the Szekeres metric. A numerical implementation has been created and some results will be presented in a subsequent paper. Future work will explore the range of possibilities.

gr-qc↗

arXiv2017

Frame Rotation in the Szekeres Spacetimes

The Szekeres metric is an inhomogeneous cosmological model without any symmetries. The standard Riemann-type coordinates can be transformed into spherical-type coordinates, but the metric is no longer diagonal, and the constant "radius" 2-spheres, 2-hyperboloids or 2-planes are known to be "non-concentric". Since the transformation into spherical-type coordinates is "radius" dependent, we question whether these coordinates have the same orientation on each 2-surface. To answer this question, we set up an orthonormal tetrad (ONT), and investigate its variation. We find that a relative rotation of the tetrad is generic, and it can increase systematically under conditions that are not very restrictive. We search for paths along which the tetrad is constant, and find they only exist under very restrictive conditions. In the process, we create a systematic method for defining an ONT with chosen properties from a given metric.

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arXiv2017

Symmetry and Equivalence in Szekeres Models

We solve for all Szekeres metrics that have a single Killing vector. For quasi hyperboloidal ($ε= -1$) metrics, we find that translational symmetries are possible, but only in metrics that have shell crossings somewhere, while metrics that can be made free of shell crossings only permit rotations. The quasi planar metrics ($ε= 0$) either have no Killing vectors or they admit full planar symmetry. Single symmetries in quasi spherical metrics ($ε= +1$) are all rotations. The rotations correspond to a known family of axially symmetric metrics, which for each $ε$ value, are equivalent to each other. We consider Szekeres metrics in which the line of dipole extrema is required to be geodesic in the 3-space, and show the same set of families emerges. We investigate when two Szekeres metrics are physically equivalent, and complete a previous list of transformations of the arbitrary functions.

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arXiv2015

Ricci Time in the Lemaitre-Tolman Model and the Block Universe

It is common to think of our universe according to the "block universe" concept, which says that spacetime consists of many "stacked" 3-surfaces, labelled by some kind of proper time, $τ$. Standard ideas do not distinguish past and future, but Ellis "evolving block universe" tries to make a fundamental distinction. One proposal for this proper time is the proper time measured along the timelike Ricci eigenlines, starting from the big bang. This work investigates the shape of the "Ricci time" surfaces relative to the the null surfaces. We use the Lemaitre-Tolman metric as our inhomogeneous spacetime model, and we find the necessary and sufficient conditions for these $\{τ$= constant$\}$ surfaces, $S(τ)$, to be spacelike or timelike. Furthermore, we look at the effect of strong gravity domains by determining the location of timelike $S$ regions relative to apparent horizons. We find that constant Ricci time surfaces are always spacelike near the big bang, while at late times (near the crunch or the extreme far future), they are only timelike under special circumstances. At intermediate times, timelike $S$ regions are common unless the variation of the bang time is restricted. The regions where these surfaces become timelike are often adjacent to apparent horizons, but always outside them, and in particular timelike $S$ regions do not occur inside black holes.

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arXiv2012

Constructing Realistic Szekeres Models from Initial and Final Data

The Szekeres family of inhomogeneous solutions, which are defined by six arbitrary metric functions, offers a wide range of possibilities for modelling cosmic structure. Here we present a model construction procedure for the quasispherical case using given data at initial and final times. Of the six arbitrary metric functions, the three which are common to both Szekeres and Lemaître-Tolman models are determined by the model construction procedure of Krasinski & Hellaby. For the remaining three functions, which are unique to Szekeres models, we derive exact analytic expressions in terms of more physically intuitive quantities - density profiles and dipole orientation angles. Using MATLAB, we implement the model construction procedure and simulate the time evolution.

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arXiv2012

A New Type of Exact Arbitrarily Inhomogeneous Cosmology: Evolution of Deceleration in the Flat Homogeneous-On-Average Case

A new method for constructing exact inhomogeneous universes is presented, that allows variation in 3 dimensions. The resulting spacetime may be statistically uniform on average, or have random, non-repeating variation. The construction utilises the Darmois junction conditions to join many different component spacetime regions. In the initial simple example given, the component parts are spatially flat and uniform, but much more general combinations should be possible. Further inhomogeneity may be added via swiss cheese vacuoles and inhomogeneous metrics. This model is used to explore the proposal, that observers are located in bound, non-expanding regions, while the universe is actually in the process of becoming void dominated, and thus its average expansion rate is increasing. The model confirms qualitatively that the faster expanding components come to dominate the average, and that inhomogeneity results in average parameters which evolve differently from those of any one component, but more realistic modelling of the effect will need this construction to be generalised.

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arXiv2011

The Metric of the Cosmos from Luminosity and Age Data

This paper presents the algorithm for determining the Lemaitre-Tolman (LT) model that best fits given datasets for maximum stellar ages, and SNIa luminosities, both as functions of redshift. It then applies it to current cosmological data. Special attention must be given to the handling of the origin, and the region of the maximum diameter distances. As with a previous combination of datasets (galaxy number counts and luminosity distances versus redshift), there are relationships that must hold at the region of the maximum diameter distance, which are unlikely to be obeyed exactly by real data. We show how to make corrections that enable a self-consistent solution to be found. We address the questions of the best way to approximate discrete data with smooth functions, and how to estimate the uncertainties of the output - the 3 free functions that determine a specific LT metric. While current data does not permit any confidence in our results, we show that the method works well, and reasonable LT models do fit with or without a cosmological constant.

astro-ph.CO↗

arXiv2010

Imitating accelerated expansion of the Universe by matter inhomogeneities - corrections of some misunderstandings

A number of misunderstandings about modeling the apparent accelerated expansion of the Universe, and about the `weak singularity' are clarified: 1. Of the five definitions of the deceleration parameter given by Hirata and Seljak (HS), only $q_1$ is a correct invariant measure of acceleration/deceleration of expansion. The $q_3$ and $q_4$ are unrelated to acceleration in an inhomogeneous model. 2. The averaging over directions involved in the definition of $q_4$ does not correspond to what is done in observational astronomy. 3. HS's equation (38) connecting $q_4$ to the flow invariants gives self-contradictory results when applied at the centre of symmetry of the Lema\^ıtre-Tolman (L-T) model. The intermediate equation (31) that determines $q_{3'}$ is correct, but approximate, so it cannot be used for determining the sign of the deceleration parameter. Even so, at the centre of symmetry of the L-T model, it puts no limitation on the sign of $q_{3'}(0)$. 4. The `weak singularity' of Vanderveld {\it et al.} is a conical profile of mass density at the centre - a perfectly acceptable configuration. 5. The so-called `critical point' in the equations of the `inverse problem' for a central observer in an L-T model is a manifestation of the apparent horizon - a common property of the past light cones in zero-lambda L-T models, perfectly manageable if the equations are correctly integrated.

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arXiv2010

The Lemaitre Model and the Generalisation of the Cosmic Mass

We consider the spherically symmetric metric with a comoving perfect fluid and non-zero pressure -- the Lemaitre metric -- and present it in the form of a calculational algorithm. We use it to review the definition of mass, and to look at the apparent horizon relations on the observer's past null cone. We show that the introduction of pressure makes it difficult to separate the mass from other physical parameters in an invariant way. Under the usual mass definition, the apparent horizon relation, that relates the diameter distance to the cosmic mass, remains the same as in the Lemaitre-Tolman case.

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arXiv2009

Modelling Inhomogeneity in the Universe

An overview of some recent developments in inhomogeneous models is presented. As the volume and precision of cosmological data improves, it will become more and more essential to understand the non-linear behaviour of the Einstein field equations. This requires the study of exact inhomogeneous solutions, including their density distributions, their evolution, their geometry, and their causal structure. Observations are strongly affected by the detailed geometry and evolution of a model, and therefore interpretation of observations depends on understanding them. It is generally assumed the universe is homogeneous if averaged over large enough scales, but to actually prove this is so, will require the assumption to be relaxed, and a rigorous inhomogeneous approach to be applied. Though the \LT metric has long been used for models of spherical inhomogeneities, there have been a number of new results, including a variety of methods for creating models with specific properties, and their application to cosmic structures on several different scales. Interest in the Szekeres metrics is on the increase, and the quasi-spherical metric was recently used to model specific cosmic structures for the first time. The quasi-planar and quasi-hyperspherical metrics have been hardly studied until recent work invesigated their physical and geometric properties. There is enormous scope for work with these metrics.

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arXiv2009

More examples of structure formation in the Lemaitre-Tolman model

In continuing our earlier research, we find the formulae needed to determine the arbitrary functions in the Lemaitre-Tolman model when the evolution proceeds from a given initial velocity distribution to a final state that is determined either by a density distribution or by a velocity distribution. In each case the initial and final distributions uniquely determine the L-T model that evolves between them, and the sign of the energy-function is determined by a simple inequality. We also show how the final density profile can be more accurately fitted to observational data than was done in our previous paper. We work out new numerical examples of the evolution: the creation of a galaxy cluster out of different velocity distributions, reflecting the current data on temperature anisotropies of CMB, the creation of the same out of different density distributions, and the creation of a void. The void in its present state is surrounded by a nonsingular wall of high density.

gr-qc↗

arXiv2009

Solving the Observer Metric

The analysis of modern cosmological data is becoming an increasingly important task as the amount of data multiplies. An important goal is to extract geometric information, i.e. the metric of the cosmos, from observational data. The observer metric is adapted to the reality of observations: information received along the past null cone, and matter flowing along timelike lines. It provides a potentially very good candidate for a developing general numerical data reduction program. As a basis for this, we elucidate the spherically symmetric solution, for which there is to date single presentation that is complete and correct. With future numerical implementation in mind, we give a clear presentation of the mathematical solution in terms of 4 arbitrary functions, the solution algorithm given observational data on the past null cone, and we argue that the evolution from one null cone to the next necessarily involves integrating down each null cone.

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arXiv2008

Determining the metric of the Cosmos: stability, accuracy, and consistency

The ultimate application of Einstein's field equations is to empirically determine the geometry of the Universe from its matter content, rather than simply assuming the Universe can be represented by a homogeneous model on all scales. Choosing an LTB model as the most convenient inhomogeneous model for the early stages of development, a data reduction procedure was recently validated using perfect test data. Here we simulate observational uncertainties and improve the previous numerical scheme to ensure that it will be usable with real data as soon as observational surveys are sufficiently deep and complete. Two regions require special treatment--the origin and the maximum in the areal radius. To minimize numerical errors near the origin, we use an LTB series expansion to provide the initial values for integrating the differential equations. We also use an improved method to match the numerical integration to the series expansion that bridges the region near the maximum in the areal radius. Because the mass enclosed within the maximum obeys a specific relationship, we show that it is possible to correct for a fixed systematic error in either the distance scale or the redshift-space mass density, such that the integrated values are consistent with the data at the maximum.

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arXiv2007

Physical and Geometrical Interpretation of the epsilon <= 0 Szekeres Models

We study the properties and behaviour of the quasi-pseudospherical and quasi-planar Szekeres models, obtain the regularity conditions, and analyse their consequences. The quantities associated with "radius" and "mass" in the quasi-spherical case must be understood in a different way for these cases. The models with pseudospherical foliation can have spatial maxima and minima, but no origins. The "mass" and "radius" functions may be one increasing and one decreasing without causing shell crossings. This case most naturally describes a snake-like, variable density void in a more gently varying inhomogeneous background, although regions that develop an overdensity are also possible. The Szekeres models with plane foliation can have neither spatial extrema nor origins, cannot be spatially flat, and they cannot have more inhomogeneity than the corresponding Ellis model, but a planar surface can be the boundary between regions of spherical and pseudospherical foliation.

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arXiv2007

Obtaining the spacetime metric from cosmological observations

Recent galaxy redshift surveys have brought in a large amount of accurate cosmological data out to redshift 0.3, and future surveys are expected to achieve a high degree of completeness out to a redshift exceeding 1. Consequently, a numerical programme for determining the metric of the universe from observational data will soon become practical; and thereby realise the ultimate application of Einstein's equations. Apart from detailing the cosmic geometry, this would allow us to verify and quantify homogeneity, rather than assuming it, as has been necessary up to now, and to do that on a metric level, and not merely at the mass distribution level. This paper is the beginning of a project aimed at such a numerical implementation. The primary observational data from our past light cone consists of galaxy redshifts, apparent luminosities, angular diameters and number densities, together with source evolution functions, absolute luminosities, true diameters and masses of sources. Here we start with the simplest case, that of spherical symmetry and a dust equation of state, and execute an algorithm that determines the unknown metric functions from this data. We discuss the challenges of turning the theoretical algorithm into a workable numerical procedure, particularly addressing the origin and the maximum in the area distance. Our numerical method is tested with several artificial data sets for homogeneous and inhomogeneous models, successfully reproducing the original models. This demonstrates the basic viability of such a scheme. Although current surveys don't have sufficient completeness or accuracy, we expect this situation to change in the near future, and in the meantime there are many refinements and generalisations to be added.

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