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Kayll Lake

Publications and source records attributed to Kayll Lake.

At least 19 recordsLinked to original sources
arXiv2026

General Expressions for Measurable Parameters in Curved Spacetime

General covariant expressions for measurable angles, distances, velocities, and accelerations are provided in terms of fundamental parameters that can be applied in any setup. The relativistic aberration of light relationship is presented in full generality, which is applicable to any orientation of observers and light rays. An expansion for the geometrical exponential map is established and used to form an expression for the physical distance between an observer and a nearby object within its extended local frame. Curvature effects on measurable distances, velocities, and accelerations are made explicit and appear in general tensorial form. The concepts of Fermi frames on timelike worldlines and the Fermi-Walker derivative are discussed in detail and used throughout; and in examining the meaning of relative stationarity between timelike observers, the Fermi-Walker derivative is established from first principles through physically meaningful consideration. A generalized type of Taylor expansion is provided for tensors of any rank in a covariant form. Expressions for the optically based angular diameter distance and luminosity distance are provided in general forms, and the reciprocity theorem is discussed and verified. A generalized version of the geodesic deviation equation, applicable to extreme relative motion, is provided as well.

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arXiv2023

Constructing maximal extensions of the Vaidya metric in Israel coordinates: II. The completeness of Israel coordinates

We present the results of an analysis of three maximal extensions of the Vaidya metric in Israel coordinates, a spherically symmetric solution to the Einstein field equations for the energy momentum tensor of pure radiation in the high-frequency approximation. This metric is necessary for various applications, such as describing the exterior geometry of a radiating star in astrophysics and studying possible formation of naked singularities in the geometry of spacetime. Contrary to the common Eddington-Finkelstein-like (EFL) coordinates, these maximal extensions, in Israel coordinates, are complete and cover the entirety of the Vaidya manifold. We develop three mass functions, one for each extension, and consider the qualitative characteristics of the three mass models and the surfaces of constant (dynamical) radius. We demonstrate that each maximal extension is null geodesically complete, which we assess by solving the radial null geodesics equation and forming the Penrose conformal diagram for each extension.

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arXiv2023

Constructing Maximal Extensions of the Vaidya Metric in Israel Coordinates: I. Integration of the Field Equations

This paper explores a complete representation of the Vaidya model, a radial flux of radiation in the eikonal approximation, used for modeling various phenomena in both classical and semi-classical General Relativity and Astrophysics. The majority of the applications of the Vaidya model have been formulated in an incomplete representation. A complete representation is obtained here by direct integration of the Einstein field equations. We present the methodology to obtain this complete representation, and its utility in the modeling of general relativistic phenomena.

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arXiv2023

Israel coordinates for all static spherically symmetric spacetimes with vanishing second Ricci invariant

Static spherically symmetric spacetimes with vanishing second Ricci invariant constitute an important class of solutions to Einstein's equations and more generally as archetypes of regular black holes. When studying completeness one is most often presented with the Kruskal - Szekeres procedure. However, this procedure only works if the spacetime admits a single non-degenerate Killing horizon (a single bifurcation two-sphere). Here we generalize the Israel procedure to examine a constructive approach to completeness based entirely on the static spherically symmetric nature of spacetimes with a vanishing second Ricci invariant. It is shown by "block gluing" that the Israel procedure can cover two bifurcation two-spheres, but can fail with three. No coordinate transformations are used in this work.

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arXiv2019

Spacetimes with a vanishing second Ricci invariant

Spacetimes with a vanishing second Ricci invariant, but which are not necessarily Ricci - flat, though common in general relativity, are seldom studied in a coordinate and symmetry independent way by actually using their Ricci invariants. Yet, and as an example, it can be shown by use of Ricci invariants that no such spacetimes can actually represent a perfect fluid by way of Einstein's equations. The widely studied Kiselev black hole is a particularly simple example of such a spacetime. Yet it is frequently, and erroneously, referred to as a perfect fluid. This paper gathers together information relevant to the study of spacetimes with vanishing a second Ricci invariant. It is shown that such spacetimes need not be stationary, a point relevant to their possible physical significance.

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arXiv2019

Boundary Orbits: 1 Static Spacetimes

The study of circular orbits in spacetime is of astrophysical importance. The identification and classification of circular orbits in both static and stationary spacetimes remains an active area of interest. Even in the simplest static spherically symmetric case, it is well known that the introduction of a cosmological constant in vacuum leads to the study of quartic polynomials in order to locate \textit{boundary orbits}, those that straddle between stable and unstable orbits. These orbits are often referred to as `marginally stable orbits' or `indifferently stable orbits'. A comprehensive study of texts offers little clarification as to the stability or instability of these boundary orbits. Here we argue that the direct use of second order perturbation theory immediately shows that these boundary orbits are unstable in the perturbative sense. Our study here includes the two-particle Curzon-Chazy solution.

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arXiv2017

Revisiting the Darmois and Lichnerowicz junction conditions

What have become known as the "Darmois" and "Lichnerowicz" junction conditions are often stated to be equivalent, "essentially" equivalent, in a "sense" equivalent, and so on. One even sees not infrequent reference to the "Darmois-Lichnerowicz" conditions. Whereas the equivalence of these conditions is manifest in Gaussian-normal coordinates, a fact that has been known for close to a century, this equivalence does not extend to a loose definition of "admissible" coordinates (coordinates in which the metric and its first order derivatives are continuous). We show this here by way of a simple, but physically relevant, example. In general, a loose definition of the "Lichnerowicz" conditions gives additional restrictions, some of which simply amount to a convenient choice of gauge, and some of which amount to real physical restrictions, away from strict "admissible" coordinates. The situation was totally confused by a very influential, and now frequently misquoted, paper by Bonnor and Vickers, that erroneously claimed a proof of the equivalence of the "Darmois" and "Lichnerowicz" conditions within this loose definition of "admissible" coordinates. A correct proof, based on a strict definition of "admissible" coordinates, was given years previous by Israel. It is that proof, generally unrecognized, that we must refer to. Attention here is given to a clarification of the subject, and to the history of the subject, which, it turns out, is rather fascinating in itself.

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arXiv2016

Relativistic Aberration and the Cosmological Constant in Gravitational Lensing I: Introduction

An analysis of null geodesics in Schwarzschild de Sitter space is presented with special attention to their global `bending angles', local measurable angles, and the involvement of the cosmological constant. We make use of a general technique which allows for finding observable intersection angles of null trajectories analytically. A general relativistic aberration relationship is established as one of its applications. The question of whether or not the cosmological constant, $Λ$, contributes to orbits of light and to related observable quantities is addressed in detail. We also discuss the ongoing debate on this issue and respond to some recent papers on the topic. The dependence of measurable quantities on the motion of observers is stressed throughout. Exact formulas for measurable intersection angles, as well as gravitational lens equations for observers in the Schwarzschild de Sitter background are provided.

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arXiv2016

An infinite class of exact static anisotropic spheres that break the Buchdal bound

An infinite class of exact static anisotropic spheres is developed. All members of the class satisfy (i) regularity (meaning no singularities), and in particular at the origin, (ii) positive but monotone decreasing energy density ($ρ(r)$), radial pressure ($p(r)$), and tangential pressure ($P(r)$), (iii) a finite value of $r=R$ such that $p(R)=0$ defining the boundary surface onto vacuum, (iv) $p \leq ρ$, and (v) $p + 2 P=3 ρ$. All standard energy conditions are satisfied except for the dominant energy condition which has an innocuous violation by the tangential stress since $ρ\leq P$ by construction. An infinite number of the solutions violate the Buchdal bound.

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arXiv2015

On the global structure of Kerr-de Sitter spacetimes

Taking advantage of the natural length scale set by the cosmological constant $Λ>0$, conditions on the parameters $(Λ, M, a^{2})$ have been found, so that a Kerr-de Sitter specetime either describes a black hole with well separated horizons, or describes degenerate configurations where two or more horizons coincide. As long as the rotation parameter $a^{2}$ is subject to the constraint $a^{2}Λ\ll 1$, while the mass parameter $M$ is subject to: $ a^{2}[1+O(a^{2}Λ)^{2})] \frac {1}{9Λ}[1+2a^{2}Λ+O(a^{2}Λ)^{2})]$ or $M^{2}< a^{2}[1+O(a^{2}Λ)^{2})]$, the Kerr-de Sitter spacetime describes a ring-like singularity enclosed by two cosmological horizons. A Kerr-de Sitter spacetime may also describe configurations where the inner, the outer and one of the cosmological horizons coincide. However, we found that this coalescence occurs provided $M^{2}Λ\sim 1$ and due to the observed smallness of $Λ$, these configurations are probably irrelevant in astrophysical settings. Extreme black holes, i.e. black holes where the inner horizon coincides with the outer black hole horizon are also admitted. We have found that in the limit $M^{2}Λ\ll 1$ and $a^{2}Λ\ll 1$, extreme black holes occur, provided $a^{2}=M^{2}(1+O(ΛM^{2}))$. Finally a coalescence between the outer and the cosmological horizon, although in principle possible, is likely to be unimportant at the astrophysical level, since this requires $M^{2}Λ\sim 1$.

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arXiv2015

Central density cusps in the Lemaître-Tolman solutions

The character of the central density profile in the Lemaître-Tolman (LT) solutions plays a fundamental role in their application as cosmological models. This same character is studied here for these solutions used to model complete gravitational collapse. A necessary condition for the development of a black hole (not even locally naked singularities) is developed. This condition allows central density cusps, the central feature of the LT solutions when used to match cosmological observations without invoking the cosmological constant.

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arXiv2015

Invariant characterization of the Kerr spacetime: Locating the horizon and measuring the mass and spin of rotating black holes using curvature invariants

We provide an invariant characterization of the physical properties of the Kerr spacetime. We introduce two dimensionless invariants, constructed out of some known curvature invariants, that act as detectors for the event horizon and ergosurface of the Kerr black hole. We also show that the mass and angular momentum can be extracted from local measurements of the curvature invariants, which in the weak field limit could be used to approximate the total angular momentum and mass of a system of merging black holes. Finally, we introduce a dimensionless invariant that gives a local measure of the "Kerrness" of the spacetime.

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arXiv2013

The Vector Volume and Black Holes

By examining the rate of growth of an invariant volume $\mathcal V$ of some spacetime region along a divergence-free vector field $v^α$, we introduce the concept of a "vector volume" $\mathcal{V}_v$. This volume can be defined in various equivalent ways. For example, it can be given as $\mathrm d \mathcal V(μ) / \mathrm d μ$, where $v^α\partial_α= \mathrm d / \mathrm d μ$, and $μ$ is a parameter distance along the integral curve of $v$. Equivalently, it can be defined as $\int v^α\mathrm d Σ_α$, where $\mathrm d Σ_α$ is the directed surface element. We find that this volume is especially useful for the description of black holes, but it can be used in other contexts as well. Moreover, this volume has several properties of interest. Among these is the fact that the vector volume is linear with respect to the the choice of vector $v^α$. As a result, for example, in stationary axially symmetric spacetimes with timelike Killing vectors $t^α$ and axial symmetric Killing vectors $ϕ^α$, the vector volume of an axially symmetric region with respect to the vector $t^α+ Ωϕ^α$ is equal for any value of $Ω$, a consequence of the additional result that $ϕ^α$ does not contribute to $\mathcal{V}_v$. Perhaps of most interest is the fact that in Kerr-Schild spacetimes the volume element for the full spacetime is equal to that of the background spacetime. We discuss different ways of using the vector volume to define volumes for black holes. Finally, we relate our work to the recent wide-spread thermodynamically motivated study of the "volumes" of black holes associated with non-zero values of the cosmological constant $Λ$.

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arXiv2013

Visualizing Spacetime Curvature via Gradient Flows III: The Kerr Metric and the Transitional Values of the Spin Parameter

The Kerr metric is one of the most important solutions to Einstein's field equations, describing the gravitational field outside a rotating black hole. We thoroughly analyze the curvature scalar invariants to study the Kerr spacetime by examining and visualizing their covariant gradient fields. We discover that the part of the Kerr geometry outside the black hole horizon changes qualitatively depending on the spin parameter, a fact previously unknown. The number of observable critical points of the curvature invariants' gradient fields along the axis of rotation changes at several transitional values of the spin parameter. These transitional values are a fundamental property of the Kerr metric. They are physically important since in general relativity these curvature invariants represent the cumulative tidal and frame-dragging effects of rotating black holes in an observer-independent way.

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arXiv2013

On the influence of the cosmological constant on trajectories of light and associated measurements in Schwarzschild de Sitter space

In this paper we review and build on the common methods used to analyze null geodesics in Schwarzschild de Sitter space. We present a general technique which allows finding measurable intersection angles of null trajectories analytically, and as one of its applications we establish a general relativistic aberration relationship. The tools presented are used to analyze some standard setups of gravitational deflection of light and gain a clear understanding of the role that the cosmological constant, $Λ$, plays in gravitational lensing phenomena. Through reviewing some recent papers on the topic with the present results in mind, we attempt to explain the major sources of disagreement in the ongoing debate on the subject, which started with Rindler and Ishak's original paper, regarding the influence of $Λ$ on lensing phenomena. To avoid ambiguities and room for misunderstanding we present clear definitions of the quantities used in the present analysis as well as in other papers we discuss.

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arXiv2012

Visualizing Spacetime Curvature via Gradient Flows II: An Example of the Construction of a Newtonian analogue

This is the first in a series of papers in which the gradient flows of fundamental curvature invariants are used to formulate a visualization of curvature. We start with the construction of strict Newtonian analogues (not limits) of solutions to Einstein's equations based on the topology of the associated gradient flows. We do not start with any easy case. Rather, we start with the Curzon - Chazy solution, which, as history shows, is one of the most difficult exact solutions to Einstein's equations to interpret physically. We show that the entire field of the Curzon - Chazy solution, up to a region very "close" to the the intrinsic singularity, strictly represents that of a Newtonian ring, as has long been suspected. In this regard, we consider our approach very successful. As regrades the local structure of the singularity of the Curzon - Chazy solution within a fully general relativistic analysis, however, whereas we make some advances, the full structure of this singularity remains incompletely resolved.

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arXiv2012

Visualizing Spacetime Curvature via Gradient Flows I: Introduction

Traditional approaches to the study of the dynamics of spacetime curvature in a very real sense hide the intricacies of the nonlinear regime. Whether it be huge formulae, or mountains of numerical data, standard methods of presentation make little use of our remarkable skill, as humans, at pattern recognition. Here we introduce a new approach to the visualization of spacetime curvature. We examine the flows associated with the gradient fields of invariants derived from the spacetime. These flows reveal a remarkably rich structure, and offer fresh insights even for well known analytical solutions to Einstein's equations. This paper serves as an overview and as an introduction to this approach.

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