arXiv subjects

Alex Simpson

Publications and source records attributed to Alex Simpson.

36 records · Page 2Linked to original sources

arXiv2021

General class of "quantum deformed" regular black holes

We discuss the "quantum deformed Schwarzschild spacetime" as originally introduced by Kazakov and Solodukhin in 1993, and investigate the precise sense in which it does and does not satisfy the desiderata for being a "regular black hole". We shall carefully distinguish (i) regularity of the metric components, (ii) regularity of the Christoffel components, and (iii) regularity of the curvature. We shall then embed the Kazakov-Solodukhin spacetime in a more general framework where these notions are clearly and cleanly separated. Finally we analyze aspects of the classical physics of these "quantum deformed Schwarzschild spacetimes". We shall discuss the surface gravity, the classical energy conditions, null and timelike geodesics, and the appropriate variant of Regge--Wheeler equation.

gr-qc↗

arXiv2020

Painleve-Gullstrand form of the Lense-Thirring spacetime

The standard Lense-Thirring metric is a century-old slow-rotation large-distance approximation to the gravitational field outside a rotating massive body, depending only on the total mass and angular momentum of the source. Although it is not an exact solution to the vacuum Einstein equations, asymptotically the Lense-Thirring metric approaches the Kerr metric at large distances. Herein we shall discuss a specific variant of the standard Lense-Thirring metric, carefully chosen for simplicity, clarity, and various forms of "improved" behaviour, (to be more carefully defined in the body of the article). In particular we shall construct an explicit unit-lapse Painleve-Gullstrand variant of the Lense-Thirring spacetime, that has flat spatial slices, a very simple and physically intuitive tetrad, and extremely simple curvature tensors. We shall verify that this variant of the Lense-Thirring spacetime is Petrov type I, (so it is not algebraically special), but nevertheless possesses some very straightforward timelike geodesics, (the "rain" geodesics). We shall also discuss on-axis and equatorial geodesics, ISCOs and circular photon orbits. Finally, we wrap up by discussing some astrophysically relevant estimates, and analyze what happens if we extrapolate down to small values of r.

gr-qc↗

arXiv2020

Photon spheres, ISCOs, and OSCOs: Astrophysical observables for regular black holes with asymptotically Minkowski cores

Classical black holes contain a singularity at their core. This has prompted various researchers to propose a multitude of modified spacetimes that mimic the physically observable characteristics of classical black holes as best as possible, but that crucially do not contain singularities at their cores. Due to recent advances in near-horizon astronomy, the ability to observationally distinguish between a classical black hole and a potential black hole mimicker is becoming increasingly feasible. Herein, we calculate some physically observable quantities for a recently proposed regular black hole with an asymptotically Minkowski core -- the radius of the photon sphere and the extremal stable timelike circular orbit (ESCO). The manner in which the photon sphere and ESCO relate to the presence (or absence) of horizons is much more complex than for the Schwarzschild black hole. We find situations in which photon spheres can approach arbitrarily close to (near extremal) horizons, situations in which some photon spheres become stable, and situations in which the locations of both photon spheres and ESCOs become multi-valued, with both ISCOs (innermost stable circular orbits) and OSCOs (outermost stable circular orbits). This provides an extremely rich phenomenology of potential astrophysical interest.

gr-qc↗

arXiv2020

Darboux diagonalization of the spatial 3-metric in Kerr spacetime

The astrophysical importance of the Kerr spacetime cannot be overstated. Of the currently known exact solutions to the Einstein field equations, the Kerr spacetime stands out in terms of its direct applicability to describing astronomical black hole candidates. In counterpoint, purely mathematically, there is an old classical result of differential geometry, due to Darboux, that all 3-manifolds can have their metrics recast into diagonal form. In the case of the Kerr spacetime the Boyer-Lindquist coordinates provide an explicit example of a diagonal spatial 3-metric. Unfortunately, as we demonstrate herein, Darboux diagonalization of the spatial 3-slices of the Kerr spacetime is incompatible with simultaneously putting the Kerr metric into unit-lapse form while retaining manifest axial symmetry. This no-go theorem is somewhat reminiscent of the no-go theorem to the effect that the spatial 3-slices of the Kerr spacetime cannot be chosen to be conformally flat.

gr-qc↗

arXiv2020

Thin-shell traversable wormhole crafted from a regular black hole with asymptotically Minkowski core

Recently, a novel model for a regular black hole was advocated which possesses an asymptotically Minkowski core implemented via an exponential suppression (in the core region) of the Misner-Sharp quasi-local mass. Using this regular black hole as a template, we shall construct a spherically symmetric thin-shell traversable wormhole using the "cut-and-paste" technique, thereby constructing yet another black hole mimicker. The surface stress-energy at the wormhole throat is calculated, and the stability of the wormhole is analyzed. An important result is that, (as compared to their Schwarzschild thin-shell counterparts), increasing the exponential suppression of the Misner-Sharp quasi-local mass by increasing the suppression parameter "a", also considerably increases the stability regions for these thin-shell wormholes, and furthermore minimizes the amount of energy condition violating exotic matter required to keep the wormhole throat open.

gr-qc↗

arXiv2020

Unit-lapse versions of the Kerr spacetime

The Kerr spacetime is perhaps the most astrophysically important of the currently known exact solutions to the Einstein field equations. Whenever spacetimes can be put in unit-lapse form it becomes possible to identify some very straightforward timelike geodesics, (the "rain" geodesics), making the physical interpretation of these spacetimes particularly clean and elegant. The most well-known of these unit-lapse formulations is the Painleve-Gullstrand form of the Schwarzschild spacetime, though there is also a Painleve-Gullstrand form of the Lense-Thirring (slow rotation) spacetime. More radically there are also two known unit-lapse forms of the Kerr spacetime -- the Doran and Natario metrics -- though these are not precisely in Painleve-Gullstrand form. Herein we shall seek to explicate the most general unit-lapse form of the Kerr spacetime. While at one level this is "merely" a choice of coordinates, it is a strategically and tactically useful choice of coordinates, thereby making the technically challenging but astrophysically crucial Kerr spacetime somewhat easier to deal with.

gr-qc↗

arXiv2020

Dynamic thin-shell black-bounce traversable wormholes

Based on the recently introduced black-bounce spacetimes, we shall consider the construction of the related spherically symmetric thin-shell traversable wormholes within the context of standard general relativity. All of the really unusual physics is encoded in one simple parameter $a$ which characterizes the scale of the bounce. Keeping the discussion as close as possible to standard general relativity is the theorist's version of only adjusting one feature of the model at a time. We shall modify the standard thin-shell traversable wormhole construction, each bulk region now being a black-bounce spacetime, and with the physics of the thin shell being (as much as possible) derivable from the Einstein equations. Furthermore, we shall apply a dynamical analysis to the throat by considering linearized radial perturbations around static solutions, and demonstrate that the stability of the wormhole is equivalent to choosing suitable properties for the exotic material residing on the wormhole throat. The construction is sufficiently novel to be interesting, and sufficiently straightforward to be tractable.

gr-qc↗

arXiv2020

ISCOs and OSCOs in the presence of positive cosmological constant

Normally one thinks of the observed cosmological constant as being so small that it can be utterly neglected on typical astrophysical scales, only affecting extremely large-scale cosmology at Gigaparsec scales. Indeed, in those situations where the cosmological constant only has a quantitative influence on the physics, a separation of scales argument guarantees the effect is indeed negligible. The exception to this argument arises when the presence of a cosmological constant qualitatively changes the physics. One example of this phenomenon is the existence of outermost stable circular orbits (OSCOs) in the presence of a positive cosmological constant. Remarkably the size of these OSCOs are of a magnitude to be astrophysically interesting. For instance: for galactic masses the OSCOs are of order the inter-galactic spacing, for galaxy cluster masses the OSCOs are of order the size of the cluster.

gr-qc↗

arXiv2019

Regular black holes with asymptotically Minkowski cores

Standard models of "regular black holes" typically have asymptotically de Sitter regions at their cores. Herein we shall consider novel "hollow" regular black holes, those with asymptotically Minkowski cores. The reason for doing so is twofold: First, these models greatly simplify the physics in the deep core, and second, one can trade off rather messy cubic and quartic polynomial equations for somewhat more elegant special functions such as exponentials and the increasingly important Lambert $W$ function. While these "hollow" regular black holes share many features with the Bardeen/Hayward/Frolov regular black holes there are also significant differences.

gr-qc↗

arXiv2019

Behavioural Equivalence via Modalities for Algebraic Effects

The paper investigates behavioural equivalence between programs in a call-by-value functional language extended with a signature of (algebraic) effect-triggering operations. Two programs are considered as being behaviourally equivalent if they enjoy the same behavioural properties. To formulate this, we define a logic whose formulas specify behavioural properties. A crucial ingredient is a collection of modalities expressing effect-specific aspects of behaviour. We give a general theory of such modalities. If two conditions, openness and decomposability, are satisfied by the modalities then the logically specified behavioural equivalence coincides with a modality-defined notion of applicative bisimilarity, which can be proven to be a congruence by a generalisation of Howe's method. We show that the openness and decomposability conditions hold for several examples of algebraic effects: nondeterminism, probabilistic choice, global store and input/output.

cs.LO↗

arXiv2019

Decomposition of total stress-energy for the generalised Kiselev black hole

We demonstrate that the anisotropic stress-energy supporting the Kiselev black hole can be mimicked by being split into a perfect fluid component plus either an electromagnetic component or a scalar field component, thereby quantifying the precise extent to which the Kiselev black hole fails to represent a perfect fluid spacetime. The perfect fluid component carries either an electric or a scalar charge, which then generates anisotropic electromagnetic or scalar fields. This in turn generates anisotropic contributions to the stress-energy. These in turn induce forces which partially (in addition to the fluid pressure gradient) support the matter content against gravity. This decomposition is carried out both for the original 1-component Kiselev black hole and for the generalized N-component Kiselev black holes. We also comment on the presence of energy condition violations (specifically for the null energy condition --- NEC) for certain sub-classes of Kiselev black holes.

gr-qc↗

arXiv2019

Vaidya spacetimes, black-bounces, and traversable wormholes

We consider a non-static evolving version of the regular "black-bounce"/traversable wormhole geometry recently introduced in JCAP02(2019)042 [arXiv:1812.07114 [gr-qc]]. We first re-write the static metric using Eddington-Finkelstein coordinates, and then allow the mass parameter $m$ to depend on the null time coordinate (a la Vaidya). The spacetime metric is \[ ds^{2}=-\left(1-\frac{2m(w)}{\sqrt{r^{2}+a^{2}}}\right)dw^{2}-(\pm 2 \,dw \,dr) +\left(r^{2}+a^{2}\right)\left(dθ^{2}+\sin^{2}θ\;dϕ^{2}\right). \] Here $w=\{u,v\}$ denotes the $\{outgoing,ingoing\}$ null time coordinate; representing $\{retarded,advanced\}$ time. This spacetime is still simple enough to be tractable, and neatly interpolates between Vaidya spacetime, a black-bounce, and a traversable wormhole. We show how this metric can be used to describe several physical situations of particular interest, including a growing black-bounce, a wormhole to black-bounce transition, and the opposite black-bounce to wormhole transition.

gr-qc↗

arXiv2019

Black-bounce to traversable wormhole

So-called "regular black holes" are a topic currently of considerable interest in the general relativity and astrophysics communities. Herein we investigate a particularly interesting regular black hole spacetime described by the line element \[ ds^{2}=-\left(1-\frac{2m}{\sqrt{r^{2}+a^{2}}}\right)dt^{2}+\frac{dr^{2}}{1-\frac{2m}{\sqrt{r^{2}+a^{2}}}} +\left(r^{2}+a^{2}\right)\left(dθ^{2}+\sin^{2}θ\;dϕ^{2}\right). \] This spacetime neatly interpolates between the standard Schwarzschild black hole and the Morris-Thorne traversable wormhole; at intermediate stages passing through a black-bounce (into a future incarnation of the universe), an extremal null-bounce (into a future incarnation of the universe), and a traversable wormhole. As long as the parameter $a$ is non-zero the geometry is everywhere regular, so one has a somewhat unusual form of "regular black hole", where the "origin" $r=0$ can be either spacelike, null, or timelike. Thus this spacetime generalizes and broadens the class of "regular black holes" beyond those usually considered.

gr-qc↗

arXiv2018

The exponential metric represents a traversable wormhole

For various reasons a number of authors have mooted an "exponential form" for the spacetime metric: \[ ds^2 = - e^{-2m/r} dt^2 + e^{+2m/r}\{dr^2 + r^2(dθ^2+\sin^2θ\, dϕ^2)\}. \] While the weak-field behaviour matches nicely with weak-field general relativity, and so also automatically matches nicely with the Newtonian gravity limit, the strong-field behaviour is markedly different. Proponents of these exponential metrics have very much focussed on the absence of horizons --- it is certainly clear that this geometry does not represent a black hole. However, the proponents of these exponential metrics have failed to note that instead one is dealing with a traversable wormhole --- with all of the interesting and potentially problematic features that such an observation raises. If one wishes to replace all the black hole candidates astronomers have identified with traversable wormholes, then certainly a careful phenomenological analysis of this quite radical proposal should be carried out.

gr-qc↗

arXiv2015

Łukasiewicz μ-calculus

The paper explores properties of the Łukasiewicz μ-calculus, or Łμ for short, an extension of Łukasiewicz logic with scalar multiplication and least and greatest fixed-point operators (for monotone formulas). We observe that Łμ terms, with $n$ variables, define monotone piecewise linear functions from $[0, 1]^n$ to $[0, 1]$. Two effective procedures for calculating the output of Łμ terms on rational inputs are presented. We then consider the Łukasiewicz modal μ-calculus, which is obtained by adding box and diamond modalities to Łμ. Alternatively, it can be viewed as a generalization of Kozen's modal μ-calculus adapted to probabilistic nondeterministic transition systems (PNTS's). We show how properties expressible in the well-known logic PCTL can be encoded as Łukasiewicz modal μ-calculus formulas. We also show that the algorithms for computing values of Łukasiewicz μ-calculus terms provide automatic (albeit impractical) methods for verifying Łukasiewicz modal μ-calculus properties of finite rational PNTS's.

cs.LO↗

arXiv2013

Łukasiewicz mu-Calculus

The paper explores properties of Łukasiewicz mu-calculus, a version of the quantitative/probabilistic modal mu-calculus containing both weak and strong conjunctions and disjunctions from Łukasiewicz (fuzzy) logic. We show that this logic encodes the well-known probabilistic temporal logic PCTL. And we give a model-checking algorithm for computing the rational denotational value of a formula at any state in a finite rational probabilistic nondeterministic transition system.

cs.LO↗

arXiv2012

Linear-use CPS translations in the Enriched Effect Calculus

The enriched effect calculus (EEC) is an extension of Moggi's computational metalanguage with a selection of primitives from linear logic. This paper explores the enriched effect calculus as a target language for continuation-passing-style (CPS) translations in which the typing of the translations enforces the linear usage of continuations. We first observe that established call-by-value and call-by name linear-use CPS translations of simply-typed lambda-calculus into intuitionistic linear logic (ILL) land in the fragment of ILL given by EEC. These two translations are uniformly generalised by a single generic translation of the enriched effect calculus into itself. As our main theorem, we prove that the generic self-translation of EEC is involutive up to isomorphism. As corollaries, we obtain full completeness results, both for the generic translation, and for the original call-by-value and call-by-name translations.

cs.LO↗

arXiv2009

Relational Parametricity for Computational Effects

According to Strachey, a polymorphic program is parametric if it applies a uniform algorithm independently of the type instantiations at which it is applied. The notion of relational parametricity, introduced by Reynolds, is one possible mathematical formulation of this idea. Relational parametricity provides a powerful tool for establishing data abstraction properties, proving equivalences of datatypes, and establishing equalities of programs. Such properties have been well studied in a pure functional setting. Many programs, however, exhibit computational effects, and are not accounted for by the standard theory of relational parametricity. In this paper, we develop a foundational framework for extending the notion of relational parametricity to programming languages with effects.

cs.PL↗