arXiv subjects

Alex Simpson

Publications and source records attributed to Alex Simpson.

At least 19 recordsLinked to original sources

arXiv2025

Design and initial results from the "Junior" Levitated Dipole Experiment

OpenStar Technologies is a private fusion company exploring the levitated dipole concept for commercial fusion energy production. OpenStar has manufactured a new generation of levitated dipole experiment, called "Junior", leveraging recent advances made in high-temperature superconducting magnet technologies. Junior houses a ~5.6 T REBCO high-temperature superconducting magnet in a 5.2 m vacuum chamber, with plasma heating achieved via < 50 kW of electron cyclotron resonance heating power. Importantly, this experiment integrates novel high temperature superconductor power supply technology on board the dipole magnet. Recently OpenStar has completed first experimental campaigns with the Junior experiment, achieving first plasmas in late 2024. Experiments conducted with the full levitated system are planned for 2025. This article provides an overview of the main results from these experiments and details improvements planned for future campaigns.

physics.plasm-ph↗

arXiv2025

Safety, Relative Tightness and the Probabilistic Frame Rule

Probabilistic separation logic offers an approach to reasoning about imperative probabilistic programs in which a separating conjunction is used as a mechanism for expressing independence properties. Crucial to the effectiveness of the formalism is the frame rule, which enables modular reasoning about independent probabilistic state. We explore a semantic formulation of probabilistic separation logic, in which the frame rule has the same simple formulation as in separation logic, without further side conditions. This is achieved by building a notion of safety into specifications, using which we establish a crucial property of specifications, called relative tightness, from which the soundness of the frame rule follows.

cs.LO↗

arXiv2025

Euclidean interval objects in categories with finite products

Based on the intuitive notion of convexity, we formulate a universal property defining interval objects in a category with finite products. Interval objects are structures corresponding to closed intervals of the real line, but their definition does not assume a pre-existing notion of real number. The universal property characterises such structures up to isomorphism, supports the definition of functions between intervals, and provides a means of verifying identities between functions. In the category of sets, the universal property characterises closed intervals of real numbers with nonempty interior. In the the category of topological spaces, we obtain intervals with the Euclidean topology. We also prove that every elementary topos with natural numbers object contains an interval object; furthermore, we characterise interval objects as intervals of real numbers in the Cauchy completion of the rational numbers within the Dedekind reals.

math.CT↗

arXiv2025

Stochastic Calculus via Stopping Derivatives

We show that a substantial portion of stochastic calculus can be developed along similar lines to ordinary calculus, with derivative-based concepts driving the development. We define a notion of stopping derivative, which is a form of right derivative with respect to stopping times. Using this, we define the drift and variance rate of a process as stopping derivatives for (generalised) conditional expectation and conditional variance respectively. Applying elementary, derivative-based methods, we derive a calculus of rules describing how drift and variance rate transform under operations on processes, culminating in a version of the multi-dimensional It\^o formula. Our approach connects with the standard machinery of stochastic calculus via a theorem establishing that continuous processes with zero drift coincide with random translations of continuous local martingales. This equivalence allows us to derive a fundamental theorem of calculus for stopping derivatives, which relates the quantities of drift and variance rate, defined as stopping derivatives, to parameters used in the description of a process as a stochastic integral.

math.PR↗

arXiv2024

An Imperative Language for Verified Exact Real-Number Computation

We introduce Clerical, a programming language for exact real-number computation that combines first-order imperative-style programming with a limit operator for computation of real numbers as limits of Cauchy sequences. We address the semidecidability of the linear ordering of the reals by incorporating nondeterministic guarded choice, through which decisions based on partial comparison operations on reals can be patched together to give total programs. The interplay between mutable state, nondeterminism, and computation of limits is controlled by the requirement that expressions computing limits and guards modify only local state. We devise a domain-theoretic denotational semantics that uses a variant of Plotkin powerdomain construction tailored to our specific version of nondeterminism. We formulate a Hoare-style specification logic, show that it is sound for the denotational semantics, and illustrate the setup by implementing and proving correct a program for computation of $\pi$ as the least positive zero of $\sin$. The modular character of Clerical allows us to compose the program from smaller parts, each of which is shown to be correct on its own. We provide a proof-of-concept OCaml implementation of Clerical, and formally verify parts of the development, notably the soundness of specification logic, in the Coq proof assistant.

cs.LO↗

arXiv2024

Equivalence and Conditional Independence in Atomic Sheaf Logic

We propose a semantic foundation for logics for reasoning in settings that possess a distinction between equality of variables, a coarser equivalence of variables, and a notion of conditional independence between variables. We show that such relations can be modelled naturally in atomic sheaf toposes. Equivalence of variables is modelled by a relation of atomic equivalence that is possessed by every atomic sheaf. We identify additional structure on the category generating the atomic topos that allows the relation of conditional independence to be interpreted in the topos. We then study the logic of equivalence and conditional independence that is induced by the internal logic of the topos. This atomic sheaf logic is a classical logic that validates a number of fundamental reasoning principles relating equivalence and conditional independence. As a concrete example of this abstract framework, we use the atomic topos over the category of surjections between finite nonempty sets as our main running example. In this category, the interpretations of equivalence and conditional independence coincide with those given by the multiteam semantics of independence logic, in which the role of equivalence is taken by the relation of mutual inclusion. A major difference from independence logic is that, in atomic sheaf logic, the multiteam semantics of the equivalence and conditional independence relations is embedded within a classical surrounding logic. We outline two other instances of our framework, to demonstrate its versatility. The first is a category of probability sheaves, in which atomic equivalence is equality-in-distribution, and the conditional independence relation is the usual probabilistic one. Our other example is the Schanuel topos where equivalence is orbit equality and conditional independence amounts to a relative form of separatedness.

cs.LO↗

arXiv2023

Excising Curvature Singularities from General Relativity

This thesis operates within the framework of general relativity without curvature singularities. The motivation for this framework is explored, and several conclusions are drawn with a look towards future research. There are many ways to excise curvature singularities from general relativity; a full list of desirable constraints on candidate geometries is presented. Several specific candidate spacetimes in both spherical symmetry and axisymmetry are rigorously analysed, typically modelling (charged or uncharged) regular black holes or traversable wormholes. Broadly, these are members of the family of black-bounce spacetimes, and the family of black holes with asymptotically Minkowski cores. Related thin-shell traversable wormhole constructions are also explored, as well as a brief look at the viability of thin-shell Dyson mega-spheres. The eye of the storm geometry is analysed, and discovered to be very close to an idealised candidate geometry within this framework. It is found to contain highly desirable features, and is not precluded by currently available measurements. For all spacetimes discussed, particular focus is placed on the extraction of (potential) astrophysical observables in principle falsifiable/verifiable by the observational and experimental communities. A cogent effort is made to streamline the discourse between theory and experiment, and to begin filling the epistemological gap, which will enable the various communities involved to optimise the advancement of physics via the newly available observational technologies (such as LIGO/Virgo, and the upcoming LISA). Furthermore, three somewhat general theorems are presented, and two new geometries are introduced for the first time to the literature.

gr-qc↗

arXiv2022

General-relativistic thin-shell Dyson mega-spheres

Loosely inspired by the somewhat fanciful notion of detecting an arbitrarily advanced alien civilization, we consider a general-relativistic thin-shell Dyson mega-sphere completely enclosing a central star-like object, and perform a full general-relativistic analysis using the Israel--Lanczos--Sen junction conditions. We focus attention on the surface mass density, the surface stress, the classical energy conditions, and the forces between hemispheres. We find that in the physically acceptable region the NEC, WEC, and SEC are always satisfied, while the DEC can be violated if the Dyson mega-sphere is sufficiently close to forming a black hole. We also demonstrate that the original quasi-local version of the maximum force conjecture, F <= {1/4} F_{Stoney} = {1/4} F_{Planck}, can easily be violated if the Dyson mega-sphere is sufficiently compact, that is, sufficiently close to forming a black hole. Interestingly there is a finite region of parameter space where one can violate the original quasi-local version of the maximum force conjecture without violating the DEC. Finally, we very briefly discuss the possibility of nested thin-shell mega-spheres (Matrioshka configurations) and thick-shell Dyson mega-spheres.

gr-qc↗

arXiv2022

Constant-$r$ geodesics in the Painleve-Gullstrand form of Lense-Thirring spacetime

Herein we explore the non-equatorial constant-$r$ ("quasi-circular") geodesics (both timelike and null) in the Painleve-Gullstrand variant of the Lense-Thirring spacetime recently introduced by the current authors. Even though the spacetime is not spherically symmetric, shells of constant-$r$ geodesics still exist. Whereas the radial motion is (by construction) utterly trivial, determining the allowed locations of these constant-$r$ geodesics is decidedly non-trivial, and the stability analysis is equally tricky. Regarding the angular motion, these constant-$r$ orbits will be seen to exhibit both precession and nutation -- typically with incommensurate frequencies. Thus this constant-$r$ geodesic motion, though integrable in the precise technical sense, is generically surface-filling, with the orbits completely covering a symmetric equatorial band which is a segment of a spherical surface, (a so-called "spherical zone"), and whose latitudinal extent is governed by delicate interplay between the orbital angular momentum and the Carter constant. The situation is qualitatively similar to that for the (exact) Kerr spacetime -- but we now see that any physical model having the same slow-rotation weak-field limit as general relativity will still possess non-equatorial constant-$r$ geodesics.

gr-qc↗

arXiv2021

Astrophysically viable Kerr-like spacetime -- into the eye of the storm

We analyse a rotating regular black hole with asymptotically Minkowski core. This Kerr-like geometry possesses the full "Killing tower" of nontrivial Killing tensor, Killing-Yano tensor, and principal tensor. The Hamilton-Jacobi equation, the Klein-Gordon equation, and Maxwell's equations are separable. Energy-condition-violating physics is pushed into an arbitrarily small region in the deep core. The geometry has a very high level of mathematical tractability; extraction of astrophysical observables falsifiable/verifiable by the observational community is straightforward.

gr-qc↗

arXiv2021

Geodesics for the Painleve-Gullstrand form of Lense-Thirring spacetime

Recently, the current authors have formulated and extensively explored a rather novel Painleve-Gullstrand variant of the slow-rotation Lense-Thirring spacetime, a variant which has particularly elegant features -- including unit lapse, intrinsically flat spatial 3-slices, and a separable Klein-Gordon equation (wave operator). This spacetime also possesses a non-trivial Killing tensor, implying separability of the Hamilton-Jacobi equation, the existence of a Carter constant, and complete formal integrability of the the geodesic equations. Herein we investigate the geodesics in some detail, in the general situation demonstrating the occurrence of "ultra-elliptic" integrals. Only in certain special cases can the complete geodesic integrability be explicitly cast in terms of elementary functions. The model is potentially of astrophysical interest both in the asymptotic large-distance limit and as an example of a "black hole mimic", a controlled deformation of the Kerr spacetime that can be contrasted with ongoing astronomical observations.

gr-qc↗

arXiv2021

The eye of the storm: A regular Kerr black hole

We present a highly tractable non-singular modification of the Kerr geometry, dubbed the "eye of the storm" -- a rotating regular black hole with an asymptotically Minkowski core. This is achieved by "exponentially suppressing" the mass parameter in the Kerr spacetime: $m\to m \; e^{-\ell/r}$. The single parameter $\ell$ quantifies the deviation from the usual Kerr spacetime. Some of the classical energy conditions are globally satisfied, whilst certain choices for $\ell$ force any energy-condition-violating physics into the deep core. The geometry possesses the full "Killing tower'" of principal tensor, Killing--Yano tensor, and nontrivial Killing tensor, with associated Carter constant; hence the Hamilton--Jacobi equations are separable, and the geodesics integrable. The Klein--Gordon equation is also separable on this candidate spacetime. The tightly controlled deviation from Kerr renders the physics extraordinarily tractable when compared with analogous results in the current literature. This spacetime will be amenable to straightforward extraction of astrophysical observables falsifiable/ verifiable by the experimental community.

gr-qc↗

arXiv2021

From black-bounce to traversable wormhole, and beyond

Key results from the literature pertaining to a class of nonsingular black hole mimickers are explored. The family of candidate spacetimes is for now labelled the `black-bounce' family, stemming from the original so-called `Simpson--Visser' spacetime in static spherical symmetry. All model geometries are analysed through the lens of standard general relativity, are globally free from curvature singularities, pass all weak-field observational tests, and smoothly interpolate between regular black holes and traversable wormholes. The discourse is segregated along geometrical lines, with candidate spacetimes each belonging to one of: static spherical symmetry, spherical symmetry with dynamics, and stationary axisymmetry.

gr-qc↗

arXiv2021

Killing tensor and Carter constant for Painleve-Gullstrand form of Lense-Thirring spacetime

Recently, the authors have formulated and explored a novel Painleve-Gullstrand variant of the Lense-Thirring spacetime, which has some particularly elegant features -- including unit-lapse, intrinsically flat spatial 3-slices, and some particularly simple geodesics, the "rain" geodesics. At linear level in the rotation parameter this spacetime is indistinguishable from the usual slow-rotation expansion of Kerr. Herein, we shall show that this spacetime possesses a nontrivial Killing tensor, implying separability of the Hamilton-Jacobi equation. Furthermore, we shall show that the Klein-Gordon equation is also separable on this spacetime. However, while the Killing tensor has a 2-form square root, we shall see that this 2-form square root of the Killing tensor is not a Killing-Yano tensor. Finally, the Killing-tensor-induced Carter constant is easily extracted, and now, with a fourth constant of motion, the geodesics become (in principle) explicitly integrable.

gr-qc↗

arXiv2021

Ringing of the regular black hole with asymptotically Minkowski core

A Regge--Wheeler analysis is performed for a novel black hole mimicker `the regular black hole with asymptotically Minkowski core', followed by an approximation of the permitted quasi-normal modes for propagating waveforms. A first-order WKB approximation is computed for spin zero and spin one perturbations of the candidate spacetime. Subsequently, numerical results analysing the respective fundamental modes are compiled for various values of the $a$ parameter (which quantifies the distortion from Schwarzschild spacetime), and for various multipole numbers $\ell$. Both electromagnetic spin one fluctuations and scalar spin zero fluctuations on the background spacetime are found to possess shorter-lived, higher-energy signals than their Schwarzschild counterparts for a specific range of interesting values of the $a$ parameter. Analysis as to what happens when one permits perturbations of the Regge--Wheeler potential itself is then conducted, first in full generality, before specialising to Schwarzschild spacetime. A general result is presented explicating the shift in quasi-normal modes under perturbation of the Regge--Wheeler potential.

gr-qc↗

arXiv2021

Traversable Wormholes, Regular Black Holes, and Black-Bounces

Various spacetime candidates for traversable wormholes, regular black holes, and `black-bounces' are presented and thoroughly explored in the context of the gravitational theory of general relativity. All candidate spacetimes belong to the mathematically simple class of spherically symmetric geometries; the majority are static, with a single dynamical (time-dependent) geometry explored. To the extent possible, the candidates are presented through the use of a global coordinate patch -- some of the prior literature (especially concerning traversable wormholes) has often proposed coordinate systems for desirable solutions to the Einstein equations requiring a multi-patch atlas. The most interesting cases include the so-called `exponential metric' -- well-favoured by proponents of alternative theories of gravity but which actually has a standard classical interpretation, and the `black-bounce' to traversable wormhole case -- where a metric is explored which represents either a traversable wormhole or a regular black hole, depending on the value of the newly introduced scalar parameter $a$. This notion of `black-bounce' is defined as the case where the spherical boundary of a regular black hole forces one to travel towards a one-way traversable `bounce' into a future reincarnation of our own universe. The metric of interest is then explored further in the context of a time-dependent spacetime, where the line element is rephrased with a Vaidya-like time-dependence imposed on the mass of the object, and in terms of outgoing\-/ingoing Eddington-Finkelstein coordinates. Analysing these candidate spacetimes extends the pre-existing discussion concerning the viability of non-singular black hole solutions in the context of general relativity, as well as contributing to the dialogue on whether an arbitrarily advanced civilization would be able to construct a traversable wormhole.

gr-qc↗

arXiv2021

Charged black-bounce spacetimes

Given the recent development of rotating black-bounce-Kerr spacetimes, for both theoretical and observational purposes it becomes interesting to see whether it might be possible to construct black-bounce variants of the entire Kerr-Newman family. Specifically, herein we shall consider black-bounce-Reissner-Nordstr\"om and black-bounce-Kerr-Newman spacetimes as particularly simple and clean everywhere-regular black hole "mimickers" that deviate from the Kerr-Newman family in a precisely controlled and minimal manner, and smoothly interpolate between regular black holes and traversable wormholes. While observationally the electric charges on astrophysical black holes are likely to be extremely low, $|Q|/m \ll 1$, introducing any non-zero electric charge has a significant theoretical impact. In particular, we verify the existence of a Killing tensor (and associated Carter-like constant) but without the full Killing tower of principal tensor and Killing-Yano tensor, also we discuss how, assuming general relativity, the black-bounce-Kerr-Newman solution requires an interesting, non-trivial matter/energy content.

gr-qc↗

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↗