Search

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.

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., Comment: PhD Thesis; 276 pages; 42 figures; 429 references. v1

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., Comment: V1: 23 pages; 6 figures. V2: Now 25 pages. (1) Minor typos fixed. (2) We now carefully distinguish quasi-local forces (forces across a surface, relevant to extended objects) from ultra-local forces (forces at a point, relevant only for point particles). This version has been accepted for publication in Physical Review D

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., Comment: Note change in article title! V1: 24 pages, 0 figures. V2: added 1 reference; no physics changes; V3: now 26 pages; 4 figures; 2 more references; terminology and presentation improved; some bug fixes, but physics conclusions unchanged. This version accepted for publication in General Relativity and Gravitation

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., Comment: 33 pages; 83 references

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., Comment: V1: 5 pages, 18 references

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., Comment: V1: 27 pages, 71 references. V2: Now 28 pages, 83 references; some discourse added with relevant citations, small physics changes with inclusion of Maxwell separability

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., Comment: V1: 20 pages, 4 figures, 73 references; MG16 Conference Proceedings, 2021. V2: Still 20 pages, 4 figures, now 76 references; MG16 Conference Proceedings, 2021

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., Comment: V1: 17 pages, no figures; V2: some added comments/clarifications, 1 added reference, no physics changes

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., Comment: V1: 29 pages, 5 figures, 2 tables, 61 references. V2: Still 29 pages, 5 figures, 2 tables, now 63 references

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., Comment: Masters Thesis, 170 pages, 165 references, 19 figures

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., Comment: V1: 27 pages, 0 figures, 61 references. V2: Still 27 pages, 0 figures, no physics changes, 6 references added (now 67 references). V3: Published and final version, now 29 pages, 2 figures added, minor discussion added on page 17, 2 references added (now 69 references)

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., Comment: 32 pages; 1 figure; 2 tables

We develop a number of novel "black-bounce" spacetimes. These are specific regular black holes where the "area radius" always remains non-zero, thereby leading to a "throat" that is either timelike (corresponding to a traversable wormhole), spacelike (corresponding to a "bounce" into a future universe), or null (corresponding to a "one-way wormhole"). We shall first perform a general analysis of the regularity conditions for such a spacetime, and then consider a number of specific examples. The examples are constructed using a mass function similar to that of Fan--Wang, and fall into several particular cases, such as the original Simpson--Visser model, a Bardeen-type model, and other generalizations thereof. We shall analyse the regularity, the energy conditions, and the causal structure of these models. The main results are several new geometries, more complex than before, with two or more horizons, with the possibility of an extremal case. We shall derive a general theorem regarding static space-time regularity, and another general theorem regarding (non)-satisfaction of the classical energy conditions., Comment: V1: 26 pages, 13 figures. V2: Still 26 pages, 13 figures; 7 references added, plus some historical comments. No physics changes. V3: Now 27 pages, still 13 figures. Title change, new section on volume quantifier integral on pg 8, new calculation Eq.~(56) on pg 10, additional discussion pg 11, 2 new references added, bibliography tidied

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., Comment: 18 pages

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., Comment: V1: 24 pages. 6 figures. V2: Two references added; one reference updated; no physics changes

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., Comment: V1: 28 pages; 9 figures; V2: 7 references added; no physics changes

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., Comment: 29 pages

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., Comment: V1:19 pages. V2: now 30 pages. Significant additions: We now discuss tetrads, orthonormal components, eigenvalues of the Weyl tensor, the Petrov classification, ISCOs and circular photon orbits. Many small changes for clarity. Two references updated, and one reference added

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., Comment: 21 pages, 13 figures; V1 modified 31/03/2020, 10 references added, introduction extended; V2 modified 18/06/2020, two figures amended, conclusion extended. V3 is now aligned with the published version other than cosmetic features in the references

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., Comment: V1: 17 pages; V2: 18 pages, 8 references added, no physics changes

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., Comment: 17 pages