arXiv subjects

Matt Visser

Publications and source records attributed to Matt Visser.

At least 73 records · Page 4Linked to original sources

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

Geodesically complete black holes in Lorentz-violating gravity

We present a systematic study of the geometric structure of non-singular spacetimes describing black holes in Lorentz-violating gravity. We start with a review of the definition of trapping horizons, and the associated notions of trapped and marginally trapped surfaces, and then study their significance in frameworks with modified dispersion relations. This leads us to introduce the notion of universally marginally trapped surfaces, as the direct generalization of marginally trapped surfaces for frameworks with infinite signal velocities (Hořava-like frameworks), which then allows us to define universal trapping horizons. We find that trapped surfaces cannot be generalized in the same way, and discuss in detail why this does not prevent using universal trapping horizons to define black holes in Hořava-like frameworks. We then explore the interplay between the kinematical part of Penrose's singularity theorem, which implies the existence of incomplete null geodesics in the presence of a focusing point, and the existence of multiple different metrics. This allows us to present a complete classification of all possible geometries that neither display incomplete physical trajectories nor curvature singularities. Our main result is that not all classes that exist in frameworks in which all signal velocities are realized in Hořava-like frameworks. However, the taxonomy of geodesically complete black holes in Ho\v rava-like frameworks includes diverse scenarios such as evaporating regular black holes, regular black holes bouncing into regular white holes, and hidden wormholes.

gr-qc↗

arXiv2021

Tractor beams, pressor beams, and stressor beams within the context of general relativity

Both traversable wormholes and warp drives, concepts originally developed within the context of science fiction, have now (for some 30 odd years) been studied, debated, and carefully analyzed within the framework of general relativity. An overarching theme of the general relativistic analysis is unavoidable violations of the classical point-wise energy conditions. Another science fiction trope, now over 80 years old, is the tractor beam and/or pressor beam. We shall discuss how to formulate both tractor beams and/or pressor beams, and a variant to be called a stressor beam, within the context of reverse engineering the spacetime metric. (While such reverse engineering is certainly well beyond our civilization's current capabilities, we shall be more interested in asking what an arbitrarily advanced civilization might be able to accomplish.) We shall see that tractor beams and/or pressor beams can be formulated by suitably modifying the notion of warp drives, and that, as for wormholes and warp drives, violations of the classical point-wise energy conditions are utterly unavoidable.

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

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-Nordström 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

Explicit Baker-Campbell-Hausdorff-Dynkin formula for Spacetime via Geometric Algebra

We present a compact Baker-Campbell-Hausdorff-Dynkin formula for the composition of Lorentz transformations $e^{σ_i}$ in the spin representation (a.k.a. Lorentz rotors) in terms of their generators $σ_i$: $$ \ln(e^{σ_1}e^{σ_2}) = \tanh^{-1}\left(\frac{ \tanh σ_1 + \tanh σ_2 + \frac12[\tanh σ_1, \tanh σ_2] }{ 1 + \frac12\{\tanh σ_1, \tanh σ_2\} }\right) $$ This formula is general to geometric algebras (a.k.a. real Clifford algebras) of dimension $\leq 4$, naturally generalising Rodrigues' formula for rotations in $\mathbb{R}^3$. In particular, it applies to Lorentz rotors within the framework of Hestenes' spacetime algebra, and provides an efficient method for composing Lorentz generators. Computer implementations are possible with a complex $2\times2$ matrix representation realised by the Pauli spin matrices. The formula is applied to the composition of relativistic $3$-velocities yielding simple expressions for the resulting boost and the concomitant Wigner angle.

math-ph↗

arXiv2021

Tractor beams, pressor beams, and stressor beams in general relativity

The metrics of general relativity generally fall into two categories: Those which are solutions of the Einstein equations for a given source energy-momentum tensor, and the "reverse engineered" metrics -- metrics bespoke for a certain purpose. Their energy-momentum tensors are then calculated by inserting these into the Einstein equations. This latter approach has found frequent use when confronted with creative input from fiction, wormholes and warp drives being the most famous examples. In this paper, we shall again take inspiration from fiction, and see what general relativity can tell us about the possibility of a gravitationally induced tractor beam. We will base our construction on warp drives and show how versatile this ansatz alone proves to be. Not only can we easily find tractor beams (attracting objects); repulsor/pressor beams are just as attainable, and a generalization to "stressor" beams is seen to present itself quite naturally. We show that all of these metrics would violate various energy conditions. This will provide an opportunity to ruminate on the meaning of energy conditions as such, and what we can learn about whether an arbitrarily advanced civilization might have access to such beams.

gr-qc↗

arXiv2021

Reconsidering maximum luminosity

The suggestion that there is a maximum luminosity (maximum power) in nature has a long and somewhat convoluted history. Though this idea is commonly attributed to Freeman Dyson, he was actually much more circumspect in his views. What is certainly true is that dimensional analysis shows that the speed of light and Newton's constant of gravitation can be combined to define a quantity P_* = c^5/G_N with the dimensions of luminosity (equivalently, power). Then in any physical situation we must have P_{physical} =$\wp$ P_*, where the quantity $\wp$ is some dimensionless function of dimensionless parameters. This has lead some authors to suggest a maximum luminosity/maximum power conjecture. Working within the framework of standard general relativity, we will re-assess this conjecture, paying particular attention to the extent to which various examples and counter-examples are physically reasonable. We focus specifically on Vaidya spacetimes, and on an evaporating version of Schwarzschild's constant density star. For both of these spacetimes luminosity can be arbitrarily large. We argue that any luminosity bound must depend on delicate internal features of the radiating object.

gr-qc↗

arXiv2021

Inner horizon instability and the unstable cores of regular black holes

Regular black holes with nonsingular cores have been considered in several approaches to quantum gravity, and as agnostic frameworks to address the singularity problem and Hawking's information paradox. While in a recent work we argued that the inner core is destabilized by linear perturbations, opposite claims were raised that regular black holes have in fact stable cores. To reconcile these arguments, we discuss a generalization of the geometrical framework, originally applied to Reissner--Nordtsröm black holes by Ori, and show that regular black holes have an exponentially growing Misner--Sharp mass at the inner horizon. This result can be taken as an indication that stable nonsingular black hole spacetimes are not the definitive endpoint of a quantum gravity regularization mechanism, and that nonperturbative backreaction effects must be taken into account in order to provide a consistent description of the quantum-gravitational endpoint of gravitational stellar collapse.

gr-qc↗

arXiv2021

Novel black-bounce spacetimes: wormholes, regularity, energy conditions, and causal structure

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.

gr-qc↗

arXiv2021

Hawking-Ellis classification of stress-energy: test-fields versus back-reaction

We consider the Hawking-Ellis (Segre-Plebanski) classification of stress-energy tensors, both in the test-field limit, and in the presence of back-reaction governed by the usual Einstein equations. For test fields it is not too difficult to get a type~IV stress-energy via quantum vacuum polarization effects. (For example, consider the Unruh quantum vacuum state for a massless scalar field in the Schwarzschild background.) However, in the presence of back-reaction driven by the ordinary Einstein equations the situation is often much more constrained. For instance: (1) in any static spacetime the stress-energy is always type I in the domain of outer communication, and on any horizon that might be present; (2) in any stationary axisymmetric spacetime the stress-energy is always type I on any horizon that might be present; (3) on any Killing horizon that is extendable to a bifurcation 2-surface the stress-energy is always type I; (4) in any stationary axisymmetric spacetime the stress-energy is always type I on the axis of symmetry; (5) some of the homogeneous Bianchi cosmologies are guaranteed to be Hawking-Ellis type I (for example, all the Bianchi type I cosmologies, all the FLRW cosmologies, and all the "single mode" Bianchi cosmologies). That is, in very many physically interesting situations once one includes back-reaction the more unusual stress-energy types are automatically excluded.

gr-qc↗

arXiv2021

Counterexamples to the maximum force conjecture

Dimensional analysis shows that the speed of light and Newton's constant of gravitation can be combined to define a quantity $F_* = {c^4\over G_N}$ with the dimensions of force (equivalently, tension). Then in any physical situation we must have $F_{physical} = f \; F_*$, where the quantity $f$ is some dimensionless function of dimensionless parameters. In many physical situations explicit calculation yields $f= O(1)$, and quite often $f \leq {1\over4}$. This has lead multiple authors to suggest a (weak or strong) maximum force/maximum tension conjecture. Working within the framework of standard general relativity, we will instead focus on counter-examples to this conjecture, paying particular attention to the extent to which the counter-examples are physically reasonable. The various counter-examples we shall explore strongly suggest that one should not put too much credence into any universal maximum force/maximum tension conjecture. Specifically, fluid spheres on the verge of gravitational collapse will generically violate the weak (and strong) maximum force conjectures. If one wishes to retain any general notion of "maximum force" then one will have to very carefully specify precisely which forces are to be allowed within the domain of discourse.

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↗

arXiv2021

Lorentz boosts and Wigner rotations: self-adjoint complexified quaternions

Herein we shall consider Lorentz boosts and Wigner rotations from a (complexified) quaternionic point of view. We shall demonstrate that for a suitably defined self-adjoint complex quaternionic 4-velocity, pure Lorentz boosts can be phrased in terms of the quaternion square root of the relative 4-velocity connecting the two inertial frames. Straightforward computations then lead to quite explicit and relatively simple algebraic formulae for the composition of 4-velocities and the Wigner angle. We subsequently relate the Wigner rotation to the generic non-associativity of the composition of three 4-velocities, and develop a necessary and sufficient condition for associativity to hold. Finally, we relate the composition of 4-velocities to a specific implementation of the Baker-Campbell-Hausdorff theorem. As compared to ordinary 4x4 Lorentz transformations, the use of self-adjoint complexified quaternions leads, from a computational view, to storage savings and more rapid computations, and from a pedagogical view to to relatively simple and explicit formulae.

gr-qc↗

arXiv2021

The second Hardy-Littlewood conjecture is true

The second Hardy-Littlewood conjecture, that $π(x)+π(y) \geq π(x+y)$ for integers $x$ and $y$ with $\min\{x,y\}\geq 2$, was formulated in 1923. It continues to attract attention to this day, almost 100 years later. In 1975 Udrescu proved that this conjecture holds for $(x,y)$ sufficiently large, but without an explicit effective bound on the region of validity. We shall revisit Udrescu's result, modifying it to obtain explicit effective bounds, ultimately proving that the second Hardy-Littlewood conjecture is in fact unconditionally true. Furthermore we note that constraints on the prime counting function imply, (and are implied by), constraints on the location of the primes, and re-cast Segal's 1962 equivalent reformulation of the second Hardy-Littlewood conjecture in the more symmetric (and perhaps clearer) form that for integers $i$ and $j$ with $\min\{i,j\} \geq 2$ one has $p_{i+j-1} \geq p_i + p_j -1$.

math.NT↗

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↗