arXiv subjects

Matt Visser

Publications and source records attributed to Matt Visser.

At least 19 recordsLinked to original sources

arXiv2026

Pressure profile bounds from relaxing the TOV equation

We develop several new and quite general bounds on the internal pressure profiles of general relativistic perfect fluid spheres --- based on various ways of relaxing the TOV system of ODEs to obtain several distinct differential inequalities. There is, as usual, a trade-off between strength of the bound, weakness of the input assumptions, and tractability of the analysis. Specifically we shall develop several straightforward but nontrivial bounds that variously depend only on the positivity of density, the boundedness of density, the monotonicity of density, or the monotonicity of the average density. We shall carefully place these new bounds within the historical framework of previous efforts in this regard, and explore the ways in which they are inter-related.

gr-qc↗

arXiv2026

Semiclassical regularity of compact trapped regions: From dynamical horizons to inner extremality

In eternal black-hole spacetimes, inner horizons are Cauchy horizons and are generically unstable. For non-extremal inner horizons, this includes both the classical mass-inflation instability and a semiclassical instability associated with divergences in the renormalized stress-energy tensor (RSET). Inner-extremal geometries, for which the inner-horizon surface gravity vanishes, evade classical mass inflation, but in stationary settings still suffer from singular behavior of the RSET. In this work, we show that the dynamical case is qualitatively different. Considering spacetimes describing the formation and evaporation of a compact trapped region in finite time, and working in the $s$-wave Polyakov approximation, we compute the expectation value of the stress-energy tensor in the in-vacuum state. Given that in this case the inner horizon is not a Cauchy horizon, the RSET remains finite everywhere. For generic non-extremal inner horizons, however, the RSET grows exponentially in time at the inner horizon, with a divergence emerging only in the asymptotic limit of an ever-lasting trapped region. For inner-extremal geometries this exponential growth is replaced by a considerably milder power-law growth. Such spacetimes may therefore be considered natural candidates for classically and semiclassically meta-stable black-hole interiors.

gr-qc↗

arXiv2026

Null geodesic defocusing in dynamical black-hole-to-white-hole transitions

We investigate the defocusing of null geodesics in dynamical, non-singular black-hole-to-white-hole transitions. Working at the level of spacetime kinematics, and without assuming any specific gravitational field equations, we show that the contraction and disappearance of a trapped region, as well as the subsequent formation and expansion of an anti-trapped region, necessarily require a violation of the null convergence condition. This conclusion follows directly from the behaviour of the null expansions across the trapping and anti-trapping horizons, and is therefore independent of the microscopic mechanism responsible for singularity resolution. We then illustrate this general argument by constructing a class of explicit bouncing geometries in generalised Painlev\'e-Gullstrand coordinates, obtained by promoting static regular black holes with de Sitter cores to time-dependent black-hole-to-white-hole transition models. For a Bardeen-type mass function, we show that the required violation of the null convergence condition is localised within the intermediate dynamical phase in which the trapped region evaporates and the anti-trapped region forms. Finally, we argue that the limiting case of an instantaneous black-hole-to-white-hole transition would require an unbounded violation of the null convergence condition, signalling a breakdown of the effective continuum metric description, and the need to appeal to a full quantum-gravitational description.

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arXiv2026

Revisiting Schwarzschild's constant density star in isotropic coordinates

Herein we shall revisit the venerable 110-year-old topic of Schwarzschild's constant density star, emphasizing that for many (though not quite all) purposes it is much easier to analyze this spacetime in isotropic coordinates (\emph{versus} the more usually adopted Hilbert--Droste area coordinates). The relevant line element is particularly transparent, containing two simple rational functions of the radial coordinate, and the two physical parameters appearing in this line element are easily and readily interpretable in terms of the central density and central pressure of the star. Local properties in the stellar interior (such as the pressure profile) will be seen to be remarkably simple, though quasi-local properties like the Misner--Sharp mass are just a little bit trickier. Apart from its simplicity and clarity, the analysis is also of considerable pedagogical interest. For instance, there are a number of interesting special cases. Mathematically there is a perfectly good solution corresponding to a zero density star -- which can physically be interpreted as an explicit verification of the fact that pressure gravitates, even in the absence of mass-energy density. Additionally, there is a singular solution containing a naked singularity that satisfies all but one of the standard classical energy conditions. Furthermore you can even do both, combining zero density with a naked singularity -- so that pressure by itself can generate naked singularities -- at the cost of merely violating the dominant energy condition, the least physical of the standard energy conditions. We argue that many physically interesting features of Schwarzschild's star are very much under-appreciated.

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arXiv2026

Effective geometrostatics of spherical stars beyond general relativity

We provide a set of general tools to study the problem of stellar equilibrium in any gravitational theory in which spherically symmetric spacetimes satisfy master field equations taking the form of an equality between an identically conserved tensor, with derivatives of up to second order in the metric, and an identically conserved matter tensor. We derive the most general expression for the Tolman--Oppenheimer--Volkoff equation of stellar equilibrium that is compatible with these minimal requirements. A general discussion of the conditions that guarantee geodesic completeness at the center of symmetry is also presented. The equations of stellar equilibrium are integrated in a subset of the space of allowed deformations of general relativity proposed by Ziprick and Kunstatter, allowing us to illustrate universal aspects associated with the weakening of the strength of gravity, such as the mitigation of the Buchdahl limit obtained in general relativity or the existence of static solutions describing regular black holes with perfect fluid cores.

gr-qc↗

arXiv2026

Explicit formulae for stochastic equilibria

Finding the stochastic equilibria for finite-state stochastic matrices amounts to solving an eigen\-vector problem $\pi = \pi P$. Various techniques for doing so are known, some extremely computationally intensive. Herein we shall aim to extract a number of relatively simple analytic results that shed light on this problem. It is very easy to find an explicit general formula for the equilibrium vector (when it is unique) of a $2\times 2$ stochastic matrix. The corresponding explicit general formula for the equilibrium vector (when it is unique) of a $3\times 3$ stochastic matrix is a somewhat messier four-line result. (Though with a bit of work you can shoe-horn it into one line of text.) An explicit general formula for the equilibrium vector (when it is unique) of a $4\times 4$ stochastic matrix requires a paragraph of text. Ultimately, for $n\times n$ stochastic matrices a general and fully explicit construction of the equilibrium vector (when it is unique) can be developed in terms of a suitable adjugate (classical adjoint) matrix, and can subsequently be reduced to the computation of $n$ principal matrix minors. Finally, an application to random walks on graphs is presented.

math-ph↗

arXiv2025

Sierpinski's Hypothesis H1

Sierpinski's Hypothesis H1, formulated in 1958, is the conjecture that (provided $n\geq 2$), when the first $n^2$ counting numbers, $1, 2,3,\dots n^2$, are arranged in a square, then each row contains at least one prime. This conjecture is particularly interesting in that it subsumes and is stronger than both the Oppermann and Legrendre conjectures. Herein I shall verify Sierpinski's Hypothesis H1 for (at least) the first $n \leq \hbox{10 070 368 414} \approx 10 \hbox{ billion}$ of these Sierpinski matrices. I shall also demonstrate some partial but more general results. For example: Even for arbitrary $n\geq \hbox{10 070 368 414}$ at least one quarter of the rows of the $n$th Sierpinski matrix contain at least one prime. Furthermore, even for arbitrary $n\geq \hbox{10 070 368 414}$ at least the first $\hbox{141 618}$ rows of the $n$th Sierpinski matrix always contain at least one prime. These and related results are obtained largely by using the locations and values of the known maximal prime gaps, the pigeonhole principle, and some recent bounds on the first Chebyshev function.

math.NT↗

arXiv2025

The spacetime geodesy of perfect fluid spheres

Herein we shall argue for the utility of "spacetime geodesy", a point of view where one delays as long as possible worrying about dynamical equations, in favour of the maximal utilization of both symmetries and geometrical features. This closely parallels Weinberg's distinction between "cosmography" and "cosmology", wherein maximal utilization of both the symmetries and geometrical features of Friedmann--Lemaitre--Robertson--Walker (FLRW) spacetimes is emphasized. This "spacetime geodesy" point of view is particularly useful in those situations where, for one reason or another, the dynamical equations of motion are either uncertain or completely unknown. Several specific examples are discussed -- we shall illustrate what can be done by considering the physics implications of demanding spatially isotropic Ricci tensors as a way of automatically implementing the (isotropic) perfect fluid condition, without committing to a specific equation of state. We also consider the structure of the Weyl tensor in spherical symmetry, with and without the (isotropic) perfect fluid condition, and relate this to the notion of "complexity". In closing, we indicate some ways in which these considerations might be further generalized to more physically complicated (and technically very much more complicated) situations such as axisymmetric spacetimes.

gr-qc↗

arXiv2025

Primordial Black Holes, Charge, and Dark Matter: Rethinking Evaporation Limits

Limits on the dark matter fraction of small mass primordial black holes from Hawking radiation are predominantly derived from the assumption of a Schwarzschild black hole evaporating. However, astrophysical black holes are usually much more realistically modelled by the rotating Kerr black hole solution. Meanwhile, electromagnetically charged black holes are astrophysically of little importance due to their fast neutralisation in the present universe. Dark matter is not just a possible solution to issues of astrophysics and cosmology, but also to issues of the standard model of particle physics. Extensions of this model thus can lead to charges present in the early universe which remain preserved in the charge of primordial black holes - even when the corresponding particles have disappeared from the particle content of the present epoch of the universe. Here, we report on a thorough proof-of-concept that such charges can greatly change evaporation limits for primordial black hole dark matter. Special emphasis is placed on (near-)extremal black holes, for which this effect is especially pronounced.

gr-qc↗

arXiv2025

Timelike convergence condition in regular black-hole spacetimes with (anti-)de Sitter core

The Hawking-Penrose (1970) singularity theorem weakens the causality assumption of global hyperbolicity used in the Penrose (1965) singularity theorem, at the expense of invoking the stronger timelike (instead of null) convergence condition (TCC). We analyze the TCC for a large class of dynamical spherically symmetric spacetimes, and show that it decomposes into three independent conditions on the Misner-Sharp mass function. For stationary black holes only two of these are non-trivial. One of these conditions is already implied by the null convergence condition (NCC), whereas the other one depends explicitly on the TCC and constrains the sign of the second derivative of the mass function. We show that generic asymptotically flat regular black holes with a smooth de Sitter core locally violate this new TCC-induced condition near the core, even if they globally satisfy the other condition imposed by the NCC. Therefore, it is the violation of the TCC which ensures that regular de Sitter core black holes circumvent the Hawking-Penrose theorem. By contrast, we show that asymptotically flat regular black holes with an anti-de Sitter core locally satisfy the new TCC-induced condition near the core, but necessarily violate it at some finite distance away from it. As concrete examples for both types of spacetimes, we consider TCC violations in the Bardeen black-hole spacetime with a de Sitter core, and in a modified Bardeen black-hole spacetime with an anti-de Sitter core.

gr-qc↗

arXiv2025

Effective short intervals containing primes

95 years ago Hoheisel proved the existence of primes in the sub-linear interval \[ \left[x, x+x^{1-{1\over 33000}}\right] \qquad \hbox{for $x$ sufficiently large}. \] This was improved by Heilbronn, proving existence of primes in the sub-linear interval \[ \left[x, x+x^{1-{1\over 250}}\right] \qquad \hbox{for $x$ sufficiently large}. \] More recently Baker, Harman, Pintz proved existence of primes in the sub-linear interval \[ \left[x, x+ x^{1-{19\over 40}}\right] \qquad \hbox{for $x$ sufficiently large}. \] In the present article I will, to the extent possible, make some of these statements effective. Specifically, among other things, I shall show that \[ \forall n \geq 4, \qquad\forall x \geq \exp(3\exp(33)), \qquad \hbox{there are primes in the interval} \left[x, x+ x^{1-{1\over n}}\right]; \] \[ \forall n \geq 91, \qquad\forall x \geq [90^{90}]^{n/(n-90)} , \qquad \hbox{there are primes in the interval} \left[x, x+ x^{1-{1\over n}}\right]. \] Furthermore \[ \forall n \geq 106, \qquad\forall x \geq 1, \qquad \hbox{there are primes in the interval} \left[x, x+ x^{1-{1\over n}}\right]. \] In particular this last observation makes both the Hoheisel and Heilbronn results fully explicit and effective. This (relatively) specific observation can be extended and generalized in various manners.

math.NT↗

arXiv2025

Traversable Kaluza-Klein wormholes?

Various authors have suggested that Kaluza--Klein variants of traversable wormholes might to some extent ameliorate the defocussing properties (the curvature condition violations, and implied energy condition violations) inherent in positing the existence of a traversable wormhole throat. Unfortunately such a hope is ill-founded. We shall show that in a traditional Kaluza--Klein context the price paid for completely eliminating the defocussing properties of the wormhole throat is extremely high -- to completely eliminate curvature condition violations the 5th dimension has to become truly enormous (formally infinite) in the vicinity of the wormhole throat, in a manner that is fundamentally incompatible with the traditional Kaluza--Klein ansatz. At best, the extra dimensions allow one to move the curvature condition violations around, they cannot be eliminated except at prohibitive cost. While traversable Kaluza--Klein wormholes might be interesting for other reasons, it must be emphasized that adding a 5th dimension is not particularly useful in terms of ameliorating violations of the curvature conditions.

gr-qc↗

arXiv2025

Behaviour of the sequence $\vartheta_n = \vartheta(p_n)$

The well-known sequence $\vartheta_n = \vartheta(p_n) = \sum_{i=1}^n \ln p_i= \ln\left([p_n]\#\right)$ exhibits numerous extremely interesting properties. Since $p_n = \exp(\vartheta_n - \vartheta_{n-1})$, it is immediately clear that the two sequences $p_n \longleftrightarrow \vartheta_n$ must ultimately encode exactly the same information. But the sequence $\vartheta_n$, while being extremely closely correlated with the primes, (in fact, $\vartheta_n \sim p_n$), is very much better behaved than the primes themselves. Using numerous suitable extensions of various reasonably standard results, I shall demonstrate that the sequence $\vartheta_n$ satisfies suitably defined $\vartheta$-analogues of the usual Cramer, Andrica, Legendre, Oppermann, Brocard, Firoozbakht, Fourges, Nicholson, and Farhadian conjectures. (So these $\vartheta$-analogues are not conjectures, they are instead theorems.) The crucial key to enabling this pleasant behaviour is the regularity (and relative smallness) of the $\theta$-gaps $\mathfrak{g}_n = \vartheta_{n+1}-\vartheta_n= \ln p_{n+1}$. While superficially these results bear close resemblance to some recently derived results for the averaged primes, $\bar p_n = {1\over n} \sum_{i=1}^n p_i$, both the broad outline and the technical details of the arguments given and proofs presented are quite radically distinct.

math.NT↗

arXiv2025

On the arithmetic average of the first $n$ primes

The arithmetic average of the first $n$ primes, $\bar p_n = {1\over n} \sum_{i=1}^n p_i$, exhibits very many interesting and subtle properties. Since the transformation from $p_n \to \bar p_n$ is extremely easy to invert, $p_n = n\bar p_n - (n-1)\bar p_{n-1}$, it is clear that these two sequences $p_n \longleftrightarrow \bar p_n$ must ultimately carry exactly the same information. But the averaged sequence $\bar p_n$, while very closely correlated with the primes, ($\bar p_n \sim {1\over2} p_n$), is much "smoother'', and much better behaved. Using extensions of various standard results I shall demonstrate that the prime-averaged sequence $\bar p_n$ satisfies prime-averaged analogues of the Cramer, Andrica, Legendre, Oppermann, Brocard, Fourges, Firoozbakht, Nicholson, and Farhadian conjectures. (So these prime-averaged analogues are not conjectures, they are theorems.) The crucial key to enabling this pleasant behaviour is the "smoothing'' process inherent in averaging. Whereas the asymptotic behaviour of the two sequences is very closely correlated the local fluctuations are quite different.

math.NT↗

arXiv2025

The n-th prime exponentially

From known effective bounds on the prime counting function of the form \[ |\pi(x)-\mathrm{Li}(x)| < a \;x \;(\ln x)^{b} \; \exp\left(-{c}\; \sqrt{\ln x}\right); \qquad (x \geq x_0); \] it is possible to establish exponentially tight effective upper and lower bounds on the prime number theorem: For $x \geq x_*$ where $x_*\leq \max\{x_0,17\}$ we have: \[ {\mathrm{Li} \over 1+a\; (\ln x)^{b+1} \; \exp\left(-c\; \sqrt{\ln x}\right)} < \pi(x) < {\mathrm{Li} \over 1-a \;(\ln x)^{b+1} \; \exp\left(-c\; \sqrt{\ln x}\right)}. \] Furthermore, it is possible to establish exponentially tight effective upper and lower bounds on the location of the $n^{th}$ prime. Specifically: \[ p_n < \mathrm{Li}^{-1} \left( n \left[1+ a \;(\ln[n\ln n])^{b+1} \; \exp\left(-{c}\; \sqrt{\ln[n\ln n]}\right)\right] \right); \qquad (n\geq n_*). \] \[ p_n > \mathrm{Li}^{-1} \left( n \left[1- a \;(\ln[n\ln n])^{b+1} \; \exp\left(-{c}\; \sqrt{\ln[n\ln n]}\right)\right] \right); \qquad (n\geq n_*). \] Here the range of validity is explicitly bounded by some $n_*$ satisfying \[ n_* \leq \max\left\{\pi(x_0),\pi(17), \pi\left( (1+e^{-1}) \exp\left( \left[2(b+1)\over c\right]^2\right)\right) \right\}. \] Many other fully explicit bounds along these lines can easily be developed.

math.NT↗

arXiv2025

Immortality through the dark forces: Dark-charge primordial black holes as dark matter candidates

The fact that no Hawking radiation from the final stages of evaporating primordial black holes (PBHs) has yet been observed places stringent bounds on their allowed contribution to dark matter. Concretely, for Schwarzschild PBHs, i.e., uncharged and non-rotating black holes, this rules out black hole masses of less than $10^{-15} M_{\odot}$. In this article, we propose that by including an additional 'dark' $U(1)$ charge one can significantly lower the PBHs' Hawking temperature, slowing down their evaporation process and significantly extending their lifetimes. With this, PBHs again become a viable option for dark matter candidates over a large mass range. We will explore in detail the effects of varying the dark electron (lightest dark charged fermion) mass and charge on the evaporation dynamics. For instance, we will show that by allowing the dark electron to have a sufficiently high mass and/or low charge, our approach suppresses both Hawking radiation and the Schwinger effect, effectively extending even the lifespan of PBHs with masses smaller than $10^{-15} M_{\odot}$ beyond the current age of the universe. We will finally present a new lower bound on the allowed mass range for dark-charged PBHs as a function of the dark electron charge and mass, showing that the PBHs' mass can get to at least as low as $10^{-24}M_\odot$ depending on the dark electron properties. This demonstrates that the phenomenology of the evaporation of PBHs is ill-served by a focus solely on Schwarzschild black holes.

gr-qc↗

arXiv2025

Violations of the null convergence condition in kinematical transitions between singular and regular black holes, horizonless compact objects, and bounces

How do regular black holes evade the Penrose singularity theorem? Various models of stationary regular black holes globally satisfy the null convergence condition (NCC). At first glance this might seem puzzling, as the NCC must generically be violated to avoid the focusing point implied by the Penrose theorem. In fact, the Penrose singularity theorem depends on subtle global assumptions and does not provide information about where and when the singularity actually forms. In particular inner horizons are typically reached at finite affine parameter, before null geodesic focusing occurs, and the region inside the inner horizon is not itself a trapped region. Specifically, the Bardeen, Dymnikova, Hayward models of stationary regular black holes feature an inner Cauchy horizon which violates global hyperbolicity, hence violating one of the key assumptions of Penrose's singularity theorem, and furthermore challenging their viability as long-living end-points of gravitational collapse. In contrast, during non-stationary processes describing kinematic transitions between standard singular black holes and regular black holes or horizonless compact objects, the inner horizon -- when present -- need not act as a Cauchy horizon. This raises the intriguing possibility that the NCC might instead be violated during intermediate stages of such transitions. Our detailed analysis confirms that NCC violations occur frequently during such kinematic transitions, even when the stationary end-point spacetimes respect the NCC. We also investigate analogous transitions toward black-bounce spacetimes and their horizonless compact counterparts, wormholes, where the NCC is always violated. These findings offer new insights into how regular black holes and related objects evade the constraints imposed by the Penrose singularity theorem.

gr-qc↗

arXiv2025

Towards a Non-singular Paradigm of Black Hole Physics

The study of regular black holes and black hole mimickers as alternatives to standard black holes has recently gained significant attention, driven both by the need to extend general relativity to describe black hole interiors, and by recent advances in observational technologies. Despite considerable progress in this field, significant challenges remain in identifying and characterizing physically well-motivated classes of regular black holes and black hole mimickers. This report provides an overview of these challenges, and outlines some of the promising research directions -- as discussed during a week-long focus programme held at the Institute for Fundamental Physics of the Universe (IFPU) in Trieste from November 11th to 15th, 2024.

gr-qc↗