arXiv subjects

Michael Hobson

Publications and source records attributed to Michael Hobson.

At least 19 recordsLinked to original sources

arXiv2026

Bessel-Hagen currents for linearised Weyl-squared gravity

For the Fierz-Pauli action the Bessel-Hagen construction does not produce a preferred local gauge-invariant energy-momentum tensor; it identifies only a gauge-invariant equivalence class of Noether currents, because the action is built from first derivatives of $h_{\mu\nu}$ whereas the first local gauge-invariant spin-2 field strength, the curvature, contains two. It is natural to ask whether a theory built directly from the linearised curvature recovers the electromagnetic-like feature of a strictly gauge-invariant local representative. We examine linearised Weyl-squared (conformal) gravity, whose action is built from the linearised Weyl tensor $C^{(1)}_{\mu\nu\rho\sigma}$, the irreducible spin-2 part of the curvature and hence the curvature counterpart of the spin-1 field strength of electromagnetism. The Bessel-Hagen construction extends naturally from the Poincar\'e to the full conformal group, realised actively on the fixed Minkowski background, and the resulting Noether current is gauge invariant as a class for every conformal generator. Nevertheless there exists no nonzero strict local, polynomial, symmetric, dimension-four rank-two tensor quadratic in $h_{\mu\nu}$ that is invariant under both gauge symmetries and conserved on the Bach shell: the two gauge symmetries force any candidate to be built from $C^{(1)}$ and its derivatives, while the dimension-four and quadratic-order hypotheses leave only expressions quadratic in $C^{(1)}$ with no additional derivatives, and the four-dimensional Weyl identities collapse these to a pure trace, which cannot be conserved. Linearised Weyl-squared gravity therefore behaves like Fierz-Pauli, not electromagnetism. This suggests that a first-order gauge-covariant field strength, rather than a curvature-built action, is the natural route to an electromagnetic-like local representative.

gr-qc↗

arXiv2026

Numerical polology: towards next-generation model-building for cosmology

The dark sector need not be restricted to simple field content. Indeed, simple bosonic configurations, such as scalar-tensor or dark photon models, contrast with the much richer picture painted by many ultraviolet scenarios. Polology is the study of propagator poles, which correspond to particle states in any given theory. We outline a numerical polology framework for discovering perturbative, ghost-free models with consistent interactions, which produces theoretical model priors by sampling the coupling space. The method is tested on tensor field theories of up to rank three. Subsequent observational constraint pipelines are illustrated for black hole superradiance (M33 X-7), dynamical dark energy (DESI DR2, Pantheon and SH0ES) and gravitational waves (GWTC-3).

astro-ph.CO↗

arXiv2026

Bessel-Hagen currents for the Fierz-Pauli action

For electromagnetism in Minkowski spacetime, the Bessel-Hagen method gives a particularly direct Noetherian derivation of the standard gauge-invariant energy-momentum tensor. The key step is to supplement the form variation generated by an infinitesimal coordinate transformation with a compensating electromagnetic gauge transformation. In this paper we ask whether the same idea can be applied to the massless spin-2 field described by the Fierz-Pauli action. We first prove that no nonzero local tensor quadratic in first derivatives of the symmetric field $h_{\mu\nu}$ can be strictly invariant under the spin-2 gauge transformation $h_{\mu\nu}\mapsto h_{\mu\nu}+\partial_\mu\xi_\nu+\partial_\nu\xi_\mu$; the direct electromagnetic analogue of the Bessel-Hagen construction therefore cannot exist. Once the inexact nature of the Fierz-Pauli gauge symmetry is treated correctly, however, the Bessel-Hagen construction does produce a gauge-invariant equivalence class of Noether currents. Changing the compensating spin-2 gauge parameter changes the current only by terms proportional to the Fierz-Pauli field equations; performing an independent spin-2 gauge transformation on $h_{\mu\nu}$ changes the current only by a trivial current given by the divergence of an antisymmetric superpotential plus field-equation terms. This provides the natural spin-2 analogue of Bessel-Hagen's electromagnetic construction, but only in the quotient space of conserved currents, and not as a preferred local gauge-invariant energy-momentum tensor.

gr-qc↗

arXiv2025

The radial metric function does not identify null surfaces

We investigate the conditions under which a hypersurface becomes null through the use of coordinate transformations. We demonstrate that, in static spacetimes, the correct criterion for a surface to be null is $g_{tt} = 0$, rather than $g^{rr} = 0$, in agreement with the results of Vollick. We further show that, if a Kruskal-like coordinate exists, the proxy condition $g^{rr} = 0$ is equivalent to $g_{tt} = 0$ if $\partial_r g_{tt} \neq 0$ and both $g^{rr}$ and $g_{tt}$ vanish at the same rate near the horizon. Our method extends naturally to axisymmetric stationary spacetimes, for which we demonstrate that the condition $\det\big(h_{ab}\big) = 0$ for the induced metric on a null hypersurface is recovered. By contrast with the induced metric approach, our method provides a physical perspective that connects the general null condition with its underlying relationship to photon geodesics.

gr-qc↗

arXiv2024

Buchdahl bound, photon ring, ISCO and radial acceleration in Einstein-\ae{}ther theory

Spherically symmetric Einstein-{\ae}ther (E{\AE}) theory with a Maxwell-like kinetic term is revisited. We consider a general choice of the metric and the \ae{}ther field, finding that:~(i) there is a gauge freedom allowing one always to use a diagonal metric; and~(ii) the nature of the Maxwell equation forces the \ae{}ther field to be time-like in the coordinate basis. We derive the vacuum solution and confirm that the innermost stable circular orbit (ISCO) and photon ring are enlarged relative to general relativity (GR). Buchdahl's theorem in E\AE{} theory is derived. For a uniform physical density, we find that the upper bound on compactness is always lower than in GR. Additionally, we observe that the Newtonian and E\AE{} radial acceleration relations run parallel in the low pressure limit. Our analysis of E\AE{} theory may offer novel insights into its interesting phenomenological generalization: \AE{}ther--scalar--tensor theory ({\AE}ST).

gr-qc↗

arXiv2024

Every Poincar\'e gauge theory is conformal: a compelling case for dynamical vector torsion

The Poincar\'e gauge theory (PGT) of gravity provides a viable formulation of general relativity (Einstein-Cartan theory), and a popular model-building framework for modified gravity with torsion. Notoriously, however, the PGT terms which propagate vector torsion lead to strongly-coupled ghosts: the modern view is that only scalar torsion can propagate. To fix this, we revisit the concept of embedding explicit mass scales in scale-invariant theories, showing how the Klein-Gordon theory naturally leads to a slowly-rolling inflaton. We then show that the unique scale-invariant embedding of PGT leads to two new terms, one of which is the Maxwell term for vector torsion. We provide the full spectrum of quantum particles in the resulting theory. Our result means that every PGT is conformal and - after a two-decade hiatus - vector torsion is back on the menu.

hep-th↗

arXiv2023

Kernel-, mean- and noise-marginalised Gaussian processes for exoplanet transits and $H_0$ inference

Using a fully Bayesian approach, Gaussian Process regression is extended to include marginalisation over the kernel choice and kernel hyperparameters. In addition, Bayesian model comparison via the evidence enables direct kernel comparison. The calculation of the joint posterior was implemented with a transdimensional sampler which simultaneously samples over the discrete kernel choice and their hyperparameters by embedding these in a higher-dimensional space, from which samples are taken using nested sampling. Kernel recovery and mean function inference were explored on synthetic data from exoplanet transit light curve simulations. Subsequently, the method was extended to marginalisation over mean functions and noise models and applied to the inference of the present-day Hubble parameter, $H_0$, from real measurements of the Hubble parameter as a function of redshift, derived from the cosmologically model-independent cosmic chronometer and $\Lambda$CDM-dependent baryon acoustic oscillation observations. The inferred $H_0$ values from the cosmic chronometers, baryon acoustic oscillations and combined datasets are $H_0= 66 \pm 6\, \mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$, $H_0= 67 \pm 10\, \mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$ and $H_0= 69 \pm 6\, \mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$, respectively. The kernel posterior of the cosmic chronometers dataset prefers a non-stationary linear kernel. Finally, the datasets are shown to be not in tension with $\ln R=12.17\pm 0.02$.

astro-ph.CO↗

arXiv2023

Manifestly covariant variational principle for gauge theories of gravity

A variational principle for gauge theories of gravity is presented, which maintains manifest covariance under the symmetries to which the action is invariant, throughout the calculation of the equations of motion and conservation laws. This is performed by deriving explicit manifestly covariant expressions for the Euler--Lagrange variational derivatives and Noether's theorems for a generic action of the form typically assumed in gauge theories of gravity. The approach is illustrated by application to two scale-invariant gravitational gauge theories, namely Weyl gauge theory (WGT) and the recently proposed `extended' Weyl gauge theory (eWGT), where the latter may be considered as a novel gauging of the conformal group, but the method can be straightforwardly applied to other theories with smaller or larger symmetry groups. The approach also enables one easily to establish the relationship between manifestly covariant forms of variational derivatives obtained when one or more of the gauge field strengths is set to zero either before or after the variation is performed. This is illustrated explicitly for both WGT and eWGT in the case where the translational gauge field strength (or torsion) is set to zero before and after performing the variation, respectively.

gr-qc↗

arXiv2022

New form of the Kerr-Newman solution

A new form of the Kerr-Newman solution is presented. The solution involves a time coordinate which represents the local proper time for a charged massive particle released from rest at spatial infinity. The chosen coordinates ensure that the solution is well-behaved at horizons and enable an intuitive description of many physical phenomena. If the charge of the particle $e = 0$, the coordinates reduce to Doran coordinates for the Kerr solution with the replacement $M \to M - Q^2/(2r)$, where $M$ and $Q$ are the mass and charge of the black hole, respectively. Such coordinates are valid only for $r \ge Q^2/(2M)$, however, which corresponds to the region that a neutral particle released from rest at infinity can penetrate. By contrast, for $e \neq 0$ and of opposite sign to $Q$, the new coordinates have a progressively extended range of validity as $|e|$ increases and tend to advanced Eddington-Finkelstein (EF) null coordinates as $|e| \to \infty$, hence becoming global in this limit. The Kerr solution (i.e.\ with $Q=0$) may also be written in terms of the new coordinates by setting $eQ = -\alpha$, where $\alpha$ is a real parameter unrelated to charge; in this case the coordinate system is global for all non-negative values of $\alpha$ and the limits $\alpha = 0$ and $\alpha \to \infty$ correspond to Doran coordinates and advanced EF null coordinates, respectively, without any need to transform between them.

gr-qc↗

arXiv2022

Conformally-rescaled Schwarzschild metrics do not predict flat galaxy rotation curves

For conformally invariant gravity theories defined on Riemannian spacetime and having the Schwarzschild--de-Sitter (SdS) metric as a solution in the Einstein gauge, we consider whether one may conformally rescale this solution to obtain flat rotation curves, such as those observed in galaxies, without the need for dark matter. Contrary to recent claims in the literature, we show that if one works in terms of quantities that can be physically measured, then in any conformal frame the trajectories followed by `ordinary' matter particles are merely the timelike geodesics of the SdS metric, as one might expect. This resolves the apparent frame dependence of physical predictions and unambiguously yields rotation curves with no flat region. We also show that attempts to model rising rotation curves by fitting the coefficient of the quadratic term in the SdS metric individually for each galaxy are precluded, since this coefficient is most naturally interpreted as proportional to a global cosmological constant. We further extend our analysis beyond static, spherically-symmetric systems to show that the invariance of particle dynamics to the choice of conformal frame holds for arbitrary metrics, again as expected. Moreover, we show that this conclusion remains valid for conformally invariant gravity theories defined on more general Weyl--Cartan spacetimes, which include Weyl, Riemann--Cartan and Riemannian spacetimes as special cases.

gr-qc↗

arXiv2022

Nested sampling for physical scientists

We review Skilling's nested sampling (NS) algorithm for Bayesian inference and more broadly multi-dimensional integration. After recapitulating the principles of NS, we survey developments in implementing efficient NS algorithms in practice in high-dimensions, including methods for sampling from the so-called constrained prior. We outline the ways in which NS may be applied and describe the application of NS in three scientific fields in which the algorithm has proved to be useful: cosmology, gravitational-wave astronomy, and materials science. We close by making recommendations for best practice when using NS and by summarizing potential limitations and optimizations of NS.

stat.CO↗

arXiv2022

Note on the absence of the second clock effect in Weyl gauge theories of gravity

We reconsider the status of the so-called second clock effect in Weyl gauge theories of gravity, which are invariant both under local Poincaré transformations and local changes of scale. In particular, we revisit and extend our previous demonstration that the second clock effect does not occur in such theories, in order both to clarify our argument and confirm our original findings in response to recent counterclaims in the literature.

gr-qc↗

arXiv2022

Fresh perspective on gauging the conformal group

We consider the construction of gauge theories of gravity that are invariant under local conformal transformations. We first clarify the geometric nature of global conformal transformations, in both their infinitesimal and finite forms, and the consequences of global conformal invariance for field theories, before reconsidering existing approaches for gauging the conformal group, namely auxiliary conformal gauge theory and biconformal gauge theory, neither of which is generally accepted as a complete solution. We then demonstrate that, provided any matter fields belong to an irreducible representation of the Lorentz group, the recently proposed extended Weyl gauge theory (eWGT) may be considered as an alternative method for gauging the conformal group, since eWGT is invariant under the full set of local conformal transformations, including inversions, as well as possessing conservation laws that provide a natural local generalisation of those satisfied by field theories with global conformal invariance, and also having an `ungauged' limit that corresponds to global conformal transformations. By contrast, although standard Weyl gauge theory also enjoys the first of these properties, it does not share the other two, and so cannot be considered a valid gauge theory of the conformal group.

gr-qc↗

arXiv2021

Conformal gravity does not predict flat galaxy rotation curves

We reconsider the widely held view that the Mannheim--Kazanas (MK) vacuum solution for a static, spherically-symmetric system in conformal gravity (CG) predicts flat rotation curves, such as those observed in galaxies, without the need for dark matter. This prediction assumes that test particles have fixed rest mass and follow timelike geodesics in the MK metric in the vacuum region exterior to a spherically-symmetric representation of the galactic mass distribution. Such geodesics are not conformally invariant, however, which leads to an apparent discrepancy with the analogous calculation performed in the conformally-equivalent Schwarzschild-de-Sitter (SdS) metric, where the latter does not predict flat rotation curves. This difference arises since the mass of particles in CG must instead be generated dynamically through interaction with a scalar field. The energy-momentum of this required scalar field means that, in a general conformal frame from the equivalence class of CG solutions outside a static, spherically-symmetric matter distribution, the spacetime is not given by the MK vacuum solution. A unique frame does exist, however, for which the metric retains the MK form, since the scalar field energy-momentum vanishes despite the field being non-zero and radially dependent. Nonetheless, we show that in both this MK frame and the Einstein frame, in which the scalar field is constant, massive particles follow timelike geodesics of the SdS metric, thereby resolving the apparent frame dependence of physical predictions and unambiguously yielding rotation curves with no flat region. We also comment on how our analysis resolves the long-standing uncertainty regarding gravitational lensing in the MK metric. (Abridged)

gr-qc↗

arXiv2020

Weyl gauge theories of gravity do not predict a second clock effect

We consider Weyl gauge theories of gravity (WGTs), which are invariant both under local Poincaré transformations and local changes of scale. Such theories may be interpreted as gauge theories in Minkowski spacetime, but their gravitational interactions are most often reinterpreted geometrically in terms of a Weyl--Cartan spacetime, in which any matter fields then reside. Such a spacetime is a straightforward generalisation of Weyl spacetime to include torsion. As first suggested by Einstein, Weyl spacetime is believed to exhibit a so-called second clock effect, which prevents the existence of experimentally observed sharp spectral lines, since the rates of (atomic) clocks depend on their past history. The prevailing view in the literature is that this rules out WGTs as unphysical. Contrary to this viewpoint, we show that if one adopts the natural covariant derivative identified in the geometric interpretation of WGTs, properly takes into account the scaling dimension of physical quantities, and recognises that Einstein's original objection requires the presence of massive matter fields to represent atoms, observers and clocks, then WGTs do not predict a second clock effect.

gr-qc↗

arXiv2019

Bayesian posterior repartitioning for nested sampling

Priors in Bayesian analyses often encode informative domain knowledge that can be useful in making the inference process more efficient. Occasionally, however, priors may be unrepresentative of the parameter values for a given dataset, which can result in inefficient parameter space exploration, or even incorrect inferences, particularly for nested sampling (NS) algorithms. Simply broadening the prior in such cases may be inappropriate or impossible in some applications. Hence our previous solution to this problem, known as posterior repartitioning (PR), redefines the prior and likelihood while keeping their product fixed, so that the posterior inferences and evidence estimates remain unchanged, but the efficiency of the NS process is significantly increased. In its most practical form, PR raises the prior to some power beta, which is introduced as an auxiliary variable that must be determined on a case-by-case basis, usually by lowering beta from unity according to some pre-defined `annealing schedule' until the resulting inferences converge to a consistent solution. Here we present a very simple yet powerful alternative Bayesian approach, in which beta is instead treated as a hyperparameter that is inferred from the data alongside the original parameters of the problem, and then marginalised over to obtain the final inference. We show through numerical examples that this Bayesian PR (BPR) method provides a very robust, self-adapting and computationally efficient `hands-off' solution to the problem of unrepresentative priors in Bayesian inference using NS. Moreover, unlike the original PR method, we show that even for representative priors BPR has a negligible computational overhead relative to standard nesting sampling, which suggests that it should be used as the default in all NS analyses.

stat.CO↗

arXiv2018

Bayesian sparse reconstruction: a brute-force approach to astronomical imaging and machine learning

We present a principled Bayesian framework for signal reconstruction, in which the signal is modelled by basis functions whose number (and form, if required) is determined by the data themselves. This approach is based on a Bayesian interpretation of conventional sparse reconstruction and regularisation techniques, in which sparsity is imposed through priors via Bayesian model selection. We demonstrate our method for noisy 1- and 2-dimensional signals, including astronomical images. Furthermore, by using a product-space approach, the number and type of basis functions can be treated as integer parameters and their posterior distributions sampled directly. We show that order-of-magnitude increases in computational efficiency are possible from this technique compared to calculating the Bayesian evidences separately, and that further computational gains are possible using it in combination with dynamic nested sampling. Our approach can also be readily applied to neural networks, where it allows the network architecture to be determined by the data in a principled Bayesian manner by treating the number of nodes and hidden layers as parameters.

astro-ph.IM↗

arXiv2016

Scale-invariant gauge theories of gravity: theoretical foundations

We consider the construction of gauge theories of gravity, focussing in particular on the extension of local Poincaré invariance to include invariance under local changes of scale. We work exclusively in terms of finite transformations, which allow for a more transparent interpretation of such theories in terms of gauge fields in Minkowski spacetime. Our approach therefore differs from the usual geometrical description of locally scale-invariant Poincaré gauge theory (PGT) and Weyl gauge theory (WGT) in terms of Riemann--Cartan and Weyl--Cartan spacetimes, respectively. In particular, we reconsider the interpretation of the Einstein gauge and also the equations of motion of matter fields and test particles in these theories. Inspired by the observation that the PGT and WGT matter actions for the Dirac field and electromagnetic field have more general invariance properties than those imposed by construction, we go on to present a novel alternative to WGT by considering an `extended' form for the transformation law of the rotational gauge field under local dilations, which includes its `normal' transformation law in WGT as a special case. The resulting `extended' Weyl gauge theory (eWGT) has a number of interesting features that we describe in detail. In particular, we present a new scale-invariant gauge theory of gravity that accommodates ordinary matter and is defined by the most general parity-invariant eWGT Lagrangian that is at most quadratic in the eWGT field strengths, and we derive its field equations. We also consider the construction of PGTs that are invariant under local dilations assuming either the `normal' or `extended' transformation law for the rotational gauge field, but show that they are special cases of WGT and eWGT, respectively.

gr-qc↗