arXiv subjects

Jonathan Holland

Publications and source records attributed to Jonathan Holland.

At least 19 recordsLinked to original sources

arXiv2026

Sachs Equations and Plane Waves VI: Penrose Limits

Let $(M,g)$ be a Lorentzian manifold whose local space $\mathcal N$ of unparametrized null geodesics is smooth. We show that its Penrose limits assemble into a smooth bundle of plane-wave germs with a canonical gauge-theoretic soldering to spacetime. The incidence correspondence $\mathcal U=\{(x,[γ]):x\inγ\}$ identifies the contact space modulo the tangent space to the sky of $x$ with the screen space at $x$, and identifies the contact line with the weight-two null quotient. These evaluation maps solder the associated-graded Penrose model. A first jet of contact scale fixes the affine scale and a Reeb representative of the weight-two direction. Above the resulting pullback of $J^1\mathcal S$, full neighborhood realizations form a torsor under the group $\mathscr G_>$ of weighted diffeomorphism germs with identity principal part. We also identify the sky-parabolic reduction and its further transverse-Lagrangian reduction to Rosen gauges. The reduction fails precisely where the chosen Lagrangian meets the moving sky, which is the Rosen caustic locus. The trace of the Grassmannian Schwarzian of the sky curve defines a conformally natural projective structure on each null geodesic, and its trace-free part is the Weyl tidal profile of the Penrose germ.

gr-qc↗

arXiv2026

The Infrared Universe

We develop a stationary open-sector framework relating galactic dynamics, cosmological matter balance, and infrared radiation in an expanding universe. Stationarity means balance against Hubble dilution and therefore requires counted exchange currents. At a fixed-radius boundary, charge balance fixes the baryon return current; a joint energy ledger separates rest-mass recycling, horizon heat, and useful return free energy. Full counting statistics gives the boundary process a Berry--Sinitsyn--Nemenman connection. Pulling its curvature back along three spatial controls yields the free step-two tangent $N_{3,2}=\mathbb{R}^3\oplusΛ^2\mathbb{R}^3$. An autonomous horizon pump realizes the clock and affinity, while an event-conditioned completely positive channel selects the central character left undetermined by holonomy. In a galactic disc this response becomes an azimuthal connection. We derive its circular-orbit law, inverse reconstruction from a rotation curve, and a screened closure with a finite approximately flat branch; the baryonic Tully--Fisher normalization remains a source-matching hypothesis. The same tangent has an exact Landau--longitudinal spectrum and sky kernels. At feature level they fit the measured TT locations, reproduce the signs of twelve TE extrema and the higher TE/EE interleaving, and supply the first EE lobe without shifting TT. Coherent log-frequency translation preserves an exact Planck spectrum, while differential translation is constrained by blackbody mixing. The framework yields exact structural relations and conditional closures, leaving the common microscopic event law and full light-cone covariance open.

gr-qc↗

arXiv2026

A new hyperbolicity wedge and a joint semicircle limit for Jensen polynomials of Riemann's $ξ$-function

Let \[ ξ\!\left(\frac12+z\right) =\sum_{n\geq 0}\frac{γ(n)}{n!}z^{2n}, \qquad J^{d,n}(X) =\sum_{j=0}^{d}\binom djγ(n+j)X^j . \] The Riemann hypothesis is equivalent to the hyperbolicity of $J^{d,n}$ for every $d,n\geq0$. We prove that there is an absolute constant $K>0$ such that \[ n^3\log^2(n+2)\geq Kd^5 \quad\Longrightarrow\quad J^{d,n}\ \text{is hyperbolic}. \] Along every sequence with $n,d\to\infty$ in this region, the empirical measure of the naturally centered and scaled zeros also converges to Wigner's semicircle law. This gives a simultaneous degree--derivative version of the global semicircle consequence of the fixed-degree Hermite limit of Griffin, Ono, Rolen, and Zagier.

math.NT↗

arXiv2026

Reciprocity and the Maslov Phase

We give a metaplectic proof of Hilbert reciprocity, and hence of quadratic reciprocity, in which the local phase is the Kashiwara--Maslov phase of a triple of Lagrangians. In rank two the phase of the ordered triple $(L_\infty,L_a,L_0)$ is the one-dimensional Weil index $γ_v(a)$. The local Hilbert symbol appears as the defect of strict multiplicativity of these phases: \[ (a,b)_v = \frac{γ_v(a)γ_v(b)}{γ_v(1)γ_v(ab)}. \] The global step compares the local and adelic realizations of a single Bruhat word for the diagonal torus elements $m(a)\in \operatorname{SL}_2(\mathbb Q)$. Locally the raw Bruhat-word lift carries the normalization factor determined by the chosen quadratic convention. These operators form a projective representation of the diagonal torus with defect \[ μ_v(a,b) = \frac{γ_v(a)γ_v(b)}{γ_v(1)γ_v(ab)}. \] For rational adelic data, the normalized Bruhat word is multiplicative. The reciprocity law states that the total defect $\prod_vμ_v(a,b)$ is $1$. Combined with the local bridge above, this yields Hilbert reciprocity, while quadratic reciprocity is then the specialization to the pair of odd primes $(p,q)$.

math.NT↗

arXiv2026

Triality and the Magic Square of Hans Freudenthal

We study real triality structures through their intrinsic tensor algebra. Starting from a single triality symbol, we construct the associated Lie algebra of two-triality operators, prove the Jacobi identity, and identify the resulting algebra uniformly with the corresponding entry of the magic square. We then examine the natural invariant bilinear forms and the Clifford-theoretic structures arising from this construction. In low dimension, the triality formalism also recovers classical arithmetic data: in the $2\times2\times2$ case, the associated binary quadratic forms have a common discriminant and fit naturally into the Bhargava cube picture.

math.RA↗

arXiv2026

Sachs Equations and Plane Waves, V: Ward, Fourier, and Heisenberg Symmetry on Plane Waves

This article studies wave equations and their solutions on plane wave spacetimes of arbitrary dimension, developing the interplay among three structural layers: the Ward progressing-wave representation of solutions to the scalar wave equation, the Fourier analysis of the Heisenberg group naturally associated to the plane wave, and the Schrödinger propagator governing the evolution of initial data. The central geometric object is a positive curve in the Lagrangian Grassmannian determined by the plane wave metric, previously studied in the authors' series. The conformal tensor $H(u)$ that parametrises this curve plays a dual role: it encodes the null-cone geometry of the spacetime and simultaneously appears as the time-dependent parameter in the Schrödinger representation of the Heisenberg group acting by isometries on the plane wave. Parallel to the classical Fourier inversion theorem, convolution by Lagrangian delta distributions on the Heisenberg group furnishes an intrinsic description of the Schrödinger propagator, and the intertwining of different polarisations by this propagator is captured by a diagram that commutes up to a Maslov phase. The theta functions and Bargmann transforms that arise from imaginary polarisations complete the analytic picture, connecting the present work to the theory of the Weil representation as developed by Lion--Vergne and to Mumford's systematic treatment of theta functions.

gr-qc↗

arXiv2025

Sachs equations and plane waves IV: projective differential geometry

This article gives an invariant representation of the curvature of a plane wave spacetime in terms of the Schwarzian of a curve in the Lagrangian Grassmannian. It develops a general theory of cross ratios and Schwarzians of curves in what it terms the middle Grassmannian. Most of the theory is developed in infinite dimensions, where the middle Grassmannian is defined via cocycles on the space of complemented subspaces of a topological vector space. In the case of Hilbert spaces, the middle Grassmannian coincides with the set of closed subspaces whose Hilbert dimension and codimension are the same cardinal. We show how to define the cross ratio of four mutually complementary subspaces, and give some applications to algebraic geometry. We then study differential calculus in the middle Grassmannian of a Banach space over a complete normed field, defining the tangent vector and covector to a curve, followed by the Schwarz invariant. We give a characterization of the vanishing of the Schwarz invariant in terms of hyperbolic structures. Then, we specialize to the case of (real) Hilbert spaces, and define a symplectic structure, which allows us to consider the Grassmannian of Lagrangian subspaces and the Lagrange--Schwarzian of positive curves in the Lagrangian Grassmannian, which we show represents the curvature of a plane wave.

gr-qc↗

arXiv2024

Sachs equations and plane waves III: Microcosms

This article examines the structure of plane wave spacetimes (of signature $(1,n+1)$, $n\ge 2$) that are homogeneous (the isometry group is transitive) and geodesically complete -- which we call microcosms. In general, a plane wave is shown to determine a smooth positive curve in the Lagrangian Grassmannian associated with the $2n$ dimensional symplectic vector space of Jacobi fields. We show how to solve the Sachs equations in full generality for microcosms and, moreover, we relate the power series expansions canonical solutions to the Sachs equations on a general plane wave to Bernoulli-like recursions. It is shown that for microcosms, the curve in the Lagrangian Grassmannian associated to a microcosm is an orbit of a one parameter group in $\operatorname{Sp}(2n,\mathbb R)$. We also give an effective method for determining the orbit. Finally, we specialize to the case of $n=2$, and give analytical formulae for the solutions to the Sachs equations and the associated one-parameter group orbit.

math-ph↗

arXiv2024

Sachs equations and plane waves II: Isometries and conformal isometries

This article describes the symmetries of plane wave spacetimes in dimension four and greater. It begins with a description of the isometric automorphisms, and in particular the homogeneous plane waves. Then the article turns to describing isometries from one plane wave to another. The structure of the isometries is relevant for the problem of classifying vacuum spacetimes by observables, and the article presents an explicit example of a family of vacuum spacetimes encoding the Bernoulli shift, and so not classifiable by observables. Next, the article turns to a description of the conformal isometries. Here it is assumed that the conformal curvature does not vanish identically, the case of Minkowski space being both trivial and very degenerate. The article then classifies all conformal automorphisms and isometries of plane waves of the Rosen and Brinkmann forms.

math-ph↗

arXiv2024

Sachs equations and plane waves, I: Rosen universes

This article, the first in a series, analyzes the general theory of plane wave spacetimes. Following Dmitri Aleekseevsky, these are defined as spacetimes admitting a group of dilations leaving invariant a smooth curve. If this curve is specified as part of the structure, the spacetime is termed a Penrose limit, whose theory was developed first by Roger Penrose. The main result is that every plane wave is a Rosen universe, a generalization of the smooth metrics of Albert Einstein and Nathan Rosen, allowing for certain isolated co-ordinate singularities; the latter are characterized. We conclude with an extended example, using the techniques developed in the article to associate a vacuum plane wave in four dimensions to any hyperbolic billiard trajectory.

gr-qc↗

arXiv2020

On chromonatural inflation in string theory

Sourced gravitational waves in chromonatural inflation (CNI) can give rise to a chiral spectrum of tensor fluctuations that is considerably enhanced relative to the vacuum fluctuations. If the field content of CNI acts purely as a spectator (SCNI), the inflationary sector can be consistent with current data making SCNI very appealing in view of future observations. We investigate the prospects of embedding SCNI in string theory, in the framework of Kähler inflation in type IIB large volume string compactifications, with a spectator sector associated with gaugino condensation on multiply magnetised D7-branes. We first introduce a generalised field theory framework that describes non-trivial multifield inflation coupled to gauge fields of the form generically arising in supergravity and string theory. We then use these results to study numerically and analytically both the background evolution and the dynamics of cosmological perturbations. We show that a successful inflationary background evolution with a large enhancement of the gravitational wave spectrum and a controllable backreaction from the amplified tensor fluctuations can be achieved by considering suitable values of three parameters present in our scenario: the magnetic flux, the degree of the condensing gauge group and the wrapping number of the D7-brane. On the other hand, the required values for these quantities may present a challenge for its successful realisation within string theory. We also discuss these challenges and the model from the viewpoint of the weak gravity conjecture.

hep-th↗

arXiv2018

Anisotropic Inflation with Derivative Couplings

We study anisotropic inflationary solutions when the inflaton and its derivative couple to a vector field. This type of coupling is motivated by D-brane inflationary models, in which the inflaton, and a vector field living on the D-brane, couple disformally (derivatively). We start by studying a phenomenological model where we show the existence of anisotropic solutions and demonstrate their stability via a dynamical system analysis. Compared to the case without a derivative coupling, the anisotropy is reduced and thus can be made consistent with current limits, while the value of the slow-roll parameter remains almost unchanged. We also discuss solutions for more general cases, including D-brane like couplings.

hep-th↗

arXiv2018

The conformal, complex and non-commutative structures of the Schwarzschild solution

The generic null geodesic of the Schwarzschild--Kruskal--Szekeres geometry has a natural complexification, an elliptic curve with a cusp at the singularity. To realize that complexification as a Riemann surface without a cusp, and also to ensure conservation of energy at the singularity, requires a branched cover of the space-time over the singularity, with the geodesic being doubled as well to obtain a genus two hyperelliptic curve with an extra involution. Furthermore, the resulting space-time obtained from this branch cover has a Hamiltonian that is null geodesically complete. The full complex null geodesic can be realized in a natural complexification of the Kruskal--Szekeres metric.

math-ph↗

arXiv2014

Cosmology: macro and micro

A new approach to cosmology and space-time is developed, which emphasizes the description of the matter degrees of freedom of Einstein's theory of gravity by a family of Kähler-Einstein Fano manifolds.

gr-qc↗

arXiv2014

The cosmology of a fundamental scalar

We observe that the standard homogeneous cosmologies, those of Minkowski, de Sitter, and anti-de Sitter, which form the matrix for the Robertson--Walker scale factor, live naturally as isolated points inside a larger family of conformally flat metrics obtained by allowing a tensor containing the information of conformal symmetry breaking to be more general. So the standard cosmological metrics are parametrically unstable in this sense, and therefore unphysical. When we pass to the stable family of perturbed metrics, we immediately encounter a scalar field, which drives the conformal expansion of the universe and which automatically obeys the non-linear sine-Gordon equation. The Lagrangian for the sine-Gordon equation is a cosine potential agreeing to the fourth order with the potential used in the approach to the generation of mass in gauge theories. Accordingly we identify our geometric scalar field---actually of the type of an abelian gauge field---with the recently discovered scalar field. There are two constants in the theory: the first, named $m$, is positive and defines a mass scale for the universe; the second, named $Λ$, is the cosmological constant. For the space-time to be everywhere non-singular, equivalently for the (strict) dominant energy condition to hold, these constants must obey the inequality $Λ> m^2/4$.

gr-qc↗

arXiv2012

Notes on symmetric spaces

These are notes from a lecture course on symmetric spaces by the second author given at the University of Pittsburgh in the fall of 2010.

math.DG↗

arXiv2012

A note on the ultrahyperbolic wave equation in 3+3 dimensions

The article concerns the relationship between the ultrahyperbolic wave operator and the Xi transform for flat space, introduced by Sparling (2007). The main theorem is that the wave operator and the Xi transform form an exact couple: the image of the wave operator is the kernel of the Xi transform and, vice versa, the image of the Xi transform is the kernel of the wave operator.

math-ph↗

arXiv2012

A two-dimensional $C^{2,1}$ metric with no local $C^2$ embedding in $\mathbb{R}^3$, following Pogorelov

This article presents a proof of Pogorelov's result that there exists a $C^{2,1}$ metric with no local $C^2$ realization in $\mathbb{R}^3$. It also construct in a very elementary way a $C^{1,1}$ realization of this metric. Pogorelov's result is somewhat controversial among the community of researchers that study isometric immersions. This in part owes to the lack of details in Pogorelov's original paper. The chief aim of the paper is therefore to provide the missing details. The construction is the same as Pogorelov's, although the verification differs in some important respects.

math.DG↗