arXiv subjects

George Sparling

Publications and source records attributed to George Sparling.

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 $\mathcal U=\{(x,[\gamma]):x\in\gamma\}$ 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

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 Schr\"odinger 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 Schr\"odinger 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 Schr\"odinger 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↗

arXiv2023

Incompleteness Theorems for Observables in General Relativity

The quest for complete observables in general relativity has been a longstanding open problem. We employ methods from descriptive set theory to show that no complete observable on rich enough collections of spacetimes is Borel definable. In fact, we show that it is consistent with the Zermelo-Fraenkel and Dependent Choice axioms that no complete observable for rich collections of spacetimes exists whatsoever. In a nutshell, this implies that the Problem of Observables is to 'analysis' what the Delian Problem was to 'straightedge and compass'. Our results remain true even after restricting the space of solutions to vacuum solutions. In other words, the issue can be traced to the presence of local degrees of freedom. We discuss the next steps in a research program that aims to further uncover this novel connection between theoretical physics and descriptive set theory.

gr-qc↗

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

Asymptotically shearfree congruences in (2,2) spacetimes and Burgers' equation

The paper proves that any asymptotically shearfree congruence at the conformal infinity (scri) in a (2,2)-signature spacetime is determined locally by a solution to the pair of forced inviscid Burgers' equations L_u+LL_x=σ(u,x,y,L) and M_u+MM_y=σ'(u,x,y,M) where u,x,y are Bondi coordinates of scri. The functions σ and σ' are determined naturally by the projective structure on the α and β surfaces that foliate scri.

math.DG↗

arXiv2012

Null electromagnetic fields and relative CR embeddings

This paper applies the notion of relative CR embeddings to study two related questions. First, it answers negatively the question posed by Penrose whether every shear-free null rotating congruence is analytic. Second, it proves that given any shear-free null rotating congruence, there exists a null electromagnetic field which is null with respect to the given congruence. In the course of answering these questions, we introduce some new techniques for studying null electromagnetic fields and shear-free congruences in general based on the notion of a relative CR embedding.

math-ph↗

arXiv2011

Causal geometries, null geodesics, and gravity

The authors study a generalized notion of null geodesic defined by the Legendrian dynamics of a regular conical subbundle of the tangent bundle on a manifold. A natural extension of the Weyl tensor is shown to exist, and to depend only on this conical subbundle. Given a suitable defining function of the conical bundle, the Raychaudhuri--Sachs equations of general relativity continue to hold, and give rise to the same phenomenon of covergence of null geodesics in regions of positive energy that underlies the theory of gravitation.

math.DG↗

arXiv2010

Causal geometries and third-order ordinary differential equations

We discuss contact invariant structures on the space of solutions of a third-order ordinary differential equation. Associated to any third-order differential equation modulo contact transformations, Chern introduced a degenerate conformal Lorentzian metric on the space of 2-jets of functions of one variable. When the Wuenschmann invariant vanishes, the degenerate metric descends to a proper conformal Lorentzian metric on the space of solutions. In the general case, when the Wuenschmann invariant is not zero, we define the notion of a causal geometry, and show that the space of solutions supports one. The Wuenschmann invariant is then related to the projective curvature of the indicatrix curve cut out by the causal geometry in the projective tangent space. When the Wuenschmann vanishes, the causal structure is then precisely the sheaf of null geodesics of the Chern conformal structure. We then introduce a Lagrangian and associated Hamiltonian from which the degenerate conformal Lorentzian metric are constructed. Finally, necessary and sufficient conditions are given for a rank three degenerate conformal Lorentzian metric in four dimensions to correspond to a third-order differential equation.

math.DG↗

arXiv2009

Bilinear Forms and Fierz Identities for Real Spin Representations

Given a real representation of the Clifford algebra corresponding to $R^{p+q}$ with metric of signature $(p,q)$, we demonstrate the existence of two natural bilinear forms on the space of spinors. With the Clifford action of $k$-forms on spinors, the bilinear forms allow us to relate two spinors with elements of the exterior algebra. From manipulations of a rank four spinorial tensor, we are able to find a general class of identities which, upon specializing from four spinors to two spinors and one spinor in signatures (1,3) and (10,1), yield some well-known Fierz identities. We will see, surprisingly, that the identities we construct are partly encoded in certain involutory real matrices that resemble the Krawtchouk matrices.

gr-qc↗

arXiv2009

Space-time and G_2

A Weyl structure is a bundle over space-time, whose fiber at each space-time point is a space of maximally isotropic complex tangent planes. We develop the theory of Weyl connections for Weyl structures and show that the requirement that the connection be torsion-free fixes the Weyl connection uniquely. Further we show that to each such Weyl connection, there is naturally associated a (2, 3, 5)-Pfaffian system, as first analyzed by Cartan. We determine the associated G_2-conformal structure and calculate it explicitly in the cases of the Kapadia family of space-times and of the Schwarzschild solution

gr-qc↗

arXiv2007

Fedosov Observables on Constant Curvature Manifolds and the Klein-Gordon Equation

In this paper we construct the set of quantum mechanical observables in the Fedosov *-formalism (a coordinate invariant way to do quantum mechanics on any manifold M) of a single free particle that lives on a constant curvature manifold with metric signature (p,q). This was done for most but not all constant curvature manifolds. We show that the algebra of all observables in n=p+q dimensions is SO(p+1,q+1) in a nonperturbative calculation. A subgroup of this group is identified as the analogue of the Poincare group in Minkowski space i.e. it is the space of symmetries on the manifolds considered. We then write down a Klein-Gordon (KG) equation given by the the equation p^2|phi>=m^2|phi> for the set of allowed physical states. This result is consistent with previous results on AdS. Furthermore we lay out the standard scheme for the free KG field from the single particle theory. Furthermore we argue that this scheme will work on a general space-time.

gr-qc↗