arXiv subjects

Gabriel Herczeg

Publications and source records attributed to Gabriel Herczeg.

At least 19 recordsLinked to original sources

arXiv2026

The Penrose Transform and the Kerr-Schild double copy

There are a number of classical double copies, each providing a prescription for generating solutions to the Maxwell and scalar wave equations from exact solutions of Einstein's equations. Two such prescriptions are the Kerr-Schild and twistorial double copies. We argue that for a broad class of self-dual vacuum solutions of the Kerr-Schild form, which we refer to as twistorial Kerr-Schild spacetimes, these two prescriptions are in fact equivalent. The approach is elementary, utilizing null Lorentz transformations, with homogenous functions on twistor space playing a central role. The equivalence is illustrated explicitly for the example of the self-dual (Kerr)-Taub-NUT spacetime. A detailed proof and several more examples will be presented in a long-form companion to this letter.

hep-th↗

arXiv2026

Potential Carroll Structures and Special Carrollian Manifolds

It is well-known that unlike space-like and time-like hypersurfaces, null hypersurfaces in Lorentzian manifolds do not naturally inherit an affine connection from the spacetime in which they are embedded. On the other hand, recent developments in flat-space holography motivate the study of the intrinsic geometry of null hypersurfaces such as null infinity and black hole event horizons. Here we initiate the study of potential Carroll structures, a candidate for an intrinsic description of null hypersurfaces which may be particularly useful in settings where conformal isometries are of interest, and we explore their relationship to another such candidate intrinsic geometry, the special Carrollian manifolds.

math.DG↗

arXiv2024

Isometries and the double copy

In the standard derivation of the Kerr-Schild double copy, the geodicity of the Kerr-Schild vector and the stationarity of the spacetime are presented as assumptions that are necessary for the single copy to satisfy Maxwell's equations. However, it is well known that the vacuum Einstein equations imply that the Kerr-Schild vector is geodesic and shear-free, and that the spacetime possesses a distinguished vector field that is simultaneously a Killing vector of the full spacetime and the flat background, but need not be timelike with respect to the background metric. We show that the gauge field obtained by contracting this distinguished Killing vector with the Kerr-Schild graviton solves the vacuum Maxwell equations, and that this definition of the Kerr-Schild double copy implies the Weyl double copy when the spacetime is Petrov type D. When the Killing vector is taken to be timelike with respect to the background metric, we recover the familiar Kerr-Schild double copy, but the prescription is well defined for any vacuum Kerr-Schild spacetime and we present new examples where the Killing vector is null or spacelike. While most examples of physical interest are type D, vacuum Kerr-Schild spacetimes are generically of Petrov type II. We present a straightforward example of such a spacetime and study its double copy structure. Our results apply to real Lorentzian spacetimes as well as complex spacetimes and real spacetimes with Kleinian signature, and provide a simple correspondence between real and self-dual vacuum Kerr-Schild spacetimes. This correspondence allows us to study the double copy structure of a self-dual analog of the Kerr spacetime. We provide evidence that this spacetime may be diffeomorphic to the self-dual Taub-NUT solution.

gr-qc↗

arXiv2024

Taub-NUT Instanton as the Self-dual Analog of Kerr

It was recently conjectured that a certain vacuum Kerr-Schild spacetime, which may be regarded as a self-dual analog of the Kerr metric, is equivalent to the self-dual Taub-NUT instanton. We confirm this conjecture by applying the Cartan-Karlhede algorithm to each metric and showing that for suitable choices of null tetrad, the algorithm leads to the same invariants and linear isotropy groups for both, establishing their equivalence. While it is well-known that the Taub-NUT solution and its self-dual version admit a double Kerr-Schild form, the observation that the self-dual Taub-NUT instanton admits a single Kerr-Schild form has only been made very recently. The two metrics we compare may be regarded as either complex metrics with Lorentzian (1,3) signature or real metrics with Kleinian (2,2) signature; here we take the latter view. Significant simplifications occur when the null tetrads are chosen to consist of two pairs of complex conjugate null vectors rather than four real independent ones. As a bonus, our work provides the first example of applying the Cartan-Karlhede algorithm using a null tetrad of this type.

hep-th↗

arXiv2023

Double Kerr-Schild spacetimes and the Newman-Penrose map

The Newman-Penrose map, which is closely related to the classical double copy, associates certain exact solutions of Einstein's equations with self-dual solutions of the vacuum Maxwell equations. Here we initiate an extension of the Newman-Penrose map to a broader class of spacetimes. As an example, we apply principles from the Newman-Penrose map to associate a self-dual gauge field to the Kerr-Taub-NUT-(A)dS spacetime and we show that the result agrees with previously studied examples of classical double copies. The corresponding field strength exhibits a discrete electric-magnetic duality that is distinct from its (Hodge star) self-dual property.

hep-th↗

arXiv2023

Dynamical Henneaux-Teitelboim Gravity

We consider a modified gravity model which we call "dynamical Henneaux-Teitelboim gravity" because of its close relationship with the Henneaux-Teitelboim formulation of unimodular gravity. The latter is a fully diffeomorphism-invariant formulation of unimodular gravity, where full diffeomorphism invariance is achieved by introducing two additional non-dynamical fields: a scalar, which plays the role of a cosmological constant, and a three-form whose exterior derivative is the spacetime volume element. Dynamical Henneaux-Teitelboim gravity is a generalization of this model that includes kinetic terms for both the scalar and the three-form with arbitrary couplings. We study the field equations for the cases of spherically symmetric and homogeneous, isotropic configurations. In the spherically symmetric case, we solve the field equations analytically for small values of the coupling to obtain an approximate black hole solution. In the homogeneous and isotropic case, we perturb around de Sitter space to find an approximate cosmological background for our model.

gr-qc↗

arXiv2023

An Inner Product for 4D Quantum Gravity and the Chern-Simons-Kodama State

We demonstrate that reality conditions for the Ashtekar connection imply a non-trivial measure for the inner product of gravitational states in the polarization where the Ashtekar connection is diagonal, and we express the measure as the determinant of a certain first-order differential operator. This result opens the possibility to perform a non-perturbative analysis of the quantum gravity scalar product. In this polarization, the Chern-Simons-Kodama state, which solves the constraints of quantum gravity for a certain factor ordering, and which has de Sitter space as a semiclassical limit, is perturbatively non-normalizable with respect to the naive ve inner product. Our work reopens the question of whether this state might be normalizable when the correct non-perturbative inner product and choice of integration contour are taken into account. As a first step, we perform a semi-classical treatment of the measure by evaluating it on the round three-sphere, viewed as a closed spatial slice of de Sitter. The result is a simple, albeit divergent, infinite product that might serve as a regulator for a more complete treatment of the problem. Additionally, our results suggest deep connections between the problem of computing the norm of the CSK state in quantum gravity and computing the Chern-Simons partition function for a complex group.

hep-th↗

arXiv2022

Twistor Space Origins of the Newman-Penrose Map

Recently, we introduced the "Newman-Penrose map," a novel correspondence between a certain class of solutions of Einstein's equations and self-dual solutions of the vacuum Maxwell equations, which we showed was closely related to the classical double copy. Here, we give an alternative definition of this correspondence in terms of quantities that are defined naturally on twistor space, and a shear-free null geodesic congruence on Minkowski space whose twistorial character is articulated by the Kerr theorem. The advantage of this reformulation is that it is purely geometrical in nature, being manifestly invariant under both spacetime diffeomorphisms and projective transformations on twistor space. While the original formulation of the map may be more convenient for most explicit calculations, the twistorial formulation we present here may be of greater theoretical utility.

hep-th↗

arXiv2022

Torsion and the Probability of Inflation

We revisit the problem of the "probability of inflation" from the point of view of the Einstein-Cartan theory, where torsion can be present off-shell even in the absence of spinorial currents. An informal estimate suggests that the barrier for tunneling from "nothing" into a classical universe becomes thinner and lower, should torsion be present even if only off-shell. This is confirmed by a detailed calculation, where the usual assumptions are re-evaluated and repurposed in our situation. Interestingly some approximations used in the literature (such as the WKB approximation) are not needed in general, and in particular in our case. When we consider wave packets centered around zero torsion, however, the conclusion depends crucially how these are built. With a Klein-Gordon current prescription for the measure and probability, for small torsion variance, σ_c, we recover the plane wave results. Nonetheless, for large σc, higher order corrections could reverse this conclusion.

gr-qc↗

arXiv2021

Black hole and cosmological analysis of BF sequestered gravity

We study a minimal extension of a recently proposed modification of general relativity that draws on concepts from topological field theory to resolve aspects of the cosmological constant problem. In the original model, the field content of general relativity was augmented to include a gauge field and an adjoint-valued two-form without modifying the classical gravitational dynamics. Here we include a kinetic term for the gauge field which sources fluctuations of the cosmological constant. We then study spherically symmetric black holes and a simple homogeneous, isotropic cosmological model predicted by the extended theory. For the black hole case, we observe deviations near the event horizon as well as a ``charge"-like behavior induced by the gauge field. In the cosmological case, $\dot{H}$ is always positive and some solutions asymptote to a constant $H$.

gr-qc↗

arXiv2021

Chiral Symmetry and the Cosmological Constant

In this work we provide a link between a nearly vanishing cosmological constant and chiral symmetry. This is accomplished with a modification of General Relativity coupled to a topological field theory, namely BF theory by introducing fermions charged under the BF theory gauge group. We find that the cosmological constant sources a chiral anomaly for the fermions, providing a `technical naturalness' explanation for the smallness of the observed cosmological constant. Applied to the early universe, we show that production of fermions during inflation can provide all the dark matter in the universe today, in the form of superheavy dark baryons.

gr-qc↗

arXiv2021

A generalized Hartle-Hawking wavefunction

The Hartle-Hawking wave function is known to be the Fourier dual of the Chern-Simons or Kodama state reduced to mini-superspace, using an integration contour covering the whole real line. But since the Chern-Simons state is a general solution of the Hamiltonian constraint (with a given ordering), its Fourier dual should provide the general solution (i.e. beyond mini-superspace) of the Wheeler DeWitt equation representing the Hamiltonian constraint in the metric representation. We write down a formal expression for such a wave function, to be seen as the generalization beyond mini-superspace of the Hartle-Hawking wave function. Its explicit evaluation (or simplification) depends only on the symmetries of the problem, and we illustrate the procedure with anisotropic Bianchi models and with the Kantowski-Sachs model. A significant difference of this approach is that we may leave the torsion inside the wave functions when we set up the ansatz for the connection, rather than setting it to zero before quantization. This allows for quantum fluctuations in the torsion, with far reaching consequences.

gr-qc↗

arXiv2021

Dynamical Quantization of Contact Structures

We construct a dynamical quantization for contact manifolds in terms of a flat connection acting on a Hilbert tractor bundle. We show that this contact quantization, which is independent of the choice of contact form, can be obtained by quantizing the Reeb dynamics of an ambient strict contact manifold equivariantly with respect to an R+-action. The contact quantization further determines a certain contact tractor connection whose parallel sections determine a distinguished choice of Reeb dynamics and their quantization. This relationship relies on tractor constructions from parabolic geometries and mirrors the tight relationship between Einstein metrics and conformal geometries. Finally, we construct in detail the dynamical quantization of the unique tight contact structure on the 3-sphere, where the Holstein-Primakoff transformation makes a surprising appearance.

math-ph↗

arXiv2020

The Newman-Penrose Map and the Classical Double Copy

Gauge-gravity duality is arguably our best hope for understanding quantum gravity. Considerable progress has been made in relating scattering amplitudes in certain gravity theories to those in gauge theories---a correspondence dubbed the "double copy". Recently, double copies have also been realized in a classical setting, as maps between exact solutions of gauge theories and gravity. We present here a novel map between a certain class of real, exact solutions of Einstein's equations and self-dual solutions of the flat-space vacuum Maxwell equations. This map, which we call the "Newman-Penrose map", is well-defined even for non-vacuum, non-stationary spacetimes, providing a systematic framework for exploring gravity solutions in the context of the double copy that have not been previously studied in this setting. To illustrate this, we present here the Newman-Penrose map for the Schwarzschild and Kerr black holes, and Kinnersley's photon rocket.

hep-th↗

arXiv2018

Contact Geometry and Quantum Mechanics

We present a generally covariant approach to quantum mechanics in which generalized positions, momenta and time variables are treated as coordinates on a fundamental "phase-spacetime." We show that this covariant starting point makes quantization into a purely geometric flatness condition. This makes quantum mechanics purely geometric, and possibly even topological. Our approach is especially useful for time-dependent problems and systems subject to ambiguities in choices of clock or observer. As a byproduct, we give a derivation and generalization of the Wigner functions of standard quantum mechanics.

hep-th↗

arXiv2017

A Tour Through Shape Dynamic Black Holes

Shape dynamics is a classical theory of gravity which agrees with general relativity in many important cases, but possesses different gauge symmetries and constraints. Rather than spacetime diffeomorphism invariance, shape dynamics takes spatial diffeomorphism invariance and spatial Weyl invariance as the fundamental gauge symmetries associated with the gravitational field. Despite these differences, shape dynamics and general relativity generically predict the same dynamics---there exist gauge-fixings of each theory that ensure agreement with the other. However, these gauge-fixing conditions are not necessarily globally well-defined and it is therefore possible to find solutions of the shape dynamics equations of motion that agree with general relativity on some open neighborhoods, but which have different global structures. In particular, the black hole solutions of the two theories disagree globally. Understanding these novel "shape dynamic black holes" is the primary goal of this thesis.

gr-qc↗

arXiv2017

Parity Horizons in Shape Dynamics

I introduce the notion of a parity horizon, and show that many simple solutions of shape dynamics possess them. I show that the event horizons of the known asymptotically flat black hole solutions of shape dynamics are parity horizons and that this notion of parity implies that these horizons possess a notion of CPT invariance that can in some cases be extended to the solution as a whole. I present three new solutions of shape dynamics with parity horizons and find that not only event horizons become parity horizons in shape dynamics, but observer-dependent horizons and Cauchy horizons do as well. The fact that Cauchy horizons become (singular) parity horizons suggests a general chronology protection mechanism in shape dynamics that prevents the formation of closed time-like curves.

gr-qc↗

arXiv2016

Towards Black Hole Entropy in Shape Dynamics

Shape dynamics is classical theory of gravity which agrees with general relativity in many important cases, but possesses different gauge symmetries and constraints. Rather than spacetime diffeomorphism invariance, shape dynamics takes spatial diffeomorphism invariance and spatial Weyl invariance as the fundamental gauge symmetries associated with the gravitational field. Since the area of the event horizon of a black hole transforms under a generic spatial Weyl transformation, there has been some doubt that one can speak sensibly about the thermodynamics of black holes in shape dynamics. The purpose of this paper is to show that by treating the event horizon of a black hole as an interior boundary, one can recover familiar notions of black hole thermodynamics in shape dynamics and define a gauge invariant entropy that agrees with general relativity.

gr-qc↗