Harmonic Gauge Research Articles

Brane-like solutions and other non-supersymmetric vacua

After recasting the standard charged and uncharged brane profiles in the harmonic gauge, we explore solutions with the same isometries where the potentials V = T eγ ϕ of ten-dimensional non-supersymmetric strings are taken into account. Combining a detailed catalog of the possible asymptotics with some numerical results suggests that these spherically symmetric backgrounds terminate at singularities within finite proper distances.

Quadrupolar radiation in de Sitter: displacement memory and Bondi metric

We obtain the closed form expression for the metric perturbation around de Sitter spacetime generated by a matter source below Hubble scale both in generalized harmonic gauge and in Bondi gauge up to quadrupolar order in the multipolar expansion, including both parities (i.e. both mass and current quadrupoles). We demonstrate that such a source causes a displacement memory effect close to future infinity that originates, in the even-parity sector, from a Λ-BMS transition between the two non-radiative regions of future infinity.

Toward a covariant framework for post-Newtonian expansions for radiative sources

We consider the classic problem of a compact fluid source that behaves nonrelativistically and that radiates gravitational waves. The problem consists of determining the metric close to the source as well as far away from it. The nonrelativistic nature of the source leads to a separation of scales resulting in an overlap region where both the 1/c and (multipolar) G expansions are valid. Standard approaches to this problem (the Blanchet-Damour and the DIRE approach) use the harmonic gauge. We define a “post-Newtonian” class of gauges that admit a Newtonian regime in inertial coordinates. In this paper we set up a formalism to solve for the metric for any post-Newtonian gauge choice. Our methods are based on previous work on the covariant theory of nonrelativistic gravity (a 1/c expansion of general relativity that uses post-Newton-Cartan variables). At the order of interest in the 1/c and G expansions we split the variables into two sets: transverse and longitudinal. We show that for the transverse variables the problem can be reduced to inverting Laplacian and d’Alembertian operators on their respective domains subject to appropriate boundary conditions. The latter are regularity in the interior and asymptotic flatness with a Sommerfeld no-incoming radiation condition imposed at past null infinity. The longitudinal variables follow from the gauge choice. The full solution is then obtained by the method of matched asymptotic expansion. We show that our methods reproduce existing results in harmonic gauge to 2.5PN order. Published by the American Physical Society 2024

The nonlinear stability of (n + 1)-dimensional FLRW spacetimes

We prove nonlinear Lyapunov stability of a family of “([Formula: see text])”-dimensional cosmological models of general relativity locally isometric to the Friedman–Lemaître–Robertson–Walker (FLRW) spacetimes including a positive cosmological constant. In particular, we show that the perturbed solutions to the Einstein–Euler field equations around a class of spatially compact FLRW metrics (for which the spatial slices are compact negative Einstein spaces in general and hyperbolic for the physically relevant [Formula: see text] case) arising from regular Cauchy data remain uniformly bounded and decay to a family of metrics with constant negative spatial scalar curvature. To accomplish this result, we employ an energy method for the coupled Einstein–Euler field equations in constant mean extrinsic curvature spatial harmonic (CMCSH) gauge. In order to handle Euler’s equations, we construct energy from a current that is similar to the one derived by Christodoulou [The Formation of Shocks in 3-dimensional Fluids, EMS Monographs in Mathematics, Vol. 2 (European Mathematical Society, 2007)] (and which coincides with Christodoulou’s current on the Minkowski space) and show that this energy controls the desired norm of the fluid degrees of freedom. The use of a fluid energy current together with the CMCSH gauge condition casts the Einstein–Euler field equations into a coupled elliptic–hyperbolic system. Utilizing the estimates derived from the elliptic equations, we first show that the gravity-fluid energy functional remains uniformly bounded in the expanding direction. Using this uniform boundedness property, we later obtain sharp decay estimates if a positive cosmological constant [Formula: see text] is included, which confirms that the accelerated expansion of the physical universe that is induced by the positive cosmological constant is sufficient to control the nonlinearities in the case of small data. A few physical consequences of this stability result are discussed.

Polyhomogeneous spin-0 fields in Minkowski space-time.

The asymptotic behaviour of massless spin-0 fields close to spatial and null infinity in Minkowski space-time is studied by means of Friedrich's cylinder at spatial infinity. The results are applied to a system of equations called the good-bad-ugly which serves as a model for the Einstein field equations in generalized harmonic gauge. The relation between the logarithmic terms (polyhomogeneity) appearing in the solution obtained using conformal methods and those obtained by means of a heuristic method based on Hörmander's asymptotic system is discussed. This review article is based on Duarte et al. (Duarte et al. 2023 Class. Quantum Gravity 40, 055002. (doi:10.1088/1361-6382/acb47e)); Gasperín & Pinto (Gasperín & Pinto 2023 Spin-0 fields and the NP-constants close to spatial infinity in Minkowski spacetime. J. Math. Phys. 64, 082502. (doi:10.1063/5.0158746)). This article is part of a discussion meeting issue 'At the interface of asymptotics, conformal methods and analysis in general relativity'.

Fully consistent rotating black holes in the cubic Galileon theory

Configurations of rotating black holes in the cubic Galileon theory are computed by means of spectral methods. The equations are written in the formalism and the coordinates are based on the maximal slicing condition and the spatial harmonic gauge. The black holes are described as apparent horizons in equilibrium. It enables the first fully consistent computation of rotating black holes in this theory. Several quantities are extracted from the solutions. In particular, the vanishing of the mass is confirmed. A link is made between that and the fact that the solutions do not obey the zeroth-law of black hole thermodynamics.

Exterior Stability of Minkowski Space in Generalized Harmonic Gauge

We give a short proof of the existence of a small piece of null infinity for (3+1)-dimensional spacetimes evolving from asymptotically flat initial data as solutions of the Einstein vacuum equations. We introduce a modification of the standard wave coordinate gauge in which all non-physical metric degrees of freedom have strong decay at null infinity. Using a formulation of the gauge-fixed Einstein vacuum equations which implements constraint damping, we establish this strong decay regardless of the validity of the constraint equations. On a technical level, we use notions from geometric singular analysis to give a streamlined proof of semiglobal existence for the relevant quasilinear hyperbolic equation.

Interference in Quantum Field Theory: Detecting Ghosts with Phases

AbstractThis study discusses the implications of the principle of locality for interference in quantum field theory. As an example, it considers the interaction of two charges via a mediating quantum field and the resulting interference pattern in the Lorenz gauge. Using the Heisenberg picture, it is proposed that detecting relative phases or entanglement between two charges in an interference experiment is equivalent to accessing empirically the gauge degrees of freedom associated with the so‐called ghost (scalar) modes of the field in the Lorenz gauge. These results imply that ghost modes are measurable and hence physically relevant, contrary to what is usually thought. They also raise interesting questions about the relation between the principle of locality and the principle of gauge‐invariance. This analysis also applies to linearized quantum gravity in the harmonic gauge, and hence has implications for the recently proposed entanglement‐based witnesses of non‐classicality in gravity.

Linearised conformal Einstein field equations

The linearisation of a second-order formulation of the conformal Einstein field equations (CEFEs) in generalised harmonic gauge (GHG), with trace-free matter is derived. The linearised equations are obtained for a general background and then particularised for the study linear perturbations around a flat background—the inversion (conformal) representation of the Minkowski spacetime—and the solutions discussed. We show that the generalised Lorenz gauge (defined as the linear analogue of the GHG-gauge) propagates. Moreover, the equation for the conformal factor can be trivialised with an appropriate choice for the gauge source functions; this permits a scri-fixing strategy using gauge source functions for the linearised wave-like CEFE-GHG, which can in principle be generalised to the nonlinear case. As a particular application of the linearised equations, the far-field and compact source approximation is employed to derive quadrupole-like formulae for various conformal fields such as the perturbation of the rescaled Weyl tensor.

Global Stability for Charged Scalar Fields in an Asymptotically Flat Metric in Harmonic Gauge

We prove global stability for the charged scalar field system on a background spacetime, which is close to 1+3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1+3$$\\end{document}-dimensional Minkowski space and whose outward light cones converge to those for the Schwarzs-child metric at null infinity. The key technique to this proof is the use of a modified null frame, depending only on the mass M of the metric, which captures the asymptotic behavior of the metric at future null infinity. Our results are analogous to results obtained in Minkowski space by Lindblad and Sterbenz in (IMRP Int Math Res Pap 2006:52976, 2006) up to a change in coordinates, and will in the sequel be used to prove the full structure of the Einstein–charge scalar field system in these modified harmonic coordinates.

Multipole expansion of gravitational waves: memory effects and Bondi aspects

In our previous work, we proposed an algorithm to transform the metric of an isolated matter source in the multipolar post-Minkowskian approximation in harmonic (de Donder) gauge to the Newman-Unti gauge. We then applied this algorithm at linear order and for specific quadratic interactions known as quadratic tail terms. In the present work, we extend this analysis to quadratic interactions associated with the coupling of two mass quadrupole moments, including both instantaneous and hereditary terms. Our main result is the derivation of the metric in Newman-Unti and Bondi gauges with complete quadrupole-quadrupole interactions. We rederive the displacement memory effect and provide expressions for all Bondi aspects and dressed Bondi aspects relevant to the study of leading and subleading memory effects. Then we obtain the Newman-Penrose charges, the BMS charges as well as the second and third order celestial charges defined from the known second order and novel third order dressed Bondi aspects for mass monopole-quadrupole and quadrupole-quadrupole interactions.

The good-bad-ugly system near spatial infinity on flat spacetime

A model system of equations that serves as a model for the Einstein field equation in generalised harmonic gauge called the good-bad-ugly system is studied in the region close to null and spatial infinity in Minkowski spacetime. This analysis is performed using H. Friedrich’s cylinder construction at spatial infinity and defining suitable conformally rescaled fields. The results are translated to the physical set up to investigate the relation between the polyhomogeneous expansions arising from the analysis of linear fields using the i 0-cylinder framework and those obtained through a heuristic method based on Hörmander’s asymptotic system.

Regularizing dual-frame generalized harmonic gauge at null infinity

The dual-frame formalism leads to an approach to extend numerical relativity simulations in generalized harmonic gauge (GHG) all the way to null infinity. A major setback is that without care, even simple choices of initial data give rise to logarithmically divergent terms that would result in irregular variables and equations on the compactified domain, which would in turn prevent accurate numerical approximation. It has been shown, however, that a suitable choice of gauge and constraint addition can be used to prevent their appearance. Presently we give a first order symmetric hyperbolic reduction of general relativity in GHG on compactified hyperboloidal slices that exploits this knowledge and eradicates these log-terms at leading orders. Because of their effect on the asymptotic solution space, specific formally singular terms are systematically chosen to remain. Such formally singular terms have been successfully treated numerically in toy models and result in a formulation with the desirable property that unphysical radiation content near infinity is suppressed.

BRST formalism of Weyl conformal gravity

We present the BRST formalism of a Weyl conformal gravity in Weyl geometry. Choosing the extended de Donder gauge-fixing condition (or harmonic gauge condition) for the general coordinate invariance and the new scalar gauge-fixing for the Weyl invariance we find that there is a Poincar${\rm{\acute{e}}}$-like $\IOSp(10|10)$ supersymmetry as in a Weyl invariant scalar-tensor gravity in Riemann geometry. We also point out that there is a gravitational conformal symmetry in quantum gravity although there is a massive Weyl gauge field as a result of spontaneous symmetry breakdown of Weyl gauge symmetry and account for how the gravitational conformal symmetry is spontaneously broken to the Poincar\'e symmetry. The corresponding massless Nambu-Goldstone bosons are the graviton and the dilaton. We also prove the unitarity of the physical S-matrix on the basis of the BRST quartet mechanism.

Peeling in generalized harmonic gauge

It is shown that a large class of systems of non-linear wave equations, based on the good–bad–ugly model, admit formal solutions with polyhomogeneous expansions near null infinity. A particular set of variables is introduced which allows us to write the Einstein field equations in generalized harmonic gauge (GHG) as a good–bad–ugly system and the functional form of the first few orders in such an expansion is found by applying the aforementioned result. Exploiting these formal expansions of the metric components, the peeling property of the Weyl tensor is revisited. The question addressed is whether or not the use of GHG, by itself, causes a violation of peeling. Working in harmonic gauge, it is found that log-terms that prevent the Weyl tensor from peeling do appear. The impact of gauge source functions and constraint additions on the peeling property is then considered. Finally, the special interplay between gauge and constraint addition, as well as its influence on the asymptotic system and the decay of each of the metric components, is exploited to find a particular gauge which suppresses this specific type of log-term to arbitrarily high order.

Note on the strong hyperbolicity of f(R) gravity with dynamical shifts

The well-posedness of the gravitational equations of $f(R)$ gravity are studied in this paper. Three formulations of the $f(R)$ gravity with dynamical shifts (which are all based on the Arnowitt-Deser-Misner (ADM) formalism of the equations) are investigated. These three formulations are all proved to be strongly hyperbolic by pseudodifferential reduction. The first one is the Baumagarte-Shapiro-Shibata-Nakamura (BSSN) formulation with the so-called "hyperbolic $K$-driver" condition and the "hyperbolic Gamma driver" condition. The second one is the ADM formulation with modified harmonic gauge conditions. We find that the equations are not strong hyperbolic in traditional Z4 formulation for $f(R)$ gravity. So, in the third formulation, we improve the Z4 formulation, and show these equations are strong hyperbolic with modified harmonic gauge conditions.

Quantum theory of Weyl-invariant scalar-tensor gravity

We perform a manifestly covariant quantization of a Weyl-invariant (i.e., a locally scale-invariant) scalar-tensor gravity in the extended de Donder gauge condition (or harmonic gauge condition) for general coordinate invariance and a new scalar gauge for Weyl invariance within the framework of the BRST formalism. We show that choral symmetry, which is a Poincar\'e-like $IOSp(8|8)$ supersymmetry in the case of Einstein gravity, is extended to a Poincar\'e-like $IOSp(10|10)$ supersymmetry. We point out that there is a gravitational conformal symmetry in quantum gravity and account for how conventional conformal symmetry in a flat Minkowski space-time is related to the gravitational conformal symmetry. Moreover, we examine the mechanism of the spontaneous symmetry breaking of the choral symmetry, and show that the gravitational conformal symmetry is spontaneously broken to the Poincar\'e symmetry and the corresponding massless Nambu-Goldstone bosons are the graviton and the dilaton. We also prove the unitarity of the physical $S$ matrix on the basis of the BRST quartet mechanism.

Boundary conditions for stationary black holes: Application to Kerr, Martínez-Troncoso-Zanelli, and hairy black holes

This work proposes a set of equations that can be used to numerically compute spacetimes containing a stationary black hole. The formalism is based on the $3+1$ decomposition of General Relativity with maximal slicing and spatial harmonic gauge. The presence of the black hole is enforced using the notion of apparent horizon in equilibrium. This setting leads to the main result of this paper: a set of boundary conditions describing the horizon and that must be used when solving the $3+1$ equations. Those conditions lead to a choice of coordinates that is regular even on the horizon itself. The whole procedure is validated with three different examples chosen to illustrate the great versatility of the method. First, the single rotating black holes are recovered up to very high values of the Kerr parameter. Second, nonrotating black holes coupled to a real scalar field, in the presence of a negative cosmological constant (the so-called Martinez-Troncoso-Zanelli black holes), are obtained. Last, black holes with complex scalar hairs are computed. Eventually, prospects for future work, in particular in contexts where stationarity is only approximate, are discussed.

Quantum scale invariant gravity in the de Donder gauge

We perform the manifestly covariant quantization of a scale invariant gravity with a scalar field, which is equivalent to the well-known Brans-Dicke gravity via a field redefinition of the scalar field, in the de Donder gauge condition (or harmonic gauge condition) for general coordinate invariance. First, without specifying the expression of a gravitational theory, we write down various equal-time (anti-)commutation relations (ETCRs), in particular, those involving the Nakanishi-Lautrup field, the FP ghost, and the FP antighost only on the basis of the de Donder gauge condition. It is shown that choral symmetry, which is a Poincar${\rm{\acute{e}}}$-like $IOSp(8|8)$ supersymmetry, can be derived from such a general action with the de Donder gauge. Next, taking the scale invariant gravity with a scalar field as a classical theory, we derive the ETCRs for the gravitational sector involving the metric tensor and scalar fields. Moreover, we account for how scale symmetry is spontaneously broken in quantum gravity, thereby showing that the dilaton is a massless Nambu-Goldstone particle.

Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation on Schwarzschild into Scalar Regge-Wheeler Equations

We consider the vector and the Lichnerowicz wave equations on the Schwarzschild spacetime, which correspond to the Maxwell and linearized Einstein equations in harmonic gauges (or, respectively, in Lorenz and de Donder gauges). After a complete separation of variables, the radial mode equations form complicated systems of coupled linear ODEs. We outline a precise abstract strategy to decouple these systems into sparse triangular form, where the diagonal blocks consist of spin-s scalar Regge-Wheeler equations (for spins s=0,1,2). Building on the example of the vector wave equation, which we have treated previously, we complete a successful implementation of our strategy for the Lichnerowicz wave equation. Our results go a step further than previous more ad-hoc attempts in the literature by presenting a full and maximally simplified final triangular form. These results have important applications to the quantum field theory of and the classical stability analysis of electromagnetic and gravitational perturbations of the Schwarzschild black hole in harmonic gauges.