Radiation Gauge Research Articles

The quantization of Maxwell theory in the Cauchy radiation gauge: Hodge decomposition and Hadamard states

AbstractThe aim of this paper is to prove the existence of Hadamard states for the Maxwell equations on any globally hyperbolic spacetime. This will be achieved by introducing a new gauge fixing condition, the Cauchy radiation gauge, that will allow to suppress all the unphysical degrees of freedom. The key ingredient for achieving this gauge is a new Hodge decomposition for differential ‐forms in Sobolev spaces on complete (possibly noncompact) Riemannian manifolds.

Spin-2 Green’s functions on Kerr in radiation gauge

Abstract We construct retarded and advanced Green’s functions for gravitational perturbations in Kerr in an ingoing radiation gauge. Our Green’s functions have a frequency domain piece that has previously been obtained by Ori (2003 Phys. Rev. D 67) based on the Chrzanowski-Cohen-Kegeles metric reconstruction method. As is well known, this piece by itself is not sufficient to obtain an actual Green’s function. We show how to complete it with a piece based on a method by Green et al (2020 Class. Quantum Grav. 37). The completion piece has a completely explicit form in the time-domain and is supported on pairs of points on the same outgoing principal null geodesic which are in the appropriate causal order. We expect our Green’s functions to be useful for gravitational self-force calculations and other perturbation problems on Kerr spacetime.

Gravitational waves on Kerr black holes: I. Reconstruction of linearized metric perturbations

Abstract The gravitational perturbations of a rotating Kerr black hole are notoriously complicated, even at the linear level. In 1973, Teukolsky showed that their physical degrees of freedom are encoded in two gauge-invariant Weyl curvature scalars that obey a separable wave equation. Determining these scalars is sufficient for many purposes, such as the computation of energy fluxes. However, some applications -- such as second-order perturbation theory -- require the reconstruction of metric perturbations. In principle, this problem was solved long ago, but in practice, the solution has never been worked out explicitly. Here, we do so by writing down the metric perturbation (in either ingoing or outgoing radiation gauge) that corresponds to a given mode of either Weyl scalar. Our formulas make no reference to the Hertz potential (an intermediate quantity that plays no fundamental role) and involve only the radial and angular Kerr modes, but not their derivatives, which can be altogether eliminated using the Teukolsky--Starobinsky identities. We expect these analytic results to prove useful in numerical studies and for extending black hole perturbation theory beyond the linear regime.

Can a radiation gauge be horizon-locking?

In this short Note, I answer the titular question: yes, a radiation gauge can be horizon-locking. Radiation gauges are very common in black hole perturbation theory. It’s also very convenient if a gauge choice is horizon-locking, i.e. the location of the horizon is not moved by a linear metric perturbation. Therefore it is doubly convenient that a radiation gauge can be horizon-locking, when some simple criteria are satisfied. Though the calculation is straightforward, it seemed useful enough to warrant writing this Note. Finally I show an example: the ℓ vector of the Hartle–Hawking tetrad in Kerr satisfies all the conditions for ingoing radiation gauge to keep the future horizon fixed.

Self-forced inspirals with spin-orbit precession

We develop the first model for extreme mass-ratio inspirals with misaligned angular momentum and primary spin, and zero eccentricity—also known as quasispherical inspirals—evolving under the influence of the first-order in mass-ratio gravitational self-force. The forcing terms are provided by an efficient spectral interpolation of the first-order gravitational self-force in the outgoing radiation gauge. In order to speed up the calculation of the inspiral, we apply a near-identity (averaging) transformation to eliminate all dependence of the orbital phases from the equations of motion while maintaining all secular effects of the first-order gravitational self-force at postadiabatic order. The resulting solutions are defined with respect to “Mino time”; thus, we perform a second averaging transformation, so the inspiral is parametrized in terms of Boyer-Lindquist time, which is more convent for LISA data analysis. We also perform a similar analysis using the two-timescale expansion and find that using either approach yields self-forced inspirals that can be evolved to subradian accuracy in less than a second. The dominant contribution to the inspiral phase comes from the adiabatic contributions, so we further refine our self-force model using information from gravitational wave flux calculations. The significant dephasing we observe between the lower and higher accuracy models highlights the importance of accurately capturing adiabatic contributions to the phase evolution. Published by the American Physical Society 2024

Computation of the complex energy shift of decaying states in intense laser fields

We present a specific calculation concerning laser-assisted decaying systems with non-Hermitian spectral calculations merged with the (t,t’)-formalism to estimate the possible intense laser-field-induced corrections to the lifetime of quasi-stationary states. The perturbed complex energy shift caused by the external vector potential which represents the linearly and circularly polarized laser field is calculated in details. This wavefunction-centered method implies the unsuitability of the radiation gauge and length gauge when quasi-stationary states are considered and described within the framework of non-hermitian quantum mechanics since in this case the A2 term cannot be transformed out.

Subcycle-resolved strong-field tunneling ionization: Identification of magnetic dipole and electric quadrupole effects

Interaction of a strong laser pulse with matter transfers not only energy but also linear momentum of the photons. Recent experimental advances have made it possible to detect the small amount of linear momentum delivered to the photoelectrons in strong-field ionization of atoms. Linear momentum transfer is a unique signature of the laser-atom interaction beyond its dipolar limit. Here, we present a decomposition of the subcycle time-resolved linear momentum transfer in term of its multipolar components. We show that the magnetic dipole contribution dominates the linear momentum transfer during the dynamical tunneling process while the post-ionization longitudinal momentum transfer in the field-driven motion of the electron in the continuum is primarily governed by the electric quadrupole interaction. Alternatively, exploiting the radiation gauge, we identify nondipole momentum transfer effects that scale either linearly or quadratically with the coupling to the laser field. The present results provide detailed insights into the physical mechanisms underlying the subcycle linear momentum transfer induced by nondipole effects.

Nonlinear Radiation Gauge for Near Kerr Spacetimes

In this paper, we introduce and explore the properties of a new gauge choice for the vacuum Einstein equation inspired by the ingoing and outgoing radiation gauges (IRG, ORG) for the linearized vacuum Einstein equation introduced by Chrzanowski in his work on metric reconstruction (Chrzanowski in Phys Rev D 11:2042–2062, 1975) on the Kerr background. It has been shown by Price et al. (Class Quantum Gravity 24:2367–2388, 2007) that the IRG/ORG are consistent gauges for the linearized vacuum Einstein equation on Petrov type II backgrounds. In (Andersson et al. Stability for linearized gravity on the Kerr spacetime, 2019), the ORG was used in proving linearized stability for the Kerr spacetime, and the new non-linear radiation gauge introduced here is a direct generalization of that gauge condition, and is intended to be used to study the stability of Kerr black holes under the evolution generated by the vacuum Einstein equation.

Eccentric self-forced inspirals into a rotating black hole

We develop the first model for extreme mass-ratio inspirals (EMRIs) into a rotating massive black hole driven by the gravitational self-force (GSF). Our model is based on an action angle formulation of the method of osculating geodesics for eccentric, equatorial (i.e., spin-aligned) motion in Kerr spacetime. The forcing terms are provided by an efficient spectral interpolation of the first-order GSF in the outgoing radiation gauge. We apply a near-identity (averaging) transformation to eliminate all dependence of the orbital phases from the equations of motion, while maintaining all secular effects of the first-order GSF at post-adiabatic order. This implies that the model can be evolved without having to resolve all orbit cycles of an EMRI, yielding an inspiral model that can be evaluated in less than a second for any mass-ratio. In the case of a non-rotating central black hole, we compare inspirals evolved using self-force data computed in the Lorenz and radiation gauges. We find that the two gauges generally produce differing inspirals with a deviation of comparable magnitude to the conservative self-force correction. This emphasizes the need for including the (currently unknown) dissipative second order self-force to obtain gauge independent, post-adiabatic waveforms.

Gravitational waves from fully general relativistic oscillon preheating

As long-lived quasisolitons from the fragmentation of a scalar condensate, oscillons may dominate the preheating era after inflation. During this period, stochastic gravitational waves can also be generated. We quantify the gravitational-wave production in this period with simulations accounting for full general relativity to capture all possible nonperturbative effects. We compute the gravitational-wave spectra across a range of choices of the oscillon preheating models and compare our results to a conventional perturbative approach on a Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) background. We clarify the gauge ambiguities in computing induced gravitational waves from scenarios where dense nonperturbative objects such as oscillons are being formed. In particular, we find that the synchronous gauge tends to contain large artificial enhancements in the gravitational-wave spectrum due to gauge modes if gravity plays an important role in the formation of the oscillons, while other gauge choices, such as the radiation gauge or a suitably chosen ``1+log'' gauge, can efficiently reduce the contributions of gauge modes. The full general-relativistic simulations indicate that gravitational-wave spectra obtained from the perturbative approach on the FLRW background are fairly accurate, except when oscillon formation induces strong gravitational effects, for which case there can be an order unity enhancement.

New metric reconstruction scheme for gravitational self-force calculations

Inspirals of stellar-mass objects into massive black holes will be important sources for the space-based gravitational-wave detector LISA. Modelling these systems requires calculating the metric perturbation due to a point particle orbiting a Kerr black hole. Currently, the linear perturbation is obtained with a metric reconstruction procedure that puts it in a ‘no-string’ radiation gauge which is singular on a surface surrounding the central black hole. Calculating dynamical quantities in this gauge involves a subtle procedure of ‘gauge completion’ as well as cancellations of very large numbers. The singularities in the gauge also lead to pathological field equations at second perturbative order. In this paper we re-analyze the point-particle problem in Kerr using the corrector-field reconstruction formalism of Green, Hollands, and Zimmerman (GHZ). We clarify the relationship between the GHZ formalism and previous reconstruction methods, showing that it provides a simple formula for the ‘gauge completion’. We then use it to develop a new method of computing the metric in a more regular gauge: a Teukolsky puncture scheme. This scheme should ameliorate the problem of large cancellations, and by constructing the linear metric perturbation in a sufficiently regular gauge, it should provide a first step toward second-order self-force calculations in Kerr. Our methods are developed in generality in Kerr, but we illustrate some key ideas and demonstrate our puncture scheme in the simple setting of a static particle in Minkowski spacetime.

Tidal Love numbers of Kerr black holes

The open question of whether a Kerr black hole can become tidally deformed or not has profound implications for fundamental physics and gravitational-wave astronomy. We consider a Kerr black hole embedded in a weak and slowly varying, but otherwise arbitrary, multipolar tidal environment. By solving the static Teukolsky equation for the gauge-invariant Weyl scalar ${\ensuremath{\psi}}_{0}$ and by reconstructing the corresponding metric perturbation in an ingoing radiation gauge, for a general harmonic index $\ensuremath{\ell}$, we compute the linear response of a Kerr black hole to the tidal field. This linear response vanishes identically for a Schwarzschild black hole and for an axisymmetric perturbation of a spinning black hole. For a nonaxisymmetric perturbation of a spinning black hole, however, the linear response does not vanish, and it contributes to the Geroch-Hansen multipole moments of the perturbed Kerr geometry. As an application, we compute explicitly the rotational black hole tidal Love numbers that couple the induced quadrupole moments to the quadrupolar tidal fields, to linear order in the black hole spin, and we introduce the corresponding notion of a tidal Love tensor. Finally, we show that those induced quadrupole moments are closely related to the well-known physical phenomenon of tidal torquing of a spinning body interacting with a tidal gravitational environment.

On the radiation gauge for spin-1 perturbations in Kerr–Newman spacetime

We extend previous work (2020 Class. Quantum Grav. 37 075001) to the case of Maxwell’s equations with a source. Our work shows how to construct a vector potential for the Maxwell field on the Kerr–Newman background in a radiation gauge. The vector potential has a ‘reconstructed’ term obtained from a Hertz potential solving Teukolsky’s equation with a source, and a ‘correction’ term which is obtainable by a simple integration along outgoing principal null rays. The singularity structure of our vector potential is discussed in the case of a point particle source.

Detweiler’s redshift invariant for extended bodies orbiting a Schwarzschild black hole

We compute the first-order self-force contribution to Detweiler's redshift invariant for extended bodies endowed with both dipolar and quadrupolar structure (with spin-induced quadrupole moment) moving along circular orbits on a Schwarzschild background. Our analysis includes effects which are second order in spin, generalizing previous results for purely spinning particles. The perturbing body is assumed to move on the equatorial plane, the associated spin vector being orthogonal to it. The metric perturbations are obtained by using a standard gravitational self-force approach in a radiation gauge. Our results are accurate through the 6.5 post-Newtonian order, and are shown to reproduce the corresponding post-Newtonian expression for the same quantity computed by using the available Hamiltonian from an effective field theory approach for the dynamics of spinning binaries.

Gravitational Deflection of Massive Particles by a Schwarzschild Black Hole in Radiation Gauge**Supported by the National Natural Science Foundation of China under Grant No. 11647314 and the Research Foundation of Education Department of Hunan Province under Grant No. 18C0427

The exact metric of a Schwarzschild black hole in the true radiation gauge was recently reported. In this work, we base on this gravity and calculate the gravitational deflection of relativistic massive particles up to the fourth post-Minkowskian order. It is found that the result is consistent with the previous formulations for both the case of dropping the fourth-order contribution and the case of light deflection. Our result might be helpful for future high-accuracy observations.

Electromagnetic fields on Kerr spacetime, Hertz potentials, and Lorenz gauge

We review two procedures for constructing the vector potential of the electromagnetic field on Kerr spacetime, namely, the classic method of Cohen & Kegeles, yielding $A^\mu$ in a radiation gauge, and the newer method of Frolov et al., yielding $A^\mu$ in Lorenz gauge. We demonstrate that the vector potentials are related by straightforward gauge transformations, which we give in closed form. We obtain a new result for a separable Hertz potential $H^{\mu \nu}$ such that $A_{\text{lor}}^\mu = \nabla_\nu H^{\mu \nu}$.

Gravitational self-force on generic bound geodesics in Kerr spacetime

In this work we present the first calculation of the gravitational self-force on generic bound geodesics in Kerr spacetime to first order in the mass-ratio. That is, the local correction to equations of motion for a compact object orbiting a larger rotating black hole due to its own impact on the gravitational field. This includes both dissipative and conservative effects. Our method builds on and extends earlier methods for calculating the gravitational self-force on equatorial orbits. In particular we reconstruct the local metric perturbation in the outgoing radiation gauge from the Weyl scalar $\psi_4$, which in turn is obtained by solving the Teukolsky equation using semi-analytical frequency domain methods. The gravitational self-force is subsequently obtained using (spherical) $l$-mode regularization. We test our implementation by comparing the large $l$-behaviour against the analytically known regularization parameters. In addition we validate our results be comparing the long-term average changes to the energy, angular momentum, and Carter constant to changes to these constants of motion inferred from the gravitational wave flux to infinity and down the horizon.

Detweiler’s redshift invariant for spinning particles along circular orbits on a Schwarzschild background

We study the metric perturbations induced by a classical spinning particle moving along a circular orbit on a Schwarzschild background, limiting the analysis to effects which are first order in spin. The particle is assumed to move on the equatorial plane and has its spin aligned with the $z$-axis. The metric perturbations are obtained by using two different approaches, i.e., by working in two different gauges: the Regge-Wheeler gauge (using the Regge-Wheeler-Zerilli formalism) and a radiation gauge (using the Teukolsky formalism). We then compute the linear-in-spin contribution to the first-order self-force contribution to Detweiler's redshift invariant up to the 8.5 post-Newtonian order. We check that our result is the same in both gauges, as appropriate for a gauge-invariant quantity, and agrees with the currently known 3.5 post-Newtonian results.

Canonical field anticommutators in the extended gauged Rarita-Schwinger theory

We reexamine canonical quantization of the gauged Rarita-Schwinger theory using the extended theory, incorporating a dimension $\frac{1}{2}$ auxiliary spin-$\frac{1}{2}$ field $\Lambda$, in which there is an exact off-shell gauge invariance. In $\Lambda=0$ gauge, which reduces to the original unextended theory, our results agree with those found by Johnson and Sudarshan, and later verified by Velo and Zwanziger, which give a canonical Rarita-Schwinger field Dirac bracket that is singular for small gauge fields. In gauge covariant radiation gauge, the Dirac bracket of the Rarita-Schwinger fields is nonsingular, but does not correspond to a positive semi-definite anticommutator, and the Dirac bracket of the auxiliary fields has a singularity of the same form as found in the unextended theory. These results indicate that gauged Rarita-Schwinger theory is somewhat pathological, and cannot be canonically quantized within a conventional positive semi-definite metric Hilbert space. We leave open the questions of whether consistent quantizations can be achieved by using an indefinite metric Hilbert space, by path integral methods, or by appropriate couplings to conventional dimension $\frac{3}{2}$ spin-$\frac{1}{2}$ fields.

Effect of two-center interference on molecular ionization in multiphoton ionization regime.

Using solution of the full three-dimensional time-dependent Schrödinger equation (TDSE) in prolate spheroidal coordinates, we investigate the orientation dependence of ionization of H2+ in near-infrared laser fields. It is found that, the ionization probability decreases as a function of the alignment angle in tunneling ionization regime, while it ascends with the increase of orientation angle in multiphoton ionization regime for the internuclear distance R=2 a.u. Furthermore, the result obtained by the length gauge strong-field approximation theory is in qualitative agreement with that calculated by the TDSE but the radiation gauge strong-field approximation and molecular ADK theories fail to reproduce the TDSE result. Analysis indicates that the above intriguing feature can be ascribed to the interference between the partial electron wave packets emitted from different molecular cores, which becomes evident at low laser intensity due to increased width of the initial mechanical momentum of the photoelectron at ionization moment. In addition, when the internuclear distance increases to R=4 a.u., the ionization yields decrease vs alignment angle in both tunneling and multiphoton regimes since the electron wavefunction of the 1σg orbit is more concentrated in the molecular axis than that of R=2 a.u.