Linearized Einstein Equations Research Articles

Metric reconstruction in Kerr spacetime

Abstract Metric reconstruction is the general problem of parameterizing GR in terms of its two “true degrees of freedom” e.g., by a complex scalar “potential”—in practice mostly with the aim of simplifying the Einstein equation (EE) within perturbative approaches. In this paper, we re-analyze the metric reconstruction procedure by Green, Hollands, and Zimmerman (GHZ) [Class. Quant. Grav. 37, 075001 (2020)], which is a generalization of the Chrzanowski-Cohen-Kegeles (CCK) approach. Contrary to the CCK method, that by GHZ is applicable not only to the vacuum, but also to the sourced linearized Einstein equation (EE) on Kerr. Our main innovation is a version of the GHZ integration scheme that is suitable for the initial value problem of the sourced linear EE. By iteration, our scheme gives the metric to as high an order in perturbation theory as one might wish, in principle. At each order, the metric perturbation is a sum of a corrector, obtained by solving a triangular system of transport equations, a reconstructed piece, obtained from a Hertz potential as in the CCK approach, and an algebraically special perturbation, determined by the ADM quantities. As a byproduct, we determine the precise relations between the asymptotic tail of the Hertz potential in the GHZ and CCK schemes, and the quantities relevant for gravitational radiation, namely, the energy flux, news- and memory tensors, and their associated BMS-supertranslations. We also discuss ways of transforming the metric perturbation to Lorenz gauge.

Mode analysis for the linearized Einstein equations on the Kerr metric: the large $\mathfrak{a}$ case

We give a complete analysis of mode solutions for the linearized Einstein equations and the 1 -form wave operator on the Kerr metric in the large \mathfrak{a} case. By mode solutions we mean solutions of the form e^{-it_*\sigma}\tilde{h}(r,\theta,\varphi) where t_{*} is a suitable time variable. The corresponding Fourier-transformed 1 -form wave operator and linearized Einstein operator are shown to be Fredholm between suitable function spaces and \tilde{h} has to lie in the domain of these operators. These spaces are constructed following the general framework of Vasy (2013, 2021) along the lines of Häfner et al. (2021). No mode solutions exist for \Im \sigma\ge 0,\, \sigma\neq 0 . For \sigma=0 mode solutions are Coulomb solutions for the 1 -form wave operator and linearized Kerr solutions plus pure gauge terms in the case of the linearized Einstein equations. If we fix a De Turck/wave map gauge, then the zero mode solutions for the linearized Einstein equations lie in a fixed seven-dimensional space. The proof relies on the absence of modes for the Teukolsky equation shown by Whiting (1989) and Anderson et al. (2019) and a complete classification of the gauge invariants of linearized gravity on the Kerr spacetime by Aksteiner and Bäckdahl (2018) and Aksteiner et al. (2021).

Pole-skipping and chaos in D3-D7 brane systems

In this paper, we analyze the pole-skipping phenomena of finite temperature Yang-Mills theory with quark flavors, which is dual to D3-D7 brane systems in bulk. We also consider the external electric field in the boundary field theory, which is dual to the world volume electric field on the D7 brane. We will work in the probe limit where the D7 branes do not backreact to the D3 brane background. In this scenario, we decode the characteristic parameters of the chaos, namely, Lyapunov exponent λL and butterfly velocity vb from the pole-skipping points by performing the near effective horizon analysis of the linearized Einstein equations. Unlike pure Yang-Mills, once charged quarks with a background electric field are added into the system, the characteristic parameters of the chaos show nontrivial dependence on the quark mass and external electric field. We have observed that λL and vb decreases with increasing electric field. We further perform the pole-skipping analysis for the gauge invariant sound, shear, and tensor modes of the perturbation in the bulk and discuss their physical importance in the holographic context. Published by the American Physical Society 2024

Linearized gravity and soft graviton theorem in de Sitter spacetime

We study the linearized gravity theory in the Newman-Unti gauge in the near horizon region of the de Sitter spacetime. The linearized Einstein equation involves the cosmological constant. The near horizon symmetry consists of near horizon supertranslation and near horizon superrotation. We compute the near horizon supertranslation charge and find the proper near horizon falloff conditions that uncover a soft graviton theorem from the Ward identity of the near horizon supertranslation. Published by the American Physical Society 2024

Cosmological observatories

Abstract We study the static patch of de Sitter space in the presence of a timelike boundary. We impose that the conformal class of the induced metric and the trace of the extrinsic curvature, K, are fixed at the boundary. We present the thermodynamic structure of de Sitter space subject to these boundary conditions, for static and spherically symmetric configurations to leading order in the semiclassical approximation. In three spacetime dimensions, and taking K constant on a toroidal Euclidean boundary, we find that the spacetime is thermally stable for all K. In four spacetime dimensions, the thermal stability depends on the value of K. It is established that for sufficiently large K, the de Sitter static patch subject to conformal boundary conditions is thermally stable. This contrasts the Dirichlet problem for which the region encompassing the cosmological horizon has negative specific heat. We present an analysis of the linearised Einstein equations subject to conformal boundary conditions. In the worldline limit of the timelike boundary, the underlying modes are linked to the quasinormal modes of the static patch. In the limit where the timelike boundary approaches the cosmological event horizon, the linearised modes are interpreted in terms of the shear and sound modes of a fluid dynamical system. Additionally, we find modes with a frequency of positive imaginary part. Measured in a local inertial reference frame, and taking the stretched cosmological horizon limit, these modes grow at most polynomially.

The linearized Einstein equations with sources

On vacuum spacetimes of general dimension, we study the linearized Einstein vacuum equations with a spatially compactly supported and (necessarily) divergence-free source. We prove that the vanishing of appropriate charges of the source, defined in terms of Killing vector fields on the spacetime, is necessary and sufficient for solvability within the class of spatially compactly supported metric perturbations. The proof combines classical results by Moncrief with the solvability theory of the linearized constraint equations with control on supports developed by Corvino–Schoen and Chruściel–Delay.

On boundary conditions for linearised Einstein’s equations

We investigate the properties of a fairly large class of boundary conditions for the linearised Einstein equations in the Riemannian setting, ones which generalise the linearised counterpart of boundary conditions proposed by Anderson. Through the prism of the quest to quantise gravitational waves in curved spacetimes, we study their properties from the point of view of ellipticity, gauge invariance, and the existence of a spectral gap.

Gravitational waves effects in a Lorentz–violating scenario

This paper focuses on how the production and polarization of gravitational waves are affected by spontaneous Lorentz symmetry breaking, which is driven by a self–interacting vector field. Specifically, we examine the impact of a smooth quadratic potential and a non–minimal coupling, discussing the constraints and causality features of the linearized Einstein equation. To analyze the polarization states of a plane wave, we consider a fixed vacuum expectation value (VEV) of the vector field. Remarkably, we verify that a space–like background vector field modifies the polarization plane and introduces a longitudinal degree of freedom. In order to investigate the Lorentz violation effect on the quadrupole formula, we use the modified Green function. Finally, we show that the space–like component of the background field leads to a third–order time derivative of the quadrupole moment, and the bounds for the Lorentz–breaking coefficients are estimated as well.

Holographic Euclidean thermal correlator

In this paper, we compute holographic Euclidean thermal correlators of the stress tensor and U(1) current from the AdS planar black hole. To this end, we set up perturbative boundary value problems for Einstein’s gravity and Maxwell theory in the spirit of Gubser-Klebanov-Polyakov-Witten, with appropriate gauge fixing and regularity boundary conditions at the horizon of the black hole. The linearized Einstein equation and Maxwell equation in the black hole background are related to the Heun equation of degenerate local monodromy. Leveraging the connection relation of local solutions of the Heun equation, we partly solve the boundary value problem and obtain exact two-point thermal correlators for U(1) current and stress tensor in the scalar and shear channels.

Characteristic gluing with $\Lambda$: 1. Linearised Einstein equations on four-dimensional spacetimes

Characteristic gluing with $\Lambda$: 1. Linearised Einstein equations on four-dimensional spacetimes

On the dynamics of a test particle in the field of the de Broglie gravitational waves

The de Broglie gravitational waves are a solution of the linearized Einstein equations with properties that are absent in the standard gravity waves because they are also longitudinal waves and obey a dispersion relation that leads to an effective mass. Furthermore, they represent a classical realization of a form of dynamics, proposed for quantum particles by de Broglie in 1927. In this paper, we discuss the dynamics of a massive particle in presence of the de Broglie gravity wave. We will compute the analytical expression for the linear and angular momentum of the corpuscle. As an application, we will consider the case in which these oscillations of spacetime are associated to an electron travelling with a velocity equal to [Formula: see text] of the light speed and we will estimate the order of magnitude of the quantities involved. We will show that a nearby positron oscillates and radiates an energy of 1[Formula: see text]eV in [Formula: see text][Formula: see text]s, an effect that is, in principle, measurable.

Gravity of gluonic fluctuations and the value of the cosmological constant

We analyze the classical linear gravitational effect of idealized pion-like dynamical systems, consisting of light quarks connected by attractive gluonic material with a stress–energy in one or more dimensions. In one orbit of a system of total mass M, quarks of mass expand apart initially with , slow due to the gluonic attraction, reach a maximum size , then recollapse. We solve the linearized Einstein equations and derive the effect on freely falling bodies for two systems: a gluonic bubble model where uniform gluonic stress–energy fills a spherical volume bounded by a 2D surface comprising the quarks’ rest mass, and a gluonic string model where a thin string connects two pointlike quarks. The bubble model is shown to produce a secular mean outward residual velocity of test particles that lie within its orbit. It is shown that the mean gravitational repulsion of bubble-like virtual-pion vacuum fluctuations agrees with the measured value of the cosmological constant, for a bubble with a radius equal to about twice the pion de Broglie length. These results support the view that the gravity of standard QCD vacuum fluctuations is the main source of cosmic acceleration.

Principal Problems of Relativistic Mechanics of Solids

The article deals with a discussion of fundamental problems arising in relativistic mechanics (general relativity) in relation to the determination of stresses generated by gravity in a deformable solid. Three such problems are considered. The first one is associated with the incompleteness of Einstein’s system of equations that includes six mutually independent equations with ten unknown coefficients of the metric tensor. The second one arises when determining the stresses in a solid generated by gravity. For a static problem, three equations of the law of conservation of theory (equilibrium equations) include six unknown stresses, which, unlike Newton’s theory, does not allow determining gravitational stresses. The third problem is related to the reduction of linearized Einstein equations to the equations of Newton’s gravitational theory. Such a reduction turns out to be possible only for empty space and is not valid for a solid body. The noted contradictions can be eliminated by limiting the scope of the theory to a special space, which is Euclidean with respect to spatial coordinates and Riemannian only with respect to time. The discussion is illustrated by a spherically symmetric problem that reduces to ordinary differential equations.

Spectral method for the gravitational perturbations of black holes: Schwarzschild background case

We develop a novel technique through spectral decompositions to study the gravitational perturbations of a black hole, without needing to decouple the linearized field equations into master equations and separate their radial and angular dependence. We first spectrally decompose the metric perturbation in a Legendre and Chebyshev basis for the angular and radial sectors respectively, using input from the asymptotic behavior of the perturbation at spatial infinity and at the black hole event horizon. This spectral decomposition allows us to then transform the linearized Einstein equations (a coupled set of partial differential equations) into a linear matrix equation. By solving the linear matrix equation for its generalized eigenvalues, we can estimate the complex quasinormal frequencies of the fundamental mode and various overtones of the gravitational perturbations simultaneously and to high accuracy. We apply this technique to perturbations of a nonspinning, Schwarzschild black hole in general relativity and find the complex quasinormal frequencies of two fundamental modes and their first two overtones. We demonstrate that the technique is robust and accurate, in the Schwarzschild case leading to relative fractional errors of $\leq 10^{-10} - 10^{-8}$ for the fundamental modes, $\leq 10^{-7} - 10^{-6}$ for their first overtones, $\leq 10^{-7} - 10^{-4}$ for their second overtones. This method can be applied to any black hole spacetime, irrespective of its Petrov type, making the numerical technique extremely powerful in the study of black hole ringdown in and outside general relativity.

Critical phenomena in the collapse of quadrupolar and hexadecapolar gravitational waves

We report on numerical simulations of critical phenomena near the threshold of black hole formation in the collapse of axisymmetric gravitational waves in vacuum. We discuss several new features of our numerical treatment, and then compare results obtained from families of quadrupolar and hexadecapolar initial data. Specifically, we construct (nonlinear) initial data from quadrupolar and hexadecapolar, time-symmetric wavelike solutions to the linearized Einstein equations (often referred to as Teukolsky waves), and evolve these using a shock-avoiding slicing condition. While our degree of fine-tuning to the onset of black-hole formation is rather modest, we identify several features of the threshold solutions formed for the two families. Both threshold solutions appear to display an at least approximate discrete self-similarity with an accumulation event at the center, and the characteristics of the threshold solution for the quadrupolar data are consistent with those found previously by other authors. The hexadecapolar threshold solution appears to be distinct from the quadrupolar one, providing further support to the notion that there is no universal critical solution for the collapse of vacuum gravitational waves.

Effective Lagrangian and stability analysis in warped space

In the warped space model, the inter-brane distance can be stabilized by the Goldberger-Wise mechanism. Of particular importance, the stabilization potential calls for a proper identification of the dynamical degree of freedom. In this paper, we provided a complete calculation of the effective Lagrangian till the quadratic order that is generic for the Randall-Sundrum model and its N-brane (N ≥ 4) extensions. By applying the variation principle to a specific perturbation field, we derived the equations of motion and orthogonal conditions for decoupling the graviton. This approach is demonstrated to be equivalent to the analysis using the linearized Einstein equation. Our derivation clarifies that in the N-brane set up, just one degree of freedom for the radion field is dynamical, with the other modes eliminated by the gauge fixings. Thus we can directly generalize the GW stabilization to the N-brane model in a way similar to the RS1 scenario.

How Inflationary Gravitons Affect the Force of Gravity

We employ an unregulated computation of the graviton self-energy from gravitons on the de Sitter background to infer the renormalized result. This is used to quantum-correct the linearized Einstein equation. We solve this equation for the potentials that represent the gravitational response to a static, point mass. We find large spatial and temporal logarithmic corrections to the Newtonian potential and to the gravitational shift. Although suppressed by a minuscule loop-counting parameter, these corrections cause perturbation theory to break down at large distances and late times. Another interesting fact is that gravitons induce up to three large logarithms, whereas a loop of massless, minimally coupled scalars produces only a single large logarithm. This is in line with corrections to the graviton mode function: a loop of gravitons induces two large logarithms, whereas a scalar loop gives none.

How inflationary gravitons affect gravitational radiation.

We include the single graviton loop contribution to the linearized Einstein equation. Explicit results are obtained for one loop corrections to the propagation of gravitational radiation. Although suppressed by a minuscule loop-counting parameter, these corrections are enhanced by the square of the number of inflationary [Formula: see text]-foldings. One consequence is that perturbation theory breaks down for a very long epoch of primordial inflation. Another consequence is that the one loop correction to the tensor power spectrum might be observable, in the far future, after the full development of 21 cm cosmology. This article is part of the theme issue 'The future of mathematical cosmology, Volume 2'.

Construction of bulk solutions for towers of pole-skipping points

The pole-skipping phenomenon has been proposed as a connection between chaotic properties of black hole geometries and special points where regular solutions of linearized Einstein equations at horizons have extra free parameters. In this work, we pursue the special points in the near-horizon analysis of integer spin-$\ensuremath{\ell}$ fields on the Rindler-AdS black hole. We construct linear combinations of field components to simplify coupled equations of massive fields and investigate towers of the special points along with imaginary Matsubara frequencies $i\ensuremath{\omega}=2\ensuremath{\pi}(n+1\ensuremath{-}\ensuremath{\ell})T$ with a non-negative integer $n$ and the Hawking temperature $T$. We also propose that integrals of spin-$\ensuremath{\ell}$ bulk propagators over horizons of static black holes capture behaviors at the special points, which are generalizations of integrals of graviton propagators for shock wave geometries. Their interpretation is provided in terms of four-point amplitudes with the spin-$\ensuremath{\ell}$ exchange.

A Scattering Theory for Linearised Gravity on the Exterior of the Schwarzschild Black Hole I: The Teukolsky Equations

We construct a scattering theory for the spin pm ,2 Teukolsky equations on the exterior of the Schwarzschild spacetime, as a first step towards developing a scattering theory for the linearised Einstein equations in double null gauge. This is done by exploiting a physical-space version of the Chandrasekhar transformation used by Dafermos et al. in (Acta Math 222(1):1–214, 2019. https://doi.org/10.4310/acta.2019.v222.n1.a1) to prove the linear stability of the Schwarzschild solution. We also address the Teukolsky–Starobinsky correspondence and construct an isomorphism between scattering data for the +,2 and -,2 Teukolsky equations. This will allow us to state an additional mixed scattering statement for a pair of curvature components satisfying the spin +,2 and -,2 Teukolsky equations and connected via the Teukolsky–Starobinsky identities, completely determining the radiating degrees of freedom of solutions to the linearised Einstein equations.