Linearized Einstein Equations Research Articles

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.

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.

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.

Effect of peculiar velocities of inhomogeneities on the shape of gravitational potential in spatially curved universe

We investigate the effect of peculiar velocities of inhomogeneities and the spatial curvature of the universe on the shape of the gravitating potential. To this end, we consider scalar perturbations of the FLRW metric. The gravitational potential satisfies a Helmholtz-type equation which follows from the system of linearized Einstein equations. We obtain analytical solutions of this equation in the cases of open and closed universes, filled with cold dark matter in presence of the cosmological constant. We demonstrate that, first, peculiar velocities significantly affect the screening length of the gravitational interaction and, second, the form of the gravitational potential depends on the sign of the spatial curvature.

Accurate relativistic observables from postprocessing light cone catalogs

We introduce and study a new scheme to construct relativistic observables from post-processing light cone data. This construction is based on a novel approach, LC-Metric, which takes general light cone or snapshot output generated by arbitrary N-body simulations or emulations and solves the linearized Einstein equations to determine the spacetime metric on the light cone. We find that this scheme is able to determine the metric to high precision, and subsequently generate accurate mock cosmological observations sensitive to effects such as post-Born lensing and nonlinear ISW contributions. By comparing to conventional methods in quantifying those general relativistic effects, we show that this scheme is able to accurately construct the lensing convergence signal. We also find the accuracy of this method in quantifying the ISW effects in the highly nonlinear regime outperforms conventional methods by an order of magnitude. This scheme opens a new path for exploring and modeling higher-order and nonlinear general relativistic contributions to cosmological observables, including mock observations of gravitational lensing and the moving lens and Rees-Sciama effects.

Energy–momentum tensor and duality symmetry of linearized gravity in the Fierz formalism

A formulation of linearized gravity in flat background, based on the Fierz tensor as a counterpart of the electromagnetic field strength, is discussed in detail and used to study fundamental properties of the linearized gravitational field. In particular, the linearized Einstein equations are written as first order partial differential equations in terms of the Fierz tensor, in analogy with the first order Maxwell equations. An energy–momentum tensor with favourable properties and exhibiting remarkable similarity to the standard energy–momentum tensor of the electromagnetic field is found for the linearized gravitational field. is quadratic in the Fierz tensor (which is constructed from the first derivatives of the linearized metric), traceless, and satisfies the dominant energy condition in a gauge that contains the transverse traceless (TT) gauge. It is further shown that in suitable gauges, including the TT gauge, linearized gravity in the absence of matter has a duality symmetry that maps the Fierz tensor, which is antisymmetric in its first two indices, into its dual. Conserved currents associated with the gauge and duality symmetries of linearized gravity are also determined. These currents show good analogy with the corresponding currents in electrodynamics.

Exact solution of Kerr black hole perturbations viaCFT2and instanton counting: Greybody factor, quasinormal modes, and Love numbers

We give explicit expressions for the finite frequency greybody factor, quasinormal modes, and Love numbers of Kerr black holes by computing the exact connection coefficients of the radial and angular parts of the Teukolsky equation. This is obtained by solving the connection problem of the confluent Heun equation in terms of the explicit expression of irregular Virasoro conformal blocks as sums over partitions via the Alday, Gaiotto, and Tachikawa correspondence. In the relevant approximation limits our results are in agreement with existing literature. The method we use can be extended to solve the linearized Einstein equation in other interesting gravitational backgrounds.

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.

Glueballs at high temperature within the hard-wall holographic model

In this investigation an holographic description of the deconfined phase transition of scalar and tensor glueballs is presented within the so called hard-wall model. The spectra of these bound states of gluons have been calculated from the linearized Einstein equations for a graviton propagating from a thermal AdS_5 space to an AdS Black-Hole. In this framework, the deconfined phase is reached via a two steps mechanism. We propose that the transition between the AdS thermal sector to the BH is described via a first order phase transition, with discontinuous masses at the critical temperature, which has been determined by Herzog’s method of regulating the free energy densities. Then, the glueball masses diverge with increasing T in the BH phase and thus lead to deconfined states à la Hagedorn.

Chaos and pole-skipping in rotating black holes

We study the connection between many-body quantum chaos and energy dynamics for the holographic theory dual to the Kerr-AdS black hole. In particular, we determine a partial differential equation governing the angular profile of gravitational shock waves that are relevant for the computation of out-of-time ordered correlation functions (OTOCs). Further we show that this shock wave profile is directly related to the behaviour of energy fluctuations in the boundary theory. In particular, we demonstrate using the Teukolsky formalism that at complex frequency ω∗ = i2πT there exists an extra ingoing solution to the linearised Einstein equations whenever the angular profile of metric perturbations near the horizon satisfies this shock wave equation. As a result, for metric perturbations with such temporal and angular profiles we find that the energy density response of the boundary theory exhibit the signatures of “pole-skipping” — namely, it is undefined, but exhibits a collective mode upon a parametrically small deformation of the profile. Additionally, we provide an explicit computation of the OTOC in the equatorial plane for slowly rotating large black holes, and show that its form can be used to obtain constraints on the dispersion relations of collective modes in the dual CFT.

Proposal of a gauge-invariant treatment of l = 0, 1-mode perturbations on Schwarzschild background spacetime

A gauge-invariant treatment of the monopole (l = 0) and dipole (l = 1) modes in linear perturbations of the Schwarzschild background spacetime is proposed. Through this gauge-invariant treatment, we derived the solutions to the linearized Einstein equation for these modes with a generic matter field. In the vacuum case, these solutions include the Kerr parameter perturbations in the l = 1 odd modes and the additional mass parameter perturbations of the Schwarzschild mass in the l = 0 even modes. The linearized version of Birkhoff’s theorem is also confirmed in a gauge-invariant manner. In this sense, our proposal is reasonable.

On the geometry of Petrov type II spacetimes

In general, geometries of Petrov type II do not admit symmetries in terms of Killing vectors or spinors. We introduce a weaker form of Killing equations which do admit solutions. In particular, there is an analog of the Penrose–Walker Killing spinor. Some of its properties, including associated conservation laws, are discussed. Perturbations of Petrov type II Einstein geometries in terms of a complex scalar Debye potential yield complex solutions to the linearized Einstein equations. The complex linearized Weyl tensor is shown to be half Petrov type N. The remaining curvature component on the algebraically special side is reduced to a first order differential operator acting on the potential.

The first-order symmetry operator on gravitational perturbations in the 5D Myers–Perry spacetime with equal angular momenta

Abstract It has been revealed that the first-order symmetry operator for the linearized Einstein equation on a vacuum spacetime can be constructed from a Killing–Yano 3-form. This might be used to construct all or part of the solutions to the field equation. In this paper, we perform a mode decomposition of a metric perturbation on the Schwarzschild spacetime and the Myers–Perry spacetime with equal angular momenta in 5 dimensions, and investigate the action of the symmetry operator on specific modes concretely. We show that, on such spacetimes, there is no transition between the modes of a metric perturbation by the action of the symmetry operator, and it ends up being the linear combination of the infinitesimal transformations of isometry.