arXiv subjects

Jibril Ben Achour

Publications and source records attributed to Jibril Ben Achour.

At least 37 records · Page 2Linked to original sources

arXiv2021

Proper time reparametrization in cosmology: Mobius symmetry and Kodama charges

It has been noticed that for a large class of cosmological models, the gauge fixing of the time-reparametrization invariance does not completely fix the clock. Instead, the system enjoys a surprising residual Noether symmetry under a Möbius reparametrization of the proper time, which maps gauge-inequivalent solutions to the Friedman equations onto each other. In this work, we provide a unified treatment of this hidden conformal symmetry and its realization in the homogeneous and isotropic sector of the Einstein-Scalar-$Λ$ system. We consider the flat Friedmann-Robertson-Walker (FRW) model, the (A)dS cosmology and provide a first treatment of the model with spatial constant curvature. We derive the general condition relating the choice of proper time and the conformal weight of the scale factor, and give a detailed analysis of the conserved Noether charges generating this physical symmetry. Our approach allows us to identify new realizations of this symmetry while recovering previous results in a unified manner. We also present the general mapping onto the conformal particle and discuss the solution-generating nature of the transformations beyond the Möbius symmetry. Finally, we show that, at least in a restricted context, this hidden conformal symmetry is intimately related to the Kodama charges of spherically symmetric gravity. This new connection suggests that the Möbius invariance of cosmology is only the corner of a larger symmetry structure which could be relevant beyond cosmological models.

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arXiv2020

Bouncing compact objects. Part II: Effective theory of a pulsating Planck star

This article presents an effective quantum extension of the seminal Oppenheimer-Snyder (OS) collapse in which the singularity resolution is modeled using the effective dynamics of the spatially closed loop quantum cosmology. Imposing the minimal junction conditions, namely the Israel-Darmois conditions, we glue this bouncing LQC geometry to the classical vacuum exterior Schwarzschild geometry across a time-like thin-shell. Consistency of the construction leads to several major deviations from the classical OS collapse model. Firstly, no trapped region can form and the bounce occurs always above, or at most at the Schwarzschild radius. Secondly, the bouncing star discussed here admits an IR cut-off, additionally to the UV cut-off and corresponds therefore to a pulsating compact object. Thirdly, the scale at which quantum gravity effects become non-negligible is encoded in the ratio between the UV cut-off of the quantum theory and the IR cut-off, which in turn, encodes the minimal energy density $ρ_{\text{min}}$ of the star prior to collapse. This energy density is no more fixed by the mass and maximal radius as in the classical OS model, but is now a free parameter of the model. In the end, while the present model cannot describe a black-to-white hole bounce as initially suggested by the Planck star model, it provides a concrete realization of a pulsating compact object based on LQC techniques. Consistency of the model shows that its regime of applicability is restricted to Planckian relics while macroscopic stellar objects are excluded. This first minimal construction should serve as a platform for further investigations in order to explore the physics of bouncing compact objects within the framework of loop quantum cosmology.

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arXiv2020

On Rotating Black Holes in DHOST Theories

Using the disformal solution-generating method, we construct new axisymmetric solutions in Degenerate Higher Order Scalar Tensor (DHOST) theories. The method consists in first considering a "seed" known solution in DHOST theories and then performing a disformal transformation of the metric to obtain a new solution. In vacuum, the two solutions are equivalent but they become physically inequivalent when one considers coupling to matter. In that way, we "disform" the stealth Kerr black hole solution and we obtain a first analytic rotating non-stealth solution in DHOST theories, while the associated scalar field is time-dependent with a constant kinetic density. The new solution is characterized by three parameters: the mass, the spin and the disformal parameter which encodes the deviation with respect to the Kerr geometry. We explore some geometrical properties of the novel disformed Kerr geometry which is no more Ricci flat, has the same singularity as the Kerr metric, admits an ergoregion, and is asymptotically flat. Moreover, the hidden symmetry of the Kerr solution is broken, providing an example of a non-circular geometry in a higher order theory of gravity. We also discuss geodesic motions and compute its (disformed) null directions which are interesting tools to understand the causal structure of the geometry. In addition, to illustrate again the potentiality of the disformal solution-generating method, we present another axisymmetric solution for DHOST theories obtained from a disformal transformation of the generalized Kerr solution of Einstein-Scalar gravity.

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arXiv2020

The Cosmological Constant from Conformal Transformations: Möbius Invariance and Schwarzian Action

The homogeneous Friedman-Lema\^ıtre-Robertson-Walker (FLRW) cosmology of a free scalar field with vanishing cosmological constant was recently shown to be invariant under the one-dimensional conformal group $\textrm{SL}(2,\mathbb{R})$ acting as M{ö}bius transformations on the proper time. Here we generalize this analysis to arbitrary transformations of the proper time, $τ\mapsto \tildeτ=f(τ)$, which are not to be confused with reparametrizations of the time coordinate. First, we show that the FLRW cosmology with a non-vanishing cosmological constant $Λ\ne 0$ is also invariant under a $\textrm{SL}(2,\mathbb{R})$ group of conformal transformations. The associated conformal Noether charges form a $\mathfrak{sl}(2,\mathbb{R})$ Lie algebra which encodes the cosmic evolution. Second, we show that a cosmological constant can be generated from the $Λ=0$ case through particular conformal transformations, realizing a compactification or de-compactification of the proper time depending on the sign of $Λ$. Finally, we propose an extended FLRW cosmological action invariant under the full group $\textrm{Diff}({\cal S}^1)$ of conformal transformations on the proper time, by promoting the cosmological constant to a gauge field for conformal transformations or by modifying the scalar field action to a Schwarzian action. Such a conformally-invariant cosmology leads to a renewed problem of time and to the necessity to re-think inflation in purely time-deparameterized terms.

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arXiv2020

Towards consistent black-to-white hole bounces from matter collapse

This article presents a new model-independent constraint for bouncing black hole geometries. Using the thin shell formalism, this constraint sets a bound on the minimal allowed radius of the time-like surface of the collapsing star at the bounce. It follows from this bound that the shell always bounces in an untrapped region or precisely on a trapping horizon. This constraint is of purely kinematical origin as it descends from the continuity of the metric between the geometries describing the interior and exterior of the collapsing object so that it is completely model-independent. The second part of this work investigates the conditions under which an effective extension of the Oppenheimer-Snyder collapse can describe consistent bouncing black hole solutions. We show that on top of the previous kinematical constraint, an additional dynamical condition has to be satisfied in order for the model to admit well-defined black-to-white hole solutions. As expected, this second condition turns out to be model-dependent. The resulting class of models describing bouncing compact objects are characterized by three parameters for the star (mass, initial radius and density) and two quantum parameters descending from the UV-completion of the exterior and interior geometries. The solution space contains (1) bouncing stars and (2) bouncing black holes and (3) a new class of astrophysical objects which alternate between a bouncing star and a bouncing black hole. Finally, we provide an explicit construction of a black-to-white hole bounce using the techniques of spatially closed LQC and discuss the novel properties induced by the presence of the inner horizon in this new framework. This generic framework lays down the foundation for interesting phenomenological investigations concerning the astrophysical properties of these objects, as well as a novel platform for further developments.

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arXiv2020

Bouncing compact objects I: Quantum extension of the Oppenheimer-Snyder collapse

This article proposes a generalization of the Oppenheimer-Snyder model which describes a bouncing compact object. The corrections responsible for the bounce are parameterized in a general way so as to remain agnostic about the specific mechanism of singularity resolution at play. It thus develops an effective theory based on a thin shell approach, inferring generic properties of such a UV complete gravitational collapse. The main result comes in the form of a strong constraint applicable to general UV models : if the dynamics of the collapsing star exhibits a bounce, it always occurs below, or at most at the energy threshold of horizon formation, so that only an instantaneous trapping horizon may be formed while a trapped region never forms. This conclusion relies solely on i) the assumption of continuity of the induced metric across the time-like surface of the star and ii) the assumption of a classical Schwarzschild geometry describing the (vacuum) exterior of the star. In particular, it is completely independent of the choice of corrections inside the star which leads to singularity-resolution. The present model provides thus a general framework to discuss bouncing compact objects, for which the interior geometry is modeled either by a classical or a quantum bounce. In the later case, our no-go result regarding the formation of trapped region suggests that additional structure, such as the formation of an inner horizon, is needed to build consistent models of matter collapse describing black-to-white hole bounces. Indeed, such additional structure is needed to keep quantum gravity effects confined to the high curvature regime, in the deep interior region, providing thus a new challenge for current constructions of quantum black-to-white hole bounce models.

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arXiv2020

The Cosmological Spinor

We build upon previous investigation of the one-dimensional conformal symmetry of the Friedman-Lema\^ itre-Robertson-Walker (FLRW) cosmology of a free scalar field and make it explicit through a reformulation of the theory at the classical level in terms of a manifestly $\textrm{SL}(2,\mathbb{R})$-invariant action principle. The new tool is a canonical transformation of the cosmological phase space to write it in terms of a spinor, i.e. a pair of complex variables that transform under the fundamental representation of $\textrm{SU}(1,1)\sim\textrm{SL}(2,\mathbb{R})$. The resulting FLRW Hamiltonian constraint is simply quadratic in the spinor and FLRW cosmology is written as a Schrödinger-like action principle. Conformal transformations can then be written as proper-time dependent $\textrm{SL}(2,\mathbb{R})$ transformations. We conclude with possible generalizations of FLRW to arbitrary quadratic Hamiltonian and discuss the interpretation of the spinor as a gravitationally-dressed matter field or matter-dressed geometry observable.

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arXiv2020

Deformed General Relativity and Quantum Black Holes Interior

Effective models of black holes interior have led to several proposals for regular black holes. In the so-called polymer models, based on effective deformations of the phase space of spherically symmetric general relativity in vacuum, one considers a deformed Hamiltonian constraint while keeping a non-deformed vectorial constraint. In this article, we revisit and study further the question of covariance in these deformed gravity models. In particular, we propose a Lagrangian formulation for these deformed gravity models where polymer-like deformations are introduced at the level of the full theory prior to the symmetry reduction and prior to the Legendre transformation. This enables us to test whether the concept of deformed covariance found in spherically symmetric vacuum gravity can be extended to the full theory, and we show that, in the large class of models we are considering, the deformed covariance can not be realized beyond spherical symmetry in the sense that the only deformed theory which leads to a closed constraints algebra is general relativity. Hence, we focus on the spherically symmetric sector, where there exist non-trivial deformed but closed constraints algebras. We investigate the possibility to deform the vectorial constraint as well and we prove that non-trivial deformations of the vectorial constraint with the condition that the constraints algebra remains closed do not exist. Then, we compute the most general deformed Hamiltonian constraint which admits a closed constraints algebra and thus leads to a well-defined effective theory associated with a notion of deformed covariance. Finally, we study homogeneous solutions of these effective theories and, remarkably, we solve explicitly and in full generality the corresponding modified Einstein equations, even for the effective theories which do not satisfy the closeness condition.

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arXiv2020

Conformal structure of FLRW Cosmology: Spinorial representation and the so(3,2) algebra of observables

It was recently shown that the homogeneous and isotropic cosmology of a massless scalar field coupled to general relativity exhibits a new hidden conformal invariance under Mobius transformation of the proper time, additionally to the invariance under time-reparamterization. The resulting Noether charges form a $sl(2,\mathbb{R})$ Lie algebra, which encapsulates the whole kinematics and dynamics of the geometry. This allows to map FLRW cosmology onto conformal mechanics and formulate quantum cosmology in $\text{CFT}_1$ terms. Here, we show that this conformal structure is embedded in a larger $so(3,2)$ algebra of observables, which allows to present all the Dirac observables for the whole gravity plus matter sectors in a unified picture. Not only this allows one to quantize the system and its whole algebra of observables as a single irreducible representation of $so(3,2)$, but this also gives access to a scalar field operator $\hatϕ$ opening the door to the inclusion of non-trivial potentials for the scalar field. As such, this extended conformal structure might allow to perform a group quantization of inflationary cosmological backgrounds.

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arXiv2020

Hairy black holes in DHOST theories: Exploring disformal transformation as a solution-generating method

Solutions-generating methods based on field redefinitions, such as conformal mapping, play an important role in investigating exact solutions in modified gravity. In this work, we explore the possibility to use disformal field redefinitions to investigate new regions of the solution space of DHOST theories. The crucial ingredient is to find suitable seed solutions to generate new exact ones for DHOST theories. We first consider a seed solution of the Einstein-Scalar system describing a naked singularity. Under suitable assumptions, we derive a no-go result showing that no black hole solution can be constructed from such a seed GR solution. Then, taking into account the stability of each degeneracy classes of quadratic DHOST theories under a general disformal mapping, we consider two kinds of known black hole solutions as seed configurations: the Schwarzschild stealth solution and the non-stealth Reissner-Nordstrom like solution. Restricting our considerations to invertible disformal transformations, we show that building new hairy black hole solutions from the stealth solution associated to a constant kinetic term is also quite constrained. We obtain new solutions which either are stealth or describe asymptotically locally flat black holes with a deficit solid angle. However, starting from the non-stealth seed solution associated to a non constant kinetic term as well as a time-dependent scalar profile, we show that a disformal transformation can introduce rather general modifications of the exterior geometry. Finally, we consider the construction of minimally modified hairy black hole solutions using a small disformal transformation. Further applications of this solution generating method should allow to provide new hairy rotating black hole solutions beyond the stealth sector useful for phenomenological investigations of compact objects beyond GR.

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arXiv2019

Cosmology as a CFT$_1$

We show that the simplest FLRW cosmological system consisting in the homogeneous and isotropic massless Einstein-Scalar system enjoys a hidden conformal symmetry under the 1D conformal group $SL(2,\mathbb{R})$ acting as Mobius transformations in proper time. This invariance is made explicit through the mapping of FLRW cosmology onto conformal mechanics. On the one hand, we identify the corresponding conformal Noether charges, as combinations of the Hamiltonian scalar constraint, the extrinsic curvature and the 3D volume, which form a closed $sl(2,\mathbb{R})$ Lie algebra. On the other hand, this approach allows to write FLRW cosmology in terms of a AdS${}_{2}$ phase space and a Schwarzian action. Preserving this conformal structure at the quantum level fixes the ordering ambiguities in the Wheeler-de Witt quantization and allows to formulate FLRW quantum cosmology as a CFT${}_{1}$. We show that the CFT two-points correlator is realized as the overlap of the evolution in proper time of cosmological coherent wave-packets. In particular, the two-points function is built from a vacuum state which, although not conformally invariant, coincides with the cosmological vacuum annihilated by the scalar constraint. These results suggest new perspectives in classical and quantum cosmology, among which the possibility to apply the conformal bootstrap program to quantize cosmological backgrounds.

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arXiv2019

Protected $SL(2,\mathbb{R})$ Symmetry in Quantum Cosmology

The polymer quantization of cosmological backgrounds provides an alternative path to the original Wheeler-de Witt (WdW) quantum cosmology, based on a different representation the commutation relations of the canonical variables. This polymer representation allows to capture the lattice like structure of the quantum geometry and leads to a radically different quantum cosmology compared to the WdW construction. This new quantization scheme has attracted considerable attention due to the singularity resolution it allows in a wide class of symmetry reduced gravitational systems, in particular where the WdW scheme fails. However, as any canonical quantization scheme, ambiguities in the construction of the quantum theory, being regularization or factor-ordering ones, can drastically modify the resulting quantum dynamics. In this work, we propose a new criteria to restrict the quantization ambiguities in the simplest model of polymer quantum cosmology, for homogeneous and isotropic General Relativity minimally coupled to a massless scalar field. This new criteria is based on an underlying $\mathfrak{sl}(2,\mathbb{R})$ structure present in the phase space of this simple cosmological model. By preserving the symmetry of this cosmological system under this 1d conformal group, we derive a new regularization of the phase space. We perform both its polymer quantization and a quantization scheme directly providing a representation of the SL$(2,\mathbb{R})$ group action. The resulting quantum cosmology can be viewed as a lattice-like quantum mechanics with an SL$(2,\mathbb{R})$ invariance. This provides a new version of Loop Quantum Cosmology consistent with the conformal symmetry. This alternative construction opens new directions, among which a possible mapping with the conformal quantum mechanics as well as with recent matrix or tensor models constructions for quantum cosmological space-time.

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arXiv2019

Polymer Quantum Cosmology: Lifting quantization ambiguities using a $SL(2,\mathbb{R})$ conformal symmetry

In this letter, we stress that the simplest cosmological model consisting in a massless scalar field minimally coupled to homogeneous and isotropic gravity has an in-built $\SL(2,\mathbb{R})$ symmetry. Protecting this symmetry naturally provides an efficient way to constrain the quantization of this cosmological system whatever the quantization scheme and allows in particular to fix the quantization ambiguities arising in the canonical quantization program. Applying this method to the loop quantization of the FLRW cosmology leads to a new loop quantum cosmology model which preserves the $\SL(2,\mathbb{R})$ symmetry of the classical system. This new polymer regularization consistent with the conformal symmetry can be derived as a non-linear canonical transformation of the classical FLRW phase space, which maps the classical singular dynamics into a regular effective bouncing dynamics. This improved regularization preserves the scaling properties of the volume and Hamiltonian constraint. 3d scale transformations, generated by the dilatation operator, are realized as unitary transformations despite the minimal length scale hardcoded in the theory. Finally, we point out that the resulting cosmological dynamics exhibits an interesting duality between short and long distances, reminiscent of the T-duality in string theory, with the near-singularity regime dual to the semi-classical regime at large volume. The technical details of the construction of this model are presented in a longer companion paper \cite{BenAchour:2019ywl}.

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arXiv2019

Polymer Schwarzschild black hole: An effective metric

We consider the modified Einstein equations obtained in the framework of effective spherically symmetric polymer models inspired by Loop Quantum Gravity. When one takes into account the anomaly free point-wise holonomy quantum corrections, the modification of Einstein equations is parametrized by a function $f(x)$ of one phase space variable. We solve explicitly these equations for a static interior black hole geometry and find the effective metric describing the trapped region, inside the black hole, for any $f(x)$. This general resolution allows to take into account a standard ambiguity inherent to the polymer regularization: namely the choice of the spin $j$ labelling the SU$(2)$-representation of the holonomy corrections. When $j=1/2$, the function $f(x)$ is the usual sine function used in the polymer litterature. For this simple case, the effective exterior metric remains the classical Schwarzschild's one but acquires modifications inside the hole. The interior metric describes a regular trapped region and presents strong similarities with the Reissner-Nordström metric, with a new inner horizon generated by quantum effects. We discuss the gluing of our interior solution to the exterior Schwarzschild metric and the challenge to extend the solution outside the trapped region due to covariance requirement. By starting from the anomaly free polymer regularization for inhomogeneous spherically symmetric geometry, and then reducing to the homogeneous interior problem, we provide an alternative treatment to existing polymer interior black hole models which focus directly on the interior geometry, ignoring covariance issue when introducing the polymer regularization.

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arXiv2019

Hairy Schwarzschild-(A)dS black hole solutions in DHOST theories beyond shift symmetry

We derive the general conditions for a large family of shift symmetry breaking degenerate higher order scalar-tensor (DHOST) theories to admit stealth black hole solutions. Such black hole configurations correspond to vacuum solutions of General Relativity and admit a scalar hair which does not gravitate, revealing itself only at the perturbative level. We focus our investigation on hairy Schwarzschild-(A)dS or pure Schwarzschild solutions, dressed with a linear time-dependent scalar hair, and assuming a constant kinetic term. We also discuss subclasses of this family which satisfy the observational constraint $c_{\text{grav}}=c_{\text{light}}$, as well as the recent constraint ensuring the absence of graviton decay. We provide at the end concrete examples of DHOST lagrangians satisfying our conditions. This work provides a first analysis of exact black hole solutions in shift symmetry breaking DHOST theories beyond Horndeski.

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arXiv2018

Non-singular black holes and the Limiting Curvature Mechanism: A Hamiltonian perspective

We revisit the non-singular black hole solution in (extended) mimetic gravity with a limiting curvature from a Hamiltonian point of view. We introduce a parameterization of the phase space which allows us to describe fully the Hamiltonian structure of the theory. We write down the equations of motion that we solve in the regime deep inside the black hole, and we recover that the black hole has no singularity, due to the limiting curvature mechanism. Then, we study the relation between such black holes and effective polymer black holes which have been introduced in the context of loop quantum gravity. As expected, contrary to what happens in the cosmological sector, mimetic gravity with a limiting curvature fails to reproduce the usual effective dynamics of spherically symmetric loop quantum gravity which are generically not covariant. Nonetheless, we exhibit a theory in the class of extended mimetic gravity whose dynamics reproduces the general shape of the effective corrections of spherically symmetric polymer models, but in an undeformed covariant manner. These covariant effective corrections are found to be always metric dependent, i.e. within the $\barμ$-scheme, underlying the importance of this ingredient for inhomogeneous polymer models. In that respect, extended mimetic gravity can be viewed as an effective covariant theory which naturally implements a covariant notion of point wise holonomy-like corrections. The difference between the mimetic and polymer Hamiltonian formulations provides us with a guide to understand the deformation of covariance in inhomogeneous polymer models.

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arXiv2018

Covariance in self dual inhomogeneous models of effective quantum geometry: Spherical symmetry and Gowdy systems

When applying the techniques of Loop Quantum Gravity (LQG) to symmetry-reduced gravitational systems, one first regularizes the scalar constraint using holonomy corrections, prior to quantization. In inhomogeneous system, where a residual spatial diffeomorphism symmetry survives, such modification of the gauge generator generating time reparametrization can potentially lead to deformations or anomalies in the modified algebra of first class constraints. When working with self-dual variables, it has already been shown that, for spherically symmetric geometry coupled to a scalar field, the holonomy-modified constraints do not generate any modifications to general covariance, as one faces in the real variables formula- tion, and can thus accommodate local degrees of freedom in such inhomogeneous models. In this paper, we extend this result to Gowdy cosmologies in the self-dual Ashtekar formulation. Furthermore, we show that the introduction of a $\barμ$-scheme in midisuperspace models, as is required in the `improved dynamics' of LQG, is possible in the self-dual formalism while being out of reach in the current effective models suing real-valued Ashtekar-Barbero variables. Our results indicate the advantages of using the self-dual variables to obtain a covariant loop regularization prior to quantization in inhomogeneous symmetry reduced polymer models, implementing additionally the crucial $\barμ$-scheme, and thus a consistent semiclassical limit.

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arXiv2017

The Thiemann Complexifier and the CVH algebra for Classical and Quantum FLRW Cosmology

In the context of Loop Quantum Gravity (LQG), we study the fate of Thiemann complexifier in homogeneous and isotropic FRW cosmology. The complexifier is the dilatation operator acting on the canonical phase space for gravity and generates the canonical transformations shifting the Barbero-Immirzi parameter. We focus on the closed algebra consisting in the complexifier, the 3d volume and the Hamiltonian constraint, which we call the CVH algebra. In standard cosmology, for gravity coupled to a scalar field, the CVH algebra is identified as a su(1,1) Lie algebra, with the Hamiltonian as a null generator, the complexifier as a boost and the su(1,1) Casimir given by the matter density. The loop gravity cosmology approach introduces a regularization length scale $λ$ and regularizes the gravitational Hamiltonian in terms of SU(2) holonomies. We show that this regularization is compatible with the CVH algebra, if we suitably regularize the complexifier and inverse volume factor. The regularized complexifier generates a generalized version of the Barbero's canonical transformation which reduces to the classical one when $λ\rightarrow 0$. This structure allows for the exact integration of the actions of the Hamiltonian constraints and the complexifier. This straightforwardly extends to the quantum level: the cosmological evolution is described in terms of SU(1,1) coherent states and the regularized complexifier generates unitary transformations. The Barbero-Immirzi parameter is to be distinguished from the regularization scale $λ$, it can be rescaled unitarily and the Immirzi ambiguity ultimately disappears from the physical predictions of the theory. Finally, we show that the complexifier becomes the effective Hamiltonian when deparametrizing the dynamics using the scalar field as a clock, thus underlining the deep relation between cosmological evolution and scale

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