Covariant Loop Quantum Gravity Research Articles

Characteristic time scales for the geometry transition of a black hole to a white hole from spinfoams

Abstract Quantum fluctuations of the metric may provide a decay mechanism for black holes through a transition to a white hole geometry. 
Previous studies formulated Loop Quantum Gravity amplitudes with a view to describe this process. We identify two timescales to be extracted which we call the crossing time and the lifetime and complete a calculation that gives explicit estimates using the asymptotics of the EPRL model. The crossing time is found to scale linearly in the mass, in agreement with previous results by Ambrus and H'aj'i\v{c}ek and more recent results by Barcel'o, Carballo--Rubio and Garay. The lifetime is found to depend instead on the spread of the quantum state, and thus its dependence on the mass can take a large range of values. This indicates that the truncation/approximation used here is not appropriate to estimate this observable with any certainty. The simplest choice of a balanced semiclassical state is shown to yield an exponential scaling of the lifetime in the mass squared. Our analysis only considers 2-complexes without bulk faces, a significant limitation. In particular it is not clear how our estimates will be affected under refinements. This work should be understood as a step towards a fuller calculation in the context of covariant Loop Quantum Gravity.

Time and Quantum Clocks: A Review of Recent Developments

In this review we present the problem of time in quantum physics, including a short history of the problem and the known objections about considering time as a quantum observable. The need to deal with time as an observable is elaborated through some unresolved problems. The lack of a consistent theory of time is currently hindering the formulation of a full-fledged theory of quantum gravity. It is argued that the proposal set forth by several authors of considering an intrinsic measurement of quantum time, besides having the conventional external time, is compelling. Recently several suggestions have been put forward to revive the proposal of Page and Wootters (1983), elaborating and resolving some of the main ambiguities of the original proposal and opening new scope for understanding its content. The approach followed in these new contributions exposes the need to go beyond the limitations enforced by the conventional approach of quantum physics. The attitude of covariant loop quantum gravity, in which it is called to completely ignore time, is also discussed. This review could be a step forward in an endeavour to reform our outlook of the unification of the theory of relativity and quantum physics by furnishing the conceptual ground needed for this goal. Intentionally, some technical details are avoided since we aim to present the approaches to resolve the problem in a simple way with the clearest possible outlook. These can be looked up in the original references provided.

Numerical analysis of the self-energy in covariant loop quantum gravity

We study numerically the first order radiative corrections to the self-energy, in covariant loop quantum gravity. We employ the recently developed 'sl2cfoam-next' spinfoam amplitudes library, and some original numerical methods. We analyze the scaling of the divergence with the infrared cutoff, for which previous analytical estimates provided widely different lower and upper bounds. Our findings suggest that the divergence is approximately linear in the cutoff. We also investigate the role of the Barbero-Immirzi parameter in the asymptotic behavior, the dependence of the scaling on some boundary data and the expectation values of boundary operators.

The accidental flatness constraint does not mean a wrong classical limit

We shed some light on the reason why the accidental flatness constraint appears in certain limits of the amplitudes of covariant loop quantum gravity. We show why this constraint is harmless, by displaying how analogous accidental constraints appear in transition amplitudes of simple systems, when certain limits are considered.

End of a black hole’s evaporation. II.

We derive an explicit expression for the transition amplitude from black to white hole horizon at the end of Hawking evaporation using covariant loop quantum gravity.

Quantum fate of timelike naked singularity with scalar hair

We study the quantum fate of a naked curvature singularity sourced by a scalar field via several methods and compare the results obtained. The first method relies on relativistic quantum mechanics on a fixed background employing the Klein–Gordon and the Dirac equations for a static spacetime. We show that both the Klein–Gordon and the Dirac particles feel this singularity therefore this method does not provide its resolution. For comparison, we subsequently employ methods for quantizing the geometry itself. We selected the canonical quantization via conditional symmetries and as a last approach we use a maximal acceleration derivation in the covariant loop quantum gravity. In both of these approaches the singularity is resolved at the quantum level. We discuss these conflicting results bearing in mind that quantum particles probe classical geometry in the first approach while the last two methods quantize the geometry itself.

Emergent four-dimensional linearized gravity from a spin foam model

Spin Foam Models (SFMs) are covariant formulations of Loop Quantum Gravity (LQG) in 4 dimensions. This work studies the perturbations of SFMs on a flat background. It demonstrates for the first time that smooth curved spacetime geometries satisfying Einstein equation can emerge from discrete SFMs under an appropriate low energy limit, which corresponds to a semiclassical continuum limit of SFMs. In particular, we show that the low energy excitations of SFMs on a flat background give all smooth solutions of linearized Einstein equations (spin-2 gravitons). This indicates that at the linearized level, classical Einstein gravity is indeed the low energy effective theory from SFMs. Thus our result heightens the confidence that covariant LQG is a consistent theory of quantum gravity. As a key technical tool, a regularization/deformation of the SFM is employed in the derivation. The deformation parameter $\delta$ becomes a coupling constant of a higher curvature correction term to Einstein gravity from SFM.

Simplicity constraints: A 3D toy model for loop quantum gravity

In Loop Quantum Gravity, tremendous progress has been made using the Ashtekar-Barbero variables. These variables, defined in a gauge-fixing of the theory, correspond to a parametrization of the solutions of the so-called simplicity constraints. Their geometrical interpretation is however unsatisfactory as they do not constitute a space-time connection. It would be possible to resolve this point by using a full Lorentz connection or, equivalently, by using the self-dual Ashtekar variables. This leads however to simplicity constraints or reality conditions which are notoriously difficult to implement in the quantum theory. We explore in this paper the possibility of imposing such constraints at the quantum level in the context of canonical quantization. To do so, we define a simpler model, in 3d, with similar constraints by extending the phase space to include an independent vielbein. We define the classical model and show that a precise quantum theory by gauge-unfixing can be defined out of it, completely equivalent to the standard 3d euclidean quantum gravity. We discuss possible future explorations around this model as it could help as a stepping stone to define full-fledged covariant Loop Quantum Gravity.

Einstein equation from covariant loop quantum gravity in semiclassical continuum limit

In this paper we explain how 4-dimensional general relativity and in particular, the Einstein equation, emerge from the spinfoam amplitude in loop quantum gravity. We propose a new limit which couples both the semiclassical limit and continuum limit of spinfoam amplitudes. The continuum Einstein equation emerges in this limit. Solutions of Einstein equation can be approached by dominant configurations in spinfoam amplitudes. A running scale is naturally associated to the sequence of refined triangulations. The continuum limit corresponds to the infrared limit of the running scale. An important ingredient in the derivation is a regularization for the sum over spins, which is necessary for the semiclassical continuum limit. We also explain in this paper the role played by the so-called flatness in spinfoam formulation, and how to take advantage of it.

Spinfoams near a classical curvature singularity

We apply the technique of spinfoam to study the space-time which, classically, contains a curvature singularity. We derive from the full covariant Loop Quantum Gravity (LQG) that the region near curvature singularity has to be of strong quantum gravity effect. We show that the spinfoam configuration describing the near-singularity region has to be of small spins $j$, in order that its contribution to the full spinfoam amplitude is nontrivial. The spinfoams in low and high curvature regions of the space-time may be viewed as in two different {phases} of covariant LQG. There should be a phase transition as the space-time described by spinfoam becomes more and more curved. A candidate of order parameter is proposed for understanding the phase transition. Moreover, we also analyze the spin-spin correlation function of spinfoam, and show the correlation is of long-range in the low curvature phase. This work is a first step toward understanding the physics of black hole and early universe from the full covariant LQG theory.

Holographic bound in covariant loop quantum gravity

We investigate puncture statistics based on the covariant area spectrum in loop quantum gravity. First, we consider Maxwell-Boltzmann statistics with a Gibbs factor for punctures. We establish formulae which relate physical quantities such as horizon area to the parameter characterizing holographic degrees of freedom. We also perform numerical calculations and obtain consistency with these formulae. These results tell us that the holographic bound is satisfied in the large area limit and correction term of the entropy-area law can be proportional to the logarithm of the horizon area. Second, we also consider Bose-Einstein statistics and show that the above formulae are also useful in this case. By applying the formulae, we can understand intrinsic features of Bose-Einstein condensate which corresponds to the case when the horizon area almost consists of punctures in the ground state. When this phenomena occurs, the area is approximately constant against the parameter characterizing the temperature. When this phenomena is broken, the area shows rapid increase which suggests the phase transition from quantum to classical area.

Encoding Curved Tetrahedra in Face Holonomies: Phase Space of Shapes from Group-Valued Moment Maps

We present a generalization of Minkowski's classic theorem on the reconstruction of tetrahedra from algebraic data to homogeneously curved spaces. Euclidean notions such as the normal vector to a face are replaced by Levi-Civita holonomies around each of the tetrahedron's faces. This allows the reconstruction of both spherical and hyperbolic tetrahedra within a unified framework. A new type of hyperbolic simplex is introduced in order for all the sectors encoded in the algebraic data to be covered. Generalizing the phase space of shapes associated to flat tetrahedra leads to group valued moment maps and quasi-Poisson spaces. These discrete geometries provide a natural arena for considering the quantization of gravity including a cosmological constant. A concrete realization of this is provided by the relation with the spin-network states of loop quantum gravity. This work therefore provides a bottom-up justification for the emergence of deformed gauge symmetries and quantum groups in 3+1 dimensional covariant loop quantum gravity in the presence of a cosmological constant.

Analog of the Peter-Weyl expansion for Lorentz group

The expansion of a square integrable function on SL(2, C) into the sum of the principal series matrix coefficients with the specially selected representation parameters was recently used in the Loop Quantum Gravity [C. Rovelli and F. Vidotto, Covariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory (Cambridge University Press, Cambridge, 2014) and C. Rovelli, Classical Quantum Gravity 28(11), 114005 (2011)]. In this paper, we prove that the sum used originally in the Loop Quantum Gravity: ∑j=0∞∑m≤j∑n≤jDjm,jn(j,τj)(g), where j, m, n ∈ Z, τ ∈ C is convergent to a function on SL(2, C); however, the limit is not a square integrable function; therefore, such sums cannot be used for the Peter-Weyl like expansion. We propose the alternative expansion and prove that for each fixed m: ∑j=m∞Djm,jm(j,τj)(g) is convergent and that the limit is a square integrable function on SL(2, C). We then prove the analog of the Peter-Weyl expansion: any ψ(g) ∈ L2(SL(2, C)) can be decomposed into the sum: ψ(g)=∑j=m∞j2(1+τ2)cjmmDjm,jm(j,τj)(g), with the Fourier coefficients cjmm=∫SL(2,C)ψ(g)Djm,jmj,τj(g)¯dg, g ∈ SL(2, C), τ ∈ C, τ ≠ i, − i, j, m ∈ Z, m is fixed. We also prove convergence of the sums ∑j=p∞∑m≤j∑n≤jdpmj2Djm,jn(j,τj)(g), where dpmj2=(j+1)12∫SU(2)ϕ(u)Dpmj2(u)¯du is ϕ(u)’s Fourier transform and p, j, m, n ∈ Z, τ ∈ C, u ∈ SU(2), g ∈ SL(2, C), thus establishing the map between the square integrable functions on SU(2) and the space of the functions on SL(2, C). Such maps were first used in Rovelli [Class. Quant. Grav. 28, 11 (2011)].

Covariant loop quantum gravity, low-energy perturbation theory, and Einstein gravity with high-curvature UV corrections

A low-energy perturbation theory is developed from the nonperturbative framework of covariant loop quantum gravity (LQG) by employing the background-field method. The resulting perturbation theory is a two-parameter expansion in the semiclassical and low-energy regime. The two expansion parameters are the large spin and small curvature. The leading-order effective action coincides with the Regge action, which well approximates the Einstein-Hilbert action in the regime. The subleading corrections organized by the two expansion parameters give the modifications of the Regge action in the quantum and high-energy regime from LQG. The perturbation theory developed here shows for the first time that covariant LQG produces the high-curvature corrections to Einstein-Regge gravity. This result means that LQG is not a naive quantization of Einstein gravity; rather, it provides the UV modification. The result of the paper may be viewed as the first step toward understanding the UV completeness of LQG.

Evidence for Maximal Acceleration and Singularity Resolution in Covariant Loop Quantum Gravity

A simple argument indicates that covariant loop gravity (spin foam theory) predicts a maximal acceleration and hence forbids the development of curvature singularities. This supports the results obtained for cosmology and black holes using canonical methods.

Discrete symmetries in covariant loop quantum gravity

We study time-reversal and parity ---on the physical manifold and in internal space--- in covariant loop gravity. We consider a minor modification of the Holst action which makes it transform coherently under such transformations. The classical theory is not affected but the quantum theory is slightly different. In particular, the simplicity constraints are slightly modified and this restricts orientation flips in a spinfoam to occur only across degenerate regions, thus reducing the sources of potential divergences.

Spin foam model from canonical quantization

We suggest a modification of the Barrett-Crane spin foam model of four-dimensional Lorentzian general relativity motivated by the canonical quantization. The starting point is Lorentz covariant loop quantum gravity. Its kinematical Hilbert space is found as a space of the so-called projected spin networks. These spin networks are identified with the boundary states of a spin foam model and provide a generalization of the unique Barrett-Crane intertwiner. We propose a way to modify the Barrett-Crane quantization procedure to arrive at this generalization: the $B$ field (bivectors) should be promoted not to generators of the gauge algebra, but to their certain projection. The modification is also justified by the canonical analysis of the Plebanski formulation. Finally, we compare our construction with other proposals to modify the Barrett-Crane model.