Einstein-Hilbert Truncation Research Articles

Asymptotic safety in Lorentzian quantum gravity

A recently introduced functional renormalization group (RG) provides a new tool to explore nonperturbative and covariant RG flows in Lorentzian spacetimes. We apply it for the first time to investigate the ultraviolet limit of quantum gravity. While the RG flow is state dependent, it is possible to evaluate state- and background-independent contributions to the flow. Taking into account only these universal terms, the RG flow exhibits a nontrivial fixed point in the Einstein-Hilbert truncation, providing a mechanism for asymptotic safety in Lorentzian quantum gravity. Published by the American Physical Society 2024

Formation and evaporation of quantum black holes from the decoupling mechanism in quantum gravity

We propose a new method to account for quantum-gravitational effects in cosmological and black hole spacetimes. At the core of our construction is the “decoupling mechanism”: when a physical infrared scale overcomes the effect of the regulator implementing the Wilsonian integration of fluctuating modes, the renormalization group flow of the scale-dependent effective action freezes out, so that at the decoupling scale the latter approximates the standard quantum effective action. Identifying the decoupling scale allows to access terms in the effective action that were not part of the original truncation and thus to study leading-order quantum corrections to field equations and their solutions. Starting from the Einstein-Hilbert truncation, we exploit for the first time the decoupling mechanism in quantum gravity to investigate the dynamics of quantum-corrected black holes from formation to evaporation. Our findings are in qualitative agreement with previous results in the context of renormalization group improved black holes, but additionally feature novel properties reminiscent of higher-derivative operators with specific non-local form factors.

Nonperturbative quantum correction to the Reissner-Nordström spacetime with running Newton’s constant

We study the consequences of the running Newton's constant on several key aspects of spherically symmetric charged black holes by performing a renormalization group improvement of the classical Reissner-Nordstr\"om metric within the framework of the Einstein-Hilbert truncation in quantum Einstein gravity. In particular, we determine that the event horizon surfaces are stable except for the extremal case and we corroborate the appearance of a new extremality condition at the Planck scale that contributes to hints about the possible existence of a final stage after the black hole evaporation process. We find explicit expressions for the area and surface gravity of the event horizon and we show the existence of an exact form $dS$ with the surface gravity as integrating factor, in contrast to a previous no-go result for axially symmetric spacetimes with null charge. As a consequence, we are able to derive a formula for the state function $S$ as the sum of the area of the classical event horizon plus a quantum correction linear in $\ensuremath{\hbar}$ and logarithmic in the classical area, in accordance with other approaches. We finally calculate an explicit formula for the Komar mass at the event horizon, and we compare it with the total mass at infinity for a wide domain of values of the black hole parameters $M$, $Q$ and $\overline{w}$, showing a loss of mass which can be interpreted as a consequence of the antiscreening effect of the gravitational field between the event horizon and infinity.

Asymptotic freedom and safety in quantum gravity

We compute non-perturbative flow equations for the couplings of quantum gravity in fourth order of a derivative expansion. The gauge invariant functional flow equation for arbitrary metrics allows us to extract β-functions for all couplings. In our truncation we find two fixed points. One corresponds to asymptotically free higher derivative gravity, the other is an extension of the asymptotically safe fixed point in the Einstein-Hilbert truncation or extensions thereof. The infrared limit of the flow equations entails only unobservably small modifications of Einstein gravity coupled to a scalar field. Quantum gravity can be asymptotically free, based on a flow trajectory from the corresponding ultraviolet fixed point to the infrared region. This flow can also be realized by a scaling solution for varying values of a scalar field. As an alternative possibility, quantum gravity can be realized by asymptotic safety at the other fixed point. There may exist a critical trajectory between the two fixed points, starting in the extreme ultraviolet from asymptotic freedom. We compute critical exponents and determine the number of relevant parameters for the two fixed points. Evaluating the flow equation for constant scalar fields yields the universal gravitational contribution to the effective potential for the scalars.

Geometric Operators in the Einstein–Hilbert Truncation

We review the study of the scaling properties of geometric operators, such as the geodesic length and the volume of hypersurfaces, in the context of the Asymptotic Safety scenario for quantum gravity. We discuss the use of such operators and how they can be embedded in the effective average action formalism. We report the anomalous dimension of the geometric operators in the Einstein–Hilbert truncation via different approximations by considering simple extensions of previous studies.

Geometric operators in the asymptotic safety scenario for quantum gravity

We consider geometric operators, such as the geodesic length and the volume of hypersurfaces, in the context of the Asymptotic Safety scenario for quantum gravity. We discuss the role of these operators from the Asymptotic Safety perspective, and compute their anomalous dimensions within the Einstein-Hilbert truncation. We also discuss certain subtleties arising in the definition of such geometric operators. Our results hint to an effective dimensional reduction of the considered geometric operators.

Asymptotic safety and field parametrization dependence in the f(R) truncation

We study the dependence on field parametrization of the functional renormalization group equation in the $f(R)$ truncation for the effective average action. We perform a systematic analysis of the dependence of fixed points and critical exponents in polynomial truncations. We find that, beyond the Einstein-Hilbert truncation, results are qualitatively different depending on the choice of parametrization. In particular, we observe that there are two different classes of fixed points, one with three relevant directions and the other with two. The computations are performed in the background approximation. We compare our results with the available literature and analyze how different schemes in the regularizations can affect the fixed point structure.

Composite operators in asymptotic safety

We study the role of composite operators in the Asymptotic Safety program for quantum gravity. By including in the effective average action an explicit dependence on new sources we are able to keep track of operators which do not belong to the exact theory space and/or are normally discarded in a truncation. Typical examples are geometric operators such as volumes, lengths, or geodesic distances. We show that this set-up allows to investigate the scaling properties of various interesting operators via a suitable exact renormalization group equation. We test our framework in several settings, including Quantum Einstein Gravity, the conformally reduced Einstein-Hilbert truncation, and two dimensional quantum gravity. Finally, we briefly argue that our construction paves the way to approach observables in the Asymptotic Safety program.

Search of scaling solutions in scalar–tensor gravity

We write new functional renormalization group equations for a scalar nonminimally coupled to gravity. Thanks to the choice of the parametrization and of the gauge fixing they are simpler than older equations and avoid some of the difficulties that were previously present. In three dimensions these equations admit, at least for sufficiently small fields, a solution that may be interpreted as a gravitationally dressed Wilson-Fisher fixed point. We also find for any dimension d>2 two analytic scaling solutions which we study for d=3 and d=4. One of them corresponds to the fixed point of the Einstein-Hilbert truncation, the others involve a nonvanishing minimal coupling.

RG flows of Quantum Einstein Gravity in the linear-geometric approximation

We construct a novel Wetterich-type functional renormalization group equation for gravity which encodes the gravitational degrees of freedom in terms of gauge-invariant fluctuation fields. Applying a linear-geometric approximation the structure of the new flow equation is considerably simpler than the standard Quantum Einstein Gravity construction since only transverse-traceless and trace part of the metric fluctuations propagate in loops. The geometric flow reproduces the phase-diagram of the Einstein–Hilbert truncation including the non-Gaussian fixed point essential for Asymptotic Safety. Extending the analysis to the polynomial f(R)-approximation establishes that this fixed point comes with similar properties as the one found in metric Quantum Einstein Gravity; in particular it possesses three UV-relevant directions and is stable with respect to deformations of the regulator functions by endomorphisms.

Towards a C-function in 4D quantum gravity

We develop a generally applicable method for constructing functions, $C$, which have properties similar to Zamolodchikov's $C$-function, and are geometrically natural objects related to the theory space explored by non-perturbative functional renormalization group (RG) equations. Employing the Euclidean framework of the Effective Average Action (EAA), we propose a $C$-function which can be defined for arbitrary systems of gravitational, Yang-Mills, ghost, and bosonic matter fields, and in any number of spacetime dimensions. It becomes stationary both at critical points and in classical regimes, and decreases monotonically along RG trajectories provided the breaking of the split-symmetry which relates background and quantum fields is sufficiently weak. Within the Asymptotic Safety approach we test the proposal for Quantum Einstein Gravity in $d>2$ dimensions, performing detailed numerical investigations in $d=4$. We find that the bi-metric Einstein-Hilbert truncation of theory space introduced recently is general enough to yield perfect monotonicity along the RG trajectories, while its more familiar single-metric analog fails to achieve this behavior which we expect on general grounds. Investigating generalized crossover trajectories connecting a fixed point in the ultraviolet to a classical regime with positive cosmological constant in the infrared, the $C$-function is shown to depend on the choice of the gravitational instanton which constitutes the background spacetime. For de Sitter space in 4 dimensions, the Bekenstein-Hawking entropy is found to play a role analogous to the central charge in conformal field theory. We also comment on the idea of a `$\Lambda$-$N$ connection' and the `$N$-bound' discussed earlier.

Renormalization group scale-setting from the action—a road to modified gravity theories

The renormalization group (RG) corrected gravitational action in Einstein–Hilbert and other truncations is considered. The running scale of the RG is treated as a scalar field at the level of the action and determined in a scale-setting procedure recently introduced by Koch and Ramirez for the Einstein–Hilbert truncation. The scale-setting procedure is elaborated for other truncations of the gravitational action and applied to several phenomenologically interesting cases. It is shown how the logarithmic dependence of the Newton's coupling on the RG scale leads to exponentially suppressed effective cosmological constant and how the scale-setting in particular RG-corrected gravitational theories yields the effective f(R) modified gravity theories with negative powers of the Ricci scalar R. The scale-setting at the level of the action at the non-Gaussian fixed point in Einstein–Hilbert and more general truncations is shown to lead to universal effective action quadratic in the Ricci tensor.

Running boundary actions, Asymptotic Safety, and black hole thermodynamics

Previous explorations of the Asymptotic Safety scenario in Quantum Einstein Gravity (QEG) by means of the effective average action and its associated functional renormalization group (RG) equation assumed spacetime manifolds which have no boundaries. Here we take a first step towards a generalization for non-trivial boundaries, restricting ourselves to action functionals which are at most of second order in the derivatives acting on the metric. We analyze two examples of truncated actions with running boundary terms: full fledged QEG within the single-metric Einstein-Hilbert (EH) truncation, augmented by a scale dependent Gibbons-Hawking (GH) surface term, and a bi-metric truncation for gravity coupled to scalar matter fields. The latter contains 17 running couplings, related to both bulk and boundary terms, whose beta-functions are computed in the induced gravity approximation. We find that the bulk and the boundary Newton constant, pertaining to the EH and GH term, respectively, show opposite RG running; proposing a scale dependent variant of the ADM mass we argue that the running of both couplings is consistent with gravitational anti-screening. We describe a simple device for counting the number of field modes integrated out between the infrared cutoff scale and the ultraviolet. This method makes it manifest that, in an asymptotically safe theory, there are effectively no field modes integrated out while the RG trajectory stays in the scaling regime of the underlying fixed point. As an application, we investigate how the semiclassical theory of Black Hole Thermodynamics gets modified by quantum gravity effects and compare the new picture to older work on `RG-improved black holes' which incorporated the running of the bulk Newton constant only. We find, for instance, that the black hole's entropy vanishes and its specific heat capacity turns positive at Planckian scales.

The “tetrad only” theory space: nonperturbative renormalization flow and asymptotic safety

We set up a nonperturbative gravitational coarse graining flow and the corresponding functional renormalization group equation on the as to yet unexplored "tetrad only" theory space. It comprises action functionals which depend on the tetrad field (along with the related background and ghost fields) and are invariant under the semi-direct product of spacetime diffeomorphisms and local Lorentz transformations. This theory space differs from that of Quantum Einstein Gravity (QEG) in that the tetrad rather than the metric constitutes the fundamental variable and because of the additional symmetry requirement of local Lorentz invariance. It also differs from "Quantum Einstein Cartan Gravity" (QECG) investigated recently since the spin connection is not an independent field variable now. We explicitly compute the renormalization group flow on this theory space within the tetrad version of the Einstein-Hilbert truncation. A detailed comparison with analog results in QEG and QECG is performed in order to assess the impact the choice of a fundamental field variable has on the renormalization behavior of the gravitational average action, and the possibility of an asymptotically safe infinite cutoff limit is investigated. Implications for nonperturbative studies of fermionic matter coupled to quantum gravity are also discussed. It turns out that, in the context of functional flow equations, the "hybrid calculations" proposed in the literature (using the tetrad for fermionic diagrams only, and the metric in all others) are unlikely to be quantitatively reliable. Moreover we find that, unlike in perturbation theory, the non-propagating Faddeev-Popov ghosts related to the local Lorentz transformations may not be discarded but rather contribute quite significantly to the beta functions of Newton's constant and the cosmological constant.

Asymptotic safety goes on shell

It is well known in quantum field theory that the off-shell effective action depends on the gauge choice and field parametrization used in calculating it. Nevertheless, the typical scheme in which the scenario of asymptotically safe gravity is investigated is an off-shell version of the functional renormalization group equation. Working with the Einstein–Hilbert truncation as a test bed, we develop a new scheme for the analysis of asymptotically safe gravity in which the on-shell part of the effective action is singled out and we show that the beta function for the essential coupling has no explicit gauge dependence. In order to reach our goal, we introduce several technical novelties, including a different decomposition of the metric fluctuations, a new implementation of the ghost sector and a new cut-off scheme. We find a nontrivial fixed point, with a value of the cosmological constant that is independent of the gauge-fixing parameters.

Asymptotically safe gravity as a scalar-tensor theory and its cosmological implications

We study asymptotically safe gravity with Einstein-Hilbert truncation taking into account the renormalization group running of both gravitational and cosmological constants. We show the classical behavior of the theory is equivalent to a specific class of Jordan-Brans-Dicke theories with vanishing Brans-Dicke parameter, and potential determined by the renormalization group equation. The theory may be reformulated as an $f(R)$ theory. In the simplest cosmological scenario, we find large--field inflationary solutions near the Planck scale where the effective field theory description breaks down. Finally, we discuss the implications of a running gravitational constant to background dynamics via cosmological perturbation theory. We show that compatibility with General Relativity requires contributions from the running gravitational constant to the stress energy tensor to be taken into account in the perturbation analysis.

The universal RG machine

Functional Renormalization Group Equations constitute a powerful tool to encode the perturbative and non-perturbative properties of a physical system. We present an algorithm to systematically compute the expansion of such flow equations in a given background quantity specified by the approximation scheme. The method is based on off-diagonal heat-kernel techniques and can be implemented on a computer algebra system, opening access to complex computations in, e.g., Gravity or Yang-Mills theory. In a first illustrative example, we re-derive the gravitational $\beta$-functions of the Einstein-Hilbert truncation, demonstrating their background-independence. As an additional result, the heat-kernel coefficients for transverse vectors and transverse-traceless symmetric matrices are computed to second order in the curvature.

Ghost wavefunction renormalization in asymptotically safe quantum gravity

Motivated by Weinberg's asymptotic safety scenario, we investigate the gravitational renormalization group flow in the Einstein–Hilbert truncation supplemented by the wavefunction renormalization of the ghost fields. The latter induces non-trivial corrections to the β-functions for Newton's constant and the cosmological constant. The resulting ghost-improved phase diagram is investigated in detail. In particular, we find a non-trivial ultraviolet fixed point, in agreement with the asymptotic safety conjecture which also survives in the presence of extra dimensions. In four dimensions the ghost anomalous dimension at the fixed point is η*c = −1.8, supporting spacetime being effectively two dimensional at short distances.

Asymptotically free scalar curvature-ghost coupling in quantum Einstein gravity

We consider the asymptotic-safety scenario for quantum gravity which constructs a nonperturbatively renormalizable quantum gravity theory with the help of the functional renormalization group (RG). We verify the existence of a non-Gaussian fixed point and include a running curvature-ghost coupling as a first step towards the flow of the ghost sector of the theory. We find that the scalar curvature-ghost coupling is asymptotically free and RG relevant in the ultraviolet. Most importantly, the property of asymptotic safety discovered so far within the Einstein-Hilbert truncation and beyond remains stable under the inclusion of the ghost flow.

Background independence and asymptotic safety in conformally reduced gravity

We analyze the conceptual role of background independence in the application of the effective average action to quantum gravity. Insisting on a background independent renormalization group (RG) flow the coarse graining operation must be defined in terms of an unspecified variable metric since no rigid metric of a fixed background spacetime is available. This leads to an extra field dependence in the functional RG equation and a significantly different RG flow in comparison to the standard flow equation with a rigid metric in the mode cutoff. The background independent RG flow can possess a non-Gaussian fixed point, for instance, even though the corresponding standard one does not. We demonstrate the importance of this universal, essentially kinematical effect by computing the RG flow of quantum Einstein gravity in the ``conformally reduced'' Einstein-Hilbert approximation which discards all degrees of freedom contained in the metric except the conformal one. Without the extra field dependence the resulting RG flow is that of a simple ${\ensuremath{\phi}}^{4}$ theory. By including it one obtains a flow with exactly the same qualitative properties as in the full Einstein-Hilbert truncation. In particular it possesses the non-Gaussian fixed point which is necessary for asymptotic safety.