4D Einstein Gravity Research Articles

A Non-perturbative Proof of the Asymptotic Safety of 4D Einstein Gravity, With or Without Matter

A Non-perturbative Proof of the Asymptotic Safety of 4D Einstein Gravity, With or Without Matter

Continuum effective field theories, gravity, and holography

We examine effective field theories (EFTs) with a continuum sector in the presence of gravity. We first explain, via arguments based on central charge and species scale, that an EFT with a free continuum cannot consistently couple to standard (i.e. 4D Einstein) gravity. It follows that EFTs with a free or nearly-free continuum must either have a finite number of degrees of freedom or nonstandard gravity. The latter claim is realized for holographically-defined continuum models. We demonstrate this by computing the deviations from standard gravity in a specific 5D scalar-gravity system that gives rise to a gapped continuum (i.e the linear dilaton background). We find an $R^{-2}$ deviation from the Newtonian potential. At finite temperature we find an energy density with matter-like behavior in the brane Friedmann equation, holographically induced from the bulk geometry. Thus, remarkably, a brane-world living in the linear dilaton background automatically contains dark matter. We also present a slightly more evolved asymptotically-AdS linear dilaton model, for which the deviations exhibit a transition between AdS and linear dilaton behaviors.

On the associativity of 1-loop corrections to the celestial operator product in gravity

The question of whether the holomorphic collinear singularities of graviton amplitudes define a consistent chiral algebra has garnered much recent attention. We analyse a version of this question for infinitesimal perturbations around the self-dual sector of 4d Einstein gravity. The singularities of tree amplitudes in such perturbations do form a consistent chiral algebra, however at 1-loop its operator products are corrected by the effective graviton vertex. We argue that the chiral algebra can be interpreted as the universal holomorphic surface defect in the twistor uplift of self-dual gravity, and show that the same correction is induced by an anomalous diagram in the bulk-defect system. The 1-loop holomorphic collinear singularities do not form a consistent chiral algebra. The failure of associativity can be traced to the existence of a recently discovered gravitational anomaly on twistor space. It can be restored by coupling to an unusual 4th-order gravitational axion, which cancels the anomaly by a Green-Schwarz mechanism. Alternatively, the anomaly vanishes in certain theories of self-dual gravity coupled to matter, including in self-dual supergravity.

Rotating black holes in 3D Hořava gravity revisited

Abstract I revisit rotating black hole solutions in 3D Hořava gravity with z = 2 as a simpler set-up of the renormalizable quantum gravity à la Lifshitz [Zh. Eksp. Teor. Fiz. 11, 255 (1941)] and DeWitt [Phys. Rev. 160, 1113 (1967)]. The solutions have a curvature singularity at the origin for a non-vanishing rotation parameter ${\cal J}$, unlike the black holes in 3D Einstein gravity. For anti-de Sitter space, there are black hole event horizons as usual and the singularity is not naked, in agreement with cosmic censorship. On the other hand, for flat or de Sitter space, the earlier solution also has a cosmic-censorship problem because there are no conventional black hole horizons as in Einstein gravity, other than the usual cosmological horizon for the latter case, so that the singularity could be naked in Hořava gravity. However, with the help of recent corrections, I show that the solutions have a peculiar black hole horizon at the origin so that the singularity is not naked even without the conventional black hole horizons in the flat or de Sitter cases, due to Lorentz-violating higher-derivative terms. On the other hand, I also note that a new “cosmological” horizon exists even for the flat case, contrary to the usual wisdom, due to the combined effects of the higher derivatives and the angular-momentum barrier. I study a unified treatment of their unusual black hole thermodynamics for the flat and de Sitter spaces, as well as the anti-de Sitter space, which might be due to the lack of absolute horizons in Lorentz-violating gravity.

Particle motion in a space-time of a 3D Einstein gravity with torsion

Particle motion in a space-time of a 3D Einstein gravity with torsion

Ehlers transformations as EM duality in the double copy

Given a solution to 4D Einstein gravity with an isometry direction, it is known that the equations of motion are identical to those of a 3D $\sigma$-model with target space geometry $SU(1,1)/U(1)$. Thus, any transformation by $SU(1, 1) \cong SL(2,\mathbb{R})$ is a symmetry for the action and allows one to generate new solutions in 4D. Here we clarify and extend recent work on electromagnetic (EM) duality in the context of the classical double copy. In particular, for pure gravity, we identify an explicit map between the Maxwell field of the single copy and the scalars in the target space, allowing us to identify the $U(1) \subset SL(2, \mathbb{R})$ symmetry dual to EM duality in the single copy. Moreover, we extend the analysis to Einstein-Maxwell theory, where we highlight the role of Ehlers-Harrison transformations and, for spherically symmetric charged black hole solutions, we interpret the equations of motion as a truncation of the putative single copy for Einstein-Yang-Mills theory.

Third order extensions of 3d Chern–Simons interacting to gravity: Hamiltonian formalism and stability

We consider inclusion of interactions between 3d Einstein gravity and the third order extensions of Chern–Simons. Once the gravity is minimally included into the third order vector field equations, the theory is shown to admit a two-parameter series of symmetric tensors with on-shell vanishing covariant divergence. The canonical energy–momentum is included into the series. For a certain range of the model parameters, the series include the tensors that meet the weak energy condition, while the canonical energy is unbounded in all the instances. Because of the on-shell vanishing covariant divergence, any of these tensors can be considered as an appropriate candidate for the right hand side of Einstein's equations. If the source differs from the canonical energy momentum, the coupling is non-Lagrangian while the interaction remains consistent with any of the tensors. We reformulate these not necessarily Lagrangian third order equations in the first order formalism which is covariant in the sense of 1+2 decomposition. After that, we find the Poisson bracket such that the first order equations are Hamiltonian in all the instances, be the original third order equations Lagrangian or not. The brackets differ from canonical ones in the matter sector, while the gravity admits the usual PB's in terms of ADM variables. The Hamiltonian constraints generate lapse, shift and gauge transformations of the vector field with respect to these Poisson brackets. The Hamiltonian constraint, being the lapse generator, is interpreted as strongly conserved energy. The matter contribution to the Hamiltonian constraint corresponds to 00-component of the tensor included as a source in the right hand side of Einstein equations. Once the 00-component of the tensor is bounded, the theory meets the usual sufficient condition of classical stability, while the original field equations are of the third order.

Four-Dimensional Covariance of Feynman Diagrams in Einstein Gravity

It has been observed in [Park 2014] that the physical states of the ADM formulation of 4D Einstein gravity holographically reduce and can be described by a 3D language. Obviously the approach poses the 4D covariance issue; it turns out that there are two covariance issues whose address is the main theme of the present work. Although the unphysical character of the trace piece of the fluctuation metric has been long known, it has not been taken care of in a manner suitable for the Feynman diagram computations; a proper handling of the trace piece through gauge-fixing is the key to more subtler of the covariance issues. As for the second covariance issue, a renormalization program can be carried out covariantly to any loop order at intermediate steps, thereby maintaining the 4D covariance; it is only at the final stage that one should consider the 3D physical external states. With the physical external states, the 1PI effective action reduces to 3D and renormalizability is restored just as in the entirely-3D approach of [Park 2014]. We revisit the one-loop two-point renormalization with careful attention to the trace piece of the fluctuation metric and in particular outline one-loop renormalization of the Newton's constant.

Mathematical foundation of foliation-based quantization

We have recently proposed in \cite{Park:2014tia} the quantization of pure 4D Einstein gravity through hypersurface foliation, and observed that the 4D Einstein gravity becomes renormalizable once all (or most) of the unphysical degrees of freedom are removed. In this work, we confirm this observation from a more mathematical angle. In particular, we show that the physical state condition arising from the shift vector constraint connects with the requirement that the manifold admit "totally geodesic (TG) foliation". The TG foliation, in turn, makes it possible to view the 4D manifold as abelian fibration over a 3D base. Associating the abelian fibration with the 4D diffeomorphism leads to reduction of the 4D manifold to 3D, thereby realizing and generalizing the holography of 't Hooft.

Asymptotic symmetries, holography and topological hair

Asymptotic symmetries of AdS4 quantum gravity and gauge theory are derived by coupling the holographically dual CFT3 to Chern-Simons gauge theory and 3D gravity in a “probe” (large-level) limit. Despite the fact that the three-dimensional AdS4 boundary as a whole is consistent with only finite-dimensional asymptotic symmetries, given by AdS isometries, infinite-dimensional symmetries are shown to arise in circumstances where one is restricted to boundary subspaces with effectively two-dimensional geometry. A canonical example of such a restriction occurs within the 4D subregion described by a Wheeler-DeWitt wavefunctional of AdS4 quantum gravity. An AdS4 analog of Minkowski “super-rotation” asymptotic symmetry is probed by 3D Einstein gravity, yielding CFT2 structure (in a large central charge limit), via AdS3 foliation of AdS4 and the AdS3/CFT2 correspondence. The maximal asymptotic symmetry is however probed by 3D conformal gravity. Both 3D gravities have Chern-Simons formulation, manifesting their topological character. Chern-Simons structure is also shown to be emergent in the Poincare patch of AdS4, as soft/boundary limits of 4D gauge theory, rather than “put in by hand” as an external probe. This results in a finite effective Chern-Simons level. Several of the considerations of asymptotic symmetry structure are found to be simpler for AdS4 than for Mink4, such as non-zero 4D particle masses, 4D non-perturbative “hard” effects, and consistency with unitarity. The last of these in particular is greatly simplified because in some set-ups the time dimension is explicitly shared by each level of description: Lorentzian AdS4, CFT3 and CFT2. Relatedly, the CFT2 structure clarifies the sense in which the infinite asymptotic charges constitute a useful form of “hair” for black holes and other complex 4D states. An AdS4 analog of Minkowski “memory” effects is derived, but with late-time memory of earlier events being replaced by (holographic) “shadow” effects. Lessons from AdS4 provide hints for better understanding Minkowski asymptotic symmetries, the 3D structure of its soft limits, and Minkowski holography.

Butterfly effect in 3D gravity

We study the butterfly effect by considering shock wave solutions near the horizon of the AdS black brane in some of 3-dimensional Gravity models including; 3D Einstein Gravity, Minimal Massive 3D Gravity, New Massive Gravity, Generalized Massive Gravity, Born-Infeld 3D Gravity and New Bi-Gravity. We calculate the butterfly velocities of these models and also we consider the critical points and different limits in some of these models. By studying the butterfly effect in the Generalized Massive Gravity, we observe a correspondence between the butterfly velocities and right-left moving degrees of freedom or the central charges of the dual 2D Conformal Field Theories.

One-loop renormalization of a gravity-scalar system

Extending the renormalizability proposal of the physical sector of 4D Einstein gravity, we have recently proposed renormalizability of the 3D physical sector of gravity-matter systems. The main goal of the present work is to conduct systematic one-loop renormalization of a gravity-matter system by applying our foliation-based quantization scheme. In this work we explicitly carry out renormalization of a gravity-scalar system with a Higgs-type potential. With the fluctuation part of the scalar field gauged away, the system becomes renormalizable through a metric field redefinition. We use dimensional regularization throughout. One of the salient aspects of our analysis is how the graviton propagator acquires the “mass” term. One-loop calculations lead to renormalization of the cosmological and Newton constants. We discuss other implications of our results as well: time-varying vacuum energy density and masses of the elementary particles as well as the potential relevance of Neumann boundary condition for black hole information.

The mass formula for an exotic BTZ black hole

An exotic Bañados–Teitelboim–Zanelli (BTZ) black hole has an angular momentum larger than its mass in three dimension (3D), which suggests the possibility that cosmic censorship could be violated if angular momentum is extracted by the Penrose process. In this paper, we propose a mass formula for the exotic BTZ black hole and show no violation of weak cosmic censorship in the gedanken process above by understanding properly its mass formula. Unlike the other black holes, the total energy of the exotic BTZ black hole is represented by the angular momentum instead of the mass, which supports a basic point of view that the same geometry should be determined by the same energy in 3D general relativity whose equation of motion can be given either by normal 3D Einstein gravity or by exotic 3D Einstein gravity. However, only the mass of the exotic black hole is related to the thermodynamics and other forms of energy are “dumb”, which is consistent with the earlier thermodynamic analysis about exotic black holes.

Hypersurface foliation approach to renormalization of ADM formulation of gravity

We carry out ADM splitting in the Lagrangian formulation and establish a procedure in which (almost) all of the unphysical components of the metric are removed by using the 4D diffeomorphism and the measure-zero 3D symmetry. The procedure introduces a constraint that corresponds to the Hamiltonian constraint of the Hamiltonian formulation, and its solution implies that the 4D dynamics admits an effective description through 3D hypersurface physics. As far as we can see, our procedure implies potential renormalizability of {the ADM formulation of} 4D Einstein gravity for which a complete gauge-fixing in the ADM formulation and hypersurface foliation of geometry are the key elements. If true, this implies that the alleged unrenormalizability of 4D Einstein gravity may be due to the presence of the unphysical fields. The procedure can straightforwardly be applied to quantization around a flat background; the Schwarzschild case seems more subtle. We discuss a potential limitation of the procedure when applying it to explicit time-dependent backgrounds.

Lagrangian constraints and renormalization of 4D gravity

It has been proposed in \cite{Park:2014tia} that 4D Einstein gravity becomes effectively reduced to 3D after solving the Lagrangian analogues of the Hamiltonian and momentum constraints of the Hamiltonian quantization. The analysis in \cite{Park:2014tia} was carried out at the classical/operator level. We review the proposal and make a transition to the path integral account. We then set the stage for explicitly carrying out the two-loop renormalization procedure of the resulting 3D action. We also address a potentially subtle issue in the gravity context concerning whether renormalizability does not depend on the background around which the original action is expanded.

Natural braneworld inflation in light of recent results from Planck and BICEP2

In this paper we report on a major theoretical observation in cosmology. We present a concrete cosmological model for which inflation has a natural beginning and natural ending. Inflation is driven by a cosine-form potential, $V(\ensuremath{\phi})={\mathrm{\ensuremath{\Lambda}}}^{4}(1\ensuremath{-}\mathrm{cos}(\ensuremath{\phi}/f))$, which begins at $\ensuremath{\phi}\ensuremath{\lesssim}\ensuremath{\pi}f$ and ends at $\ensuremath{\phi}={\ensuremath{\phi}}_{\text{end}}\ensuremath{\lesssim}5f/3$. The distance traversed by the inflaton field $\ensuremath{\phi}$ is sub-Planckian. The Gauss-Bonnet term ${\mathcal{R}}^{2}$ arising as leading curvature corrections in the action $S=\ensuremath{\int}{d}^{5}x\sqrt{\ensuremath{-}{g}_{5}}{M}^{3}(\ensuremath{-}6\ensuremath{\lambda}{M}^{2}+R+\ensuremath{\alpha}{M}^{\ensuremath{-}2}{\mathcal{R}}^{2})+\ensuremath{\int}{d}^{4}x\sqrt{\ensuremath{-}{g}_{4}}({\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{\ensuremath{\phi}}}^{2}/2\ensuremath{-}V(\ensuremath{\phi})\ensuremath{-}\ensuremath{\sigma}+{\mathcal{L}}_{\text{matter}})$ (where $\ensuremath{\alpha}$ and $\ensuremath{\lambda}$ are constants and $M$ is the five-dimensional Planck mass) plays a key role to terminate inflation. The model generates appropriate tensor-to-scalar ratio $r$ and inflationary perturbations that are consistent with Planck and BICEP2 data. For example, for ${N}_{*}=50--60$ and ${n}_{s}\ensuremath{\sim}0.960\ifmmode\pm\else\textpm\fi{}0.005$, the model predicts that $M\ensuremath{\sim}5.64\ifmmode\times\else\texttimes\fi{}1{0}^{16}\text{ }\text{ }\mathrm{GeV}$ and $r\ensuremath{\sim}(0.14--0.21)$ [${N}_{*}$ is the number of $e$-folds of inflation and ${n}_{s}$ (${n}_{t}$) is the scalar (tensor) spectrum spectral index]. The ratio $\ensuremath{-}{n}_{t}/r$ is (13%--24%) less than its value in 4D Einstein gravity, $\ensuremath{-}{n}_{t}/r=1/8$. The upper bound on the energy scale of inflation ${V}^{1/4}=2.37\ifmmode\times\else\texttimes\fi{}1{0}^{16}\text{ }\text{ }\mathrm{GeV}$ ($r<0.27$) implies that $(\ensuremath{-}\ensuremath{\lambda}\ensuremath{\alpha})\ensuremath{\gtrsim}75\ifmmode\times\else\texttimes\fi{}1{0}^{\ensuremath{-}5}$ and $\mathrm{\ensuremath{\Lambda}}<2.17\ifmmode\times\else\texttimes\fi{}1{0}^{16}\text{ }\text{ }\mathrm{GeV}$, which thereby rule out the case $\ensuremath{\alpha}=0$ (Randall-Sundrum model). The true nature of gravity is holographic as implied by the braneworld realization of string and M theory. The model correctly predicts a late-epoch cosmic acceleration with the dark energy equation of state ${\mathrm{w}}_{\mathrm{D}E}\ensuremath{\approx}\ensuremath{-}1$.

Thermodynamics of "exotic" Bañados-Teitelboim-Zanelli black holes.

A number of three-dimensional (3D) gravity models, such as 3D conformal gravity, admit "exotic" black hole solutions: the metric is the same as the Bañados-Teitelboim-Zanelli metric of 3D Einstein gravity but with reversed roles for mass and angular momentum, and an entropy proportional to the length of the inner horizon instead of the event horizon. Here we show that the Bañados-Teitelboim-Zanelli solutions of the exotic 3D Einstein gravity (with parity-odd action but Einstein field equations) are exotic black holes, and we investigate their thermodynamics. The first and second laws of black hole thermodynamics still apply, and the entropy still has a statistical interpretation.

Gravity on a little warped space

We investigate the consistent inclusion of 4D Einstein gravity on a truncated slice of ${\mathrm{AdS}}_{5}$ whose bulk-gravity and UV scales are much less than the 4D Planck scale, ${M}_{*}\ensuremath{\ll}{M}_{\mathrm{Pl}}$. Such ``Little Warped Spaces'' have found phenomenological utility and can be motivated by string realizations of the Randall-Sundrum framework. Using the interval approach to brane-world gravity, we show that the inclusion of a large UV-localized Einstein-Hilbert term allows one to consistently incorporate 4D Einstein gravity into the low-energy theory. We detail the spectrum of Kaluza-Klein metric fluctuations and, in particular, examine the coupling of the little radion to matter. Furthermore, we show that Goldberger-Wise stabilization can be successfully implemented on such spaces. Our results demonstrate that realistic low-energy effective theories can be constructed on these spaces, and have relevance for existing models in the literature.

TS-LIKE GRAVITATIONAL WAVES IN 4D EINSTEIN GRAVITY

We propose TS-like class of gravitational wave solutions in 4D Einstein gravity. TS1-like gravitational wave solution is analyzed in detail. On the axis, the gauge potential changes from a finite value to zero at t = τ. The spacetime on the axis approaches the flat one as t → ∞. It is found that by an appropriate parameter substitution and coordinate transformation TS-like gravitational wave solutions in 4D Einstein gravity may be obtained from TS solutions.

STRINGS AND MEMBRANES FROM EINSTEIN GRAVITY, MATRIX MODELS AND W∞ GAUGE THEORIES AS PATHS TO QUANTUM GRAVITY

It is shown how w∞, w1+∞ gauge field theory actions in 2D emerge directly from 4D gravity. Strings and membranes actions in 2D and 3D originate as well from 4D Einstein gravity after recurring to the nonlinear connection formalism of Lagrange–Finsler and Hamilton–Cartan spaces. Quantum gravity in 3D can be described by a W∞ matrix model in D = 1 that can be solved exactly via the collective field theory method. We describe why a quantization of 4D gravity could be attained via a 2D quantum W∞ gauge theory coupled to an infinite-component scalar-multiplet. A proof that noncritical W∞ (super)strings are devoid of BRST anomalies in dimensions D = 27(D = 11), respectively, follows and which coincide with the critical (super)membrane dimensions D = 27(D = 11). We establish the correspondence between the states associated with the quasifinite highest weights irreducible representations of W∞, [Formula: see text] algebras and the quantum states of the continuous Toda molecule. Schrödinger-like quantum mechanics wave functional equations are derived and solutions are found in the zeroth-order approximation. Since higher-conformal spin W∞ symmetries are very relevant in the study of 2DW∞ gravity, the quantum Hall effect, large N QCD, strings, membranes, … it is warranted to explore further the interplay among all these theories.