We present the first direct and non-perturbative computation of the graviton spectral function in quantum gravity. This is achieved with the help of a novel Lorentzian renormalisation group approach, combined with a spectral representation of correlation functions. We find a positive graviton spectral function, showing a massless one-graviton peak and a multi-graviton continuum with an asymptotically safe scaling for large spectral values. We also study the impact of a cosmological constant. Further steps to investigate scattering processes and unitarity in asymptotically safe quantum gravity are indicated.
It has long been known that photons and gravitons may appear as vector and tensor Goldstone modes caused \ by spontaneous Lorentz invariance violation (SLIV). Usually this approach is considered for photons and gravitons separately. We develop the emergent electrogravity theory consisting of the ordinary QED and the tensor field gravity model which mimics the linearized general relativity in Minkowski spacetime. In this theory, Lorentz symmetry appears incorporated into higher global symmetries of the length-fixing constraints put on the vector and tensor fields involved, $A_{μ}^{2}=\pm M_{A}^…
A family of spherically symmetric, static and self--dual Lorentzian wormholes is obtained in n--dimensional Einstein gravity. This class of solutions includes the n--dimensional versions of the Schwarzschild black hole and the spatial--Schwarzschild traversable wormhole. Using isotropic coordinates we study the geometrical structure of the solution, and delineate the domains of the free parameters for which wormhole, naked singular geometries and the Schwarzschild black hole are obtained. It is shown that, in the lower dimensional Einstein gravity without cosmological constant, we can not have…
The quantization of Lorentzian or Euclidean 2+1 gravity by canonical methods is a well-studied problem. However, the constraints of 2+1 gravity are those of a topological field theory and therefore resemble very little those of the corresponding Lorentzian 3+1 constraints. In this paper we canonically quantize Euclidean 2+1 gravity for arbitrary genus of the spacelike hypersurface with new, classically equivalent constraints that maximally probe the Lorentzian 3+1 situation. We choose the signature to be Euclidean because this implies that the gauge group is, as in the 3+1 case, SU(2) rather t…
The gravitational asymptotic safety program strives for a consistent and predictive quantum theory of gravity based on a non-trivial ultraviolet fixed point of the renormalization group (RG) flow. We investigate this scenario by employing a novel functional renormalization group equation which takes the causal structure of space-time into account and connects the RG flows for Euclidean and Lorentzian signature by a Wick-rotation. Within the Einstein-Hilbert approximation, the $β$-functions of both signatures exhibit ultraviolet fixed points in agreement with asymptotic safety. Surprisingly, th…
We uncover a surprising correspondence between a non-perturbative formulation of three-dimensional Lorentzian quantum gravity and a hermitian two-matrix model with ABAB-interaction. The gravitational transfer matrix can be expressed as the logarithm of a two-matrix integral, and we deduce from the known structure of the latter that the model has two phases. In the phase of weak gravity, well-defined two-dimensional universes propagate in proper time, whereas in the strong-coupling phase the spatial hypersurfaces disintegrate into many components connected by wormholes.
We study the spectrum of the length and area operators in Lorentzian loop quantum gravity, in 2+1 spacetime dimensions. We find that the spectrum of spacelike intervals is continuous, whereas the spectrum of timelike intervals is discrete. This result contradicts the expectation that spacelike intervals are always discrete. On the other hand, it is consistent with the results of the spin foam quantization of the same theory.
A special class of higher curvature theories of gravity, Ricci Cubic Gravity (RCG), in general d dimensional space-time has been investigated in this paper. We have used two different approaches, the linearized equations of motion and auxiliary field formalism to study the massive and massless graviton propagating modes of the AdS background. Using the auxiliary field formalism, we have found the renormalized boundary stress tensor to compute the mass of Schwarzschild AdS and Lifshitz black holes in RCG theory.
We consider a way of eliminating the unwanted scalar graviton from Horava-Lifshitz gravity. That is achieved via introduction of certain additional constraints. We perform canonical analysis of both projectable and non-projectable versions of the theory. We obtain the structure of constraints in each case, and analyze its dependence on the values of the coupling constants involved in the additional constraints. In the non-projectable theory, the scalar graviton is absent when the coupling constants have certain values, while for other values the scalar graviton appears. The projectable theory …
We compare the standard Higgs mechanism corresponding to the scalar field, with the dynamical origin of the graviton mass inside the scenario of the dRGT theory of massive gravity. We demonstrate that the effective mass perceived locally by different observers, depends on the way how they define the local time with respect to the preferred notion of time defined by the Stückelberg function $T_0(r,t)$.
In two space-time dimensions, there is a theory of Lorentzian quantum gravity which can be defined by a rigorous, non-perturbative path integral and is inequivalent to the well-known theory of (Euclidean) quantum Liouville gravity. It has a number of appealing features: i) its quantum geometry is non-fractal, ii) it remains consistent when coupled to matter, even beyond the c=1 barrier, iii) it is closer to canonical quantization approaches than previous path-integral formulations, and iv) its construction generalizes to higher dimensions.
De Sitter Chern-Simons gravity in D = 1 + 2 spacetime is known to possess an extension with a Barbero-Immirzi like parameter. We find a partial gauge fixing which leaves a compact residual gauge group, namely SU(2). The compacticity of the residual gauge group opens the way to the usual LQG quantization techniques. We recall the exemple of the LQG quantization of SU(2) CS theory with cylindrical space topology, which thus provides a complete LQG of a Lorentzian gravity model in 3-dimensional space-time.
Due to seminal works of Weinberg, Cachazo and Strominger we know that tree level quantum gravity amplitudes satisfy three factorization constraints. Building on previous works which relate two of these constraints to symmetries of quantum gravity at null infinity, we present rather strong evidence that the third constraint is also equivalent to a new set of symmetries of (perturbative) quantum gravity. Our analysis implies that the symmetry group of quantum gravity may be even richer than the BMS group (or infinite dimensional extension thereof) previously considered.
The graviton 1-loop partition function in Euclidean topologically massive gravity (TMG) is calculated using heat kernel techniques. The partition function does not factorize holomorphically, and at the chiral point it has the structure expected from a logarithmic conformal field theory. This gives strong evidence for the proposal that the dual conformal field theory to TMG at the chiral point is indeed logarithmic. We also generalize our results to new massive gravity.
We construct an n-dimensional Born-Infeld type gravity theory that has the same properties as Einstein's gravity in terms of the vacuum and particle content: Namely, the theory has a unique viable vacuum (maximally symmetric solution) and a single massless unitary spin-2 graviton about this vacuum. The BI gravity, in some sense, is the most natural, minimal generalization of Einstein's gravity with a better UV behavior, and hence, is a potentially viable proposal for low energy quantum gravity. The Gauss-Bonnet combination plays a non-trivial role in the construction of the theory. As an extre…
Making the Lorentzian path integral for quantum gravity well-defined and computable has been a long standing challenge. In this work we adopt the recently proposed effective spin foam models to the Lorentzian case. This defines a path integral over discrete Lorentzian quantum geometric configurations, which include metric and torsion degrees of freedom. The torsion degrees of freedom arise due to an anomaly, which is parametrized by the Barbero--Immirzi parameter. Requiring a semi-classical regime constrains this parameter, but the precise bound has to be determined by probing the dynamics. Th…
We study the generalized massive gravity in three dimensional flat spacetime. A massive logarithmic mode is propagating in the flat spacetime at the critical point where two masses degenerate. Furthermore, we discuss the logarithmic extension of the Galilean conformal algebra (GCA) which may arise from the exotic and standard rank-2 logarithmic conformal field theory (LCFT) on the boundary of AdS3 spacetime.
Motivated by the interest raised by the problem of Lorenz-symmetry violating gauge theories in connetion with gravity models, this contribution sets out to provide a general method to systematically study the excitation spectrum of gravity actions which include a Lorentz-symmetry breaking Chern-Simons-type action term for the spin connection. A complete set of spin-type operators is found which accounts for the (Lorentz) violation parameter to all orders and graviton propagators are worked out in a number of different situations.
We investigate propagations of graviton and additional scalar on four-dimensional anti de Sitter (AdS$_4$) space using $f(R)$ gravity models with external sources. It is shown that there is the van Dam-Veltman-Zakharov (vDVZ) discontinuity in $f(R)$ gravity models because $f(R)$ gravity implies GR with additional scalar. This indicates a difference between general relativity and $f(R)$ gravity clearly.
The asymmetric ABAB-matrix model describes the transfer matrix of three-dimensional Lorentzian quantum gravity. We study perturbatively the scaling of the ABAB-matrix model in the neighbourhood of its symmetric solution and deduce the associated renormalization of three-dimensional Lorentzian quantum gravity.
Recently there has been a surge of interest in studying Lorentzian quantum cosmology using Picard-Lefschetz methods. The present paper aims to explore the Lorentzian path-integral of Gauss-Bonnet gravity in four spacetime dimensions with metric as the field variable. We employ mini-superspace approximation and study the variational problem exploring different boundary conditions. It is seen that for mixed boundary conditions non-trivial effects arise from Gauss-Bonnet sector of gravity leading to additional saddle points for lapse in some case. As an application of this we consider the No-boun…
We study graviton propagations of scalar, vector, and tensor modes in the deformed Hořava-Lifshitz gravity ($λR$-model) without projectability condition. The quadratic Lagrangian is invariant under diffeomorphism only for $λ=1$ case, which contradicts to the fact that $λ$ is irrelevant to a consistent Hamiltonian approach to the $λR$ model. In this case, as far as scalar propagations are concerned, there is no essential difference between deformed Hořava-Lifshitz gravity ($λR$-model) and general relativity. This implies that there are two degrees of freedom for a massless graviton without Hořa…
We revisit the problem of constraining the weak field limit of the gravitational lagrangian from S-matrix properties. From unitarity and Lorentz invariance of the S-matrix of massless gravitons, we derive on-shell gauge invariance to consist on the transverse part of the linearised diffeomorphisms group. Moreover, by looking to the interaction between sources, we conclude that there exist only two possible lagrangians that lead to a well-defined covariant interaction, corresponding to the weak field limits of General Relativity and Unimodular Gravity. Additionally, this result confirms the equ…
Starting from 2D Euclidean quantum gravity, we show that one recovers 2D Lorentzian quantum gravity by removing all baby universes. Using a peeling procedure to decompose the discrete, triangulated geometries along a one-dimensional path, we explicitly associate with each Euclidean space-time a (generalized) Lorentzian space-time. This motivates a map between the parameter spaces of the two theories, under which their propagators get identified. In two dimensions, Lorentzian quantum gravity can therefore be viewed as a ``renormalized'' version of Euclidean quantum gravity.
A Wheeler-Dewitt quantum constraint operator for four-dimensional, non-perturbative Lorentzian vacuum quantum gravity is defined in the continuum. The regulated Wheeler-DeWitt constraint operator is densely defined, does not require any renormalization and the final operator is anomaly-free and at least symmmetric. The technique introduced here can also be used to produce a couple of other completely well-defined regulated operators including but not exhausting a) the Euclidean Wheeler-DeWitt operator, b)the generator of the Wick rotation transform that maps solutions to the Euclidean Hamilton…
We reconstruct the Lorentzian graviton propagator in asymptotically safe quantum gravity from Euclidean data. The reconstruction is applied to both the dynamical fluctuation graviton and the background graviton propagator. We prove that the spectral function of the latter necessarily has negative parts similar to, and for the same reasons, as the gluon spectral function. In turn, the spectral function of the dynamical graviton is positive. We argue that the latter enters cross sections and other observables in asymptotically safe quantum gravity. Hence, its positivity may hint at the unitarity…
We study the graviton propagator in euclidean loop quantum gravity, using the spinfoam formalism. We use boundary-amplitude and group-field-theory techniques, and compute one component of the propagator to first order, under a number of approximations, obtaining the correct spacetime dependence. In the large distance limit, the only term of the vertex amplitude that contributes is the exponential of the Regge action: the other terms, that have raised doubts on the physical viability of the model, are suppressed by the phase of the vacuum state, which is determined by the extrinsic geometry of …
We compute some components of the graviton propagator in loop quantum gravity, using the spinfoam formalism, up to some second order terms in the expansion parameter.
In the context of gravity in the strong-coupling regime, the propagation amplitude of gravity coupled to relativistic particles undergoing geodesic separation is calculated exactly. Geodesic separation gives rise to boundary terms associated with the h_\times and h_+ graviton components. At low temperatures the propagation amplitude vanishes, implying no graviton propagation in this regime.
We consider an extension of the Hořava-Lifshitz gravity with extra conformal symmetry by introducing a scalar field with higher order curvature terms. Relaxing the exact local Weyl symmetry, we construct an action with three free parameters which breaks local anisotropic Weyl symmetry but still preserves residual global Weyl symmetry. At low energies, it reduces to a Lorentz-violating scalar-tensor gravity. With a constant scalar field background and particular choices of the parameters, it reduces to the Hořava-Lifshitz (HL) gravity, but any perturbation from these particular configurations p…
In this paper, we introduce the $n$-dimensional Lorentzian wormhole solutions of third order Lovelock gravity. In contrast to Einstein gravity and as in the case of Gauss-Bonnet gravity, we find that the wormhole throat radius, $r_0$, has a lower limit that depends on the Lovelock coefficients, the dimensionality of the spacetime and the shape function. We study the conditions of having normal matter near the throat, and find that the matter near the throat can be normal for the region $r_0 \leq r \leq r_{\max}$, where $r_{\max}$ depends on the Lovelock coefficients and the shape function. We …
We construct a positive complexifier, differentiable almost everywhere on the classical phase space of real triads and $SU(2)$ connections, which generates a Wick Transform from Euclidean to Lorentzian gravity everywhere except on a phase space set of measure zero. This Wick transform assigns an equal role to the self dual and anti-self dual Ashtekar variables in quantum theory. We argue that the appropriate quantum arena for an analysis of the properties of the Wick rotation is the diffeomorphism invariant Hilbert space of Loop Quantum Gravity (LQG) rather than its kinematic Hilbert space. We…
We consider the generalization of the "spinor approach" to the Lorentzian case, in the context of 3d loop quantum gravity with cosmological constant $Λ=0$. The key technical tool that allows this generalization is the recoupling theory between unitary infinite-dimensional representations and non-unitary finite-dimensional ones, obtained in the process of generalizing the Wigner-Eckart theorem to SU(1,1). We use SU(1,1) tensor operators to build observables and a solvable quantum Hamiltonian constraint, analogue of the one introduced by V. Bonzom and his collaborators in the Euclidean case (wit…
We calculate graviton multi-point amplitudes in an anti-de Sitter black brane background for higher-derivative gravity of arbitrary order in numbers of derivatives. The calculations are performed using tensor graviton modes in a particular regime of comparatively high energies and large scattering angles. The regime simplifies the calculations but, at the same time, is well suited for translating these results into the language of the dually related gauge theory. After considering theories of up to eight derivatives, we generalize to even higher-derivative theories by constructing a "basis" fo…
We suggest a generalization of the dynamical triangulation approach to quantum gravity with both timelike and spacelike edges, which can serve as a toy model for quantum gravity in the Lorentz sector in two dimensions. It is possible to consider the model in a purely Lorentzian sector or to relax this constraint and allow local signature changing moves. We show that, with suitable conventions, the model is equivalent to an Ising model coupled to 2D Euclidean quantum gravity and conduct a preliminary numerical simulation of the Lorentz sector.
We discuss the production and evolution of cosmological gravitons showing how the cosmological background affects their dynamics. Besides, the detection of cosmological gravitons could constitute an extremely important signature to discriminate among different cosmological models. Here we consider the cases of scalar-tensor gravity and $f(R)$ gravity where it is demonstrated the amplification of graviton amplitude changes if compared with General Relativity. Possible observational constraints are discussed.
Evaluating gravitational path integrals in the Lorentzian has been a long-standing challenge due to the numerical sign problem. We show that this challenge can be overcome in simplicial quantum gravity. By deforming the integration contour into the complex, the sign fluctuations can be suppressed, for instance using the holomorphic gradient flow algorithm. Working through simple models, we show that this algorithm enables efficient Monte Carlo simulations for Lorentzian simplicial quantum gravity. In order to allow complex deformations of the integration contour, we provide a manifestly holomo…
We analyze scattering amplitudes with one soft external graviton and arbitrary number of other finite energy external states carrying arbitrary mass and spin to sub-subleading order in the momentum of the soft graviton. Our result can be expressed as the sum of a universal part that depends only on the amplitude without the soft graviton and not the other details of the theory and a non-universal part that depends on the amplitude without the soft graviton, and the two and three point functions of the theory. For tree amplitudes our results are valid in all space-time dimensions while for loop…
We propose a definition for the Lorentzian Jackiw-Teitelboim (JT) gravity path integral that includes Lorentzian topology changing configurations. The construction is inspired by the bosonic string genus expansion on singular Lorentzian worldsheets, with geometries known as lightcone diagrams playing a prominent role. The Lorentzian path integral is defined through a suitable analytic continuation of the Euclidean path integral, and includes metrics that become degenerate at isolated points allowing for Lorentzian topology changing transitions. We discuss the relation between Euclidean JT ampl…
We study the cosmological sector of the Lorentzian Barrett-Crane (BC) model coupled to a free massless scalar field in its Group Field Theory (GFT) formulation, corresponding to the mean-field hydrodynamics obtained from coherent condensate states. The relational evolution of the condensate with respect to the scalar field yields effective dynamics of homogeneous and isotropic cosmologies, similar to those previously obtained in $\text{SU}(2)$-based EPRL-like models. Also in this manifestly Lorentzian setting, in which only continuous $\text{SL}(2,\mathbb{C})$-representations are used, we obta…
The model of Lorentzian three-dimensional dynamical triangulations provides a non-perturbative definition of three-dimensional quantum gravity. The theory has two phases: a weak-coupling phase with quantum fluctuations around a ``semiclassical'' background geometry which is generated dynamically despite the fact that the formulation is explicitly background-independent, and a strong-coupling phase where ``classical'' space disintegrates into a foam of baby universes.
We construct Born-Infeld (BI) type gravity theories which describe tree-level unitary (non-ghost and non-tachyonic) massless spin-2 modes around their maximally symmetric vacua in four dimensions. Building unitary BI actions around flat vacuum is straightforward; but, this is a complicated task around (anti)-de Sitter backgrounds. In this work, we solve the issue and give details of constructing perturbatively viable determinantal BI theories. It is interesting that the Gauss-Bonnet combination, which is a total derivative in four dimensions, plays an important role in the construction of viab…
We employ the curvature expansion of the quantum effective action for gravity-matter systems to construct graviton-mediated scattering amplitudes for non-minimally coupled scalar fields in a Minkowski background. By design, the formalism parameterises all quantum corrections to these processes and is manifestly gauge-invariant. The conditions resulting from UV-finiteness, unitarity, and causality are analysed in detail and it is shown by explicit construction that the quantum effective action provides sufficient room to meet these structural requirements without introducing non-localities or h…
In the gauge-theoretic formulation of gravity the cubic vertex becomes simple enough for some graviton scattering amplitudes to be computed using Berends-Giele-type recursion relations. We present such a computation for the current with all same helicity on-shell gravitons. Once the recursion relation is set up and low graviton number cases are worked out, a natural guess for the solution in terms of a sum over trees presents itself readily. The solution can also be described either in terms of the half-soft function familiar from the 1998 paper by Bern, Dixon, Perelstein and Rozowsky or as a …
A well-defined regularized path integral for Lorentzian quantum gravity in three and four dimensions is constructed, given in terms of a sum over dynamically triangulated causal space-times. Each Lorentzian geometry and its associated action have a unique Wick rotation to the Euclidean sector. All space-time histories possess a distinguished notion of a discrete proper time. For finite lattice volume, the associated transfer matrix is self-adjoint and bounded. The reflection positivity of the model ensures the existence of a well-defined Hamiltonian. The degenerate geometric phases found previ…
We explore the relationship between the quantum state of a compact matter source and of its asymptotic graviton field. For a matter source in an energy eigenstate, the graviton state is determined at leading order by the energy eigenvalue. Insofar as there are no accidental energy degeneracies there is a one to one map between graviton states on the boundary of spacetime and the matter source states. A typical semiclassical matter source results in an entangled asymptotic graviton state. We exhibit a purely quantum gravitational effect which causes the subleading asymptotic behavior of the gra…
A new set of projection operators is constructed to suitably handle non-relativistic theories of gravity with anisotropic scaling, including the ones with parity-violating terms. This alternative procedure allows us to discuss unitarity and spectral properties for different formulations of the Horava-Lifshitz gravity. This task, that generally involves lengthy algebraic steps, becomes more systematic and greatly simplified in terms of the projectors we work out. Moreover, this procedure allows us to fix the number of propagating degrees of freedom and the structure of gauge symmetries is readi…
Within the Asymptotic Safety scenario, we discuss whether Quantum Einstein Gravity (QEG) can give rise to a semi-classical regime of propagating physical gravitons (gravitational waves) governed by an effective theory which complies with the standard rules of local quantum field theory. According to earlier investigations based on single-metric truncations there is a tension between this requirement and the condition of Asymptotic Safety since the former (latter) requires a positive (negative) anomalous dimension of Newton's constant. We show that the problem disappears using the bi-metric ren…
