In recent years, a growing momentum has been gained by the emergent gravity framework. Within the latter, the very concepts of geometry and gravitational interaction are not seen as elementary aspects of Nature but rather as collective phenomena associated to the dynamics of more fundamental objects. In this paper we want to further explore this possibility by proposing a model of emergent Lorentzian signature and scalar gravity. Assuming that the dynamics of the fundamental objects can give rise in first place to a Riemannian manifold and a set of scalar fields we show how time (in the sense of hyperbolic equations) can emerge as a property of perturbations dynamics around some specific class of solutions of the field equations. Moreover, we show that these perturbations can give rise to a spin-0 gravity via a suitable redefinition of the fields that identifies the relevant degrees of freedom. In particular, we find that our model gives rise to Nordstrom gravity. Since this theory is invariant under general coordinate transformations, this also shows how diffeomorphism invariance (albeit of a weaker form than the one of general relativity) can emerge from much simpler systems.
We present evidence that recent numerical results from the reduced classical equations of the Lorentzian IIB matrix model can be interpreted as corresponding to the emergence of an expanding universe. In addition, we propose an effective metric to describe the emerging (3+1)-dimensional spacetime. This metric gives, at all times, finite values for the Ricci and Kretschmann curvature scalars. With these results, we are able to give a heuristic discussion of the origin of the Universe in the context of the IIB matrix model.
The Lorentzian type IIB matrix model is a promising candidate for a non-perturbative formulation of superstring theory. However, it was found recently that a Euclidean space-time appears in the conventional large-$N$ limit. In this work, we add a Lorentz invariant mass term to the original model and consider a limit, in which the coefficient of the mass term vanishes at large $N$. By performing complex Langevin simulations to overcome the sign problem, we observe the emergence of expanding space-time with the Lorentzian signature.
The type IIB matrix model is a promising candidate for a nonperturbative formulation of superstring theory. As such, it is expected to explain the origin of space--time and matter at the same time. This has been partially demonstrated by the previous Monte Carlo studies on the Lorentzian version of the model, which suggested the emergence of (3+1)-dimensional expanding space--time. Here we investigate the same model by solving numerically the classical equation of motion, which is expected to be valid at late times since the action becomes large due to the expansion of space. Many solutions ar…
The Lorentzian type IIB matrix model is a promising candidate for a non-perturbative formulation of superstring theory. In previous studies, Monte Carlo calculations provided interesting results indicating the spontaneous breaking of SO(9) to SO(3) and the emergence of (3+1)-dimensional space-time. However, an approximation was used to avoid the sign problem, which seemed to make the space-time structure singular. In this talk, we report our results obtained by using the complex Langevin method to overcome the sign problem instead of using this approximation. In particular, we discuss the emer…
Holographic entanglement entropy was recently recast in terms of Riemannian flows or 'bit threads'. We consider the Lorentzian analog to reformulate the 'complexity=volume' conjecture using Lorentzian flows -- timelike vector fields whose minimum flux through a boundary subregion is equal to the volume of the homologous maximal bulk Cauchy slice. By the nesting of Lorentzian flows, holographic complexity is shown to obey a number of properties. Particularly, the rate of complexity is bounded below by conditional complexity, describing a multi-step optimization with intermediate and final targe…
We expand on the idea that spacetime signature should be treated as a dynamical degree of freedom in quantum field theory. It has been argued that the probability distribution for signature, induced by massless free fields, is peaked at the Lorentzian value uniquely in D=4 dimensions. This argument is reviewed, and certain consistency constraints on the generalized signature (i.e. the tangent space metric $η_{ab}(x)=\mbox{diag}[e^{iθ(x)},1,1,1]$) are derived. It is shown that only one dynamical "Wick angle" $θ(x)$ can be introduced in the generalized signature, and the magnitude of fluctuation…
It is suggested that not only the curvature, but also the signature of spacetime is subject to quantum fluctuations. A generalized D-dimensional spacetime metric of the form $g_{μν}=e^a_μη_{ab} e^b_ν$ is introduced, where $η_{ab} = diag\{e^{iθ},1,...,1\}$. The corresponding functional integral for quantized fields then interpolates from a Euclidean path integral in Euclidean space, at $θ=0$, to a Feynman path integral in Minkowski space, at $θ=π$. Treating the phase $e^{iθ}$ as just another quantized field, the signature of spacetime is determined dynamically by its expectation value. The comp…
We introduce a new spinorial, BF-like action for the Einstein gravity. This is a first, up to our knowledge, 2-form action which describes the real, Lorentzian gravity and uses only the self-dual connection. In the generic case, the corresponding classical canonical theory is equivalent to the Einstein-Ashtekar theory plus the reality conditions.
Shadow signatures and energy accumulation in Lorentzian-Euclidean black holes
Istituto Nazionale di Fisica Nucleare（意大利国家核物理研究所）；Università degli Studi di Napoli “Federico II”（那不勒斯费德里科二世大学）；Scuola Superiore Meridionale（南方高等学院）；RIKEN iTHEMS（理化学所 iTHEMS）
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…
The Lorentzian metric structure used in any field theory allows one to implement the relativistic notion of causality and to define a notion of time dimension. This article investigates the possibility that at the microscopic level the metric is Riemannian, i.e. locally Euclidean, and that the Lorentzian structure, that we usually consider as fundamental, is in fact an effective property that emerges in some regions of a 4-dimensional space with a positive definite metric. In such a model, there is no dynamics nor signature flip across some hypersurface; instead, all the fields develop a Loren…
General hypersurface layers are considered in order to describe brane-worlds and shell cosmologies. No restriction is placed on the causal character of the hypersurface which may thus have internal changes of signature. Strengthening the results in our previous letter [1], we confirm that a good, regular and consistent description of signature change is achieved in these brane/shells scenarios, while keeping the hypersurface and the bulk completely regular. Our formalism allows for a unified description of the traditional timelike branes/shells together with the signature-changing, or pure nul…
We consider an empty (4+1) dimensional Kaluza-Klein universe with a negative cosmological constant and a Robertson-Walker type metric. It is shown that the solutions to Einstein field equations have degenerate metric and exhibit transitioins from a Euclidean to a Lorentzian domain. We then suggest a mechanism, based on signature transition which leads to compactification of the internal space in the Lorentzian region as $a \sim |Λ|^{1/2}$. With the assumption of a very small value for the cosmological constant we find that the size of the universe $R$ and the internal scale factor $a$ would be…
The emergence of (3+1)-dimensional expanding spacetime from complex Langevin simulations of the Lorentzian type IIB matrix model with deformations
National Technical University of Athens（雅典国立技术大学）；Setsunan University（摂南大学）；Università degli Studi di Milano-Bicocca（米兰比可卡大学）；Istituto Nazionale di Fisica Nucleare (INFN)（意大利国家核物理研究所）；High Energy Accelerator Research Organization（高能加速器研究机构）；SOKENDAI（日本综合研究大学院大学）；Shizuoka University（静冈大学）
The constructive method of conformal blocks is developed for the construction of global solutions for two-dimensional metrics having one Killing vector. The method is proved to yeild a smooth universal covering space with a smooth pseudo-Riemannian metric. The Schwarzschild, Reisner--Nordstrom solutions, extremal black hole, dilaton black hole, and constant curvature surfaces are considered as examples.
Local flatness is a property shared by all the spin foam models. It ensures that the theory's fundamental building blocks are flat by requiring locally trivial parallel transport. In the context of simplicial Lorentzian spin foam theory, we show that local flatness is the main responsible for the emergence of geometry independently of the details of the spin foam model. We discuss the asymptotic analysis of the EPRL spin foam amplitudes in the large quantum number regime, highlighting the interplay with local flatness.
I suggest in this letter a new strategy to attack the problem of the reality conditions in the Ashtekar approach to classical and quantum general relativity. By writing a modified Hamiltonian constraint in the usual $SO(3)$ Yang-Mills phase space I show that it is possible to describe space-times with Lorentzian signature without the introduction of complex variables. All the features of the Ashtekar formalism related to the geometrical nature of the new variables are retained; in particular, it is still possible, in principle, to use the loop variables approach in the passage to the quantum t…
Thermal partition functions for gravitational systems have traditionally been studied using Euclidean path integrals. But in Euclidean signature the gravitational action suffers from the conformal factor problem, which renders the action unbounded below. This makes it difficult to take the Euclidean formulation as fundamental. However, despite their familiar association with periodic imaginary time, thermal gravitational partition functions can also be described by real-time path integrals over contours defined by real Lorentzian metrics. The one caveat is that we should allow certain codimens…
We study in this paper a new approach to the problem of relating solutions to the Einstein field equations with Riemannian and Lorentzian signatures. The procedure can be thought of as a "real Wick rotation". We give a modified action for general relativity, depending on two real parameters, that can be used to control the signature of the solutions to the field equations. We show how this procedure works for the Schwarzschild metric and discuss some possible applications of the formalism in the context of signature change, the problem of time, black hole thermodynamics...
Given the class of Finsler spaces with Lorentzian signature $(M,L)$ on a manifold $M$ endowed with a timelike vector field $\mathcal{X}$ satisfying $g_{(x,y)}(\mathcal{X},\mathcal{X})<0$ at any point $(x,y)$ of the slit tangent bundle, a pseudo-Riemannian metric defined on $M$ of signature $n-1$ is associated to the fundamental tensor $g$. Furthermore, an affine, torsion free connection is associated to the Chern connection determined by $L$. The definition of the average connection does not make use of $\mathcal{X}$. Therefore, there is no direct relation between these two averaged objects.
A key insight used in developing the theory of Causal Dynamical Triangulations (CDTs) is to use the causal (or light-cone) structure of Lorentzian manifolds to restrict the class of geometries appearing in the Quantum Gravity (QG) path integral. By exploiting this structure the models developed in CDTs differ from the analogous models developed in the Euclidean domain, models of (Euclidean) Dynamical Triangulations (DT), and the corresponding Lorentzian results are in many ways more "physical". In this paper we use this insight to formulate a Lorentzian signature model that is analogous to the…
Four-dimensional manifolds with changing signature are obtained by taking the large $N$ limit of fuzzy $CP^2$ solutions to a Lorentzian matrix model. The regions of Lorentzian signature give toy models of closed universes which exhibit cosmological singularities. These singularities are resolved at finite $N$, as the underlying $CP^2$ solutions are expressed in terms of finite matrix elements.
We study a model of free massless scalar fields on a two dimensional cylinder with metric that admits a change of signature between Lorentzian and Euclidean type (ET), across the two timelike hypersurfaces (with respect to Lorentzian region). Considering a long strip-shaped region of the cylinder, denoted by an angle θ, as the signature changed region it is shown that the energy spectrum depends on the angle θand in a sense differs from ordinary one for low energies. Morever diffeomorphism algebra of corresponding infinite conserved charges is different from '' Virasoro'' algebra and approache…
A regular vacuum solution in 5D gravity on the principal bundle with the U(1) structural group is proposed as a 4D wormhole. This solution has two null hypersurfaces where an interchange of the sign of some 5D metric components happens. For a 4D observer living on the base of this principal bundle this is a wormhole with two asymptotically flat Lorentzian (Euclidean) spacetimes connected by a Euclidean (Lorentzian) throat. The 4D Lorentzian observer sees these two null hypersurfaces as electric charges.
In this paper we study some aspects of classical and quantum cosmology in the novel-Gauss-Bonnet (nGB) gravity in four space-time dimensions. Starting with a generalised Friedmann-Lemaître-Robertson Walker (FLRW) metric respecting homogeneity and isotropicity in arbitrary space-time dimension $D$, we find the action of theory in four spacetime dimension where the limit $D\to4$ is smoothly obtained after an integration by parts. The peculiar rescaling of Gauss-Bonnet coupling by factor of $D-4$ results in a non-trivial contribution to the action. We study the system of equation of motion to fir…
We give a relativistic spin network model for quantum gravity based on the Lorentz group and its q-deformation, the Quantum Lorentz Algebra. We propose a combinatorial model for the path integral given by an integral over suitable representations of this algebra. This generalises the state sum models for the case of the four-dimensional rotation group previously studied in gr-qc/9709028. As a technical tool, formulae for the evaluation of relativistic spin networks for the Lorentz group are developed, with some simple examples which show that the evaluation is finite in interesting cases. We c…
We present the construction of the partition function of 3-dimensional gravity in the Lorentzian regime as a state sum model over a triangulation. This generalize the work of Ponzano and Regge to the case of Lorentzian signature.
We study the classical and quantum evolution of a universe in which the matter source is a massive Dirac spinor field and the universe is described by a Bianchi type I metric. We focus attention on those classical solutions that admit a degenerate metric in which the scale factors have smooth behavior in transition from a Euclidean to a Lorentzian domain and show that this transition happens when the cosmological constant, $Λ$, is negative. The resulting quantum cosmology and the corresponding Wheeler-DeWitt equation are also studied and closed form expressions for the wave functions of the un…
We proposes an alternative model of duality symmetry, based on the previously obtained divergence theory, including an scalar field, an internal vector and a metric signature. At some small scale an effective scalar field equation has appeared whose potential acts like a Higgs one, where the metric signature plays the role of an order parameter. Non-vanishing Vacuum condensation of this Higgs field occurs once a signature change from Euclidean to Lorentzian is formed. The mass scale of Higgs field excitations around this vacuum may contribute, in the Lorentzian sector, to the cosmological cons…
We study the classical and quantum cosmology of a 4+1-dimensional space-time with a non-zero cosmological constant coupled to a self interacting massive spinor field. We consider a spatially flat Robertson-Walker universe with the usual scale factor $R(t)$ and an internal scale factor $a(t)$ associated with the extra dimension. For a free spinor field the resulting equations admit exact solutions, whereas for a self interacting spinor field one should resort to a numerical method for exhibiting their behavior. These solutions give rise to a degenerate metric and exhibit signature transition fr…
We consider general relativity with cosmological constant minimally coupled to the electromagnetic field and assume that the four-dimensional space-time manifold is a warped product of two surfaces with Lorentzian and Euclidean signature metrics. Field equations imply that at least one of the surfaces must be of constant curvature leading to the symmetry of the metric (``spontaneous symmetry emergence''). We classify all global solutions in the case when the Lorentzian surface is of constant curvature. These solutions are invariant with respect to the Lorentz SO(1,2) or Poincare IO(1,1) groups…
We study manifolds with Lorentzian signature and prove that all scalar curvature invariants of all orders vanish in a higher-dimensional Lorentzian spacetime if and only if there exists an aligned non-expanding, non-twisting, geodesic null direction along which the Riemann tensor has negative boost order.
We generalize a model recently proposed for Euclidean quantum gravity to the case of Lorentzian signature. The main features of the Euclidean model are preserved in the Lorentzian one. In particular, the boundary Hilbert space matches the one of SU(2) loop quantum gravity. As in the Euclidean case, the model can be obtained from the Lorentzian Barrett-Crane model from a flipping of the Poisson structure, or alternatively, by adding a topological term to the action and taking the small Barbero-Immirzi parameter limit.
Covariant models of loop quantum gravity generically imply dynamical signature change at high density. This article presents detailed derivations that show the fruitful interplay of this new kind of signature change with wave-function proposals of quantum cosmology, such as the no-boundary and tunneling proposals. In particular, instabilities of inhomogeneous perturbations found in a Lorentzian path-integral treatment are naturally cured. Importantly, dynamical signature change does not require Planckian densities when off-shell instantons are relevant.
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.
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…
We suggest a new scenario of gravitation in which gravity at the fundamental level is described by a Riemannian (i.e. locally Euclidean) theory without the notion of time. The Lorentzian metric structure and the notion of time emerge as effective properties at long distances. On the other hand, at short distances, higher derivative terms compatible with the Riemannian diffeomorphism become important and the system is described by a power-counting renormalizable Riemannian theory.
In this paper we study the Lorentzian path-integral of Gauss-Bonnet gravity in the mini-superspace approximation in four spacetime dimensions and investigate the transition amplitude from one configuration to another. Past studies motivate us on imposing Neumann boundary conditions on initial boundary as they lead to stable behaviour of fluctuations. The transition amplitude is computed exactly while incorporating the non-trivial contribution coming from the Gauss-Bonnet sector of gravity. A saddle-point analysis involving usage of Picard-Lefschetz methods allow us to gain further insight of t…
In this article we use the idea of algorithmic complexity (AC) to study various cosmological scenarios, and as a means of quantizing the gravitational interaction. We look at 5D and 7D cosmological models where the Universe begins as a higher dimensional Planck size spacetime which fluctuates between Euclidean and Lorentzian signatures. These fluctuations are governed by the AC of the two different signatures. At some point a transition to a 4D Lorentzian signature Universe occurs, with the extra dimensions becoming ``frozen'' or non-dynamical. We also apply the idea of algorithmic complexity …
Recent developments in quantum gravity have shown the Lorentzian treatment to be a fruitful approach towards the emergence of macroscopic spacetimes. In this paper, we discuss another related aspect of the Lorentzian treatment: we argue that collective quantum coherence may provide a simple mechanism for highlighting symmetric configurations over generic non-symmetric ones. After presenting the general framework of the mechanism, we show the phenomenon in some concrete simple examples in the randomly connected tensor network, which is tightly related to a certain model of quantum gravity, i.e.…
In a previous article [JHEP 1111 (2011) 072; arXiv:1108.4965] we have developed a Lorentzian version of the Quantum Regge Calculus in which the significant differences between simplices in Lorentzian signature and Euclidean signature are crucial. In this article we extend a central result used in the previous article, regarding the realizability of Lorentzian triangles, to arbitrary dimension. This technical step will be crucial for developing the Lorentzian model in the case of most physical interest: 3+1 dimensions. We first state (and derive in an appendix) the realizability conditions on t…
We explore potential uses of physics formulated in Kleinian (i.e., $2+2$) signature spacetimes as a tool for understanding properties of physics in Lorentzian (i.e., $3+1$) signature. Much as Euclidean (i.e., $4+0$) signature quantities can be used to formally construct the ground state wavefunction of a Lorentzian signature quantum field theory, a similar analytic continuation to Kleinian signature constructs a state of low particle flux in the direction of analytic continuation. There is also a natural supersymmetry algebra available in $2+2$ signature, which serves to constrain the structur…
In this paper we perform a full gauge-fixing of the phase space of four dimensional General Relativity (GR) of Lorentzian signature for the time symmetric case, using the CDJ variables. In particular, the Gauss' law constraint in the chosen gauge meets the conditions of the Cauchy-Kovalevskaya theorem for first order, quasilinear PDEs. This implies the existence of a unique analytic solution to the initial value constraints problem in some region of 3-space, featuring four free functions per spacetime point. This result constitutes a step toward addressal of the reduced phase space problem of …
Gravitational model of non-minimally coupled Brans Dicke (BD) scalar field $ϕ$ with dynamical unit time-like four vector field is used to study flat Robertson Walker (RW) cosmology in the presence of variable cosmological parameter $V(ϕ)=Λϕ.$ Aim of the paper is to seek cosmological models which exhibit metric signature transition. The problem is studied in both classical and quantum cosmological approach with large values of BD parameter $ω>>1$. Scale factor of RW metric is obtained as $R(t)=6\sqrt{\frac{3}Λ}\cosh\big(\frac{t}{4}\sqrt{\fracΛ{3}}\big)$ which describes nonsingular inflationary …
New non trivial solutions to the Lorentzian-signature symmetry reduced Bianchi VIII Wheeler DeWitt equation for Hartle Hawking ordering parameters $ \pm 2\sqrt{33} $ are obtained using an Euclidean-signature semi classical method. Using the aforementioned method we define 'excited' states for the Bianchi VIII wave function and construct leading order 'excited' states which appear to be a hybrid of scattering and bound states. Also six new solutions for the $Λ\ne 0$ Bianchi VIII Euclidean-signature Hamilton Jacobi equation are presented which can be used to construct semi-classical states corre…
We apply in a novel fashion a modified semi-classical method to the Bianchi II and VII$_{h=0}$ models when a cosmological constant, aligned electromagnetic field and stiff matter are present. Additionally we study the non-commutative quantum Bianchi II models when an aligned electromagnetic field is included. Through the use of the Euclidean-signature semi classical method we find a plethora of new solutions to these model's corresponding Lorentzian signature Wheeler DeWitt equations which we can interpret qualitatively. These new solutions for the aforementioned models involving matter source…
We consider a wide class of two-dimensional metrics having one Killing vector. The method is proposed for the construction of maximally extended surfaces with the given Riemannian metric which is the analog of the conformal block method for two-dimensional Lorentzian signature metrics. The Schwarzschild solution is considered as an example.
We present an example of emergent spacetime as the hydrodynamic limit of a more fundamental microscopic theory. The low-energy, long-wavelength limit in our model is dominated by collective variables that generate an effective Lorentzian metric. This system naturally exhibits a microscopic mechanism allowing us to perform controlled signature change between Lorentzian and Riemannian geometries. We calculate the number of particles produced from a finite-duration Euclidean-signature event, where we take the position that to a good approximation the dynamics is dominated by the evolution of the …
