Timelike Extra Dimension Research Articles

Equivalence of EFTs and timelike extra dimensions

A flat 3-brane probing a five-dimensional anti–de Sitter space has the same symmetries and symmetry breaking pattern as one probing a five-dimensional de Sitter space with two time directions. Despite the seemingly different physical setups, we show that the effective field theory of the brane bending mode is identical in both cases. In particular, despite “wrong” signs in the Dirac-Born-Infeld (DBI) action for the two-time de Sitter case, the theories have the same S-matrix. This is consistent with the expectation that effective field theories are determined solely by their degrees of freedom and pattern of symmetry breaking, even in the case of spacetime symmetries. We comment further on the equivalence between the Weyl/dilaton and DBI representations of the EFT of broken conformal symmetry. Published by the American Physical Society 2024

Traversable Lorentzian wormhole on the Shtanov-Sahni braneworld with matter obeying the energy conditions

In this paper we have explored the possibility of constructing a traversable wormhole on the Shtanov-Sahni braneworld with a timelike extra dimension. We find that the Weyl curvature singularity at the throat of the wormhole can be removed with physical matter satisfying the NEC ρ + p ≥ 0, even in the absence of any effective Λ-term or any type of charge source on the brane. (The NEC is however violated by the effective matter description on the brane arising due to effects of higher dimensional gravity.) Besides satisfying NEC the matter constituting the wormhole also satisfies the Strong Energy Condition (SEC), ρ + 3p ≥ 0, leading to the interesting possibility that normal matter on the brane may be harnessed into a wormhole. Incidentally, these conditions also need to be satisfied to realize a non-singular bounce and cyclic cosmology on the brane [1] where both past and future singularities can be averted. Thus, such a cyclic universe on the brane, constituted of normal matter can naturally contain wormholes. The wormhole shape function on the brane with a time-like extra dimension represents the tubular structure of the wormhole spreading out at large radial distances much better than in wormholes constructed in a braneworld with a spacelike extra dimension and have considerably lower mass resulting in minimization of the amount of matter required to construct a wormhole. Wormholes in the Shtanov-Sahni (SS) braneworld also have sufficiently low tidal forces, facilitating traversability. Additionally they are found to be stable and exhibit a repulsive geometry. We are left with the intriguing possibility that both types of curvature singularity can be resolved with the SS model, which we discuss at the end of the concluding section.

Complexifying the Spacetime Algebra by Means of an Extra Timelike Dimension: Pin, Spin and Algebraic Spinors

Because of the isomorphism \(C \ell _{1,3}({\mathbb {C}})\cong C \ell _{2,3}({\mathbb {R}})\), it is possible to complexify the spacetime Clifford algebra \(C \ell _{1,3}({\mathbb {R}})\) by adding one additional timelike dimension to the Minkowski spacetime. In a recent work we showed how this treatment provide a particular interpretation of Dirac particles and antiparticles in terms of the new temporal dimension. In this article we thoroughly study the structure of the real Clifford algebra \(C \ell _{2,3}({\mathbb {R}})\) paying special attention to the isomorphism \(C \ell _{1,3}({\mathbb {C}})\cong C \ell _{2,3}({\mathbb {R}})\) and the embedding \(C \ell _{1,3}({\mathbb {R}})\subseteq C \ell _{2,3}({\mathbb {R}})\). On the first half of this article we analyze the Pin and Spin groups and construct an injective mapping \({\text {Pin}}(1,3)\hookrightarrow {\text {Spin}}(2,3)\), obtaining in particular elements in \({\text {Spin}}(2,3)\) that represent parity and time reversal. On the second half of this paper we study the spinor space of the algebra and prove that the usual structure of complex spinors in \(C \ell _{1,3}({\mathbb {C}})\) is reproduced by the Clifford conjugation inner product for real spinors in \(C \ell _{2,3}({\mathbb {R}})\).

Particle-antiparticle duality from an extra timelike dimension

It is a well known fact that the usual complex structure on the real Clifford Algebra (CA) of Minkowski spacetime can be obtained by adding an extra time-like dimension, instead of the usual complexification of the algebra. In this article we explore the consequences of this approach and reinterpret known results in this new context. We observe that Dirac particles and antiparticles at rest can be interpreted as eigenstates of the generator of rotations in the plane formed by the two time-like coordinates and find an effective finite scale for the extra dimension when no EM fields are present (without postulating compactness). In the case of non-vanishing EM fields, we find a gauge condition to preserve such a scale.

New asymptotic Anti-de Sitter solution with a timelike extra dimension in 5D relativity

In 5D relativity, the usual 4D cosmological constant is determined by the extra dimension. If the extra dimension is spacelike, one can get a positive cosmological constant Λ and a 4D de Sitter (dS) space. In this paper we present that, if the extra dimension is timelike oppositely, the negative Λ will be emerged and the induced 4D space will be an asymptotic Anti-de Sitter (AdS). Under the minimum assumption, we solve the Kaluza–Klein equation RAB=0 in a canonical system and obtain the AdS solution in a general case. The result shows that an AdS space is induced naturally from a Kaluza–Klein manifold on a hypersurface (brane). The Lagrangian of test particle indicates the equation of motion can be geodesics if the 4D metric is independent of extra dimension. The causality is well respected because it is appropriately defined by a null higher dimensional interval. In this 5D relativity, the holographic principle can be used safely because the brane is asymptotic Euclidean AdS in the bulk. We also explore some possible holographic duality implications about the field/operator correspondence and the two-points correlation functions.

Stability of Einstein static state universe in the spatially flat branemodels

Abstract With the assumption that a perfect fluid with a constant equation of state is the only energy component on the brane, we study the stability of Einstein static state solution under homogeneous and inhomogeneous scalar perturbations in both spatially flat Randall–Sundrum (RS) and Shtanov–Sahni (SS) braneworlds. We find that if the perfect fluid has a phantom-like property and the “Weyl fluid” originating from the projection of the bulk Weyl tensor onto the brane behaves like a radiation with positive energy density, the Einstein static state solution is stable in the SS braneworld, but unstable in the RS one. Furthermore, we demonstrate that the static state solution is also stable in the bulk with a timelike extra dimension. Thus, in the model where the extra dimension is timelike, our universe can stay at the Einstein static state past-eternally, which means that the big bang singularity might be resolved successfully by an emergent scenario.

Bianchi IX dynamics in bouncing cosmologies: homoclinic chaos and the BKL conjecture

We examine the dynamics of a Bianchi IX model with three scale factors on a 4-dim Lorentzian brane embedded in a 5-dim conformally flat empty bulk with a timelike extra dimension. The matter content is a pressureless perfect fluid restricted to the brane, with the embedding consistently satisfying the Gauss–Codazzi equations. The 4-dim Einstein equations on the brane reduce to a 6-dim Hamiltonian dynamical system with additional terms (due to the bulk–brane interaction) that avoid the singularity and implement nonsingular bounces in the model. We examine the complex Bianchi IX dynamics in its approach to the neighborhood of the bounce which replaces the cosmological singularity of general relativity. The phase space of the model presents (i) two critical points (a saddle-center–center and a center–center–center) in a finite region of phase space, (ii) two asymptotic de Sitter critical points at infinity, one acting as an attractor to late-time acceleration and (iii) a 2-dim invariant plane, which together organize the dynamics of the phase space. The saddle-center–center engenders in the phase space the topology of stable and unstable 4-dim cylinders R × S3, where R is a saddle direction and S3 is the center manifold of unstable periodic orbits, the latter being the nonlinear extension of the center–center sector. By a proper canonical transformation the degrees of freedom of the dynamics are separated into one degree connected with the expansion/contraction of the scales of the model, and two rotational degrees of freedom associated with the center manifold S3. The typical dynamical flow is thus an oscillatory mode about the orbits of the invariant plane. The stable and unstable cylinders are spanned by oscillatory orbits about the separatrix towards the bounce, leading to the homoclinic transversal intersection of the cylinders, as shown numerically in two distinct simulations. The homoclinic intersection manifold has the topology of R × S2 consisting of homoclinic orbits biasymptotic to the center manifold S3. This behavior defines a chaotic saddle associated with S3, indicating that the intersection points of the cylinders have the nature of a Cantor set with compact support S2. This is an invariant signature of chaos in the model. We discuss the connection between these properties of the dynamics, namely the oscillatory approach to the bounce together with its chaotic behavior, and analogous features present in the BKL conjecture in general relativity.

Bounded scalar perturbations in bouncing brane world cosmologies

We examine the dynamics of scalar perturbations in closed Friedmann-Lema\^itre-Robertson- Walker (FLRW) universes in the framework of Brane World theory with a timelike extra dimension. In this scenario, the unperturbed Friedmann equations contain additional terms arising from the bulk-brane interaction that implement non-singular bounces in the models with a cosmological constant and non-interacting perfect fluids. The structure of the phase-space of the models allows for two basic configurations, namely, one bounce solutions or eternal universes. Assuming that the matter content of the model is given by dust and radiation, we derive the dynamical field equations for scalar hydrodynamical perturbations considering either a conformally flat (de Sitter) bulk or a perturbed bulk. We perform a numerical analysis which can shed some light on the study of cosmological scalar perturbations in bouncing brane world models. From a mathematical point of view we show that although the bounce enhances the amplitudes of scalar perturbations for one bounce models in the case of a de Sitter bulk, the amplitudes of the perturbations remain sufficiently small and bounded relative to the background values up to a certain scale. For one bounce models in the case of a perturbed bulk the amplitudes of all perturbations (apart from the Weyl fluid energy density) remain sufficiently small and bounded relative to the background values for any scale of the perturbations. We also discuss and compare the stability and bounded behaviour of the perturbations in the late accelerated phase of one bounce solutions. For eternal universes we argue that some of these features are maintained only for early times (typically of the order of the first bounce). In this sense we show that eternal solutions are highly unstable configurations considering the background model of this paper.

Bouncing model in brane world theory

We examine the nonlinear dynamics of a closed Friedmann-Robertson-Walker universe in the framework of Brane World formalism with a timelike extra dimension. In this scenario, the Friedmann equations contain additional terms arising from the bulk-brane interaction which provide a concrete model for nonsingular bounces in the early phase of the Universe. We construct a nonsingular cosmological scenario sourced with dust, radiation and a cosmological constant. The structure of the phase space shows a nonsingular orbit with two accelerated phases, separated by a smooth transition corresponding to a decelerated expansion. Given observational parameters we connect such phases to a primordial accelerated phase, a soft transition to Friedmann (where the classical regime is valid), and a graceful exit to a de Sitter accelerated phase.

Friedmann-Like Equations for High Energy Area of Universe

In this paper, evolution of the high energy area of universe, through the scenario of 5 dimensional (5D) universe, has been studied. For this purpose, we solve Einstein equations for 5D metric and 5D perfect fuid to derive Friedmann-like equations. Then we obtain the evolution of scale factor and energy density with respect to both space-like and time-like extra dimensions. We obtain the novel equations for the space-like extra dimension and show that the matter with zero pressure cannot exist in the bulk. Also, for dark energy fuid and vacuum fluid, we have both accelerated expansion and contraction in the bulk.

GRAVITATIONAL COLLAPSE AND BLACK HOLE THERMODYNAMICS IN BRANEWORLD SCENARIO

We examine the dynamics of the gravitational collapse in a 4-dim Lorentzian brane embedded in a 5-dim bulk with an extra timelike dimension. By considering the collapse of pure dust on the brane we derive a bouncing FLRW interior solution and match it with a corrected Schwarzschild exterior geometry. In the physical domain considered for the parameters of the solution, the analytical extension is built, exhibiting an exterior event horizon and a Cauchy horizon, analogous to the Reissner–Nordstrom solution. For such an exterior geometry we examine the effects of the bulk-brane corrections in the Hawking radiation. In this scenario the model extends Bekenstein's black hole geometrical thermodynamics for quasi-extremal configurations, with an extra work term in the laws associated with variations of the brane tension. We also propose a simple statistical mechanics model for the entropy of the bouncing collapsed matter by quantizing its fluctuations and constructing the associated partition function. This entropy differs from the geometrical entropy by an additive constant proportional to the area of the extremal black hole and satisfies an analogous first law of thermodynamics. A possible connection between both entropies is discussed.

Bulk Scale Factor at Very Early Universe

In this paper we propose a higher dimensional Cosmology based on FRW model and brane-world scenario. We consider the warp factor in the brane-world scenario as a scale factor in 5-dimensional generalized FRW metric, which is called as bulk scale factor, and obtain the evolution of it with space-like and time-like extra dimensions. It is then showed that, additional space-like dimensions can produce exponentially bulk scale factor under repulsive strong gravitational force in the empty universe at a very early stage.

Alternative Inflationary Scenario Due to Compact Extra Dimensions

The main goal of this paper is to give an alternative interpretation of space-like and time-like extra dimensions as a primary factor for inflation in the early universe. We introduce the 5-dimensional perfect fluid and compare the energy-momentum tensor for the bulk scalar field with space-like and time-like extra dimensions. It is shown, that additional dimensions can imply to negative pressure in the slow roll regime in the early higher-dimensional world.

Erratum: Braneworld dynamics in Einstein-Gauss-Bonnet gravity [Phys. Rev. D76, 104028 (2007)

We discuss the cosmological evolution of a braneworld in five dimensional Gauss-Bonnet gravity. Our discussion allows the fifth (bulk) dimension to be space-like as well as time-like. The resulting equations of motion have the form of a cubic equation in the (H^2,(\rho+\sigma)^2) plane, where \sigma is the brane tension and \rho is the matter density. This allows us to conduct a comprehensive pictorial analysis of cosmological evolution for the Gauss-Bonnet brane. The many interesting properties of this braneworld include the possibility of accelerated expansion at late times. For a finite region in parameter space the accelerated expansion can be phantom-like so that w < -1. At late times, this branch approaches de Sitter space (w = -1) and avoids the big-rip singularities usually present in phantom models. For a time-like extra dimension the Gauss-Bonnet brane can bounce and avoid the initial singularity.

Braneworlds with a timelike extra dimension

In this work, we consider a braneworld model with a timelike extra dimension. There are strong constraints to the parameter values of such a model resulting from the claim that there must be a physical solution to the Friedmann equation at least between now and the time of recombination. We fitted the model to supernova type Ia data and checked the consistency of the result with other observations. For parameter values that are consistent with observations, the braneworld model is indistinguishable from a $\ensuremath{\Lambda}\mathrm{CDM}$ universe as far as the considered cosmological tests are concerned.

Braneworld dynamics in Einstein-Gauss-Bonnet gravity

We discuss the cosmological evolution of a braneworld in five-dimensional Gauss-Bonnet gravity. Our discussion allows the fifth (bulk) dimension to be spacelike as well as timelike. The resulting equations of motion have the form of a cubic equation in the $({H}^{2},(\ensuremath{\rho}+\ensuremath{\sigma}{)}^{2})$ plane, where $\ensuremath{\sigma}$ is the brane tension and $\ensuremath{\rho}$ is the matter density. This allows us to conduct a comprehensive pictorial analysis of cosmological evolution for the Gauss-Bonnet brane. The many interesting properties of this braneworld include the possibility of accelerated expansion at late times. For a finite region in parameter space the accelerated expansion can be phantomlike so that $wl\ensuremath{-}1$. At late times, this branch approaches de Sitter space ($w=\ensuremath{-}1$) and avoids the big-rip singularities usually present in phantom models. For a timelike extra dimension the Gauss-Bonnet brane can bounce and avoid the initial singularity.

Non-linear resonance in bouncing braneworld universes and initial conditions forinflation

We examine the phase space dynamics of closed Friedmann–Robertson–Walker universeswith a massive inflaton field, where the Friedmann equations contain additional termsarising from high energy corrections to cosmological scenarios. The model is basedupon a Randall–Sundrum type of action, with an extra timelike dimension, andthe corrections implement non-singular bounces in the early evolution of theuniverse. In narrow windows of the parameter space of the models non-linearresonance phenomena of Kolmogorov–Arnold–Moser tori are shown to occur, leadingto the destruction of tori that trap the inflaton. As a consequence the escapeinto inflation takes place. These resonance windows are labeled with an integern≥2,where n is related to the ratio of the frequencies in the scale factor to those in the scalar field degrees offreedom. We examine the constraints imposed by non-linear resonance in the physicaldomain of parameters of the model so that inflation may be realized. The larger the ordern of the resonance, the stronger the gravitational interaction in the braneworld universeinflated from initial conditions connected with the resonance considered. We also discussthe structural stability of the resonance pattern, the complex dynamics arising inthis pre-inflationary phase and some of its possible imprints in the physics ofinflation.

Germ of a synthesis: space–time is spinorial, extra dimensions are time-like

A pressing issue for modern physics is the possibility of extra dimensions of space–time. Here, a novel approach to this question is put forward, with three facets: First, an integral transform is introduced into Einstein's general relativity that is non-local and spinorial. For Minkowskian space–time, the transform intertwines three spaces of six dimensions, which a priori are on an equal footing, linked by the octavic triality of Cartan. Two of these spaces are interpreted as null twistor spaces; the third may be regarded as giving space–time two extra time-like dimensions, for which the ordinary space–time is an axis of symmetry. Second, it is suggested that the extra dimensions perdure for a general space–time: the overall structure is controlled by a generalized Fefferman tensor. Accordingly, it is posited that the additional time-like dimensions arise naturally and constitute an aspect of space–time reality that ultimately will be amenable to experimental investigation. Conceivably, devices such as the Large Hadron Collider will uncover this reality. Third, it is argued that the structure hints at a synthesis of ideas deriving from general relativity, string theory, condensed matter physics, category theory and non-commutative geometry.

Reconsidering extra time-like dimensions

In this study we reconsider the phenomenological problems related to tachyonic modes in the context of extra time-like dimensions. First we reconsider a lower bound on the size of extra time-like dimensions and improve on the conclusion in the literature. Next we discuss the issues of spontaneous decay of stable fermions through tachyonic decays and disappearance of fermions due to tachyonic contributions to their self-energies. We find that the tachyonic modes due to extra time-like dimensions are less problematic than the tachyonic modes in the usual 4-dimensional setting because the most troublesome Feynman diagrams are forbidden once the conservation of momentum in the extra time-like dimensions is imposed.

Perturbations in a regular bouncing universe

We consider a simple toy model of a regular bouncing universe. The bounce is caused by an extra timelike dimension, which leads to a sign flip of the ${\ensuremath{\rho}}^{2}$ term in the effective four dimensional Randall Sundrum-like description. We find a wide class of possible bounces: big bang avoiding ones for regular matter content, and big rip avoiding ones for phantom matter. Focusing on radiation as the matter content, we discuss the evolution of scalar, vector and tensor perturbations. We compute a spectral index of ${n}_{s}=\ensuremath{-}1$ for scalar perturbations and a deep blue index for tensor perturbations after invoking vacuum initial conditions, ruling out such a model as a realistic one. We also find that the spectrum (evaluated at Hubble crossing) is sensitive to the bounce. We conclude that it is challenging, but not impossible, for cyclic/ekpyrotic models to succeed, if one can find a regularized version.