Black Holes and Black String-like Solutions in Codimension-2 Braneworlds
Eleftherios Papantonopoulos
备注：
9 pages, no figures, plenary talk given at the 7th Friedmann International Seminar on Gravitation and Cosmology, 29 June-5 July 2008, Joao Pessoa, Brazil, to appear in the proceedings
We discuss black hole solutions with a Gauss-Bonnet term in the bulk and an induced gravity term on a thin brane of codimension-2. We show that these black holes can be localized on the brane, and they can be extended further into the bulk by a warp function. These solutions have regular horizons and no other curvature singularities appear apart from the string-like ones. The projection of the Gauss-Bonnet term on the brane imposes a constraint relation which dictates the form of matter on the brane and in the bulk.
It has recently been argued that codimension-two braneworlds offer a promising line of attack on the cosmological constant problem, since in such models the Hubble rate is not directly related to the brane tension. We point out challenges to building more general models where the brane content is not restricted to pure tension. In order to address these challenges, we construct a thick brane model which we linearize around a well known static solution. We show that the model's cosmology does reduce to standard FRW behaviour, but find no hint of a self-tuning mechanism which might help solve th…
We derive the Lichnerowicz equation in the presence of the Gauss-Bonnet term. Using the modified Lichnerowicz equation we study the metric perturbations of Gauss-Bonnet black strings in Codimension-2 Braneworlds.
We consider maximally symmetric 3-branes embedded in a six-dimensional bulk spacetime with Lovelock dynamics. We study the properties of the solutions with respect to their induced curvature, their vacuum energy and their effective compactness in the extra dimensions. Some simple solutions are shown to give rise to self-accelerating braneworlds, whereas several others solutions have self-tuning properties. For the case of geometric self-acceleration we argue that the cross-over scale in between four-dimensional and higher-dimensional gravity and the scale of late-time geometric acceleration, f…
The quantum creation probability and entropy of a 2-codimensional braneworld are calculated in the framework of no-boundary universe. The entropy can take an arbitrarily large value as the brane tensions increase, in violation of the conjectured "N-bound" in quantum gravity, even for a 4-dimensional ordinary universe.
We study the matching conditions of intersecting brane worlds in Lovelock gravity in arbitrary dimension. We show that intersecting various codimension 1 and/or codimension 2 branes one can find solutions that represent energy-momentum densities localized in the intersection, providing thus the first examples of infinitesimally thin higher codimension braneworlds that are free of singularities and where the backreaction of the brane in the background is fully taken into account.
We consider five-dimensional gravity with a Gauss-Bonnet term in the bulk and an induced gravity term on a 2-brane of codimension-2. We show that this system admits BTZ-like black holes on the 2-brane which are extended into the bulk with regular horizons.
We study a codimension 2 braneworld in the Einstein Gauss-Bonnet gravity. We carefully examine the structure of possible singularities in the system which characterize the braneworld through matching conditions. Consequently, we find that the thickness of the brane can be incorporated as the distributional source, which we dub quasi-thickness. On the basis of our formalism, we analyze the linearized gravity and show the conventional Einstein gravity can be recovered on the brane. In the nonlinear regime, however, we find corrections due to the thickness and the bulk geometry. We also point out…
Recently, in (Eur.Phys.J.C 80 (2020) 5, 432), the present authors obtained general stringent conditions on the localization of fields in braneworlds by imposing that its zero-mode must satisfy Einstein's equations (EE). Here, we continue this study by considering free $p$-form. These fields present an on-shell equivalency relation between a $p$-form and a $(D-p-2)$-form, provided by Hodge duality (HD). This symmetry will impose a new consistency condition, namely, confinement of a $p$-form must imply the localization of its dual. We apply the above conditions to $6$D braneworlds. With this, we…
Lagrangian description of lightlike p-branes is presented in two equivalent forms - a Polyakov-type formulation and a dual to it Nambu-Goto-type formulation. Next, the properties of lightlike brane dynamics in generic gravitational backgrounds of spherically symmetric and axially symmetric type are discussed in some detail: "horizon straddling" and "mass inflation" effects for codimension-one lightlike branes and ground state behavior of codimension-two lightlike "braneworlds".
We consider five-dimensional gravity with a Gauss-Bonnet term in the bulk and an induced gravity term on a 2-brane of codimension-2. We show that this system admits BTZ black holes on the 2-brane which are extended into the bulk with regular horizons.
Building on earlier work on football shaped extra dimensions, we construct a compact codimension two braneworld with precisely one brane. The two extra dimensions topologically represent a 2-torus which is stabilized by a bulk cosmological constant and magnetic flux. The torus has positive constant curvature almost everywhere, except for a single conical singularity at the location of the brane. In contradistinction to the football shaped case, there is no fine-tuning required for the brane tension. We also present some plausibility arguments why the model should not suffer from serious stabil…
We consider four-dimensional de Sitter, flat and anti de Sitter branes embedded in a six-dimensional bulk spacetime whose dynamics is dictated by Lovelock theory. We find, applying a generalised version of Birkhoff's theorem, that all possible maximally symmetric braneworld solutions are embedded in Wick-rotated black hole spacetimes of Lovelock theory. These are warped solitonic spaces, where the horizons of the black hole geometries correspond to the possible positions of codimension-2 branes. The horizon temperature is related via conical singularities to the tension or vacuum energy of the…
In this work, we compute the corrections in the Newton's law of gravitation due to Kaluza-Klein gravitons in codimension-1 warped thick braneworld scenarios. We focus in some models recently proposed in the literature, the so-called asymmetric hybrid brane and compact brane. Such models are deformations of the $ϕ^4$ and sine-Gordon topological defects, respectively. Therefore we consider the branes engendered by such defects and we also compute the corrections in their cases. We use suitable numerical techniques to attain the mass spectrum and its corresponding eigenfunctions which are the ess…
We consider a thick p-brane embedded in an n-dimensional spacetime possessing radial symmetry in the directions orthogonal to the brane. We first consider a static brane, and find a general fine tuning relationship between the brane and bulk parameters required for the brane to be flat. We then consider the cosmology of a time dependent brane in a static bulk, and find the Friedmann equation for the brane scale factor a(t). The singularities that would ordinarily arise when considering arbitrary codimensions are avoided by regularizing the brane, giving it a finite profile in the transverse di…
We consider the propagation of gravitational waves in six dimensions induced by sources living on 3-branes in the context of a recent exact solution hep-th/0506050. The brane geometries are de Sitter and the bulk is a warped geometry supported by a positive cosmological constant as well as a 2-form flux. We show that at low energies ordinary gravity is reproduced, and explicitly compute the leading corrections from six dimensional effects. After regulating the brane we find a logarithmic dependence on the cutoff scale of brane physics even for modes whose frequency is much less than this energ…
We give a short review of the recent development in the investigation of the vacuum expectation values of the bulk and surface energy-momentum tensors generated by quantum fluctuations of a massive scalar field with general curvature coupling parameter subject to Robin boundary conditions on two codimension one parallel branes located on $(D+1)$-dimensional anti-de Sitter (AdS) bulk. An application to the Randall-Sundrum braneworld with arbitrary mass terms on the branes is discussed.
We study axially symmetric codimension-2 cosmology for a distributional braneworld fueled by a localised 4D perfect fluid, in a 6D Lovelock theory. We argue that only the matching conditions (dubbed topological) where the extrinsic curvature on the brane has no jump describe a pure codimension-2 brane. If there is discontinuity in the extrinsic curvature on the brane, this induces inevitably codimension-1 distributional terms. We study these topological matching conditions, together with constraints from the bulk equations evaluated at the brane position, for two cases of regularisation of the…
We propose here a static and axisymmetric braneworld in six dimensions as a string-like model extension. For a subtle warp function, this scenario provides near brane corrections. By varying the bulk cosmological constant, we obtain a source which passes through different phases. The solution is defined both for the interior as for the exterior of the string and satisfies the weak energy condition. Smooth gravitational massless mode is localized on the brane which core is displaced from the origin. In contrast to the thin string model, the massive solutions have high amplitude near the brane. …
We present a comprehensive study of the cosmological solutions of 6D braneworld models with azimuthal symmetry in the extra dimensions, moduli stabilization by flux or a bulk scalar field, and which contain at least one 3-brane that could be identified with our world. We emphasize an unusual property of these models: their expansion rate depends on the 3-brane tension either not at all, or in a nonstandard way, at odds with the naive expected dimensional reduction of these systems to 4D general relativity at low energies. Unlike other braneworld attempts to find a self-tuning solution to the c…
Braneworld scenarios appear in dynamical string tension theories, where string tension is positive between two surfaces separated by a short distance and at the two surfaces themselves the string tensions become infinite, therefore producing an effective confinement of the strings and therefore of all matter and gravity to the space between these to surfaces. The specific model studied is in the context of the modified measure formulation the string where tension appear as an additional dynamical degree of freedom and these tensions are not universal, but rather each string generates its own t…
In the context of six-dimensional conical braneworlds we consider a simple and explicit model that incorporates long distance modification of gravity and regularization of codimension-2 singularities. To resolve the conical singularities we replace the codimension-2 branes with ring-like codimension-1 branes, filling in the interiors with regular caps. The six-dimensional Planck scale in the cap is assumed to be much greater than the bulk Planck scale, which gives rise to the effect analogous to brane-induced gravity. Weak gravity on the regularized brane is studied in the case of a sharp coni…
Following up on a recent paper by two of us (DM and SS), demonstrating the large enhancement in observable optical activity in radiation from high redshift sources arising from the string-based coupling of bulk Kalb-Ramond field to the Maxwell Chern Simons three-form on the brane in a Randall-Sundrum braneworld, we exhibit here a similar enhancement in parity-violating temperature-polarization correlations, yet unseen, in the CMB anisotropy due to a generalized parity-violating Kalb-Ramond axion-photon interaction proposed earlier by one of us (PM). The non-observation of such correlations in …
In this work we deal with the spacetime shaped by a straight cosmic string, emerging from local gauge theories, in the braneworld gravity context. We search for physical consequences of string features due to the modified gravitational scenario encoded in the projected gravitational equations. It is shown that cosmic strings in braneworld gravity may present significant differences when compared to the general relativity predictions since its linear density is modified and the deficit angle produced by the cosmic string is attenuated. Furthermore, the existence of cosmic strings in that scenar…
It has been suggested that codimension-two braneworlds might naturally explain the vanishing of the 4D effective cosmological constant, due to the automatic relation between the deficit angle and the brane tension. To investigate whether this cancellation happens dynamically, and within the context of a realistic cosmology, we study a codimension-two braneworld with spherical extra dimensions compactified by magnetic flux. Assuming Einstein gravity, we show that when the brane contains matter with an arbitrary equation of state, the 4D metric components are not regular at the brane, unless the…
We look at general braneworlds in six-dimensional Einstein-Gauss-Bonnet gravity. We find the general matching conditions for the Einstein-Gauss-Bonnet braneworld, which remarkably turn out to give precisely the four-dimensional Einstein equations for the induced metric and matter on the brane, even when the extra dimensions are non-compact and have infinite volume. We also show that relaxing regularity of the curvature in the vicinity of the brane, or alternatively having a finite width brane, gives rise to an additional possible correction to the Einstein equations, which contains information…
We discuss various realizations of the four dimensional braneworld inflation in warped geometries of string theory. In all models the inflaton field is represented by a Dp probe brane scalar specifying its position in the warped throat of the compactification manifold. We study existing inflationary throat local geometries, and construct a new example. The inflationary brane is either a D3- or a D5-brane of type IIB string theory. In the latter case the inflationary brane is wrapping a two-cycle of the compactification manifold. We discuss some phenomenological aspects of the model where slow-…
This paper aims to evince the corrections on the black string warped horizon in the braneworld paradigm, and their drastic physical consequences, as well as to provide subsequent applications in astrophysics. Our analysis concerning black holes on the brane departs from the Schwarzschild case, where the black string is unstable to large-scale perturbation. The cognizable measurability of the black string horizon corrections due to braneworld effects is investigated, as well as their applications in the variation of quasars luminosity. We delve into the case wherein two solutions of Einstein's …
We explore the problem of braneworld black holes in the heterotic braneworld scenario of Lukas, Ovrut, Stelle and Waldram (LOSW). We show that black string solutions are unstable, and demonstrate some unusual asymptotics in the linearized metric. We also solve the fully coupled brane and bulk Einstein equations, finding an exact, though singular, solution which corresponds to a brane black hole in which the branes spike apart at the Schwarzschild radius.
We study the Friedmann equation for the warped codimension-two braneworld background which most closely resembles the Randall-Sundrum model. Extra matter on the (Planck) 4-brane, with equation of state p_θ=(α-1)ρfor the azimuthal pressure, is required to satisfy the junction conditions. For 1 < α< 5, we show that there are two static solutions to the Einstein equations for given values of the brane stress-energies. Close to the static solutions, the relation between Hubble expansion rate H and brane tension reproduces the standard 4D result for small H, but exhibits unusual deviations when H i…
To understand the time-dependent quantum correlation in expanding universes, we study the time-dependent entanglement entropy in the braneworld model. If we take into account a generalized string cloud geometry caused by uniformly distributed open strings, cosmologies on the braneworld result in the standard Friedmann-Lemaître-Robertson-Walker cosmologies with various matter contents. On the dual field theory side, open strings are reinterpreted as a fundamental matter, while the black hole mass corresponds to the excitation energy of massless gauge bosons. In this work, we show how the string…
This work investigates the influence of a probe string on the complexity of braneworld according to the CA (Complexity equals action) conjecture within the Horndeski gravity. In the current study, it is considered that scalar fields that source Horndeski gravity has a spatial dependence. In addition, our system contains a particle moving on the boundary, which corresponds to the insertion of a fundamental string in the higher dimensional bulk. Such an effect is given by the Nambu-Goto term, which also incorporates the time-dependence and evolution in our system. Both warp factor, scalar field,…
We generalise the RS braneworld model by taking into account a general stringy bulk containing the scalar dilaton field and the two-form Kalb-Ramond field, apart from gravity. Assuming small fluctuations around a RS background, the back-reacted warp factor is obtained. It is shown that the fine tuning problem in connection with the Higgs mass reappears in a new guise and the effective modular potential fails to stabilise the braneworld.
We study gravitational waves in the black string Randall-Sundrum braneworld. We present a reasonably self-contained and complete derivation of the equations governing the evolution of gravitational perturbations in the presence of a brane localized source, and then specialize to the case of spherical radiation from a pointlike body in orbit around the black string. We solve for the resulting gravitational waveform numerically for a number of different orbital parameters.
We continue the investigation of a recent proposal on alternative matching conditions for self-gravitating defects which generalize the standard matching conditions. The reasoning for this study is the need for consistency of the various codimension defects and the existence of a meaningful equation of motion at the probe limit, things that seem to lack from the standard approach. These matching conditions arise by varying the brane-bulk action with respect to the brane embedding fields (and not with respect to the bulk metric at the brane position) in a way that takes into account the gravita…
On the way towards quantum gravity and the unification of interaction, several ideas have been rejected and avenues avoided because they were perceived as physically unviable. But in the literature there are works in which it was found the contrary, namely that those rejected topics make sense after all. Such topics, reviewed in this article, are negative energies occurring in higher derivative theories and ultrahyperbolic spaces, ordering ambiguity of operators in curved spaces, the vast landscape of possible compactifications of extra dimensions in string theory, and quantization of a 3-bran…
We present some solutions for lumps in two dimensions in level-expanded string field theory, as well as in two tachyonic theories: pure tachyonic string field theory and pure $ϕ^3$ theory. Much easier to handle, these theories might be used to help understanding solitonic features of string field theory. We compare lump solutions between these theories and we discuss some convergence issues.
In the Randall-Sundrum two-brane model (RS1), a Kerr black hole on the brane can be naturally identified with a section of rotating black string. To estimate Kaluza-Klein (KK) corrections on gravitational waves emitted by perturbed rotating black strings, we give the effective Teukolsky equation on the brane which is separable equation and hence numerically manageable. In this process, we derive the master equation for the electric part of the Weyl tensor $E_{μν}$ which would be also useful to discuss the transition from black strings to localized black holes triggered by Gregory-Laflamme inst…
A covariantly constant dynamical two-form is exploited on a $D_3$-brane to obtain its gravity dual action, governing an $S^3$ deformed $AdS_5$ black hole, in a type IIB string theory on $S^1\times K3$. We invoke the Kaluza-Klein compactification to work out the open/closed string duality. Interestingly, the Reissner-Nordstrom black hole is obtained on the "non-Reimannian" braneworld.
In this work, we propose a new class of smooth thick string-like braneworld in six dimensions. The brane exhibits a varying brane-tension and an $AdS$ asymptotic behavior. The brane-core geometry is parametrized by the Bulk cosmological constant, the brane width and by a geometrical deformation parameter. The source satisfies the dominant energy condition for the undeformed solution and has an exotic asymptotic regime for the deformed solution. This scenario provides a normalized massless Kaluza-Klein mode for the scalar, gravitational and gauge sectors. The near-brane geometry allows massive …
We study the localization of gravity on string-like defects in codimension two. We point out that the gravity-localizing `local cosmic string' spacetime has an orbifold singularity at the horizon. The supergravity embedding and the AdS/CFT correspondence suggest ways to resolve the singularity. We find two resolutions of the singularity that have a semiclassical gravity description and study their effect on the low-energy physics on the defect. The first resolution leads, at long distances, to a codimension one Randall-Sundrum scenario. In the second case, the infrared physics is like that of …
In this work the gravitational Kaluza-Klein (KK) modes are analysed on string-like braneworlds in six dimensions, namely the thin string-like Gherghetta-Shaposhnikov (GS) and the thick string-cigar model. We find a new massless mode in both modes satisfying the respective Schroedinger equation. By means of numerical analysis for both models, a complete graviton spectrum and eigenfunctions is attained. Besides the linear regime, the KK spectrum exhibits a decreasing behavior with an asymmetric mass gap with respect to the massless mode. The discontinuity is bigger in the GS model, and it is inv…
We proposed a six dimensional string-like braneworld built from a warped product between a 3-brane and the Hamilton cigar soliton space, the string-cigar braneworld. This transverse manifold is a well-known steady solution of the Ricci flow equation that describes the evolution of a manifold. The resulting bulk is an interior and exterior metric for a thick string. This is a physical and feasible scenario since the source satisfies the dominant energy condition. It is possible to realize the geometric flow as a result of variations of the matter content of the brane, actually, as its tensions.…
We discuss a particular higher order gravity theory, Lovelock theory, that generalises in higher dimensions, general relativity. After briefly motivating modifications of gravity, we will introduce the theory in question and we will argue that it is a unique, mathematically sensible, and physically interesting extension of general relativity. We will see, by using the formalism of differential forms, the relation of Lovelock gravity to differential geometry and topology of even dimensional manifolds. We will then discuss a generic staticity theorem, which will give us the charged static black …
String tension fundamentally determines the properties of strings; yet its value is often assigned arbitrarily, creating a fine-tuning problem. We describe a mechanism for dynamically generating string tension in a flat or almost flat spacetime, using the modified measures formalism, which in turn naturally generates a new type of stringy brane-world scenario. Such a scenario allows strings to achieve near infinite tension confining the strings to two very close expanding surfaces, but the infinite tensions also threatens to distort the near-flat embedding spacetime through large back reaction…
Braneworld models are interesting theoretical and phenomenological frameworks to search for new physics beyond the standard model of particles and cosmology. In this work, we discuss braneworld models whose gravitational dynamics are governed by teleparallel $f(T)$ gravities. Here, we emphasize a codimension two axisymmetric model, also known as a string-like brane. Likewise, in the 5D domain-walls models, the $f(T)$ gravitational modification leads to a phase transition on the perfect fluid source providing a brane-splitting mechanism. Furthermore, the torsion changes the gravitational pertur…
We consider cosmological inflation driven by the rolling tachyon in the context of the braneworld scenario. We show that sufficient inflation consistent with the observational constraints can be achieved for well defined upper limits on the five-dimensional mass scale, string mass scale and the string coupling for the bosonic string.
We revisit D-term inflation with the bounds of the cosmic string tension from gravitational wave observations and consider the possible deviation of the spectral index compared with the $Λ$CDM model in light of prerecombination resolutions of the Hubble tension. D-term inflation requires very small coupling constants under these constraints. We show that natural coupling constants ($g=λ=0.1$) can be achieved in the case of D-term inflation on the brane.
In this work we investigate the properties of an Abelian gauge vector field in a thin and in a smoothed string-like braneworld, the so-called string-cigar model. This thick brane scenario satisfies the regularity conditions and it can be regarded as an interior and exterior string-like solution. The source undergoes a geometric Ricci flow which is connected to a variation of the bulk cosmological constant. The Ricci flow changes the width and amplitude of the massless mode at the brane core and recover the usual thin string-like behavior at large distances. By numerical means we obtain the Kal…
We consider braneworld scenarios including the leading correction to the Einstein-Hilbert action suggested by superstring theory, the Gauss-Bonnet term. We obtain and study the complete set of equations governing the cosmological dynamics. We find they have the same form as those in Randall-Sundrum scenarios but with time-varying four-dimensional gravitational and cosmological constants. Studying the bulk geometry we show that this variation is produced by bulk curvature terms parameterized by the mass of a black hole. Finally, we show there is a coupling between these curvature terms and matt…
Braneworld models with variable tension are investigated, and the corrections on the black string horizon along the extra dimension are provided. Such corrections are encrypted in additional terms involving the covariant derivatives of the variable tension on the brane, providing profound consequences concerning the black string horizon variation along the extra dimension, near the brane. The black string horizon behavior is shown to be drastically modified by the terms corrected by the brane variable tension. In particular, a model motivated by the phenomenological interesting case regarding …
We construct a familly of exact solutions of Lovelock equations describing codimension four branes with discrete symmetry in the transverse space. Unlike what is known from pure Einstein gravity, where such brane solutions of higher codimension are singular, the solutions we find, for the complete Lovelock theory, only present removable singularities. The latter account for a localised tension-like energy-momentum tensor on the brane, in analogy with the case of a codimension two self-gravitating cosmic string in pure Einstein gravity. However, the solutions we discuss present two main distinc…
We study the spin 1/2 and spin 3/2 fermion fields in a thick braneworld scenario in six dimensions called string-cigar model. This smooth string-like model has a source that satisfies the dominant energy condition and undergoes a Ricci flow. We propose a new coupling for the fermions with a background gauge field which allows a smooth and normalized massless mode in the brane with positive tension. By numerical methods the mass spectrum and the massive eigenfunctions are obtained. The Kaluza-Klein massive tower exhibits the usual increasing pattern and, in this scenario, the coupling term does…
We investigate the effects of a brane and magnetic-flux-carrying cosmic string on the vacuum expectation value (VEV) of the current density for a charged fermionic field in the background geometry of 4+1 dimensional anti-de Sitter (AdS) spacetime. The brane is parallel to the AdS boundary and the cosmic string is orthogonal to the brane. Two types of boundary conditions are considered on the brane that include the MIT bag boundary condition and the boundary conditions in Z2-symmetric braneworld models. The brane divides the space into two regions with different properties of the vacuum state. …
If the graviton possesses an arbitrarily small (but nonvanishing) mass, perturbation theory implies that cosmic strings have a nonzero Newtonian potential. Nevertheless in Einstein gravity, where the graviton is strictly massless, the Newtonian potential of a cosmic string vanishes. This discrepancy is an example of the van Dam--Veltman--Zakharov (VDVZ) discontinuity. We present a solution for the metric around a cosmic string in a braneworld theory with a graviton metastable on the brane. This theory possesses those features that yield a VDVZ discontinuity in massive gravity, but nevertheless…
We describe a concise general scheme for constructing solutions of Einstein-Maxwell-Kalb-Ramond gravity-matter system in bulk space-time interacting self-consistently with one or more (widely separated) codimension-one electrically charged lightlike branes. The lightlike brane dynamics is explicitly given by manifestly reparametrization invariant world-volume actions. We present several explicit classes of solutions with different physical interpretation as wormhole-like space-times with one, two or more "throats", singularity-free black holes, brane worlds and space-times undergoing a sequenc…
We study linearized gravity in a six-dimensional Einstein-Maxwell model of warped braneworlds, where the extra dimensions are compactified by a magnetic flux. It is difficult to construct a strict codimension two braneworld with matter sources other than pure tension. To overcome this problem we replace the codimension two defect by an extended brane, with one spatial dimension compactified on a Kaluza-Klein circle. Our background is composed of a warped, axisymmetric bulk and one or two branes. We find that weak gravity sourced by arbitrary matter on the brane(s) is described by a four-dimens…
