We investigate the validity of the generalized second law of thermodynamics on the $(n-1)$-dimensional brane embedded in the $(n+1)$-dimensional bulk. We examine the evolution of the apparent horizon entropy extracted through relation between gravitational equation and the first law of thermodynamics together with the matter field entropy inside the apparent horizon. We find that the apparent horizon entropy extracted through connection between gravity and the first law of thermodynamics satisfies the generalized second law of thermodynamics. This result holds regardless of whether there is the intrinsic curvature term on the brane or a cosmological constant in the bulk. The observed satisfaction of the generalized second law provides further support on the thermodynamical interpretation of gravity based on the profound connection between gravity and thermodynamics.
We study dynamics of equation of state parameter for a non-minimally coupled quintom dark energy component on the warped DGP brane. We investigate crossing of the cosmological constant line in this scenario. This crossing occurs in both DGP$^{\pm}$ branches of the model.
We construct an inflation model with inflaton non-minimally coupled to gravity on a warped DGP brane. Using an exponential potential, we calculate scalar power spectrum, spectral index and the running of the spectral index. We show that for a suitable range of non-minimal coupling it is possible to exit the inflationary phase spontaneously and without any mechanism, even in the case that minimal inflation can not exit spontaneously. By a detailed analysis of parameter space, we study the constraints imposed on the non-minimal coupling from recent observational data.
We solve for the behaviour of cosmological perturbations in the Dvali-Gabadadze-Porrati (DGP) braneworld model using a new numerical method. Unlike some other approaches in the literature, our method uses no approximations other than linear theory and is valid on large scales. We examine the behaviour of late-universe density perturbations for both the self-accelerating and normal branches of DGP cosmology.
We construct a quintom dark energy model with two non-minimally coupled scalar fields, one quintessence and the other phantom field, confined on the warped DGP brane. We study some important issues such as phantom divide line crossing, existence of the bouncing solutions and the stability of the solutions in this framework. We show that this model accounts for crossing of the phantom divide line and realization of the bouncing solutions. This model allows for stability of the solutions in separate regions of the $ω$-$ω'$ phase-plane.
The self-accelerating branch of the Dvali-Gabadadze-Porrati (DGP) five-dimensional braneworld has provided a compelling model for the current cosmic acceleration. Recent observations, however, have not favored it so much. We discuss the solutions which contain a de Sitter 3-brane in the cascading DGP braneworld model, which is a kind of higher-dimensional generalizations of the DGP model,where a $p$-dimensional brane is placed on a $(p+1)$-dimensional one and the $p$-brane action contains the $(p+1)$-dimensional induced scalar curvature term. In the simplest six-dimensional model, we derive th…
We solve for the behaviour of cosmological perturbations in the Dvali-Gabadadze-Porrati (DGP) braneworld model using a new numerical method. Unlike some other approaches in the literature, our method uses no approximations other than linear theory and is valid on large scales. We examine the behaviour of late-universe density perturbations for both the self-accelerating and normal branches of DGP cosmology. Our numerical results can form the basis of a detailed comparison between the DGP model and cosmological observations.
We present here the general transformation that leaves unchanged the form of the field equations for perfect fluid cosmologies in the DGP braneworld model. Specifically, a prescription for relating exact solutions with different equations of state is provided, and the symmetries found can be used as algorithms for generating new cosmological models from previously known ones. We also present, implicitly, the first known exact DGP perfect fluid spacetime. A particular case of the general transformation is used to illustrate the crucial role played both by the number of scalar fields and the ext…
In this paper we find analytical solutions for the scalar and gauge fields in the Freedman-Robertson-Walker multiply warped braneworld scenario. With this we find the precise mass spectra for these fields. We compare these spectra with that previously found in the literature for the static case.
We review various key issues in connection with the warped braneworld models which provide us with new insights and explanations of physical phenomena through interesting geometrical features of such extra dimensional theories. Starting from the original Randall-Sundrum two brane models, we have discussed the stability, hierarchy and other important issues in connection with such braneworld. The role of higher derivative terms in the bulk for modulus stabilization has been explained. Implications of the existence of various bulk fields have been discussed and it has been shown how a warped bra…
The braneworld model of Dvali, Gabadadze and Porrati (DGP) provides an intriguing modification of gravity at large distances and late times. By embedding a three-brane in an uncompactified extra dimension with separate Einstein-Hilbert terms for both brane and bulk, the DGP model allows for an accelerating universe at late times even in the absence of an explicit vacuum energy. We examine the evolution of cosmological perturbations on large scales in this theory. At late times, perturbations enter a DGP regime in which the effective value of Newton's constant increases as the background densit…
We study cosmological dynamics of a canonical bulk scalar field in the DGP setup within a superpotential approach. We show that the normal branch of this DGP-inspired model realizes a late-time de Sitter expansion on the brane. We extend this study to the case that the bulk contains a phantom scalar field. Our detailed study in the supergravity-style analysis reveals some yet unexplored aspects of cosmological dynamics of bulk scalar field in the normal DGP setup. Some clarifying examples along with numerical analysis of the model parameter space are presented in each case.
We study the static, analytical solution of black holes in the warped DGP braneworld scenario. We show that the linearized field equations and matching conditions lead to solutions that are not compatible with Schwarzschild-(A)dS$_{(4)}$ solutions on the brane. This incompatibility is similar to vDVZ discontinuity in massive gravity theory. Following the standard procedure to remove this discontinuity, which firstly was proposed by Vainshtein, we keep some appropriate nonlinear terms in the field equations. This strategy has its origin in the fact that the spatial extrinsic curvature of the br…
Many dark energy (DE) models have been proposed, in recent years, to explain acceleration of the Universe expansion. It seems necessary to discriminate the various DE models in order to check the viability of each model. Statefinder diagnostic is a useful method which can differentiate various DE models. In this paper, we investigate the statefinder diagnosis parameters for the holographic dark energy (HDE) model in two cosmological setup. First, we study statefinder diagnosis for HDE in the context of flat Friedmann-Robertson-Walker (FRW) Universe in Einstein gravity. Then, we extend our stud…
We study quasi-static perturbations in a cosmological background in the Dvali-Gabadadze-Porrati (DGP) braneworld model. We identify the Vainshtein radius at which the non-linear interactions of the brane bending mode become important in a cosmological background. The Vainshtein radius in the early universe is much smaller than the one in the Minkowski background, but in a self-accelerating universe it is the same as the Minkowski background. Our result shows that the perturbative approach is applicable beyond the Vainshtein radius for weak gravity by taking into account the second order effect…
In this talk I review three fatal flaws of the DGP braneworld model, which has been put forward as a possible model for late time acceleration without a cosmological constant: Ghosts, Cosmological Crashes, and Instability of the 5D vacuum. The talk is based on work in collaboration with Charmousis, Kaloper, Myers and Padilla.
We construct a DGP inspired braneworld scenario where a scalar field non-minimally coupled to the induced Ricci curvature is present on the brane. We show that this model allows for an embedding of the standard Friedmann cosmology in the sense that the cosmological evolution of the background metric on the brane can be described by the standard Friedmann equation. The relation between our framework and the dark-energy formalism is explored.
Braneworld models with induced gravity have the potential to replace dark energy as the explanation for the current accelerating expansion of the Universe. The original model of Dvali, Gabadadze and Porrati (DGP) demonstrated the existence of a ``self--accelerating'' branch of background solutions, but suffered from the presence of ghosts. We present a new large class of braneworld models which generalize the DGP model. Our models have negative curvature in the bulk, allow a second brane, and have general brane tensions and localized curvature terms. We exhibit three different kinds of ghosts,…
We construct a warm inflation model with inflaton field non-minimally coupled to induced gravity on a warped DGP brane. We incorporate possible modification of the induced gravity on the brane in the spirit of $f(R)$-gravity. We study cosmological perturbations in this setup. In the case of two field inflation such as warm inflation, usually entropy perturbations are generated. While it is expected that in the case of one field inflation these perturbations to be removed, we show that even in the absence of the radiation field, entropy perturbations are generated in our setup due to non-minima…
We consider the holographic dark energy (HDE) model in the framework of DGP braneworld with Granda-Oliveros infrared (IR) cutoff, $L=(α\dot{H}+βH^2)^{-1/2}$. With this choice for IR cutoff, we are able to derive evolution of the cosmological parameters such as the equation of state and the deceleration parameters, $w$ and $q$, as the functions of the redshift parameter $z$. As far as we know, most previous models of HDE presented in the literatures, do not gives analytically $ω=ω(z)$ and $q=q(z)$. We plot the evolution of these parameters versus $z$ and discuss that the results are compatible …
As a follow-up to our previous paper arxiv: 1309.4244[hep-th], we determine radion induced spherically symmetric solution using the gradient approximation scheme, when two warped $3$-branes are slant with respect to each other such that the radion field in this case is a radial co-ordinate varying function. The slanting between the branes is assumed to be small. In the absence of any on-brane matter except that induced effects of radion field, the resulting black hole is found to be extremal Reissner-Nordstrom whose mass is proportional to the slanting between the $3$-branes. It is however see…
We study cosmological aspects of braneworld models with a warped dimension and an arbitrary number of compact dimensions. With a stabilized radion, a number of different cosmological bulk solutions are found in a general case. Both one and two brane models are considered. The Friedmann equation is calculated in each case. Particular attention is paid to six dimensional models where we find that the usual Friedmann equation can typically be recovered without fine-tuning.
We construct exact solutions with the bubble of nothing in the Dvali-Gabadadze-Porrati braneworld model. The configuration with a single brane can be constructed, unlike in the Randall-Sundrum braneworld model. The geometry on the single brane looks like the Einstein-Rosen bridge. We also discuss the junction of multibranes. Surprisingly, even without any artificial matter fields on the branes such as three-dimensional tension of the codimension-two objects, two branes can be connected in certain configurations. We investigate solutions of multibranes too. The presence of solutions may indicat…
A geometric understanding of the origin of mass hierarchy among the standard model fermions have been addressed in a multiple warped spacetime. We show that the requirement of $10^{16}$ order warping between the Planck and the visible branes without creating any significant hierarchy among the moduli results into splitting of fermion masses in the standard model. Values of the various parameters of the extra dimensional model and the corresponding fermion masses are determined.
We discuss a Higgs inflation model in the warped DGP braneworld background. It generates reasonable primordial perturbations. At the same time with enhanced non-minimal coupling it overcomes the severe problem in the Higgs inflation in 4 dimension, which says that the effective field theory become invalid at an energy scale far below the energy scale for inflation exit. Furthermore we present the constraints for the parameters confront to PLANCK and related observations. PLANCK low-l data almost fixes the inflation energy scale in this Higgs inflation model with specific brane parameters.
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-…
We find an exact solution of Kerr-Newman-de Sitter type on the braneworld(4D) of the DGP model. When a constant 4D Ricci scalar is assumed, only zero(flat) and a positive(de-Sitter) values satisfy the Hamiltonian constraint equation coming from the extra dimension. With a Z_2-symmetry across the brane and a stationary and axisymmetric metric ansatz on the brane, we solve the constraint equation exactly in the Kerr-Schild form with de-Sitter background. In the de-Sitter background this Kerr-Schild solution is well behaved under Boyer-Lindquist transformation: the constraint equation is preserve…
We generalize the Randall Sundrum warped braneworld model in six and higher dimension and propose a resolution to the mass hierarchy among the standard model fermions. The fine tuning problem in connection with the scalar mass however is shown to reappear in a new guise in five dimensional warped model when the two form antisymmetric tensor fields, a massless string mode, propagates in the bulk. Finally the issue of modulus stabilization is re-examined in presence of a bulk scalar by considering it's back-reaction on the background geometry. The role of higher derivative term of the bulk scala…
Thermodynamic properties of Schwarzschild-Anti de-Sitter (Sch-AdS) and Reissner-Nordström-Anti de-Sitter (RN-AdS) blackholes in 3+1 dimensional spacetime are studied critically with special reference to the warped braneworld black holes with non-vanishing cosmological constant on the brane. Explicit dependence of the thermodynamic variables on the parameters of the braneworld model such as the induced three brane cosmological constant as well as the bulk cosmological constant have been determined. Hawking-Page phase transition has been discussed for both Sch-AdS and RN-AdS black holes. At the …
In this paper we first obtain Friedmann equations for the $(n-1)$-dimensional brane embedded in the $(n+1)$-dimensional bulk, with intrinsic curvature term of the brane included in the action (DGP model). Then, we show that one can always rewrite the Friedmann equations in the form of the first law of thermodynamics, $dE=TdS+WdV$, at apparent horizon on the brane, regardless of whether there is the intrinsic curvature term on the brane or a cosmological constant in the bulk. Using the first law, we extract the entropy expression of the apparent horizon on the brane. We also show that in the ca…
The Kaluza-Klein (KK) modes of a massive scalar field on a 3-brane embedded in six dimensional multiply warped spacetime are determined. Due to the presence of warping along both the extra dimensions the KK mass spectrum splits into two closely spaced branches which is a distinct feature of this model compared to the five dimensional Randall-Sundrum model. This new cluster of the KK mode spectrum is expected to have interesting phenomenological implications for the upcoming collider experiments. Such a scenario may also be extended for even larger number of orbifolded extra dimensions.
An impressed feature of inflation on warped Dvali-Gabadadze-Porrati (DGP) brane is that the inflationary phase exits spontaneously for a scalar inflaton field with exponential potential, which presents a graceful exit mechanism for the inflation. But its reheating mechanism leaves open. We investigate the curvaton reheating in inflation on warped DGP brane model. The reheating may occur in effctively 5 dimensional or 4 dimensional stage. We study the permitted parameter space of the curvaton field in detail. We demonstrate how the inflation model of the warped DGP brane is improved by the curv…
We study the spectrum of an asymmetric warped braneworld model with different AdS curvatures on either side of the brane. In addition to the RS-like modes we find a resonance state. Its mass is proportional to the geometric mean of the two AdS curvature scales, while the difference between them determines the strength of the resonance peak. There is a complementarity between the RS zero-mode and the resonance: making the asymmetry stronger weakens the zero-mode but strengthens the resonance, and vice versa. We calculate numerically the braneworld gravitational potential and discuss the hologra…
The role of a bulk graviton in predicting the signature of extra dimensions through collider-based experiments is explored in the context of a multiply warped spacetime. In particular it is shown that in a doubly warped braneworld model, the presence of the sixth dimension, results in enhanced concentration of graviton Kaluza Klein (KK) modes compared to that obtained in the usual 5-dimensional Randall-Sundrum model. Also, the couplings of these massive graviton KK modes with the matter fields on the visible brane turn out to be appreciably larger than that in the corresponding 5- dimensional …
The stability of the modular field in a warped brane world scenario has been a subject of interest for a long time. Goldberger \& Wise ( GW ) proposed a mechanism to achieve this by invoking a massive scalar field in the bulk space-time neglecting the back-reaction. In this work, we examine the possibility of stabilizing the modulus without bringing in any external scalar field. We show that instead of flat 3-branes as considered in Randall-Sundrum ( RS ) warped braneworld model, if one considers a more generalized version of warped geometry with de-Sitter 3-brane, then the brane vacuum energy…
We investigate the vacuum energy density induced by quantum fluctuations of a bulk scalar field with general curvature coupling parameter on two codimension one parallel branes in a $(D+1)$-dimensional background spacetime ${\mathrm{AdS}}_{D1+1}\times Σ$ with a warped internal space $Σ$. It is assumed that on the branes the field obeys Robin boundary conditions. Using the generalized zeta function technique in combination with contour integral representations, the surface energies on the branes are presented in the form of the sums of single brane and second brane induced parts. For the geomet…
Hubble expansion in warped braneworld model is addressed in presence of more than one warped extra dimensions. It is shown that while the expansion depends on all the moduli, an exponential nature of the expansion of the scale factor emerges as a generic feature which is independent of the number of extra dimensions. Expression for the effective brane cosmological constant in such model has been derived. It is shown that similar to the 5-dimensional Randall-Sundrum model a fine tuning between the bulk cosmological constant and brane tension is required to obtain the desired cosmological consta…
A new family of five dimensional, R=0 braneworlds with asymmetric warp factors is proposed. Beginning with the invariance of the Ricci scalar for the general class of asymmetrically warped spacetimes we, subsequently specialise to the R=0 case. Solutions are obtained by choosing a particular relation (involving a parameter $ν$) between the warp factors. Symmetric warping arises as a special case (particular value of $ν$). Over a range of values of $ν$ the energy conditions for the matter stress energy are found to hold good. It turns out that the energy density and pressures required to suppor…
We generalize nested multiply warped braneworld models by incorporating non-zero brane curvature caused by an effective cosmological constant Ω induced on the 3-branes. Starting with the doubly warped model, we first analyze the case where the maximally warped brane is identified as the visible brane. For Ω < 0, resolution of the gauge hierarchy problem imposes a small upper bound on |Ω|, and can possibly lead to positivity of all the 3-brane tensions. For Ω > 0, the latter is not possible but the tuning of the cosmological constant to its tiny observed value is linked to the tuning of the ext…
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 show that quantum mechanical supersymmetries are emerged in Kaluza-Klein spectrum of linearized gravity in several warped backgrounds as a consequence of higher-dimensional general coordinate invariance. These emergent supersymmetries play an essential role for the spectral structure of braneworld gravity. We show that for the case of braneworld models with two codimension-1 branes the spectral pattern is completely determined only through the supersymmetries.
The braneworld model of Dvali-Gabadadze-Porrati realizes the self-accelerating universe. However, it is known that this cosmological solution contains a spin-2 ghost. We study the possibility of avoiding the appearance of the ghost by slightly modifying the model, introducing the second brane. First we consider a simple model without stabilization of the separation of the brane. By changing the separation between the branes, we find we can erase the spin-2 ghost. However, this can be done only at the expense of the appearance of a spin-0 ghost instead. We discuss why these two different types …
We study the behavior of the four$-$dimensional Newton's law in warped braneworlds. The setup considered here is a $(3+n)$-brane embedded in $(5+n)$ dimensions, where $n$ extra dimensions are compactified and a dimension is infinite. We show that the wave function of gravity is described in terms of the Bessel functions of $(2+n/2)$-order and that estimate the correction to Newton's law. In particular, the Newton's law for $n=1$ can be exactly obtained.
We consider the 6D Cascading DGP model, a braneworld model which is a promising candidate to realize the phenomenon of the degravitation of vacuum energy. Focusing on a recently proposed thin limit description of the model, we study solutions where the induced metric on the codimension-2 brane is of the de Sitter form. While these solutions have already been recovered in the literature imposing by hand the bulk to be flat, we show that it is possible to derive them without making this assumption, by solving a suitably chosen subset of the bulk equations.
Braneworld theories are often described as low-energy effective field theories (EFTs) featuring an infinitely thin 3-brane and 4D fields exactly localized on it. We investigate whether an exactly localized braneworld can arise as a limit of a theory of 5D fields. Using a holographic formalism we argue that such limit does not exist in the presence of gravity, therefore implying a discontinuity in the space of EFTs. We then present specific models involving exactly localized fields in which inconsistencies appear, which are solved when fields are taken as quasilocalized. Part of our arguments r…
We examine the modulus stabilization mechanism of a warped geometry model with nested warping. Such a model with multiple moduli is known to offer a possible resolution of the fermion mass hierarchy problem in the standard model. A six dimensional doubly warped braneworld model under consideration admits two distinct moduli, with the associated warp factors dynamically generating different physical mass scales on four 3-branes. In order to address the hierarchy problem related to the Higgs mass, both moduli need to be stabilized around their desired values without any extreme fine tuning of pa…
We readdress the problems associated with bulk Higgs and the gauge fields in a 5-dimensional Randall-Sundrum model by extending the model to six dimensions with double warping along the two extra spatial dimensions. In this 6-dimensional model we have a freedom of two moduli scales as against one modulus in the 5-dimensional model. With a little hierarchy between these moduli we can obtain the right magnitude for $W$ and $Z$ boson masses from the Kaluza-Klein modes of massive bulk gauge fields where the spontaneous symmetry breaking is triggered by bulk Higgs . We also have determined the gaug…
A generalized Randall sundrum model in six dimensional bulk is studied in presence of non-flat 3-branes at the orbifold fixed points. The warp factors for this model is determined in terms of multiple moduli and brane cosmological constant. We show that the requirements of a vanishingly small cosmological constant on the visible brane along with non-hierarchical moduli, each with scale close to Planck length, lead to a scenario where the 3-branes can not have any intermediate scale and have energy scales either close to Tev or close to Planck scale. Such a scenario can address both the gauge h…
In this paper, we investigate gravitational waves as metric perturbations around a general warped 5-dimensional background. We find an analytical solution in Randall-Sundrum braneworld model and analyze the implications of braneworld models in the gravitational waves propagation.
The `braneworld' (described by the usual worldvolume action) is a D dimensional timelike surface embedded in a N dimensional ($N>D$) warped, nonfactorisable spacetime. We first address the conditions on the warp factor required to have an extremal flat brane in a five dimensional background. Subsequently, we deal with normal deformations of such extremal branes. The ensuing Jacobi equations are analysed to obtain the stability condition. It turns out that to have a stable brane, the warp factor should have a minimum at the location of the brane in the given background spacetime. To illustrate …
We consider a generalisation of the DGP model, by adding a second brane with localised curvature, and allowing for a bulk cosmological constant and brane tensions. We study radion and graviton fluctuations in detail, enabling us to check for ghosts and tachyons. By tuning our parameters accordingly, we find bigravity models that are free from ghosts and tachyons. These models will lead to large distance modifications of gravity that could be observable in the near future.
We study cosmological inflation on a warped DGP braneworld where inflaton field is non-minimally coupled to induced gravity on the brane. We present a detailed calculation of the perturbations and inflation parameters both in Jordan and Einstein frame. We analyze the parameters space of the model fully to justify about the viability of the model in confrontation with recent observational data. We compare the results obtained in these two frames also in order to judge which frame gives more acceptable results in comparison with observational data.
The braneworld model of Dvali-Gabadadze-Porrati (DGP) provides an interesting alternative to a positive cosmological constant by modifying gravity at large distances. We investigate the asymptotic behavior of homogeneous and anisotropic cosmologies on the DGP brane. It is shown that all Bianchi models except type IX isotropize, as in general relativity, if the so called $E_{μν}$ term satisfies some energy condition. Isotropization can proceed slower in DGP gravity than in general relativity.
Warm inflationary universe models on a warped Dvali-Gabadadze-Porrati brane are studied. General conditions required for these models to be realizable are derived and discussed. By using an effective exponential potential we develop models for constant and variable dissipation coefficient ratio $r=\fracΓ{3 H}$. We use recent astronomical observations for constraining the parameters appearing in our models.
We have investigated the features of the gauge vector field in a braneworld scenario built as a warped product between a 3-brane and a 2-cycle of the resolved conifold. This scenario allowed us to study how the gauge field behaves when the transverse manifold evolves upon a geometric flow that controls the singularity at the origin. Besides, since the transverse manifold has a cylindrical symmetry according to the 3-brane, this geometry can be regarded as a near brane correction of the string-like branes. Indeed, by means of a new warp function and the angular metric component of the resolved …
We investigate the stability of the extra dimensions in a warped, codimension two braneworld that is based upon an Einstein-Maxwell-dilaton theory with a non-vanishing scalar field potential. The braneworld solution has two 3-branes, which are located at the positions of the conical singularities. For this type of brane solution the relative positions of the branes (the shape modulus) is determined via the tension-deficit relations, if the brane tensions are fixed. However, the volume of the extra dimensions (the volume modulus) is not fixed in the context of the classical theory, implying we …
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…
