Regular Bulk Solutions in Brane-worlds with Inhomogeneous Dust and Generalized Dark Radiation

From the dynamics of a brane-world with matter fields present in the bulk, the bulk metric and the black string solution near the brane are generalized, when both the dynamics of inhomogeneous dust/generalized dark radiation on the brane-world and inhomogeneous dark radiation in the bulk as well are considered -- as exact dynamical collapse solutions. Based on the analysis on the inhomogeneous static exterior of a collapsing sphere of homogeneous dark radiation on the brane, the associated black string warped horizon is studied, as well as the 5D bulk metric near the brane. Moreover, the black string and the bulk are shown to be more regular upon time evolution, for suitable values for the dark radiation parameter in the model, by analyzing the physical soft singularities.


I. INTRODUCTION
Brane-world models with a single extra dimension [1,2] are decidedly a 5D phenomenological realization of Hořava-Witten supergravity solutions [3], if the moduli effects from compact extra dimensions can be ignored (for a review, see e. g. [4]). The Hořava-Witten solution [3] can be thought of as being effectively 5D, with an extra dimension that can be large, when collated to the fundamental scale. They provide the basis for the well-known Randall-Sundrum (RS) braneworld models [1,2], that comprise the mirror symmetry and a brane with tension as well, which counter balances the leverage of the negative bulk cosmological constant on the brane, encompassing furthermore the branes self-gravity [4]. In RS brane-world scenarios, our Universe is embedded in a 5D bulk of type AdS 5 [2]. The formalism to be used hereon employs a general metric for the brane-world, instead of the Minkowski metric in the standard RS model [2].
Brane-world black holes were comprehensively studied in Randall-Sundrum like brane-world cosmologies [5][6][7], where the dynamical equations on the brane are different from the general relativity ones. In fact, the brane-world framework presents terms that handle both the effects of the free gravitational field in the bulk and of the brane embedding in the bulk as well. The imprint of the nonlocal gravitational field in the bulk on the brane provides a splitting into anisotropic stress, flux, and nonlocal energy density, where this last determines the tidal acceleration out of the brane, possibly opposing the formation of singularities [4]. Unlike the nonlocal energy density and flux, the nonlocal anisotropic stress is not ascertained by any evolution equation on the brane. In particular, isotropy of the cosmic microwave background make the existence of the FRW background be under risk. Adiabatic density perturbations are furthermore coupled to perturbations in the bulk field, making an open system on the brane [7].
Moreover, consequences of the gravitational collapse were proposed in the context of brane-world scenarios in, e. g., [24][25][26][27][28]. In addition, dark matter was investigated already in [29] as a bulk effect on the brane.
Black strings can be thought as being extended objects endowed with an event horizon, in low energy string theory [30]. The bulk metric near the brane and the black string warped horizon along the extra dimension are here reviewed, based on previous developments [4,31]. Originally, a Schwarzschild black hole on the brane-world was shown to be a black string in a higher dimensional spacetime, what leads to the usual astrophysical properties of black holes to be recovered in this scenario [31]. In this prototypical context, the Kretschmann curvature invariants diverge when the black string event horizon is approached along the axis of the black string. Several generalizations provide attempts to preclude singularities both in the bulk and on the brane-world as well.
When variable brane tension scenarios are taken into account, the brane tension can gauge the Kretschmann scalars involved. For instance, regular bulk solutions and black strings were obtained in Friedmann-Robertson-Walker brane-worlds under the Eötvös law [32], where the singularities related to the McVittie metric can be partially controlled as the cosmological time elapses. Indeed, for this type of metric the 5D physical soft singularities in the bulk are alleviated as time elapses, providing a regular 5D bulk solution, as the 5D Kretschmann invariants do not diverge. When other metrics are taken into account, for instance the Casadio-Fabbri-Mazzacurati metric one [33], black strings can be still emulated [34], however the related singularities in the bulk remain, regardless. In order to accomplish it, effective/perturbative approaches are usually employed, where the black string is made to evolve from the brane-world [4,31,35].
Brane solutions of static black hole exteriors with 5D corrections to the Schwarzschild metric have been found, for instance, in [36][37][38], and furthermore in the context where the bulk singularities can be removed [39]. The (Schwarzchild) black string is unstable near the AdS 5 horizon, defining the so called Gregory-Laflamme instability [40,41]. This scenario might be drastically altered by the inhomogeneous dust and the dark radiation. In order to accomplish this effect, we use a procedure to calculate both the metric near the brane and the 5D black string horizon [15], uniquely from a brane-world black hole metric and the associated Weyl tensor. Based on the knowledge of both the Sasaki-Shiromizu-Maeda effective field equations on the brane and upon the 5D Einstein and Bianchi equations [4,7,35,42,43], both the bulk metric near the brane and in particular the black string warped horizon can be designed, by using a Taylor expansion along the extra dimension. Such procedure provides information about all the bulk metric components [15].
Indeed, the bulk spacetime may be either given, by solving the full 5D equations or alternatively obtained by evolving the brane-world black hole metric off the brane, what encompasses the imprint from the bulk via the Weyl tensor. Numerical methods have been employed to find black hole solutions in the context of black strings and fluid/gravity correspondence [41]. Similar methods involving expansions of the metric have been used in the context of black strings [44], disposing the black string metric as the leading order solution in a Taylor expansion.
The bulk shape of the black string horizon has been merely investigated in very particular cases [4,35], and latterly the standard black string was studied in the context of a brane-world with variable tension [15]. Moreover, realistic models that take into account a post-Newtonian parameter on the Casadio-Fabbri-Mazzacurati black string [45], and the black string in a Friedmann-Robertson-Walker Eötvös brane-world [34], also represent interesting applications.
Recently regular black strings solutions associated to a dynamical brane-world have been obtained in the context of a variable brane tension [32]. The analysis of the 5D Kretschmann invariants makes us capable of attenuating the bulk physical singularities along some eras of the evolution of the Universe for the McVittie metric on an Eötvös fluid brane-world. This paper is devoted to encompass a framework with dark radiation and inhomogeneous dust. The 5D physical singularities in the bulk are shown to be inherited from the 4D brane-world and no additional singularity appears in the bulk, for some range of parameters in our model. Nevertheless, the bulk physical soft singularities can be unexpectedly controlled in the bulk upon time evolution, what makes a regular bulk 5D solution in most ranges of the dark radiation parameter. This paper is organized as follows: in Section II the dark radiation dynamics on the brane is analyzed and reviewed. Starting with the Lemaître-Tolman-Bondi (LTB) metric on the brane, the effective field equations for dark radiation on the brane are solved. The dynamical radiation model is shown to mimic a 4D cosmological constant on the brane. Both the black string solution and the bulk metric are obtained thereon. After obtaining the standard dark radiation model, a generalized framework is proposed. Both associated metrics are derived. In Section III the 5D bulk metric near the brane and the generalized black string are derived and studied, and in Section IV the black string warped horizon in the context of inhomogeneous dust and generalized dark radiation is studied. Moreover, the black string physical singularities are analyzed from the Kretschmann invariants. The 5D physical singularities in the bulk reflect the 4D brane-world physical singularities. We analyze further the Kretschmann scalars generated by higher order derivatives of the Riemann tensor, and the respective physical soft singularities show that the bulk 5D solution is regular, in some ranges of the dark radiation parameter.
In order to fix the notation hereupon µ, ν = 0, 1, 2, 3 and M, N = 0, 1, 2, 3, 5, and let n be a time-like covector field normal to the brane and y the associated Gaussian coordinate. The brane metric components g µν and the corresponding components of the bulk metricǧ µν are related by g µν + n µ n ν =ǧ µν [4]. With these choices we can write g 55 = 1 and g µ5 = 0, and thus the 5D bulk metric readsǧ where M, N = 0, 1, 2, 3 effectively.
The initial paradigm concerning a perturbative method for obtaining the black string solution consisting in assuming the Schwarzschild form for the induced brane metric, on a RS brane-world.
Subsequently a sheaf of such solutions are disposed into the extra dimension [31]: where = −6 Λ 5 denotes the curvature radius of the bulk AdS 5 , wherein the RS brane-world is embedded. Each space of constant y is a 4D Schwarzschild spacetime with a singularity along r = 0 for all y, the well-known (Schwarzschild) black string.
As it is going to be clearer in Section III, the areal radius of the sheaf of such solutions along the extra dimension is called the black string warp horizon, that shall be precisely defined in Section III.

II. DARK RADIATION DYNAMICS ON THE BRANE
Henceforward some results concerning the dark radiation dynamics on the brane will be shortly revisited [46][47][48], in order to briefly introduce the framework to get both the bulk metric near the brane and the black string encompassing the dark radiation parameter and the effective cosmological constant. New black strings solutions are here derived in the scenario provided in [46][47][48] [49][50][51]. A solution for the black string that has as limit a tidal Reissner-Nordström black hole solution on the brane was obtained in [46], in the Randall-Sundrum scenario. In order to work with the effective Einstein equations on the brane, some conditions on the projected Weyl tensor are usually assumed, in order to provide a closed system. Besides, by taking into account a system of equations where a specific state equation leads to a inhomogeneous density that has precisely the dark radiation form [52][53][54] (and its generalizations for thick branes [13,55]), and by solving the effective Einstein equations, the LTB metric can be derived. Subsequently both the bulk near the brane and the black string solution is obtained are obtained in this context.
From the dynamics of a brane-world with matter fields present in the bulk [47], the associated black string solution will be shown to present a generalized dark radiation form. The way to obtain the LTB metric is essentially different from that acquired in [46]. The inhomogeneous density is associated with conformal bulk fields, instead of being related to the electric part of the Weyl tensor. The black string solution is similarly obtained by a change in the coordinate system and its final form generalizes the first case. Thereafter two new black string solutions will be presented and subsequently used in the construction of the horizon profile in the bulk. metrics. An alternative approach for the LTB space-time is based in evolution equations of covariant objects, as the density, expansion scalar, electric Weyl tensor, shear tensor and spatial curvature. The dynamics is reduced to scalar equations, and the FRW spacetime is achieved when two of these scalars associated with the shear tensor and electric Weyl tensor are zero. This formulation is based on a 1 + 3 covariant description [56], which can be further applied to the LTB model [57,58]. In general, the applications of these models involve black holes, galaxy clusters, superclusters, cosmic voids, supernovas and redshift drift, for instance [59]. Initially found by Lemaître [49], the LTB metric describes a spherically symmetric inhomogeneous fluid with anisotropic pressurecosmological constant are present for instance in the Tolman model [50]. To derive the LTB solution, in comoving coordinates the general form for the line element is given by: where the 2-sphere area element is denoted by dΩ 2 , the energy-momentum tensor written as where ρ denotes the energy density, Λ 4 denotes the brane cosmological constant with associated energy density ρ Λ = κ −2 4 Λ 4 , and κ 4 is the 4D gravitational coupling constant. The Einstein equations, for each one of the space diagonal components, are given by the following expressions: The function A(r, t) = g(r)∂ r R satisfies Eq.(5). By setting g(r) = (1 + f (r)) −1/2 , the usual form of the LTB metric is hence obtained: where f (r) > −1. The function f can be interpreted as the energy density shell f (r) = 2E(r). The function g(r) is a geometric factor such that when g(r) = 1 the spatial sections are flat. Eqs. (4) and (5) are not independent, leading to the expression where M = M (r) is an arbitrary function of integration that gives the gravitational mass within each comoving shell of coordinate radius r. By definition of mass in [51], one can write 2dm/dr = The first term is interpreted as kinetic energy, the second stands for the Newtonian potential term and f is twice the energy of the system, when Λ 4 = 0. Hence M is the relativistic generalization of the Newtonian mass. When f is negligible, namely in the non-relativistic limit, the spatial sections are flat when g = 1. The function g provides the energy in each spatial section, and thus carries the information of curvature for each section.
where t N is known as the "bang time". Eq. (8) can be used to classify the LTB models into three classes. When Λ 4 = 0 it reads: When Λ 4 = 0 the potential V (R) = 2M R + Λ 4 3 R 2 leads to a different classification, depending on the sign of Λ 4 .

B. LTB Solution on the Brane
In this Subsection the LTB solution associated with dark radiation on the brane is reviewed, starting with the projected Einstein equations on the brane in vacuum. Unlike the Reissner-Nordström black hole, this new solution has a specific dark radiation tidal charge Q. The 4D and 5D coupling constants are related by κ 2 4 = 1 6 λκ 4 5 . The field equations in 5D Einstein theory lead to the projected equations [42,60] where Π µν is a term quadratic in the energy-momentum T µν and provides high-energy corrections arising from the extrinsic curvature of the brane, what increases the pressure and effective density of collapsing matter. The term E µν is the projection of the bulk Weyl tensor and provide Kaluza-Klein corrections originated from 5D graviton stresses [28], as the massive modes for the graviton in the linearized regime. For observers on the brane such stresses are nonlocal, in the sense that they are local density inhomogeneities on the brane generate Weyl curvature in the bulk backreacting nonlocally on the brane [4,7,16,17,53,54]. Therefore the vacuum equations are where the Weyl projected tensor can be identified with a trace-free energy- where U is the effective energy, P µν is an anisotropic stress tensor, Q µ is the effective energy flux, v µ is a 4D velocity vector satisfying v µ v µ = −1, and h µν is such that v µ h µν = 0, being thus possible A non-static spherically symmetric brane-world with P µν = 0 can be described by the line element with Q µ = 0. The anisotropic stress tensor P µν can be represented by P µν = P r µ r ν + 1 3 h µν , where P = P (r, t) is a scalar field and r µ is the unit radial vector. With these assumptions the electric part of the Weyl tensor yields with ρ = U , 3p r = U + 2P and 3p T = U − P .
By assuming the brane field equation ∇ µ E µν = 0 [4] and by considering the state equation where the constant Q is the dark radiation tidal charge [46][47][48]. As ρ = U , the energy density in this case is related to the inhomogeneous density. Thus the 4D Einstein equations (10) read It follows, by solving the above equations, that the component G tr is obtained by the expression ∂ t B∂ r R − B∂ t ∂ r R = 0, and therefore the function satisfies this relation with H = H(r). Considering thus such expression for B in the trace equation [46] it is possible to write the expression as follows which is similar to Eq.(8). Integrating Eq. (17), it reads which is analogous to Eq. (9). It is thus possible to write (12) in the LTB form given by (7).
Making the transformation of the LTB coordinates (t, r) to curvature coordinates (T, R) as the following 4D metric is finally obtained: This metric is known as the inhomogeneous static exterior of a collapsing sphere of homogeneous dark radiation [28,62]. Note that when Λ 4 = 0, the solution (20) is formally analogous to the Reissner-Nordström solution, when one identifies the electric charge to the dark radiation tidal charge.
In what follows this solution will be generalized, by considering a generalized dark radiation term with dark radiation charge Q η where η is a parameter characterizing the model of the dark radiation [47]. The dynamics of a spherically symmetric brane-world is also analyzed, when the bulk a) carries matter fields; and b) when its warp factor characterizes a global conformal transformation consistent with Z 2 symmetry. Finally, it is possible to study the bulk metric and the black string solutions with a term analogous to the black hole solution with cosmological constant on the brane.
In this framework, the energy-momentum tensor encompasses conformal bulk matter fields, whose dynamics provide a specific state equation [46][47][48].
Consider now a general conformal spherically symmetric metric ds 2 5 consistent with Z 2 symmetry along the extra dimension on the brane where z stands for the conformal extra dimensional coordinate, and A, B, R, and Ω are general functions of the coordinates (t, r, z). Ω denotes the conformal factor. The Einstein field equations are given byG whereg M N denotes the components of the is the induced metric andT µν stands for the components of the energy-momentum tensor representing the bulk fields. The brane is localized at z = z 0 .
Under the conformal transformationT M N = Ω s T M N , the energy-momentum tensor is assumed, as usual, to have weight s = −4 [46]. The conformal Einstein tensor is given bẙ By using the expressiong M N = Ω −1 g M N , Eq.(22) hence leads to the following equations: Eq. (24) Hence Eqs. (26) and (27) require that 2T z z = T µ µ . Now, by considering the energy-momentum the state equation ρ − p r − 2p T + 2p z = 0 holds. It implies that ∇ z T z z = 0 and subsequently that ∂ z p z = 0 Thus ρ, p r and p T must be independent of z.
The system of inhomogeneous dark radiation and an effective cosmological constant is defined by a conformal bulk matter with the following equations of state where η characterizes the dark radiation model and Λ is a bulk quantity and mimics a 4D cosmological constant on the brane. The components of the Einstein tensor can be thus evinced: By taking the divergence of T µ ν , it reads what consequently leads to where here Q η = const is interpreted as a generalized dark radiation tidal charge and behaves like (14) when η = −1. Regarding the following energy conditions [63] ρ ≥ 0 and ρ + p i ≥ 0, respectively known as weak, strong and dominant energy conditions, for ρ = 0 (by taking the weak energy condition) the equality ρ DR = −κ −2 5 Λ holds, such that Λ = −Q η R 2η−2 ≤ 0 for Q η ≥ 0. One can realize that ρ = −p r regards the weak condition and ρ + p T = (1 − η)ρ DR holds if η ≤ 1, where here ρ > 0 satisfies the first condition. By a similar analysis in 5D we see that η ≤ 0 (weak), and η ≤ 0 (4D and strong), η ≤ −1/3 (5D and strong), |η| ≤ 1 (4D and dominant) and what generalizes Eq. (17), being identical to it when η = −1. By performing one more integration it reads where ±t refers to expansion or collapse and τ corresponds to the evaluation of the function at t = 0. The condition R(0, r) = r is taken on the hypersurface t = 0. The radial equation leads to H = √ 1 + f [47], and thus the metric has the 4D LTB form. Hence the 5D conformal line element reads ds 2 (5) = Ω 2 RS −dt 2 + (∂ r R/H) 2 dr 2 + R 2 dΩ 2 + dz 2 , where Ω RS is the Randall-Sundrum warp factor [2]. From the dynamical dark radiation models the marginal bound f = 0 corresponds actually to static solutions. Finally, the transformation from The black string solution is therefore obtained: The models that describe the inhomogeneous static exterior of a collapsing sphere of homogeneous standard dark radiation require the value η = −1 [28,46,62]. In general the exterior spacetime is not static in the brane-world scenario [28], however the collapse of a homogeneous Kaluza-Klein energy density is static, and can be identified to the dark radiation. Hence in the case η = −1 the model of homogeneous dark radiation is recovered. It is worth to emphasize that the exterior is static solely when the system has tidal charge and cosmological constant, where the physical mass equals zero. When Λ = 0, the zero mass limit of the tidal Reissner-Nordström black hole is obtained [36]. The event horizon is determined by the solutions of the following equation: on the brane. Thus for Λ = 0 it implies that R 2η h = (2η + 1)/Q η . When Λ = 0, the exact location of the horizons can not be obtained, with exceptions for the values η = −1 and η = 1/2. Respectively, the horizons are given by: The conditions Λ < 0 and Q 1/2 > 0 imply two horizons, an inner and an outer one. For two specific values of Q 1/2 the singularity is naked, thus violating the cosmic censorship hypothesis [64]. Yet, if Λ > 0 and Q 1/2 ≷ 0, then there is a single horizon R (±) h .

III. BULK METRIC AND THE BLACK STRING
In this Section, the bulk metric near the brane as well as the black string associated to a black hole on a brane-world are briefly introduced [4,15,42]. Eq.(2) represents the black string metric.
By denoting (5) R µνσρ the components of the 5D Riemann tensor, as the 5D Kretschmann invariant e 4|y|/ is unbounded as y → ∞ [4,31], thus the Schwarzschild solution is not a good candidate neither for a brane-world black hole nor for trying to remove at least some of the bulk singularities. Hence a well-established perturbative method is employed to find both the bulk metric near the brane and, in particular, the black string warped horizon along the extra dimension. In what follows such framework is revisited. This shall be further accomplished in Section IV in the context of inhomogeneous dust and generalized dark radiation on the brane-world and inhomogeneous dark radiation in the bulk as well. There we shall evince that in such scenario the bulk (and the black string) can be regular for certain ranges of the dark radiation parameters.
In brane-worlds with Z 2 symmetry, the junction conditions imply that the extrinsic curvature of the brane is given by [4] K µν = κ 2 The trace-free and symmetric components of the bulk Weyl tensor C µνσρ are respectively given by B µνρ = g τ µ g σ ν C τ σρβ n β and E µν = C µνσρ n σ n ρ , where n α denote components of a vector field out of the braneworld. Hereupon we denote by R µνσρ = (5) R µνσρ (x α , 0) the components of the 5D Riemann tensor computed on the brane.
One thus finds the above expansion up to the fourth order is given by [15] g µν (x α , y) = g µν − κ 2 5 1 3 (λ − T )g µν + T µν |y| where H ≡ H µ µ and H 2 ≡ H ρσ H ρσ , for any tensor H of rank two.
The black string warped horizon [66] can be studied from g θθ (x α , y) in Eq. (52). In fact, regarding a spherically symmetric 4D metric modelling a brane black hole, the usual 4D areal radial coordinate r is related to the 5D metric in Eq.(52) by the expression r 2 = g θθ (x α , 0) [15,35]. The black string has a warped horizon on the brane that has radius r = g θθ (x α , 0), where r denotes the coordinate singularity which is solution of g −1 rr (r) = 0 (See Eq. (II.6) of Ref. [67]). The radius of the black string warped horizon is hence r(y) = g θθ (x α , y). The term g θθ (x α , y) in Eq. (52) for µ = ν = θ corresponds to the bulk (squared) areal radius, and includes both the black string horizon for r = r, and in particular the brane black hole horizon for y = 0.

IV. BLACK STRINGS AND DARK DUST
In this Section we investigate the black string related to the induced black hole on the brane given by Eq. (41). The black string warped horizon is well known to be the component g θθ (x α , y) in Eq. (52), evaluated at event horizons in (44). At first let us calculate such component for an arbitrary x α . In order to accomplish it, the 4D energy-momentum tensor on the brane is given by the brane-world components in Eq.(28), where and the expressions for the energy-momentum, the projected Weyl tensor components, the extrinsic curvature, and the Riemann tensor respectively given respectively by (A1a-A4c) in Appendix are thus obtained. Hence the component θθ of the metric, corresponding to the black string horizon along the bulk (52), can be written as: The last expression does not explicitly include the terms of order y 4 /4! as they are extensive and awkward, although we shall consider such terms in our subsequent analysis.
Our first analysis consists in regarding the variable T = T (r, t), which appears solely as the term exp(−αT ) at the order |y| 3 /3! in Eq. (56). The graphic below in Fig. 1 shows the dependence of the component g θθ (x α , y) on T = T (r, t). Hence, near the brane, our results can be taken to be     Besides the black string warped horizon, Eq.(52) provides more generally the bulk metric near the brane. In order to check whether the bulk is regular, we aim to study the 4D and the 5D Kretschmann invariants, related to the black hole on the brane and to the black string in the bulk.
Our analysis of the bulk physical singularities is independent on the perturbative method (52), as the curvature invariants are independent of it.
The Gauss equation which relates the 4D and 5D Riemann curvature tensors according to (5) and the 4D and 5D Kretschmann invariants are given by For the generalized dark radiation model they are related by: The physical singularities are provided by R = 0 and by the solutions R h of the Eq. (42) for the values η = −1 and η = 1/2 given by Eqs. (44). Neither new singularities are introduced nor the existing ones (R = 0 and R = R h in Eqs. (43) and (44) are removed from the bulk.
We are going to show in what follows, by the analysis of the physical soft singularities, that the bulk can be regular. Indeed, the 4D invariant (here ∇ µ denotes the covariant derivative on the brane) is very soft, since it takes invariants involving at least two derivatives of the curvature to detect it. Consequently, the 5D version of the invariant ξ reads where D a denotes the 5D covariant derivative. It is worth to point that a, b are effectively 4D spacetime indexes, as the 5D covariant derivative can be realized as D a = ∇ µ and D a = ∇ 5 , when the extra dimension y is taken into account. It implies that the 5D Kretschmann invariant (5) ξ is given by [45] Based on the values of the metric (41) and the extrinsic curvature components given in (45), the 5D Kretschmann invariants (5) ξ for the bulk can calculated. Due to the awkwardness of the expression for these invariants, we opt to analyze our results by the graphics in Figs. 8, 10, 12, and 14. Fig.   7 has to be compared to Fig. 8, that describes the 5D invariant (5) ξ. In Fig. 7 since η ∼ 0 then the 4D invariant ξ diverges, independently of the value for R. On the other hand, Fig. 8 depicts that for η ∼ 0, the 5D Kretschmann scale (5) ξ goes to infinity for most values of R, but when 0.94 R 1.06 the 5D Kretschmann invariant (5) ξ does not diverge. Hence for this range of physical soft singularities present on the brane the bulk is regular. Moreover, Fig. 7 shows that ξ → +∞ for R → 0, however Fig. 8 evinces that for 0 η 0.5 the limit ξ → +∞ on the brane alters to (5) ξ → −∞, for values of R ∼ 0. Still, for −1 η −0.5 the 5D Kretschmann invariant Now the case Λ = 1 is regarded. Fig. 9, regarding the 4D invariant ξ, must be compared respectively to Fig. 10, describing the 5D invariant (5) ξ. A similar pattern is realized in this case, where now Fig. 9 shows that ξ → +∞ for R → 0. We regard the Λ = −1 case in Figs

V. CONCLUDING REMARKS
The bulk metric near the brane was obtained and in particular the black string warped horizon, for inhomogeneous dust and generalized dark radiation on the brane-world and inhomogeneous dark radiation in the bulk are regarded. The standard dark radiation [46,[52][53][54] is a particular case analyzed, where the dark radiation parameter η = −1 and the mimicked cosmological constant Λ equals zero, provided by the 5D pressure also equal to zero.
By analyzing the 4D and 5D standard Kretschmann invariants, respectively defined by Eqs. (58) and (59), the Gauss equation is shown to imply that the bulk associated to brane-world models with inhomogeneous dust and generalized dark radiation inherits the brane-world physical singularities at R = 0 and at R = R h is the solution of Eq. (42). Although the black string warped horizon is obtained as a particular case of the 5D bulk metric near the brane via an effective method, the analysis of the 5D bulk singularities relies on an exact method provided by the Gauss equation (57).
The 4D and 5D Kretschmann invariants, respectively given by (61) and (62) For the extrinsic curvature, by computing (45) we obtain Finally, the necessary components of the Riemann tensor are given by