On analogues of black brane solutions in the model with multicomponent anisotropic fluid

doi:10.1016/j.physletb.2010.08.060

Physics Letters B

Volume 693, Issue 3, 4 October 2010, Pages 399-403

https://doi.org/10.1016/j.physletb.2010.08.060 Get rights and content

Abstract

A family of spherically symmetric solutions with horizon in the model with m-component anisotropic fluid is presented. The metrics are defined on a manifold that contains a product of $n - 1$ Ricci-flat “internal” spaces. The equation of state for any s-th component is defined by a vector $U^{s}$ belonging to $R^{n + 1}$ . The solutions are governed by moduli functions $H_{s}$ obeying non-linear differential equations with certain boundary conditions imposed. A simulation of black brane solutions in the model with antisymmetric forms is considered. An example of solution imitating $M_{2} – M_{5}$ configuration (in $D = 11$ supergravity) corresponding to Lie algebra $A_{2}$ is presented.

Introduction

In this Letter we continue our investigations of spherically-symmetric solutions with horizon (e.g., black brane ones) defined on product manifolds containing several Ricci-flat factor-spaces (with diverse signatures and dimensions). These solutions appear either in models with antisymmetric forms and scalar fields [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11] or in models with (multicomponent) anisotropic fluid [12], [13], [14], [15]. For black brane solutions with 1-dimensional factor-spaces (of Euclidean signatures) see [16], [17], [18] and references therein.

These and more general brane cosmological and spherically symmetric solutions were obtained by reduction of the field equations to the Lagrange equations corresponding to Toda-like systems [2], [19]. An analogous reduction for models with multicomponent anisotropic fluids was performed earlier in [20], [21]. For cosmological-type models with antisymmetric forms without scalar fields any brane is equivalent to an anisotropic fluid with the equations of state: ${\hat{p}}_{i} = - \hat{ρ} or {\hat{p}}_{i} = \hat{ρ},$ when the manifold $M_{i}$ belongs or does not belong to the brane worldvolume, respectively (here ${\hat{p}}_{i}$ is the effective pressure in $M_{i}$ and $\hat{ρ}$ is the effective density).

In this Letter we present spherically-symmetric solutions with horizon (e.g the analogues of intersecting black brane solutions) in a model with multicomponent anisotropic fluid (MCAF), when certain relations on fluid parameters are imposed. The solutions are governed by a set of moduli functions $H_{s}$ obeying non-linear differential master equations with certain boundary conditions imposed. These master equations are equivalent to Toda-like equations and depend upon the non-degenerate ( $m \times m$ ) matrix A. It was conjectured earlier that the functions $H_{s}$ should be polynomials when A is a Cartan matrix for some semisimple finite-dimensional Lie algebra (of rank m) [6]. This conjecture was verified for Lie algebras: $A_{m}$ , $C_{m + 1}$ , $m ⩾ 1$ [7], [8]. A special case of black hole solutions with MCAF corresponding to semisimple Lie algebra $A_{1} \oplus \dots \oplus A_{1}$ was considered earlier in [13] (for $m = 1$ see [12]).

The Letter is organized as follows. In Section 2 the model is formulated. In Section 3 spherically-symmetric MCAF solutions with horizon corresponding to black brane type solutions, are presented. In Section 4 a polynomial structure of moduli functions $H_{s}$ for semisimple finite-dimensional Lie algebras is discussed. In Section 5 a simulation of intersecting black brane solutions is considered and an analogue of $M_{2} – M_{5}$ dyonic solution is presented.

Section snippets

The model

In this Letter we deal with a family of spherically symmetric solutions to Einstein equations with an anisotropic matter source $R_{N}^{M} - \frac{1}{2} δ_{N}^{M} R = k^{2} T_{N}^{M},$ defined on the manifold $M = \underset{\binom{radial}{variable}}{R_{*}} \times \underset{\binom{spherical}{variables}}{(M_{0} = S^{d_{0}})} \times \underset{time}{(M_{1} = R)} \times \dots \times M_{n},$ with the block-diagonal metrics $d s^{2} = e^{2 γ (u)} d u^{2} + \sum_{i = 0}^{n} e^{2 β^{i} (u)} h_{m_{i} n_{i}}^{[i]} d y^{m_{i}} d y^{n_{i}} .$

Here $R_{*} \subseteq R$ is an open interval. The manifold $M_{i}$ with the metric $h^{[i]}$ , $i = 1, 2, \dots, n$ , is a Ricci-flat space of dimension $d_{i}$ : $R_{m_{i} n_{i}} [h^{[i]}] = 0,$ and $h^{[0]}$ is the standard metric on the unit sphere $S^{d_{0}}$ , so

Spherically symmetric solutions with horizon

We will make the following assumptions: $1^{\circ} . U_{0}^{s} = 0 \Leftrightarrow {\hat{p}}_{0}^{s} = {\hat{ρ}}^{s},$ $2^{\circ} . U_{1}^{s} = 1 \Leftrightarrow {\hat{p}}_{1}^{s} = - {\hat{ρ}}^{s},$ $3^{\circ} . (U^{s}, U^{s}) = U_{i}^{s} G^{i j} U_{j}^{s} > 0,$ $4^{\circ} . 2 (U^{s}, U^{l}) / (U^{l}, U^{l}) = A_{sl},$ where $A = (A_{sl})$ is non-degenerate matrix, $G^{i j} = \frac{δ^{i j}}{d_{i}} + \frac{1}{2 - D},$ are components of the matrix inverse to the matrix of the minisuperspace metric [22] $(G_{i j}) = (d_{i} δ_{i j} - d_{i} d_{j}),$ $i, j = 0, 1, \dots, n$ , and $D = 1 + \sum_{i = 0}^{n} d_{i}$ is the total dimension.

The conditions $1^{\circ}$ and $2^{\circ}$ in brane terms mean that brane “lives” in the time manifold $M_{1}$ and does not “live” in $M_{0}$ . Due to assumptions $1^{\circ}$ and $2^{\circ}$ and the equations of

Polynomial structure of $H_{s}$ for Lie algebras

Now we deal with solutions to second order non-linear differential equations (3.12) that may be rewritten as follows $\frac{d}{d z} (\frac{(1 - 2 μ z)}{H_{s}} \frac{d}{d z} H_{s}) = {\bar{B}}_{s} \prod_{l = 1}^{m} H_{l}^{- A_{s l}},$ where $H_{s} (z) > 0$ , $z = R^{- d} \in (0, {(2 μ)}^{- 1})$ ( $μ > 0$ ) and ${\bar{B}}_{s} = B_{s} / d^{2} \neq 0$ . Eqs. (3.13), (3.14) read $H_{s} ({(2 μ)}^{- 1} - 0) = H_{s 0} \in (0, + \infty),$ $H_{s} (+ 0) = 1,$ $s = 1, \dots, m$ .

The condition (3.15) reads as follows $H_{s} (z) > 0 is smooth in (0, z_{ϵ}),$ $s = 1, \dots, m$ , where $z_{ϵ} = {(2 μ)}^{- 1} e^{ϵ d}$ , $ϵ > 0$ .

It was conjectured in [6] that Eqs. (4.1), (4.2), (4.3) have polynomial solutions when $(A_{s s^{'}})$ is a Cartan matrix for some

Analogues of intersecting black brane solutions

The solution from the previous section for MCAF allows one to simulate the intersecting black brane solutions [11] in the model with antisymmetric forms without scalar fields. In this case the parameters $U_{i}^{s}$ and pressures have the following form: $U_{i}^{s} = {\begin{matrix} d_{i}, p_{i}^{s} = - ρ^{s}, & i \in I_{s}; \\ 0, ρ^{s}, & i \notin I_{s} . \end{matrix}$

Here $I_{s} = {i_{1}^{s}, \dots, i_{k_{s}}^{s}} \in {1, \dots, n}$ is the index set [11] corresponding to brane submanifold $M_{i_{1}^{s}} \times \dots \times M_{i_{k_{s}}^{s}}$ .

The relation $4^{\circ}$ (3.1) leads us to the following dimensions of intersections of brane submanifolds (“worldvolumes”) [2], [11]

Conclusions

Here we have presented a family of spherically symmetric solutions with horizon in the model with multicomponent anisotropic fluid with the equations of state (2.8) and the conditions (3.1) imposed. The metric of any solution contains $(n - 1)$ Ricci-flat “internal” space metrics.

As in [6], [7], [8] the solutions are defined up to solutions of non-linear differential equations (equivalent to Toda-like ones) with certain boundary conditions imposed. These solutions may have a polynomial structure

Acknowledgements

This work was supported in part by grant NPK-MU (PFUR) and Russian Foundation for Basic Research (Grant No. 09-02-00677-a).

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View full text

On analogues of black brane solutions in the model with multicomponent anisotropic fluid

Abstract

Introduction

Section snippets

The model

Spherically symmetric solutions with horizon

Polynomial structure of Hs for Lie algebras

Analogues of intersecting black brane solutions

Conclusions

Acknowledgements

Phys. Lett. B

Nucl. Phys. B

Phys. Lett. B

Phys. Lett. B

Phys. Lett. B

Phys. Lett. B

Grav. Cosmol.

J. Math. Phys.

Grav. Cosmol.

Grav. Cosmol.

Grav. Cosmol.

Grav. Cosmol.

Class. Quantum Grav.

J. Math. Phys.

Polynomial structure of $H_{s}$ for Lie algebras