# A class of higher-dimensional solutions of Einstein’s vacuum equation

- 126 Downloads

## Abstract

A new class of higher-dimensional exact solutions of Einstein’s vacuum equation is presented. These metrics are written in terms of the exponential of a symmetric matrix and when this matrix is diagonal the solution reduces to higher-dimensional generalizations of Kasner spacetime with a cosmological constant. On the other hand, the metrics attained when such matrix is non-diagonal have more intricate algebraic structures. In the same vein, we integrate Einstein’s equation for homogeneous spacetimes, but not necessarily isotropic, in the presence of a perfect fluid and attain exact solutions in terms of the exponential of a symmetric matrix. Such solutions have not been presented in the literature yet.

## 1 Introduction

*i*,

*j*run from 1 to

*n*. The subscripts

*x*and

*y*indicate that a function depends just on the coordinate

*x*and

*y*respectively. Thus, for instance, the components of the \(n\times n\) symmetric matrix \(H^{ij}_x\) are functions of

*x*, \(H^{ij}_x = H^{ij}_x(x)\). In particular, none of these functions depend on the coordinates \(\sigma _i\), so that these spaces are endowed with

*n*commuting vector fields, namely \(\partial _{\sigma _i}\). Note that the functions \(\Delta _x\) and \(\Delta _y\) can be easily gauged away by redefining the coordinates

*x*and

*y*. Nevertheless, instead of setting these functions to 1, we shall keep them and make a more convenient choice afterwards. As we shall see in the sequel, the general solution of Einstein’s vacuum equation can be elegantly written in terms of the exponential of a symmetric constant matrix \(\varvec{Q_0}\). In particular, the special case in which the matrix \(\varvec{Q_0}\) is diagonal leads to higher-dimensional generalizations of Kasner spacetimes [1, 2, 3, 4, 4], which are generally used to model homogeneous but anisotropic cosmological systems [5, 6]. However, the most interesting cases are the ones in which the matrix \(\varvec{Q_0}\) cannot be diagonalized, namely when \(\varvec{Q_0}\) is complex. As far as the authors know, the latter higher-dimensional solutions have not been described in the literature yet.

*n*commuting Killing vectors and a Killing tensor of rank two [10]. These spaces have inverse metric given by

*n*independent Killing vector fields \(\partial _{\sigma _i}\), and two Killing tensors, namely the metric and \(\varvec{K}\).

The case \(n=2\) of the problem treated in this article has already been considered recently in Ref. [7], where it has been proved that almost all solutions that arise from integrating Einstein’s vacuum equation for the line element (1) when \(n=2\) are already known, a particular example being the Kasner metric with cosmological constant [11, 12]. Nevertheless, it turns out that one solution for the latter problem had not been described in the literature before its appearance in Ref. [7]. Differently from Kasner spacetime, in this new solution one of the Killing vectors \(\partial _{\sigma _i}\) is not orthogonal to a family of hypersurfaces, so that the line element is non-diagonal when we use the cyclic coordinates \(\sigma _i\). Thus, besides leading to a whole class of new higher-dimensional solutions of Einstein’s vacuum equation, the problem considered in the present article serves also to shed light on the origin of the new four-dimensional solution obtained in Ref. [7].

Since the establishment of General Relativity, interest in spacetimes with dimension greater than four has being increasing, specially over the last two decades. The reasons for studying these spaces are abundant. For instance, the gravity/gauge duality provides a map between field theories in *d* dimensions and gravitational theories in \(d+1\) dimensions, linking the weak coupling regime of one side to the strong coupling regime of the other side [13, 14, 15]. Such tool has been used, for example, to obtain results on strongly coupled quantum chromodynamics, which is a field theory in four dimensions, by performing calculations with a weak gravitational field in five dimensions [16]. A particularly exciting illustration are the experimentally verified results on quark-gluon plasma [17, 18]. Another important motivation for studying higher-dimensional spacetimes is string theory, which, among other things, provides a description of quantum gravity. In order to be consistent, string theory requires spacetime to have 10 dimensions [19]. Besides these two examples, there are several other theories that seek to explain our Universe through the use of spaces with dimension greater than four, for reviews see [20, 21]. With these motivations in mind, the solutions presented here shall be of particular application to cosmological models, inasmuch as some of them have spatial homogeneity.

The outline of the article is the following. In the next section we establish the basic notation used throughout the article and introduce a frame in order to compute the components of the Ricci tensor through Cartan’s structure equations. Then, in Sect. 3 we fully integrate Einstein’s vacuum equation with a cosmological constant, namely \(R_{ab}=\Lambda g_{ab}\). In Sect. 4 we show that these spaces have integrable geodesics. The solutions obtained depend on the exponential of a symmetric matrix, which is generally hard to compute explicitly. Thus, in Sect. 5 we use some theorems of matrix theory along suitable coordinate transformations in order to provide an explicit form for the solutions obtained in the preceding section. Particularly, we stress that the solutions can have different algebraic structures depending on the canonical form of the symmetric matrix. Then, in Sect. 6 we work out some examples for the solutions found in Sect. 3. Finally, in Sect. 7 we integrate Einstein’s equation for spacetimes possessing homogeneous space-like hypersurfaces in the presence of a perfect fluid matter field. Also, the conclusions and perspectives of the article are presented in Sect. 8.

## 2 Introducing a vielbein and computing the curvature

In what follows, we shall use the frame formalism in order to compute the curvature and integrate Einstein’s equations. In this preliminary section we shall define the vielbein and then compute the spin coefficients and use them to calculate the Ricci tensor. Before proceeding, let us establish our index conventions. Indices from the beginning of the alphabet, like *a*, *b* and *c*, run over the number of spacetime dimensions, from 1 to \(n+2\), whereas indices *i*, *j* and *k*, from the middle of the alphabet, range from 1 to *n*.

## 3 Integration of Einstein’s vacuum equation

### 3.1 The case \(S'_y=0\)

*y*in such a way to eliminate the dependence on the function \(\Delta _y\) of this line element. More precisely, we will replace the coordinate

*y*by \(\tilde{y} = \int dy/\Delta _y\). Dropping the tilde after the transformation, this amounts to assuming \(\Delta _y=1\) in the line element (1). Therefore, in this subsection we shall set

*x*. Therefore, without loss of generality, let us choose \(\Delta _x\) to be the function that makes the expression enclosed by the square brackets in Eq. (8) identically zero, namely let us assume

In particular, from the expression for \(S_x\), (14), we see that the case in which \(S_x\) and \(S_y\) are both constant can be attained by setting \(a_1\) and \(a_2\) to zero. Nevertheless, going through the whole integration process assuming \(S_x' = 0\) from the very beginning, one can check that this last requirement is, actually, not necessary. Rather, the constants \(a_1\) and \(a_2\) are just constrained by the relation \(a_2 = (n+1)a_1^2\). Thus, the case \(a_2 = a_1 = 0\) is just a particular solution. So, the most general solution for this case is provided by the metric (19) with \(S_x=s_1\), \(\Delta _x\) given by (12), and \(\varvec{Q_0}\) satisfying \(tr(\varvec{Q_0})^2=tr(\varvec{Q_0}^2)\). Moreover, one can easily see from the condition \(R_{\hat{y}\hat{y}}=\Lambda \) that if \(S_x\) and \(S_y\) are both constant one must have \(\Lambda =0\) in order to attain a solution, see (7).

### 3.2 Subcase \(S'_x=0\) and \(S'_y\ne 0\)

*y*. Here we follow steps analogous to the ones taken in the previous subsection. For instance, absorbing the constant value of \(S_x\) into \(S_y\), we can assume, without any loss of generality, that

*x*and

*y*respectively. Thus, let us conveniently choose them to be such that the following equations hold

*x*along with a coordinate transformation in the cyclic coordinates. Note that no restriction has been placed over the function \(S_y\), conveying the fact that we still have a coordinate freedom to choose \(S_y\) to be any non-constant function of

*y*, each choice corresponding to a different coordinate system but representing the same physical spacetime. In the latter line element, the matrix \(\varvec{Q_0}\) is an arbitrary \(n\times n\) symmetric matrix such that

## 4 Integrability of geodesics

As mentioned in the introduction, in the class of spacetimes considered here the geodesic equation is fully integrable, due to the existence of enough symmetries. The aim of the present section is to show explicitly how one can separate the geodesic equation of the solutions that we have found. In order to do so, it is worth recalling that if \(V^a\) is a geodesic vector field with affine parametrization, \(V^a\nabla _aV^b=0\), and \(Q^a\) is a Killing vector field, it follows that \(V^a Q_a\) is constant along the geodesic flow. Likewise, if \(K_{ab}\) is a rank two Killing tensor, i. e., \(\nabla _{(a}K_{bc)}=0\), then \(V^aV^b K_{ab}\) must also be constant along the geodesic curves. In particular, since the metric \(g_{ab}\) is covariantly constant it is also a Killing tensor, so that \(V^aV^b g_{ab}\) is constant along the flow.

*s*being an affine parameter, it turns out that the following conservation laws hold:

*dy*/

*ds*and inserting the results into the last conservation law, we are lead to

*x*(

*s*). With this at hand, one can replace \(S_x\) and \(H_x^{ij}\) as functions of

*s*in the first two relations of (28) and finally find \(\sigma _i(s)\) and

*y*(

*s*), proving that the geodesic equation is separable and, therefore, integrable.

*y*(

*s*), which plugging into the third equation along with using the first conservation law yields

*x*(

*s*). With these at hand, one can integrate the first equation in (30) and finally attain \(\sigma ^i(s)\). This shows that in case (B) the geodesic equation is also separable.

## 5 Putting the solution in a treatable form

Although the solutions (19) and (26) have been expressed in an elegant and compact form, the exponential \(e^{2x\varvec{Q_0}}\) will mostly result in an extremely cumbersome matrix, specially for solutions in dimension higher than four, namely when the values of *n* are greater than two. Actually, no general expression exists for the exponential of an \(n\times n\) matrix when \(n>2\). In spite of this, the intent of the present section is to show that, by means of a suitable choice of cyclic coordinates, the term \(\varvec{d\tilde{\sigma }}^t e^{2x\varvec{Q_0}} \varvec{d\tilde{\sigma }}\) can always be explicitly written as a finite sum of terms.

Recall that the matrix \(\varvec{Q_0}\) appearing on the exponential must be symmetric. Therefore, due to the spectral theorem, it follows that if \(\varvec{Q_0}\) is real then it can be put in a diagonal form by means of an orthogonal transformation on the basis. Once put in the diagonal form, it is trivial to compute its exponential. However, the case in which \(\varvec{Q_0}\) is complex is trickier. Fortunately, there exists a complementary theorem for complex symmetric matrices. It states that if the complex symmetric matrix \(\varvec{Q_0}\) can be diagonalized this will be accomplished by a complex orthogonal transformation (see for instance theorem 4.4.13 (page 211) of [22]). Moreover, in the case in which \(\varvec{Q_0}\) cannot be diagonalized, there always exist an orthogonal complex matrix \(\varvec{M}\) such that \(\varvec{M}^{-1} \varvec{Q_0 M}\) is the sum of a nilpotent matrix plus a diagonal matrix that commutes with it, so that the exponential can also be easily computed [27, 28]. By “orthogonal complex matrix” we mean an \(n\times n\) matrix \(\varvec{M}\) with complex entries obeying the relation \(\varvec{M}^t \varvec{M}=\varvec{I_n}\), where \(\varvec{I_n}\) stands for the \(n\times n\) identity matrix. In what follows we shall consider the cases in which \(\varvec{Q_0}\) is diagonalizable or not separately.

In the context of gauge/gravity duality, another class of line-elements determined by the exponential of a matrix have been found in Ref. [29], in which case the matrix being exponentiated is the stress-energy tensor of a field theory in the boundary of the spacetime.

### 5.1 Diagonalizable case

### 5.2 Nondiagonalizable case

*k*being an integer, as the matrices whose components are given by

*i*and

*j*range from 1 to

*m*. As can easily be checked, these matrices have the following properties:

*m*. Moreover, note that \(\varvec{P_{1,0}}=0\), so that if the block \(\varvec{Q_\nu } = q_\nu \varvec{I_{m_\nu }} + \varvec{P_{m_\nu ,0}}\) is a \(1\times 1\) matrix it becomes just the eigenvalue \(q_\nu \).

Thus, now we have a recipe to explicitly construct new solutions for Einstein’s vacuum equation in arbitrary dimensions. For each value of *n* we can have solutions with different algebraic structures depending on the size of the blocks of the canonical form of \(\varvec{Q_0}\). For instance, when \(n=4\) we have five possibilities: (I) the line element is diagonal, namely the canonical form of \(\varvec{Q_0}\) is the direct sum of one-dimensional blocks; (II) the canonical form of \(\varvec{Q_0}\) is the direct sum of two \(2\times 2\) blocks; (III) the canonical form of \(\varvec{Q_0}\) is the direct sum of a \(2\times 2\) block plus two \(1\times 1\) blocks; (IV) the canonical form of \(\varvec{Q_0}\) is the direct sum of a \(3\times 3\) block plus one \(1\times 1\) block; and (V) \(\varvec{Q_0}\) cannot be broken in smaller blocks by a similarity transformation, it is a single \(4\times 4\) block. In general, for a given *n*, the number of different algebraic structures for the line element is the number of partitions of the integer *n*. In the next section we shall explore the possibilities for \(n=2\) and \(n=3\) in full detail.

## 6 Examples

In this section we work out explicit examples of the solutions found above, for both cases regarding whether the matrix \(\varvec{Q_0}\) is diagonalizable or not. Particularly, in the former case all the solutions turn out to be higher-dimensional generalizations of the Kasner metric. On the other hand, when \(\varvec{Q_0}\) is nondiagonalizable new solutions are attained. As stressed at the introduction, the special case \(n=2\) of our solutions, namely when spaces are four-dimensional, has already been addressed previously in Ref. [7]. Thus, as we will check below, all solutions for \(n=2\) should coincide with the ones of Ref. [7].

### 6.1 Diagonalizable case

In this subsection we shall deal with the case in which the complex symmetric matrix \(\varvec{Q_0}\) can be diagonalized. During the integration process of Einstein’s equation, we had to consider separately two possibilities: (A) when \(S_y'=0\); and (B) when \(S_x'=0\) and \(S_y'\ne 0\). Likewise, here we shall treat these two possibilities separately.

#### 6.1.1 Case (A), \(S_y'=0\)

*i*running from 1 to

*n*, and \(p_{n+1}\) by

#### 6.1.2 Case (B), \(S_x'=0\) and \(S_y'\ne 0\)

### 6.2 Nondiagonalizable case

Now, we shall deal with the most interesting case, namely when \(\varvec{Q_0}\) cannot be diagonalized. For \(n\ge 3\), this case leads to solutions that, as far as the authors know, have not been described in the literature yet. But, first, let us start considering the case \(n=2\) and showing that the solution obtained here coincides with the one of Ref. [7]. Then, we shall consider the case \(n=3\).

#### 6.2.1 Case \(n=2\)

*q*, and there exists an orthogonal complex matrix \(\varvec{M}\) such that \(\varvec{M}^{t} \varvec{Q_0 M} =\varvec{Q_1}\) assumes the following canonical form:

For the case (B), where \(S_x'=0\) and \(S_y\ne 0\), a similar treatment provides that the solution is a maximally symmetric space, as already anticipated in Ref. [7]. In such case, we notice that even though the part \(\varvec{d\tilde{\sigma }}^t e^{2x\varvec{Q_0}} \varvec{d\tilde{\sigma }}\) of the line element cannot be diagonalized using cyclic coordinates, the metric as a whole can.

#### 6.2.2 Case \(n=3\)

*n*, whereas the case (II) can be easily tackled by adding the term \(e^{2x q_2}d\tau _3^2\) to the line element obtained in the previous Sect. (case \(n=2\)), in accordance with Eq. (36), which yields

*q*, we have that

*q*, \(q \rightarrow \tilde{q} =-(1-i)q\), which easily leads to

*x*, it may seem that we have spoiled the expressions for \(S_x\) and \(\Delta _x\) with complex numbers. But this is not the case. Indeed, in the expressions for these functions, see Eqs. (12), (14), and (24), the coordinate

*x*appears in the combinations \(a_1 x\) and \(\sqrt{a_2 - a_1^2} x\). But, since

*n*, we can set \(s_1=1\) and \(x_0=0\) without changing the geometry of the spacetime. Likewise, we can also set \(\tilde{q}=1\) in the latter line element, which just amounts to a coordinate transformation. Thus, the only important parameter in this metric is the cosmological constant, \(\Lambda \). As far as the authors know, this solution has not been described before in the literature.

*n*, whenever the canonical form of \(\varvec{Q_0}\) is a single block. Thus, for the case \(n=3\) with \(S_x'=0\) we eventually arrive at the following solution:

## 7 Cosmological models with perfect fluid

From the physical point of view, a particularly interesting case of the class of metrics considered here, see Eq. (1), is when we assume that the function \(\Delta _x\) is imaginary, in which case the vector field \(\partial _x\) becomes time-like and *x* can be seen as a global time coordinate. Note that this possibility can be easily implemented in the above solutions by means of complexifying the integration constant \(c_1\), which appears as a multiplicative global factor in the function \(\Delta _x\). Indeed, assuming that \(c_1 = i \,\tilde{c}_1\), with \(\tilde{c}_1\) being a real constant, it follows that the sign of \(dx^2/\Delta _x^2\) becomes negative. Note also that no problem appears in the definition of the other integrated functions, since they do not depend on \(c_1\). The only other place in which \(c_1\) appears is in the constraint (15), but in this expression \(c_1\) appears squared, so that the constraint keeps its real character. In the present section, we shall assume exactly this, namely *x* will be a global time coordinate.

*p*standing respectively for the energy density and the pressure fields of the fluid, whereas \(U^a\) is the normalized velocity field of the fluid. Here we will assume that \(U^a\) is orthogonal to the space-like hyper-surfaces generated by \(\partial _{\sigma _i}\) and \(\partial _y\), so that

*t*’s in the expressions, like the one in \(\varvec{A_t}^t\), stands for transposition. Then, taking into account that the above components are the only ones that are nonvanishing, the projection of the Einstein’s equations (53) on the orthonormal frame yields immediately to the following set of equations:

Thus, in conclusion, the solution for the perfect fluid when \(\rho \) is not constant is given by the line element (52) with the matrix \(H_t^{ij}\) given by Eq. (57), while the functions \(\Delta _t\) and \(S_t\) are given by Eqs. (56) and (63) respectively. In its turn, the energy density of the fluid is given by (62).

## 8 Conclusions and perspectives

Starting with the class of \((n+2)\)-dimensional spaces of Eq. (1), we fully integrate Einstein’s vacuum equation with a cosmological constant. The solutions found in Sect. 3 turn out to depend on the exponential of an arbitrary constant symmetric matrix \(\varvec{Q_0}\). By the use of suitable coordinate transformations, this exponential can be explicitly computed and the attained line element can have different algebraic structures depending on the canonical form of \(\varvec{Q_0}\). For example, in the simplest case, when \(\varvec{Q_0}\) is diagonalizable, the solutions are higher-dimensional generalizations of Kasner metric, which represent homogeneous but non-isotropic spacetimes used in cosmological models. However, when \(\varvec{Q_0}\) cannot be diagonalized more interesting solutions are obtained. As argued in the paragraph just below Eq. (47), after thoroughly looking in the literature for solutions with the same geometrical properties of the non-diagonal class of exact solutions presented here, we are confident that these solutions have not been described before in the literature. As a further indication, note that already in the example for \(n=3\) the exponential of a non-diagonal matrix becomes quite messy when written explicitly, even when a particularly simple basis is used, see Eq. (50), let alone cases of higher *n*. Therefore, it is hard to conceive that these solutions have been found elsewhere without using the structure of the exponential of a matrix, in which case it would not be hard to track it down.

In the general solution obtained here, the part of the line element associated with the cyclic coordinates \(\tau _i\) is generally composed by the sum of smaller blocks, see Eq. (36). Thus, once we have computed the structure of the line element in the cases in which the canonical forms of \(\varvec{Q_0}\) are single \(n\times n\) blocks, for different values of *n*, we can compose these solution in order to generate Einstein spaces of higher dimensions. For instance, in Eq. (48) we have merged a \(2\times 2\) block with a \(1\times 1\) block in order to create a solution with \(n=3\).

Finally, in Sect. 7, we have also considered cosmological models with perfect fluid and integrated Einstein’s equation analytically. In these solutions the spacetimes are endowed with a global time coordinate and the space-like hyper-surfaces of constant time are homogenous, as usually assumed in cosmology, although not isotropic.

In the present work we have focused exclusively on the integration of Einstein’s equation and simplification of the algebraic structure of the solutions. The physical aspects, as well as the geometrical properties of these solutions, have not been addressed. In the future we intend to fill these gaps in order to attain a full comprehension of such spaces.

## Notes

### Acknowledgements

C. B. would like to thank Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) for the partial financial support through the research productivity fellowship. Likewise, C. B. thanks Universidade Federal de Pernambuco for the funding through Qualis A project. G. L. A. thanks CNPq for the financial support. We both thank CAPES for funding the physics graduation program at UFPE through the PROEX 534/2018, process 23038.003382/2018-39. Finally, we thank the anonymous referee of EPJC for suggesting the analysis of the cosmological model pursued in Sect. 7.

## References

- 1.J. Demaret, M. Henneaux, P. Spindel, Nonoscillatory behavior in vacuum Kaluza-klein cosmologies. Phys. Lett.
**164B**, 27 (1985)ADSCrossRefGoogle Scholar - 2.N. Deruelle, On the approach to the cosmological singularity in quadratic theories of gravity: The Kasner regimes. Nucl. Phys. B
**327**, 253 (1989)ADSMathSciNetCrossRefGoogle Scholar - 3.T. Kitaura, J.T. Wheeler, New singularity in anisotropic, time dependent, maximally Gauss-Bonnet extended gravity. Phys. Rev. D
**48**, 667 (1993)ADSMathSciNetCrossRefGoogle Scholar - 4.X .O. Camanho, N. Dadhich, A. Molina, Pure Lovelock Kasner metrics. Class. Quantum Gravity
**32**(17), 175016 (2015)ADSMathSciNetCrossRefGoogle Scholar - 5.V.A. Belinsky, I.M. Khalatnikov, E.M. Lifshitz, Oscillatory approach to a singular point in the relativistic cosmology. Adv. Phys.
**19**, 525 (1970)ADSCrossRefGoogle Scholar - 6.K. Jacobs,
*Bianchi Type I Cosmological Models*(PhD thesis) (1969)Google Scholar - 7.G.L. Almeida, C. Batista,
*A Class of Integrable Metrics II*. Accepted Phys. Rev. D arXiv:1805.09206 [hep-th] - 8.A. Anabalón, Carlos Batista, A class of integrable metrics. Phys. Rev. D
**93**, 064079 (2016)ADSMathSciNetCrossRefGoogle Scholar - 9.G .L. Almeida, C. Batista, Class of integrable metrics and gauge fields. Phys. Rev. D
**96**(8), 084003 (2017). arXiv:1707.04630 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 10.S. Benenti, M. Francaviglia, Remarks on certain separability structures and their applications to general relativity. Gen. Relat. Gravit.
**10**, 79 (1979)ADSCrossRefGoogle Scholar - 11.H. Stephani,
*Exact solutions of Einstein’s field equations*(Cambridge University Press, Cambridge, 2009)zbMATHGoogle Scholar - 12.E. Kasner, Geometrical theorems on Einstein’s cosmological equations. Am. J. Math.
**43**, 217 (1921)MathSciNetCrossRefGoogle Scholar - 13.J.M. Maldacena, The large N limit of superconformal field theories and supergravity. Int. J. Theor. Phys.
**38**, 1113 (1999). [Adv. Theor. Math. Phys.**2**(1998) 231]. arXiv:hep-th/9711200 MathSciNetCrossRefGoogle Scholar - 14.G.T. Horowitz, J. Polchinski,
*Gauge/gravity duality*, in *Oriti, D. (ed.): Approaches to quantum gravity, 169-186, (2009). arXiv:gr-qc/0602037 - 15.V.E. Hubeny, The AdS/CFT correspondence. Class. Quantum Gravity
**32**(12), 124010 (2015)ADSMathSciNetCrossRefGoogle Scholar - 16.J. Erlich, E. Katz, D.T. Son, M.A. Stephanov, QCD and a holographic model of hadrons. Phys. Rev. Lett.
**95**, 261602 (2005)ADSCrossRefGoogle Scholar - 17.S.I. Finazzo, R. Rougemont, H. Marrochio, J. Noronha, Hydrodynamic transport coefficients for the non-conformal quark-gluon plasma from holography. JHEP
**1502**, 051 (2015)ADSGoogle Scholar - 18.J. Casalderrey-Solana, H. Liu, D. Mateos, K. Rajagopal, U.A. Wiedemann,
*Gauge/String Duality, Hot QCD and Heavy Ion Collisions, book:Gauge/String Duality, Hot QCD and Heavy Ion Collisions*(Cambridge University Press, Cambridge, 2014). arXiv:1101.0618 [hep-th]CrossRefGoogle Scholar - 19.S. Mukhi, String theory: a perspective over the last 25 years. Class. Quantum Gravity
**28**, 153001 (2011)ADSMathSciNetCrossRefGoogle Scholar - 20.R. Emparan, H.S. Reall, Black holes in higher dimensions. Living Rev. Relat.
**11**, 6 (2008)ADSCrossRefGoogle Scholar - 21.C. Csáki,
*TASI lectures on extra dimensions and branes*, in Shifman, M. (ed.) et al.: From fields to strings, vol. 2, 967–1060 (2005). arXiv:hep-ph/0404096 - 22.R.A. Horn, C.R. Johnson,
*Matrix Analysis*(Cambridge University Press, Cambridge, 2013)zbMATHGoogle Scholar - 23.W. Kopczynski, A. Trautman, Simple spinors and real structures. J. Math. Phys.
**33**, 550 (1992)ADSMathSciNetCrossRefGoogle Scholar - 24.P.D. Lax, Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math.
**21**, 467 (1968)MathSciNetCrossRefGoogle Scholar - 25.O. Babelon, D. Bernard, M. Talon,
*Introduction to Classical Integrable Systems*(Cambridge University Press, Cambridge, 2003)CrossRefGoogle Scholar - 26.M. Cariglia, V.P. Frolov, P. Krtous, D. Kubiznak, Geometry of Lax pairs: particle motion and Killing-Yano tensors. Phys. Rev. D
**87**(2), 024002 (2013). arXiv:1210.3079 [math-ph]ADSCrossRefGoogle Scholar - 27.B.D. Craven, Complex symmetric matrices. J. Aust. Math. Soc.
**10**, 341 (2009)MathSciNetCrossRefGoogle Scholar - 28.F.R. Gantmacher,
*The Theory of Matrices*(Chelsea Publishing Company, london, 1984)Google Scholar - 29.P. Glorioso, Classification of certain asymptotically AdS space-times with Ricci-flat boundary. JHEP
**1612**, 126 (2016)ADSMathSciNetCrossRefGoogle Scholar - 30.S. Hervik, Discrete symmetries in translation invariant cosmological models. Gen. Relat. Gravit.
**33**, 2027 (2001)ADSMathSciNetCrossRefGoogle Scholar - 31.V.D. Ivashchuk, V.N. Melnikov, On singular solutions in multidimensional gravity. Gravit. Cosmol.
**1**, 204 (1995). gr-qc/9507056ADSzbMATHGoogle Scholar - 32.S.S. Kokarev, Multidimensional generalization of Kasner solution. Gravit. Cosmol.
**2**, 321 (1996). [gr-qc/9510059]zbMATHGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP^{3}