V. D. Ivashchuk

Exact solutions in multidimensional gravity with antisymmetric forms

This short review deals with a multidimensional gravitational model containing dilatonic scalar fields and antisymmetric forms. The manifold is chosen in the form M = M 0 × M 1 × ... × M n , where M i are Einstein spaces (i ≥ 1). The sigma-model approach and exact solutions in the model are reviewed and the solutions with p-branes (e.g. solutions with harmonic functions, “cosmological”, spherically symmetric and black-brane ones) are considered.

Billiard representation for multidimensional cosmology with multicomponent perfect fluid near the singularity

The multidimensional cosmological model describing the evolution of n Einstein spaces is considered in the presence of a multicomponent perfect fluid. When certain restrictions on the parameters of the model are imposed, the dynamics of the model near the singularity are reduced to a billiard on the (n-1)-dimensional Lobachevsky space . The geometrical criterion for the finiteness of the billiard volume and its compactness is suggested. This criterion reduces the problem to the problem of illumination of an (n-2)-dimensional sphere by point-like sources. Some generalizations of the considered scheme (including scalar field and quantum generalizations) are considered.

Solutions with intersecting p-branes related to Toda chains

Solutions in multidimensional gravity with m p-branes related to Toda-type systems (of general type) are obtained. These solutions are defined on a product of n+1 Ricci-flat spaces M0×M1×⋯×Mn and are governed by one harmonic function on M0. The solutions are defined up to the solutions of Laplace and Toda-type equations and correspond to null-geodesics of the (sigma-model) target-space metric. Special solutions relating to Am Toda chains (e.g., with m=1,2) are considered.

Sigma-model for the generalized composite p-branes

A multidimensional gravitational model containing several dilatonic scalar fields and antisymmetric forms is considered. The manifold is chosen in the form M = M0 ×M1 × . . . ×Mn, where Mi are Einstein spaces (i ≥ 1). The block-diagonal metric is chosen and all fields and scale factors of the metric are functions on M0. For the forms composite (electro-magnetic) p-brane ansatz is adopted. The model is reduced to gravitating self-interacting sigma-model with certain constraints. In pure electric and magnetic cases the number of these constraints is n1(n1 − 1)/2 where n1 is number of 1-dimensional manifolds among Mi. In the ”electro-magnetic” case for dimM0 = 1, 3 additional n1 constraints appear. A family of ”MajumdarPapapetrou type” solutions governed by a set of harmonic functions is obtained, when all factor-spaces Mν are Ricci-flat. These solutions are generalized to the case of non-Ricci-flat M0 when also some additional ”internal” Einstein spaces of non-zero curvature are added to M . As an example exact solutions for D = 11 supergravity and related 12-dimensional theory are presented. PACS number(s): 04.50.+h, 98.80.Hw, 04.60.Kz

Integrable pseudo‐Euclidean Toda‐like systems in multidimensional cosmology with multicomponent perfect fluid

The multidimensional cosmological model describing the evolution of n Einstein spaces in the presence of multicomponent perfect fluid is considered. When the vectors corresponding to the equations of state of the components are orthogonal with respect to the minisuperspace metric, the Einstein equations are integrated and a Kasner‐like form of the solutions is presented. For special sets of parameters the cosmological model is reduced to the Euclidean Toda‐like system connected with some Lie algebra G. For G=A2 exact solutions are explicitly written. A certain family of wormhole solutions is also obtained.

On anisotropic Gauss-Bonnet cosmologies in ( n + 1) dimensions, governed by an n -dimensional Finslerian 4-metric

The (n + 1)-dimensional Einstein-Gauss-Bonnet (EGB) model is considered. For diagonal cosmological metrics, the equations of motion are written as a set of Lagrange equations with the effective Lagrangian containing two “minisuperspace” metrics on ℝn: a 2-metric of pseudo-Euclidean signature and a Finslerian 4-metric proportional to the n-dimensional Berwald-Moor 4-metric. For the case of the “pure” Gauss-Bonnet model, two exact solutions are presented, those with power-law and exponential dependences of the scale factors (w.r.t. the synchronous time variable) are presented. (The power-law solution was considered earlier by N. Deruelle, A. Toporensky, P. Tretyakov, and S. Pavluchenko.) In the case of EGB cosmology, it is shown that for any nontrivial solution with an exponential dependence of scale factors, ai(τ) = Ai exp(viτ), there are no more than three different numbers among v1, …, vn.

Hyperbolic Kac–Moody algebra from intersecting p-branes

A subclass of a recently discovered class of solutions in multidimensional gravity with intersecting p-branes related to Lie algebras and governed by a set of harmonic functions is considered. This subclass in case of three Euclidean p-branes (one electric and two magnetic) contains a cosmological solution to D=11 supergravity related to hyperbolic Kac–Moody algebra F3 (of rank 3). This solution describes the non-Kasner power-law inflation.

Electric S-brane solutions corresponding to rank-2 Lie algebras: Acceleration and small variation of G

Electric S-brane solutions with two non-composite electric branes and a set of l scalar fields are considered. The intersection rules for branes correspond to Lie algebras A2, C2 and G2. The solutions contain five factor spaces. One of them, M0, is interpreted as our 3-dimensional space. It is shown that there exists a time interval where accelerated expansion of our 3-dimensional space is compatible with a small enough variation of the effective gravitational constant G(τ). This interval contains τ0, a point of minimum of the function G(τ). A special solution with two phantom scalar fields is analyzed, and it is shown that, in the vicinity of the point τ0, the time variation of G(τ) (calculated in the linear approximation) decreases in the sequence of Lie algebras A2, C2 and G2.

On the billiard approach in multidimensional cosmological models

A short overview of the billiard approach for cosmological-type models with n Einstein factor spaces is presented. We start with the billiard representation for pseudo-Euclidean Toda-like systems of cosmological origin. Then we consider cosmological models with a multicomponent perfect-fluid and with composite branes. The second case describes cosmological and spherically symmetric configurations in a theory with scalar fields and fields of forms. The conditions for appearance of asymptotic Kasnerlike and oscillating behaviors in the limits τ → +0 and τ → +∞ (where τ is the synchronous variable) are formulated (e.g., in terms of inequalities for Kasner parameters). Examples of billiards related to the hyperbolic Kac-Moody algebras E10, AE3 and A1,II are given.

Cosmological and spherically symmetric solutions with intersecting p-branes

Multidimensional model describing the cosmological evolution and/or spherically symmetric configuration with n+1 Einstein spaces in the theory with several scalar fields and forms is considered. When electro-magnetic composite p-brane ansatz is adopted, n “internal” spaces are Ricci-flat, one space M0 has a nonzero curvature, and all p-branes do not “live” in M0, a class of exact solutions is obtained if certain block-orthogonality relations on p-brane vectors are imposed. A subclass of spherically symmetric solutions (containing nonextremal p-brane black holes) is considered. Post-Newtonian parameters are calculated.