We study the five-dimensional Einstein-Yang-Mills system with a cosmological constant. Assuming a spherically symmetric spacetime, we find a new analytic black hole solution, which approaches asymptotically ''quasi-Minkowski,'' ''quasi-anti-de Sitter,'' or ''quasi-de Sitter'' spacetime depending on the sign of the cosmological constant. Since there is no singularity except for the origin that is covered by an event horizon, we regard it as a localized object. This solution corresponds to a magnetically charged black hole. We also present a singularity-free particlelike solution and a nontrivial black hole solution numerically. Those solutions correspond to the Bartnik-McKinnon solution and a colored black hole with a cosmological constant in four dimensions. We analyze their asymptotic behavior, spacetime structures, and thermodynamical properties. We show that there is a set of stable solutions if the cosmological constant is negative.

We consider n-dimensional asymptotically anti-de Sitter spacetimes in higher curvature gravitational theories with n ≥ 4, by employing the conformal completion technique. We first argue that a condition on the Ricci tensor should be supplemented to define an asymptotically anti-de Sitter spacetime in higher curvature gravitational theories and propose an alternative definition of an asymptotically anti-de Sitter spacetime. Based on that definition, we then derive a conservation law of the gravitational field and construct conserved quantities in two classes of higher curvature gravitational theories. We also show that these conserved quantities satisfy a balance equation in the same sense as in Einstein gravity and that they reproduce the results derived elsewhere. These conserved quantities are shown to be expressed as an integral of the electric part of the Weyl tensor alone and hence they vanish identically in the pure anti-de Sitter spacetime as in the case of Einstein gravity.

We investigate spherically symmetric spacetimes which contain a perfect fluid obeying the polytropic equation of state and admit a kinematic self-similar vector of the second kind which is neither parallel nor orthogonal to the fluid flow. We assume two kinds of polytropic equations of state and show in general relativity that such spacetimes must be a vacuum, which is in contrast with the result in the Newtonian case.

We classify all spherically symmetric spacetimes admitting a kinematic self-similar vector of the second, zeroth or infinite kind. We assume that the perfect fluid obeys either a polytropic equation of state or an equation of state of the form p = Kµ, where p and µ are the pressure and the energy density, respectively, and K is a constant. We study the cases in which the kinematic self-similar vector is not only "tilted" but also parallel or orthogonal to the fluid flow. We find that, in contrast to Newtonian gravity, the polytropic perfect-fluid solutions compatible with kinematic self-similarity are the Friedmann-Robertson-Walker solution and general static solutions. We find three new exact solutions, which we call the dynamical solutions (A) and (B) and the Λ-cylinder solution. §1. IntroductionThere is no characteristic scale in Newtonian gravity or general relativity. In such systems, a set of field equations is invariant under a scale transformation if we assume appropriate matter fields. This implies the existence of scale-invariant solutions to the field equations. Such solutions are called 'self-similar solutions'. Among them, spherically symmetric self-similar solutions have been widely studied in the context of both Newtonian gravity and general relativity. Although self-similar solutions are only special solutions of the field equations, it has often been supposed that they play an important role in situations where gravity is an essential ingredient in a spherically symmetric system. In particular, a self-similarity hypothesis has been proposed, which states that solutions in a variety of astrophysical and cosmological situations may naturally evolve into self-similar forms even if they are initially more complicated. 1) Self-similar solutions in Newtonian gravity have been studied in an effort to obtain realistic solutions of gravitational collapse leading to star formation. 2) -5) For an isothermal gas cloud, Larson and Penston independently found a self-similar solution, which is called the Larson-Penston solution, describing a gravitationally collapsing sphere. 2), 3) Thereafter, Hunter found a new series of self-similar solutions, and noted that the set of such solutions is infinite and discrete. 5) Recent numerical simulations and mode analyses have shown that the Larson-Penston solution gives * )

We give a classification of spherically symmetric kinematic self-similar solutions. This classification is complementary to that given in a previous work by the present authors [Prog. Theor. Phys. 108 (2002), 819]. Dust solutions of the second, zeroth and infinite kinds, perfect-fluid solutions and vacuum solutions of the first kind are treated. The kinematic selfsimilarity vector is either parallel or orthogonal to the fluid flow in the perfect-fluid and vacuum cases, while the 'tilted' case, i.e., neither parallel nor orthogonal case, is also treated in the dust case. In the parallel case, there are no dust solutions of the second (except when the self-similarity index α is 3/2), zeroth and infinite kinds, and in the orthogonal case, there are no dust solutions of the second and infinite kinds. Except in these cases, the governing equations can be integrated to give exact solutions. It is found that the dust solutions in the tilted case belong to a subclass of the Lemaître-Tolman-Bondi family of solutions for the marginally bound case. The flat Friedmann-Robertson-Walker (FRW) solution is the only dust solution of the second kind with α = 3/2 in the tilted and parallel cases and of the zeroth kind in the orthogonal case. The flat, open and closed FRW solutions with p = −µ/3, where p and µ are the pressure and energy density, respectively, are the only perfect-fluid first-kind self-similar solutions in the parallel case, while a new exact solution with p = µ, which we call the "singular stiff-fluid solution", is the only such solution in the orthogonal case. The Minkowski solution is the only vacuum first-kind self-similar solution both in the parallel and orthogonal cases. Some important corrections and complements to the authors' previous work are also presented. §1. IntroductionScale-invariance is one of the most fundamental properties of classical gravity. In the vacuum case without a cosmological constant, the gravitational constant G is the only dimensional constant that appears in the field equations in Newtonian gravity, while there appears another physical constant, the speed of light, c, in general relativity. Neither a characteristic time nor a length scale can be constructed from them. This implies that there exists a class of scale-invariant solutions, which are often referred to as self-similar solutions. There also exist self-similar solutions in scalar-tensor theories of gravity. (See Ref. 1) for an investigation of the perfect-fluid * )