A static wormhole solution for gravity in vacuum is found for odd dimensions greater than four.In five dimensions the gravitational theory considered is described by the Einstein-Gauss-Bonnet action where the coupling of the quadratic term is fixed in terms of the cosmological constant. In higher dimensions d = 2n + 1, the theory corresponds to a particular case of the Lovelock action containing higher powers of the curvature, so that in general, it can be written as a Chern-Simons form for the AdS group. The wormhole connects two asymptotically locally AdS spacetimes each with a geometry at the boundary locally given by R × S 1 × H d−3 . Gravity pulls towards a fixed hypersurface located at some arbitrary proper distance parallel to the neck. The causal structure shows that both asymptotic regions are connected by light signals in a finite time. The Euclidean continuation of the wormhole is smooth independently of the Euclidean time period, and it can be seen as instanton with vanishing Euclidean action. The mass can also be obtained from a surface integral and it is shown to vanish.

We study the stability under linear perturbations of a class of static solutions of Einstein-Gauss-Bonnet gravity in D n 2 dimensions with spatial slices of the form n R , n an n manifold of constant curvature . Linear perturbations for this class of spacetimes can be generally classified into tensor, vector and scalar types. The analysis in this paper is restricted to tensor perturbations. We show that the evolution equations for tensor perturbations can be cast in Schrödinger form, and obtain the exact potential. We use S deformations to analyze the Hamiltonian spectrum, and find an S-deformed potential that factors in a convenient way, allowing us to draw definite conclusions about stability in every case. It is found that there is a minimal mass for a D 6 black hole with a positive curvature horizon to be stable. For any D, there is also a critical mass above which black holes with negative curvature horizons are unstable.

We study the negative mass Schwarzschild spacetime, which has a naked singularity, and show that it is perturbatively unstable. This is achieved by first introducing a modification of the well known Regge -Wheeler -Zerilli approach to black hole perturbations to allow for the presence of a "kinematic" singularity that arises for negative masses, and then exhibiting exact exponentially growing solutions to the linearized Einstein's equations. The perturbations are smooth everywhere and behave nicely around the singularity and at infinity. In particular, the first order variation of the scalar invariants can be made everywhere arbitrarily small as compared to the zeroth order terms. Our approach is also compared to a recent analysis that leads to a different conclusion regarding the stability of the negative mass Schwarzschild spacetime. We also comment on the relevance of our results to the stability of more general negative mass, nakedly singular spacetimes.

We analyze the tensor mode perturbations of static, spherically symmetric solutions of the Einstein equations with a quadratic Gauss-Bonnet term in dimension D > 4. We show that the evolution equations for this type of perturbations can be cast in a Regge-Wheeler-Zerilli form, and obtain the exact potential for the corresponding Schrödinger-like stability equation. As an immediate application we prove that for D = 6 and α > 0, the sign choice for the Gauss-Bonnet coefficient suggested by string theory, all positive mass black holes of this type are stable. In the exceptional case D = 6, we find a range of parameters where positive mass asymptotically flat black holes, with regular horizon, are unstable. This feature is found also in general for α < 0. * Electronic address:

A proof is given that the space L of solutions of the linearized vacuum Einstein equation around a Schwarzschild black hole is parametrized by two scalar fields, which are gauge invariant combinations of perturbed algebraic and differential invariants of the Weyl tensor and encode the information on the odd (-) and even (+) sectors L ±. These fields measure the distortion of the geometry caused by a generic perturbation and are shown to be pointwise bounded on the outer region r ≥ 2M.

Higher dimensional Einstein gravity in vacuum admits static black hole solutions with an Einstein manifold of non constant curvature as a horizon. This gives a much richer family of static black holes than in four dimensional GR. However, as we show in this paper, the Gauss-Bonnet string theory correction to Einstein gravity poses severe limitations on the geometry of a horizon Einstein manifold. The additional stringy constraints rule out most of the known examples of exotic black holes with a horizon of non constant curvature.

We study the stability under linear perturbations of a class of static solutions of Einstein-Gauss-Bonnet gravity in D = n+2 dimensions with spatial slices of the form Σ n κ × R + , Σ n κ an n−manifold of constant curvature κ. Linear perturbations for this class of space-times can be generally classified into tensor, vector and scalar types. In a previous paper, tensor perturbations were analyzed. In this paper we study vector and scalar perturbations. We show that vector perturbations can be analyzed in general using an S-deformation approach and do not introduce instabilities. On the other hand, we show by analyzing an explicit example that, contrary to what happens in Einstein gravity, scalar perturbations may lead to instabilities in black holes with spherical horizons when the Gauss-Bonnet string corrections are taken into account.

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