Trapped surfaces, horizons and exact solutions in higher dimensions

A very simple criterion to ascertain if (D − 2)-surfaces are trapped in arbitrary D-dimensional Lorentzian manifolds is given. The result is purely geometric, independent of the particular gravitational theory, of any field equations or of any other conditions. Many physical applications arise, a few shown here: a definition of general horizon, which reduces to the standard one in black holes/rings and other known cases; the classification of solutions with a (D − 2)-dimensional abelian group of motions and the invariance of the trapping under simple dimensional reductions of the Kaluza-Klein/string/M-theory type. Finally, a stronger result involving closed trapped surfaces is presented. It provides in particular a simple sufficient condition for their absence. PACS Numbers: 04.50.+h, 04.20.Cv, 04.20.Jb, 02.40.Ky In 1965 Penrose [1] introduced in General Relativity (GR) the concept of closed trapped surface, which was crucial for the development of the singularity theorems and the study of gravitational collapse, black holes, cosmological expansion and several types of horizons, see e.g. [2, 3]. Trapped surfaces (closed or not) are 2-dimensional imbedded spatial surfaces such that any portion of them has, at least initially, a decreasing area along any future evolution direction. The term “closed” is used if the surfaces are compact without boundary [1, 2, 3]. This concept carries over to general Lorentzian manifolds (V, g) of any dimension D [4]. To fix ideas and notation, let S be a (D − 2)-dimensional surface with intrinsic coordinates {λA} (A,B, . . . = 2, . . . , D − 1) imbedded into the spacetime V by the parametric equations x = Φ(λ) (α, β . . . = 0, 1, . . . , D − 1). (1) S is alternatively locally defined by two independent relations F1(x ) = 0 and F2(x ) = 0. The tangent vectors {~eA} of S are ~eA ≡ eμA ∂ ∂xμ ∣