The Universality of Penrose Limits near Space-Time Singularities

We prove that Penrose limits of metrics with arbitrary singularities of power-law type show a universal leading u^{-2}-behaviour near the singularity provided that the dominant energy condition is satisfied and not saturated. For generic power-law singularities of this type the oscillator frequencies of the resulting homogeneous singular plane wave turn out to lie in a range which is known to allow for an analytic extension of string modes through the singularity. The discussion is phrased in terms of the recently obtained covariant characterisation of the Penrose limit; the relation with null geodesic deviation is explained in detail.Comment: 36 pages, LaTeX2e, 4 figure

The congruence theory of closure properties of regular tree languages

AbstractBoolean operations, tree homomorphisms and their converses, and forest product, in special cases σ-catenation, x-product, x-quotient and x-iteration, preserve the regularity of forests. These closure properties are proved algebraically by using congruences of term algebras which saturate the forests operated on and constructing, by means of them, a congruence which saturates the product forest. The index of the constructed congruence is finite, if the congruences saturating the forests to operate are of finite indexes. The cardinalities of ranked and frontier alphabets are arbitrary. The preservation of recognizability is a straightforward consequence of those congruence constructions and the Nerode type of congruence characterization for recognizable forests. Furthermore, the constructed congruences can also be applied directly to construct explictly tree automatas to recognize the product forests

A Covariant Entropy Conjecture

We conjecture the following entropy bound to be valid in all space-times admitted by Einstein's equation: Let A be the area of any two-dimensional surface. Let L be a hypersurface generated by surface-orthogonal null geodesics with non-positive expansion. Let S be the entropy on L. Then S does not exceed A/4. We present evidence that the bound can be saturated, but not exceeded, in cosmological solutions and in the interior of black holes. For systems with limited self-gravity it reduces to Bekenstein's bound. Because the conjecture is manifestly time reversal invariant, its origin cannot be thermodynamic, but must be statistical. Thus it places a fundamental limit on the number of degrees of freedom in nature.Comment: 41 pages, 7 figures. v2,v3: references adde