Very long chains of annihilator ideals

Here we examine one of the two right Goldie conditions: the ascending chain condition on right annihilator ideals, r. ACC^⊥. One might well ask if a ring has r. ACC^⊥, does there exist a bound on the lengths of chains of right annihilator ideals? Under certain additional hypotheses, this bound does exist. In general, however, a bound does not exist, as is shown by the two example presented here.     

Axiomatic renormalization of the stress tensor of a conformally invariant field in conformally flat spacetimes

Using only the general properties which the renormalized stress-energy tensor T_μν should satisfy—and not relying on any assumptions associated with specific renormalization techniques—we derive the expression for T_μν for conformally invariant fields in conformally flat spacetimes of two and four dimensions. In two dimensions, these arguments rederive the Davies-Fulling-Unruh expression for the stress tensor of a scalar field; in four dimensions the results agree with those of Brown and Cassidy, except that we exclude the local curvature term depending on fourth-order derivatives of the metric. The dynamics of a k = 0 Robertson-Walker universe filled with radiation of the conformally invariant field is investigated and it is found that the equations cease to admit a solution when the Planck density is reached.     

A Higher Dimensional Stationary Rotating Black Hole Must be Axisymmetric

A key result in the proof of black hole uniqueness in 4-dimensions is that a stationary black hole that is “rotating”—i.e., is such that the stationary Killing field is not everywhere normal to the horizon—must be axisymmetric. The proof of this result in 4-dimensions relies on the fact that the orbits of the stationary Killing field on the horizon have the property that they must return to the same null geodesic generator of the horizon after a certain period, P. This latter property follows, in turn, from the fact that the cross-sections of the horizon are two-dimensional spheres. However, in spacetimes of dimension greater than 4, it is no longer true that the orbits of the stationary Killing field on the horizon must return to the same null geodesic generator. In this paper, we prove that, nevertheless, a higher dimensional stationary black hole that is rotating must be axisymmetric. No assumptions are made concerning the topology of the horizon cross-sections other than that they are compact. However, we assume that the horizon is non-degenerate and, as in the 4-dimensional proof, that the spacetime is analytic.     

Black holes and quantum coherence

We attempt to gain some insight into the issue of whether pure states evolve to density matrices in the black hole evaporation process by examining the mode functions of the particles entering the black hole which are correlated with the particles which escape to infinity. We show that these particles enter the black hole singularity at relatively early times. This tends to support the view that pure states evolve to density matrices, i.e., that in this process quantum coherence is lost.     

Martinaxiom und die Beschreibung gewisser Homomorphismen in der Theorie der ℵ1-freien abelschen Gruppen

P(I) denotes the abelian group of integer-valuated functions defined on the set I, P_κ (I) its subgroup consisting of all elements, which support has size less than the infinite cardinal κ and Q_κ (I) the quotient P(I)/P_κ(I). We formulate a statement I(κ) in the following way: For an arbitrary set I you can lift every homomorphism from P (?) to Q_κ(I) to a homomorphism from P(?) to P(I). Using Martin's Axiom we show, that L(?) is independent of ZFC.