Null Surface Quantization and Quantum Field Theory in Asymptotically Flat Space-Time

A quantum theory of the free scalar, electromagnetic and gravitational fields in a curved asymptotically flat space-time is developed. It is shown that the Penrose conformal technique makes it possible to reformulate the null infinity quantization as a problem of the quantization on the proper null surface in the corresponding Penrose space. The Schwinger dynamical principle is exploited to derive the corresponding null surface commutation relations. The general covariant and gauge-independent form of the commutation relations is also given. The existence of the asymptotic symmetry (BMS) group in the asymptotically flat space-time is used to define uniquely the “in” and “out” vacuum states. The explicit expressions for the S-matrix operator and for the S-matrix elements in the asymptotically simple space-time are given. The functional integration method is used to find the expression for the density matrix describing the observations at ∮+ in the weakly asymptotically simple space-time when the information loss due to the event horizons or the existence of bare singularities is possible. The application of the developed approach to the problem of quantum evaporation of black holes (Hawking effect) is briefly discussed.     

Censorship,strong curvature,and asymptotic causal pathology

Necessary and sufficient conditions are established for a weakly asymptotically simple and empty, null convergent, generic space-time to be future asymptotically predictable. These conditions require that the causal structure of the space-time is well behaved near spatial infinity and future null infinity, and that there are no singularities of less than a certain finite strength in the future asymptotic limit.     

Asymptotic symmetry and the global structure of future null infinity

Space-times for whichI ⁺ (future null infinity) is not necessarily homeomorphic toR×S ² are considered. It is shown that, depending on the global conformal structure ofI ⁺, a given space-time either (1) possesses an asymptotic symmetry group with a normal subgroup of supertranslations, similar in structure to the BMS group, or (2) possesses a simpler kind of asymptotic symmetry group, not involving supertranslations, or (3) has no asymptotic symmetry. The setting is Newman and Unti's approach to asymptotically flat space-times and use is made of the characterization of the asymptotic symmetry transformation as a conformal motion ofI ⁺ that preserves null angles.     

The Kerr space-time near the ring singularity

We investigate the geometry of the Kerr space-time near the ring singularity. A systematic study of the mathematical and physical structure of this region reveals that the singularity in the Kerr space-time is naturally understood in terms of a subset of the immersion of the set defined byr=0 (in Boyer-Lindquist coordinates) in the Kerr space-time. It is well known that the Kerr space-time is not a differentiable manifold (due to the curvature singularity) or a topological manifold, but a well defined topological space with a structure that is manifested by the constrast in taking limits along the hypersurface atr=0 and the equatorial plane which approach singularity. We find that the ring singularity is either an edge or a self-intersection of the topological space depending on which extension of the metric throughr=0 is implemented. A major result of this analysis is the extrapolation to the general accelerating case of Carter's proof that the only nonspacelike geodesics which can reach the ring singularity are restricted to the equatorial plane. For finite magnitudes of proper acceleration, it is shown that only lightlike trajectories that asymptotically approach the null generator of the ring singularity can reach it from above or below the equatorial plane.     

Renormalization of Quantum Field Theory in Curved Space-Time and Renormalization Group Equations

The structure of quantum field theory renormalization in curved space-time is investigated. The equations allowing us to investigate the behaviour of vacuum energy and vertex functions in the limit of small distances in the external gravitational field are established. The behaviour of effective charges corresponding to the parameters of nonminimal coupling of the matter with the gravitational field is studied and the conditions under which asymptotically free theories become asymptotically conformally invariant are found. The examples of asymptotically conformally invariant theories are given. On the basis of a direct solution of renormalization group equations the effective potential in the external gravitational field and the effective action in the gravity with the high derivatives are obtained. The expression for the cosmological constant in terms of R²-gravity Lagrangian parameters is given which does not contradict the observable data. Renormalization and renormalization group equations for the theory in curved space-time with torsion are investigated.     

On the existence ofn-geodesically complete or future complete solutions of Einstein's field equations with smooth asymptotic structure

It is demonstrated that initial data sufficiently close to De-Sitter data develop into solutions of Einstein's equations Ric[g]=g with positive cosmological constant , which are asymptotically simple in the past as well as in the future, whence null geodesically complete. Furthermore it is shown that hyperboloidal initial data (describing hypersurfaces which intersect future null infinity in a space-like two-sphere), which are sufficiently close to Minkowskian hyperboloidal data, develop into future asymptotically simple whence null geodesically future complete solutions of Einstein's equations Ric[g]=0, for which future null infinity forms a regular cone with vertexi ⁺ that represents future time-like infinity.     

Real miniheaven

For the class of axisymmetric gravitational fields originally treated by Bondi, the existence of a two-dimensional family of asymptotically shear-free slices (good slices) of null infinity is exhibited. They form a real two-dimensional submanifold (miniheaven) of the four-dimensional complex manifold of good slices (heaven) constructed by Newman. The real good slices can be described as surfaces of revolution of geodesic curves. Geodesic deviation due to the Bondi news function gives rise to the geometrical properties of miniheaven. These properties are examined for the special case of axisymmetric Robinson-Trautman solutions. The Robinson-Trautman family of shear-free null hypersurfaces corresponds to a geodesic in miniheaven possessing an infinite number of angular momentum and supermomentum constants of the motion. The global properties of the Robinson-Trautman miniheaven are examined in the linear approximation in which a remarkably simple expression for the curvature of miniheaven is found. The asymptotic properties of miniheaven turn out to be identical to those of the original Robinson-Trautman space-time.Research supported by grant No. MPS74-18020 from the National Science Foundation. The work constitutes part of a doctoral dissertation submitted by R. Dubisch to the University of Pittsburgh.     

Left-flat spaces and their associated space-times

It is already known that for an asymptotically flat space-time the metric coefficients and the other Newman-Penrose variables (in a suitable frame) can be constructed, in principle, by specifying certain initial data at conformal null infinity (and one further function on another null hypersurface), integrating the Newman-Penrose equations in the conformally rescaled “unphysical” space, and then transforming the results back to the physical space-time. If this is done approximately near ?⁺, for vacuum, the well-known Newman-Unti expansion is obtained. In this paper, after complexifying null infinity ?⁺ we generate, in a similar fashion, a left-flat spaceH using as much of the initial data of a given asymptotically flat space-timeM as possible, and show that the left-flat spaceH thus constructed is, in fact, the H-space corresponding toM. The advantage of our method is that it allows a reversal of procedure. Under suitable conditions we can generate from a given left-flat spaceH a class of physical space-times whose H-space is precisely the given left-flat spaceH. We shall see that the formal procedure requires only the local but not the global properties of ?⁺.     

Einstein-Weyl space-times with geodesic and shear-free neutrino rays: Asymptotic behaviour

We consider a neutrino field with goodesic and shear-free rays, in interaction with a gravitational field according to the Einstein-Weyl field equations. Furthermore we suppose that there exists a Killing vector r^μ whose magnitude is almost everywhere bounded at the future and past endpoints of the neutrino rays. The implications of the asymptotic behavior of r^μ on the structure of space-time are investigated and a useful set of reduced equations is obtained. It is found that under these hypothesis the space-time cannot be asymptotically flat if the neutrino field is nonvanishing. All the Demianski-Kerr-NUT-like space-times as well as the space-times which admit a covariantly constant null vector are explicitly obtained.     

Null cut loci of spacelike surfaces

The null cut locus of a spacelike submanifold of codimension 2 in a space-time is defined. In globally hyperbolic space-times, it is shown that the future (past) null cut locusc _n ⁺ (H) [c _n ^- (H)] of a compact, acausal, spacelike submanifoldH of codimension 2 is a closed subset of the space-time, and each pointx c _n ⁺ (H) is either a focal point ofH along some future-directed null geodesic meetingH orthogonally or there exist at least two null geodesics fromH tox, realizing the distance betweenH andx or both. Also, it can be shown that the assumptions of the Penrose's singularity theorem for open globally hyperbolic space-times may be weakened to the space-times which are conformal to an open subset of an open globally hyperbolic space-time.This study is based on Chapter 3 of the author's Ph.D. thesis.     

The Theory of H-space

The theory of $\text{H}$-space, the four-dimensional manifold of those complex null hypersurfaces of an asymptotically flat space-time which are asymptotically shear-free, is reviewed.In addition to a discussion of the origins of the theory, we present two independent formalisms for the derivation of the basic properties of $\text{H}$-space: that it is endowed with a natural holomorphic complex Riemannian metric which satisfies the vacuum Einstein equations and whose Weyl tensor is self-dual.We show the connection of our work on $\text{H}$-space to that of Plebanski and to the theory of deformed twistor spaces, due to Penrose.Finally, there is a discussion of equations of motion in $\text{H}$-space.     

The gravitational field in the wave zone II. Consequences of asymptotic structure

We consider an asymptotically flat and empty space-time generated by a bounded source of perfect fluid. The vanishing of the conformal Weyl tensor onI ⁺ and of the Ricci tensor nearI ⁺ are used to simplify the expression obtained in the previous paper for the coefficient ofr ^– of the metric tensor after an expansion in powers ofc ^–1. The result is a very simple expression for the dominant term ofg _,0 in the radiation zone in terms of the quadrupole moment of the source. Using this expression and an invariant definition of the total energy, we calculate in the framework of full general relativity the radiated energy per unit time and prove that the first term is identical with the quadrupole radiation as given by the linearized version of general relativity.     

Post-Newtonian approximation for a rotating frame of reference

The field equations of general relativity are solved to post-Newtonian order for a rotating frame of reference. A new method of approximation is used based on a 3+1 decomposition of the equations. The results are expressed explicitly in terms of the gravitational potentials. The space-time is asymptotically flat but not locally flat. The space-time metric contains gravitational terms, inertial terms, and coupled gravitational-inertial terms. The inertial terms in the equation of motion are in agreement with terms obtained by other authors using kinematic methods. The metric and equation of motion reduce to those for an inertial frame of reference under a simple coordinate transformation. The total energy of a particle is given. For the restricted three-body problem this represents the relativistic extension of Jacobi's integral to post-Newtonian order.This article received an honorable mention from the Gravity Research Foundation for the year 1984—Ed.     

Inflation within closed friedmann universe models

For the LagrangianL = R ²,the de Sitter space-time is known to be an attractor solution. Here, we classify for closed Friedmann models in which initial conditions lead asymptotically to a de Sitter phase and what the behaviour is for the other solutions. Four types of solutions form together a generic set, and three of them are asymptotically de Sitter; the fourth one has both an initial and final singularity. Furthermore, exactly seven other solutions exist and can be given in closed form. Three of them are known, the other four are new and have a linear asymptotic behaviour of the cosmic scale factor.     

The space-time cut locus

Let (M, g) be a space-time with Lorentzian distance functiond. If (M, g) is distinguishing andd is continuous, then (M, g) is shown to be causally continuous. Furthermore, a strongly causal space-time (M, g) is globally hyperbolic iff the Lorentzian distance is always finite valued for all metricsg conformal tog. Lorentzian distance may be used to define cut points for space-times and the analogs of a number of results holding for Riemannian cut loci may be established for space-time cut loci. For instance in a globally hyperbolic space-time, any timelike (or respectively, null) cut pointq of p along the geodesicc must be either the first conjugate point ofp or else there must be at least two maximal timelike (respectively, null) geodesics fromp toq. Ifq is a closest cut point ofp in a globally hyperbolic space-time, then eitherq is conjugate top or elseq is a null cut point. In globally hyperbolic space-times, no point has a farthest nonspacelike cut point.     

The gravitational field in the wave zone I. The radiating terms of the metric tensor

We consider an asymptotically flat space-time generated by a perfect fluid source of compact spatial support. Using the de Donder gauge conditions, the Einstein equations are reduced to a new form of Poisson-type equations. A formal iterative scheme is set up to solve these equations by expanding the components of the metric tensor in powers ofc ^–1. The coefficient of each power ofc ^–1 depends on the asymptotically retarded timeu andx, y, z and satisfies a Poisson-type equation. Assuming asymptotic flatness the solution is carried out in the first orders. The results are explicit expressions of the metric up to orderc ^–4 in terms of the source functions. These expressions hold over all space-time. A further expansion in powers ofr ^–1 gives the first terms of the metric that contribute to gravitational radiation.     

The global structure of simple space-times

According to a standard definition of Penrose, a space-time admitting well-defined future and past null infinitiesI ⁺ andI ^– is asymptotically simple if it has no closed timelike curves, and all its endless null geodesics originate fromI ^– and terminate atI ⁺. The global structure of such space-times has previously been successfully investigated only in the presence of additional constraints. The present paper deals with the general case. It is shown thatI ⁺ is diffeomorphic to the complement of a point in some contractible open 3-manifold, the strongly causal regionI ₀ ⁺ ofI ⁺ is diffeomorphic to , and every compact connected spacelike 2-surface inI ⁺ is contained inI ₀ ⁺ and is a strong deformation retract of bothI ₀ ⁺ andI ⁺. Moreover the space-time must be globally hyperbolic with Cauchy surfaces which, subject to the truth of the Poincaré conjecture, are diffeomorphic to ³.     

Finite GUE Distribution with Cut-Off at a Shock

We consider the totally asymmetric simple exclusion process with initial conditions generating a shock. The fluctuations of particle positions are asymptotically governed by the randomness around the two characteristic lines joining at the shock. Unlike in previous papers, we describe the correlation in space-time without employing the mapping to the last passage percolation, which fails to exists already for the partially asymmetric model. We then consider a special case, where the asymptotic distribution is a cut-off of the distribution of the largest eigenvalue of a finite GUE matrix. Finally we discuss the strength of the probabilistic and physically motivated approach and compare it with the mathematical difficulties of a direct computation.     

Existence and structure of past asymptotically simple solutions of Einstein's field equations with positive cosmological constant

The initial value problem for Einstein's field equations with positive cosmological constant is analysed where data are prescribed at past conformal infinity. It is found that the data on past conformal infinity are given, up to arbitrary conformal rescalings, by a freely specifyble Riemannian metric and a trace-free, symmetric tensorfield of valence two, which satisfies a divergence equation. For each initial data set exists a unique (semi-global) past asymptotically simple solution of Einstein's equations. The case is discussed where in such a space-time exists a Killing vector field with a time-like trajectory which approaches a point p on conformal infinity. It is shown that in a neighbourhood of the trajectory near p the space-time is conformally flat.