The nature of singularities

Conclusions We have established, under reasonable conditions, that singularities must be either curvature singularities or primordial singularities. There is still the possibility that the Cauchy data for, say, a gravitational collapse might be compatible with or might necessitate a quasi-regular singularity that, though technically primordial, lies entirely to the future of the data-surface. Until this is ruled out, we cannot be sure that classical relativity does give curvature singularities in physically reasonable situations.The methods examined here are concerned only with whether the Riemann tensor is bounded, and so they can say nothing on the crucial question ofhow badly the Riemann tensor may be unbounded. Upper limits on the behavior on some geodesies have been set by Tipler [6], but more work needs to be done here. One would like to know whether the curvature could fail to be, say,L ¹; but to answer this we should have to use quite different methods based on integration overM rather than the behavior along curves.A revised version of material presented at the Singularities Symposium of the 8th International Conference on General Relativity and Gravitation, August 1977.     

The classification of singularities

Two definitions of the strengthof a singularity in spacetime are described and compared, and the possibility of extending the ideas to stronger singularities, using spacetimes of low differentiability, is explored.     

Space-time singularities

A set of conditions for the reasonableness of space-time is proposed and investigated. Using these, together with strong causality and an assumption of genericness, it is shown that future timelike or null geodesically incomplete space-times contain either curvature or intermediate singularities, or primordial singularities.     

Lissajous singularities

Coherent optical Lissajous states are easily created by nonlinear processes such as second-harmonic generation (SHG). Singular properties of such states are discussed and illustrated theoretically with non-phase-matched SHG of an ellipse field containing a C point.     

Recovering singularities of a potential from singularities of scattering data

In this paper we show that the leading singularities of certain potentials can be determined from the leading singularities of the backscattering (as well as other determined sets of scattering data). The potentials in question are conormal with respect to smooth surfaces of arbitrary dimension; the restrictions on their orders allow for unbounded potentials in all dimension greater than or equal to three.Partially supported by NSF Grant DMS-9101298 and an Alfred P. Sloan Research FellowshipPartially supported by NSF Grant DMS-9100178     

The nonmodular topological phase and phase singularities

Generalizing an earlier definition of the noncyclic geometric phase [R. Bhandari, Phys. Lett. A 157 (1991) 221], a nonmodular topological phase is defined with reference to a generic time-dependent two-slit interference experiment involving particles with N internal states in which the internal state of both the beams undergoes unitary evolution. A simple proof of the shorter geodesic rule for closure of the open path is presented and several useful new insights into the behavior of the dynamical and geometrical components of the phase shift presented. An effective Hamiltonian interpretation of the observable phase shifts is also presented.     

The cosmological term,compensation and singularities

The necessity for the cosmological term in the Einstein and other theories of gravitation is emphasized; in these theories a “compensation” interpretation of the gravitational field as a calibration field is of fundamental importance and it leads, in particular, to torsion and nonlinearities for spinor and other fields. In conjunction with these ideas, the importance of allowing for exposed singularities is pointed out and their analogy with the latest elementary particle models is stressed.     

The anomalous pole singularities in atomic processes

It is shown that the triangle graph has a pole anomalous singularity due to Coulomb interaction for atomic processes.     

The logic of quantum systems with diagonal singularities

The work of the Brussels-Austin groups on irreversibility over the last years has shown that Quantum Large Poincaré systems with diagonal singularity lead to an extension of the conventional formulation of dynamics at the level of mixtures which is manifestly time asymmetric. States with diagonal singularity acquire meaning as linear fractionals over the involutive Banach algebra of operators with diagonal singularity. We show in this paper that the logic of quantum systems with diagonal singularity is not the conventional logic of Hilbert space, because only finite combinations of prepositions are allowed.     

Embedding of singularities

The Friedmann and Kantowski-Sachs models of the universe are embedded in eight-dimensional pseudo-Euclidean spaces in such a way, that their boundaries have the same structure as Geroch'sg-boundary.     

The structure of singularities in space-times with torsion

An analysis of the extension of the Hawking-Penrose singularity theorem to Riemann-Cartan U₄ space-times with torsion and spin density is undertaken. The minimal coupling principle in U₄ is used to formulate a new expression for the convergence condition autoparallels in Einstein-Cartan theory. The Gödel model with torsion is given as an example.On leave of absence from Departmento de Fisica Teórica, Instituto de Física, Universidade do Estado do Rio de Janeiro, CEP:20550, RJ, Brazil.     

The Landau singularities for a double-stream instability plasma

Dispersion relation roots in a double stream instability plasma are given: the pole responsible for this instability is not the Maxwellian Landau pole. Then, we determine the analytic value of the Landau pole in a quasi-linear theory.     

Global string singularities

By analysing the fully coupled equations of motion for a U(1) global string with gravity, we show that global string spacetimes are singular. This singularity is not removable (i.e. due to a bad choice or coordinates) but is a physical curvature singularity.     

Generalized shear-free singularities

The possibility of matter singularities in a class of cosmological models is considered. The results are applied to shear-free models and suggest that in these the fluid cannot simultaneously expand and rotate.     

Emergent polarization singularities

Polarization singularities are shown to emerge spontaneously from the incoherent addition of uncorrelated optical fields that individually need not contain singularities. Examples of this phenomenon are given for both vector and ellipse fields. The incoherent addition of vector fields whose singularities have integer winding numbers is shown to yield fields whose singularities have half-integer winding numbers. These findings are used to make predictions about the singularities of the polarized component of the cosmic microwave background.     

Occurrence of whimper singularities

HOmogeneous space-times (i.e. those admitting a three-parameter group of isometries) are studied using the Newman Penrose formalism. It is found that solutions containing horizons depend on two fewer parameters than the most general solution, so that horizons and the associated whimper singularities are not stable features of homogeneous space-times. In the vacuum case, there are just three two-parameter families with horizons, two of which are the NUT solutions and certain plane waves.     

Polychromatic polarization singularities

Elliptical polarization can appear in only monochromatic optical fields. In polychromatic vector fields the polarization is a Lissajous figure, but in only commensurate fields do the figures have well-defined shapes; in other fields the shapes are undefined. Nonetheless, I show that a given paraxial polychromatic vector field has a coherency ellipse field associated with it that contains polarization singularities and stationary points that are surrogates for the corresponding critical points of the parent optical field.     

Contributions of renormalon singularities

On the basis of some simple examples we suggest a general method for discussing the contribution of renormalons to the Borel transform based on the general analytical structure of the renormalization group β function. No simple principal value or contour integral Borel representation seems to be possible. Instead a splitting of the Borel transform at each renormalon singularity seems to be necessary making the infinite sum over renormalons possibly difficult to define.