Topics on space-time topology

The orientability properties of space-times are analysed in detail using elementary algebraic methods. Time, space and charge orientability are discussed and various possible generalisations of charge orientability suggested. There is also a bundle-theoretic analysis of the first two topological properties together with a discussion of spinor-structures from the point of view of the Lorentz bundle of bases over a space-time. A section is devoted to some comments on the topologisation of certain space-times with topologies derived from their causal relations.     

Topics on space-time topology II

Closed, compact, oriented, Lorentzian four-manifolds are investigated using elementary cobordism theory. The groups of cobordism classes of such manifolds under various cobordism relations are calculated. Oriented vector cobordism classes form an infinite free cyclic group and oriented cobordism classes form a subgroup of index two in the four-dimensional oriented cobordism group. The properties of compact five-manifolds bounded by closed, compact, oriented, Lorentzian four-manifolds are investigated and some speculations made on their possible interpretation.     

Topics on space-time geometry

Several topics in the geometry of space-times are discussed, including the time and space-distributions of the space-time and a geometrical definition of singularities.     

Geodesic focusing and space-time topology

Given a space-timeM and a pointp inM, it is shown that, if the locus of first conjugate points ofp along future-directed null geodesics consists of a single point, thenM admits a compact (S ³) spacelike hypersurface. If in addition the null geodesics do not intersect before focusing, then, in a simply connected space-time, the spacelike hypersurface is a partial Cauchy surface.     

Completeness of the Alexandrov topology for space-time

We show that the Alexandrov topology for a subsetX of a space-time with nondegenerate space-time metric is complete iffX is strongly causal. Therefore, the property of the Alexandrov topology being complete and the property of being Hausdorff coincide. There is thus no physically measurable distinction between the Hausdorff nature and the completeness of the Alexandrov topology for space-time.     

Stochastic strings,topology, and space-time confinement

Stochastic space-time caused by random strings is considered. By using a conformlike transformation of the metric, we reconstruct gravitational theory and derive its consequences. Such an approach permits us to find natural quark confinement due to induced gravitation and to take into account the topological structure of space-time in any physical quantity.     

The topology of asymptotically Euclidean static perfect fluid space-time

It is shown that a geodesically complete, asymptotically Euclidean, static perfect fluid space-time satisfying the time-like convergence condition and having a connected fluid region is diffeomorphic to ³×.     

Quantum condensates classified in superconductivity with topology in the Minkowski space-time

A substantial problem in the macroscopic theory of pure superconductivity has been left forgotten for a long time since London and London in 1935. An impression survived that the Meissner effect is more substantial than the zero-resistivity. But, the London equation [I], the Newtonian equation of motion, was abandoned, whereas the London equation [II], derived from the Maxwell equations, was postulated. The London equation [II] included the logical gap [ α ] in real time, whereas the London equation [I] has been ignored without even noting the logical gap [ β ] in space. Microscopically, after the publication of F. London's book and the discovery of the isotope effect in 1950, the success of the Bardeen--Cooper--Schrieffer (BCS) theory in 1957 was likely to have finally given the definitive explanation on superconductivity by proving only the London equation [II] that claimed the coherent condensation of Cooper pairs in the momentum space. Since then, these arguments have been regarded to be a standard among various preceding theories. Meanwhile, the London equation [I] has faded away and has been long-forgotten. But we must not abandon the London equation [I], and, rather, retrieve it. We later recognized also that the DC-component of a persistent current can never be determined by using the Fourier transform analysis, because of its singularity at ω?=?0 and q ?=?0 with huge differences of space-time domain. Quite recently, in 2003, we first recognized a proper and harmonious view to simultaneously account for (i) the zero-resistivity in an open system with (i-c) the resultant persistent current in a closed system, and (ii) the perfect diamagnetism at T???0?K in the space-time aspects in terms of the gauge field theory. Here, we further clarify where and how we have lost and found a properly perspective view of the superconductivity. Here, we eliminate two logical gaps [ α ] and [ β ] by using the gauge field theory for further clarifying a position of the previous and present works. We especially classify superconductors with topology which eventually leads us such as (ii-2D) magnetic flux quantization in a ring. By projecting the 3-dimensional BCS-theory with the concept of ‘coherence’ among an enormous number of Bosons like Cooper pairs onto the (1?+?3)-dimensional Minkowski space-time [β?=?(v/c)?=?0], we clarify responses of the ground state Ψ _macro at T???0?K in a set of the basic equations, for (i) the zero-resistivity, [E _K ???qφ( R )]?=?0 at ω?=?0 and (ii) the perfect diamagnetism [?K ???qA ( R )]?=?0 at q ?=?0 as an inevitable consequence at the gauge fields in the proper theory of superconductivity.     

Quark-gluon plasma (Selected Topics)

Introductory lectures to the theory of (strongly interacting) quark-gluon plasma given at the Winter School of Physics of ITEP (Moscow, February 2010). We emphasize theoretical issues highlighted by the discovery of the low viscosity of the plasma. The topics include relativistic hydrodynamics, manifestations of chiral anomaly in hydrodynamics, superfluidity, relativistic superfluid hydrodynamics, effective stringy scalars, holographic models of Yang-Mills theories.     

Spinor waves in a space-time lattice (II)

In a previous note, an exceptional space-time lattice was found by a roundabout heuristic process. This process was far from convincing; here a more translucent characterization of the lattice is presented. A cornerstone is the consideration of pairs of reciprocal lattices, together with the basic symmetry (S ₄) of the metric tensor. The basic requirement is that one member of a pair of reciprocal lattices contains the other as a sublattice. One preferred lattice is discussed in some detail; it contains three copies of its reciprocal lattice, and it is the simplest example satisfying the requirements. In the expression of the metric tensor in terms of the lattice generators a possible topology on the lattice is suggested. By means of this topology, propagation of spinor waves can be formulated. This proposed—the simplest—propagation mechanism is inhibited, though, by the fact that the three sublattices are required to carry the two types of spinors alternatively. This inhibition can be lifted by introducing a second type of elementary propagation, to next nearest neighbors. If this inhibition is only feebly lifted, this would result in particles with mass small as compared to the inverse of the lattice constant, presumably the Planck mass. Including the propagation to next nearest neighbors leads to spinor waves with six components, two components for each sublattice. In the long-wavelength limit four of them obey a massive Dirac equation, while the remaining two obey a Weyl equation. These considerations conceivably provide a root for the lack of parity invariance in nature, and for the joint occurrence of pairs of massive and massless spinor waves. The construction, furthermore, allows one to accommodate just three different families of spinor waves of this type. Extension of the above arguments outside the realm of the long-wavelength limit forcibly makes the lattice concept independent of the original continuous Minkowski spacetime: the latter is no longer a unique embedding space for the lattice, but appears as anapproximate interpolation, valid near the long-wavelength limit. This may be the minimal requirement to be imposed on a lattice theory in the light of the empirical evidence, if the scale of the lattice structure is, compared to the empirical scales, as small as the Planck scale.     

Some exact solutions of string cosmology in Bianchi III space-time

Following the techniques used by Letelier and Stachel some exact Bianchi III cosmological solutions of massive strings in the presence of magnetic field are obtained and their physical features are discussed. Some string solutions in which magnetic fields are absent are also discussed.     

‘Hyperspace’ (the cobordism theory of space-time)

It is demonstrated that a compact space and time-orientable space-time is cobordant in the unoriented sense, that is, bounds a compact five-manifold. The bounding property is a direct consequence of the triviality of the Euler number.     

Theoretical oscillator strengths for some transitions in P(III), As(III), Sb(III) and Bi(III)

Relativistic as well as non-relativistic oscillator strengths have been calculated for transitions in the principal sharp and diffuse series of P(III), As(III), Sb(III) and Bi(III) spectra. The radial integrals were computed by employing the wave functions obtained from a semiempirical method which included exchange effects. A comparison is presented for our calculated f_ik values with experimental and other theoretical data. The influence of relativistic effects on oscillator strengths for transitions in the P(III) through Bi(III) homologous sequence is discussed.     

Stochastic evolution of the topology and geometry of space-time and the six-dimensional theory of gravitation

A theory of random processes with a suitable phase space is used to predict the variation of topology and geometry, and thus also of gravitational fields in vase regions of space—time, that is, on the macrolevel.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 59–62, August, 1995.     

Dirac equation in (1+1)-dimensional curved space-time

The Dirac equation in (1+1)-dimensional curved space-time is solved explicitly for the spatially flat Robertson-Walker space-time and the cigar metric considered by Witten.     

Gravitational (dynamic) time dilation according to absolute space-time theory

Proceeding from our absolute space-time conceptions, we obtain the formula for the gravitational frequency shift in an extremely simple way. Using our burst model for photons, we show that the different rates of clocks placed in spatial regions with different gravitational potentials appear as a direct result of the gravitational frequency shift and the axiomatic assumption that at any space point the time unit is to be defined by light clocks with equal arms, i.e., that at any space point the light velocity (in moving frames the there-and-back velocity) has the same numerical valuec. Considering the principle of equivalence, we come to the logical conclusion that the kinematic (Einstein-Lorentz) time dilation is an absolute phenomenon.     

U(4) spin gauge theory over a Riemann-Cartan space-time

A spin gauge theory based on the groupU(4) is investigated in a general relativistic context including the possibility of nonzero torsion. The language of Clifford bundles over a space-time with metric and metric compatible torsion is used as a convenient tool for the study of fields defined on space-time possessing Clifford multiplication properties. A Dirac-type representation is investigated in detail and the geometric implications for spin gauge theory are pointed out.