卷 25, 编号 2 (2019)

Starting from quantum theory (instead of general relativity) to approach quantum gravity within a minimal setting allows us here to describe the quantum space-time structure and the quantum light cone. From the classical-quantum duality and quantum harmonic oscillator (X, P) variables in global phase space, we promote the space-time coordinates to quantum noncommuting operators. The phase space instanton (X, P = iT) describes the hyperbolic quantum space-time structure and generates the quantum light cone. The classical Minkowski space-time null generators X = ±T disappear at the quantum level due to the relevant quantum [X, T] commutator which is always nonzero. A new quantum Planck scale vacuum region emerges. We describe the quantum Rindler and quantum Schwarzschild-Kruskal space-time structures. The horizons and the r = 0 space-time singularity are quantum mechanically erased. The four Kruskal regions merge inside a single quantum Planck scale “world.” The quantum space-time structure consists of hyper bolic discrete levels of odd numbers (X² — T²)_{n =} (2n + 1) (in Planck units ), n = 0,1, 2....(X_n, T_n) and the mass levels being v(2n + 1). A coherent picture emerges: large n levels are semiclassical tending towards a classical continuum space-time. Low n are quantum, the lowest mode (n = 0) being the Planck scale. Two dual (±) branches are present in the local variables (v2n + 1 ± v2n) reflecting the duality of the large and small n behaviors and covering the whole mass spectrum from the largest astrophysical objects in branch (+) to quantum elementary particles in branch (—) passing by the Planck mass. Black holes belong to both branches (+) and (—).

We first observe that the Path topology of Hawking, King and MacCarthy is an analogue, in curved space-times, of a topology that was suggested by Zeeman as an alternative topology to his so-called Fine topology in Minkowski space-time. We then review a result of a recent paper on spaces of paths and the Path topology, and see that there are at least five more topologies in the class \(\mathfrak{Z}-\mathfrak{G}\) of Zeeman-Göbel topologies which admit a countable basis, incorporate the causal and conformal structures, but the Limit Curve Theorem (LCT) fails to hold. The “problem” that the LCT does not hold can be resolved by “adding back” the light cones in the basic-open sets of these topologies, and create new basic open sets for new topologies. But, the main question is: do we really need the LCT. to hold, and why? Why is the manifold topology, under which the group of homeomorphisms of a space-time is vast and of no physical significance (Zeeman), more preferable than an appropriate topology in the class \(\mathfrak{Z}-\mathfrak{G}\) under which a homeomorphism is an isometry (Göbel)? Since topological conditions that come as a result of a causality requirement are key in the existence of singularities in general relativity, the global topological conditions that one will supply the space-time manifold might play an important role in describing the transition from a quantum nonlocal theory to a classical local theory.

A discontinuous spherically symmetric solution of the Einstein equations is found, containing an annihilation shock wave. Before the shock wave, there occurs a parabolic (with zero speed at infinity) compression of dust consisting of particles and antiparticles, which is glued on the wave of complete annihilation of antiparticles with hyperbolic expansion of a gas with an extremely stiff equation of state. Specifically, in contrast to the Friedmann homogeneous expansion. the solution behind the shock wave has a nonzero pressure gradient associated with acceleration of the continuum. Estimates are given for the parameters related to the stage of baryon-antibaryon annihilation at formation of the Universe.

We present a cosmological model constructed using a pure geometric field theory. The unification principle implies defining any physical object in the model using the building blocks of the geometry used, the Absolute Parallelism (AP) geometry. The type of AP geometry used has simultaneously nonvanishing curvature and torsion. The AP geometric structure, used for this application, satisfies the cosmological principle and switches off the electromagnetic sector of the theory automatically. The model obtained is found to be free from particle horizons and flatness problems. This model can be considered as representing a transition phase between a decelerated and accelerated Universe. Conservation in the model is guaranteed by the theory used.

We deal with the Einstein-Gauss-Bonnet model in dimension D with a cosmological constant. We obtain three stable cosmological solutions with exponential behavior (in time) of three scale factors, corresponding to subspaces of dimensions (l₀l₁l₂ = 3, 4, 4), (3, 3, 2), (3, 4, 3) and D = 12, 9, 11, respectively. Any of the solutions may describe an exponential expansion of 3D subspace governed by the Hubble parameter H. Two of them may also describe a small enough variation of the effective gravitational constant G (in Jordan’s frame) for certain values of Λ.

We study the original Chaplygin gas model in the context of Randall-Sundrum type-II braneworld with a hyperbolic potential. We consider the Chaplygin gas as a candidate for inflation by using the latest data release from Planck 2015. We found that the various inflationary spectrum parameters n_s, r and \(\frac{dn_{s}}{d\;\text{ln}\;k}\) depend only on the number of e-folds. The compatibility of these parameters with the last measurement of Planck is realized with large values of N. In this context, a suitable observational central value of n_s = 0.965 is obtained in the case of the original Chaplygin gas and a hyperbolic potential.

As a footprint of primordial perturbations in cosmological observations, the Berry phase of cosmological perturbations can serve to probe the cosmological inflation. Considering linear perturbations in two-field slow-roll inflation, we derive the Hamiltonians of the scalar and tensor Fourier modes in the form of time-dependent harmonic oscillator Hamiltonians. We find the invariant operators of the resulting Hamiltonians and use these invariants to calculate the Berry phase for sub-horizon scalar and tensor modes in the adiabatic limit.

We consider a rotating modified Hayward black hole and construct a rotating modified Bardeen black hole to study particle acceleration of two colliding particles near the horizon. These classes of black holes have new and important parameters with mass dimension, which made crucial differences with the Kerr black hole. We investigate the CM energy of two colliding neutral particles with the same rest masses falling from rest at infinit to near the horizons of the mentioned black holes. We confirm that rotational motion of these black holes is necessary to have infinite CM energy for collision of two particles near the horizon. We also investigate the range of the particles’ angular momentum and the orbit of the particle, hence find an infinite region for the case of rotating modified Bardeen black hole and a finite region for the case of a modified Hayward black hole.