Finitary-Algebraic ‘Resolution’ of the Inner Schwarzschild Singularity

A ‘resolution’ of the interior singularity of the spherically symmetric Schwarzschild solution of the Einstein equations for the gravitational field of a point-particle is carried out entirely and solely by finitistic and algebraic means. To this end, the background differential spacetime manifold and, in extenso, Differential Calculus-free purely algebraic (:sheaf-theoretic) conceptual and technical machinery of Abstract Differential Geometry (ADG) is employed. As in previous works [Mallios, A. and Raptis, I. (2001). Finitary spacetime sheaves of quantum causal sets: Curving quantum causality. International Journal of Theoretical Physics, 40, 1885 [gr-qc/0102097]; Mallios, A. and Raptis, I. (2002). Finitary Čech-de Rham cohomology. International Journal of Theoretical Physics, 41, 1857 [gr-qc/0110033]; Mallios, A. and Raptis, I. (2003). Finitary, causal and quantal vacuum Einstein gravity. International Journal of Theoretical Physics 42, 1479 [gr-qc/0209048]], which this paper continues, the starting point for the present application of ADG is Sorkin's finitary (:locally finite) poset (:partially ordered set) substitutes of continuous manifolds in their Gel'fand-dual picture in terms of discrete differential incidence algebras and the finitary spacetime sheaves thereof. It is shown that the Einstein equations hold not only at the finitary poset level of ‘discrete events,’ but also at a suitable ‘classical spacetime continuum limit’ of the said finitary sheaves and the associated differential triads that they define ADG-theoretically. The upshot of this is two-fold: On the one hand, the field equations are seen to hold when only finitely many events or ‘degrees of freedom’ of the gravitational field are involved, so that no infinity or uncontrollable divergence of the latter arises at all in our inherently finitistic-algebraic scenario. On the other hand, the law of gravity—still modelled in ADG by a differential equation proper—does not break down in any (differential geometric) sense in the vicinity of the locus of the point-mass as it is traditionally maintained in the usual manifold-based analysis of spacetime singularities in General Relativity (GR). At the end, some brief remarks are made on the potential import of ADG-theoretic ideas in developing a genuinely background-independent Quantum Gravity (QG). A brief comparison between the ‘resolution’ proposed here and a recent resolution of the inner Schwarzschild singularity by Loop QG means concludes the paper. PACS numbers: 04.60.−m, 04.20.Gz, 04.20.−q     

Logarithmic Deformations of the Rational Superpotential/Landau-Ginzburg Construction of Solutions of the WDVV Equations

The superpotential in the Landau-Ginzburg construction of solutions to the Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equations is modified to include logarithmic terms. This results in deformations - quadratic in the deformation parameters- of the normal prepotential solutions of the WDVV equations. Such solutions satisfy various pseudo-quasi-homogeneity conditions, on assigning a notional weight to the deformation parameters. These solutions originate in the so-called ‘water-bag’ reductions of the dispersionless KP hierarchy. This construction includes, as a special case, deformations which are polynomial in the flat coordinates, resulting in a new class of polynomial solutions of the WDVV equations.     

The gravitational bending of light by stars: a continuing story of curiosity,scepticism, surprise,and fascination

Driven entirely by human curiosity, the effect of the gravitational bending of light has evolved on unforeseen paths, in an interplay between shifts in prevailing paradigms and advance of technology, into the most unusual way to study planet populations. The confirmation of the bending angle predicted by Einstein with the Solar Eclipse measurements from 1919 marked the breakthrough of the theory of General Relativity, but it was not before the detection of the double image of the quasar 0957+561 that ‘gravitational lensing’ really entered the observational era. The observation of a characteristic transient brightening of a star caused by the gravitational deflection of its light by an intervening foreground star, constituting a ‘microlensing event’, required even further advance in technology before it could first emerge in 1993. While it required more patience in waiting before ‘Einstein’s blip’ for the first time revealed the presence of a planet orbiting a star other than the Sun, such detections can now be monitored live, and gravitational microlensing is not only sensitive to masses as low as that of the Moon, but can even reveal planets around stars in galaxies other than the Milky Way.     

Field theoretic treatment of gravitational interaction in electrodynamics

A theory of gravitational interaction in classical electrodynamics is developed on the basis of an earlier-proposed minimal relativistic model of gravitation. From the variation principle, a system of gaugeinvariant equations of the interacting electromagnetic and gravitational fields is deduced and their common energy-momentum tensor is constructed. A rigorous solution to the problem of regularizing the field mass of a point charge is given with consideration for the coupling energy of the gravitational interaction. The propagation of electromagnetic waves in the gravitational field is discussed. It is shown that, under the condition of the existing resonant ratio 2: 3 for the periods of Mercury’s orbital revolution and daily rotation, tidal forces cause a regular shift in the planet’s perihelion in an observable forward direction.     

Effect of boundary conditions on the invariant density of noisy maps at fully-developed chaos

The invariant density of one-dimensional maps in the regime of fully-developed chaos with uncorrelated additive noise is considered. Boundary conditions are shown to play a significant role in determining the precise form of the invariant density, via the manner in which they handle the spill-over, caused by the noise, of orbits beyond the interval. The known case of periodic boundary conditions is briefly recapitulated. Analytic solutions for the invariant density that are possible under certain conditions are presented with applications to specific well-known maps. The case of ‘sticky’ boundaries is generalized to ‘re-injection at the nearest boundary’, and the exact functional equations determining the invariant density are derived. Interesting boundary layer effects are shown to occur, that lead to significant modifications of the invariant density corresponding to the unperturbed (noise-free) case, even when the latter is a constant — as illustrated by an application of the formalism to the noisy tent map. All our results are non-perturbative, and hold good for any noise amplitude in the interval.     

Quantum information paradox: Real or fictitious?

One of the outstanding puzzles of theoretical physics is whether quantum information indeed gets lost in the case of black hole (BH) evaporation or accretion. Let us recall that quantum mechanics (QM) demands an upper limit on the acceleration of a test particle. On the other hand, it is pointed out here that, if a Schwarzschild BH exists, the acceleration of the test particle would blow up at the event horizon in violation of QM. Thus the concept of an exact BH is in contradiction with QM and quantum gravity (QG). It is also reminded that the mass of a BH actually appears as an integration constant of Einstein equations. And it has been shown that the value of this integration constant is actually zero! Thus even classically, there cannot be finite mass BHs though zero mass BH is allowed. It has been further shown that during continued gravitational collapse, radiation emanating from the contracting object gets trapped within it by the runaway gravitational field. As a consequence, the contracting body attains a quasi-static state where outward trapped radiation pressure gets balanced by inward gravitational pull and the ideal classical BH state is never formed in a finite proper time. In other words, continued gravitational collapse results in an ‘eternally collapsing object’ which is a ball of hot plasma and which is asymptotically approaching the true BH state with M = 0 after radiating away its entire mass energy. And if we include QM, this contraction must halt at a radius suggested by the highest QM acceleration. In any case no event horizon (EH) is ever formed and in reality, there is no quantum information paradox.     

Bianchi-I Space-time with Variable Gravitational and?Cosmological “Constants”

We consider Einstein’s field equations with variable gravitational and cosmological “constants” for a spatially homogeneous and anisotropic Bianchi-I space-time. A law of variation for the Hubble parameter, which is related to the average scale factor and yields a constant value of the deceleration parameter, is assumed to solve the field equations. The gravitational constant is allowed to follow a power-law form. We find that a time-increasing gravitational constant is suitable for describing the present evolution of universe. The solutions reveal the dynamics of a universe, which expands forever. The physical interpretation of the solutions is discussed in detail.     

Gravitational field of global monopole with semi-classical effects

We find an exact solution for the space-time of a global monopole by using the vacuum expectation value of the stress energy tensor due to an arbitrary collection of conformal mass less free quantum fields as a source. In a particular situation, the solution is shown to possess an interesting feature like ‘wormholes’ space-time. The monopole exerts no gravitational force on the surrounding matter.     

Gravitational collapse of phantom fluid in (2 + 1)-dimensions

In this paper, we solve Einsteins’ field equations for a circularly symmetric anisotropic fluid, with kinematic self-similarity of the first kind, in (2 + 1)-dimensional spacetimes. Considering the case where the radial pressure vanishes, we show that there exists a solution that represents the gravitational collapse of an anisotropic fluid, and the collapse will finally form a black hole, even if the fluid is constituted by phantom energy.     

Cosmology with cosmic microwave background anisotropy

Measurements of CMB anisotropy and, more recently, polarization have played a very important role in allowing precise determination of various parameters of the ‘standard’ cosmological model. The expectation of the paradigm of inflation and the generic prediction of the simplest realization of inflationary scenario in the early Universe have also been established — ‘acausally’ correlated initial perturbations in a flat, statistically isotropic Universe, adiabatic nature of primordial density perturbations. Direct evidence for gravitational instability mechanism for structure formation from primordial perturbations has been established. In the next decade, future experiments promise to strengthen these deductions and uncover the remaining crucial signature of inflation — the primordial gravitational wave background.     

Spinor Relativity

Spinor relativity is a unified field theory, which derives gravitational and electromagnetic fields as well as a spinor field from the geometry of an eight-dimensional complex and ‘chiral’ manifold. The structure of the theory is analogous to that of general relativity: it is based on a metric with invariance group GL(ℂ²), which combines the Lorentz group with electromagnetic U(1), and the dynamics is determined by an action, which is an integral of a curvature scalar and does not contain coupling constants. The theory is related to physics on spacetime by the assumption of a symmetry-breaking ground state such that a four-dimensional submanifold with classical properties arises. In the vicinity of the ground state, the scale of which is of Planck order, the equation system of spinor relativity reduces to the usual Einstein and Maxwell equations describing gravitational and electromagnetic fields coupled to a Dirac spinor field, which satisfies a non-linear equation; an additional equation relates the electromagnetic field to the polarization of the ground state condensate.     

Boundary Conditions for Coupled Quasilinear Wave Equations with Application to Isolated Systems

We consider the initial-boundary value problem for systems of quasilinear wave equations on domains of the form [0, T] × Σ, where Σ is a compact manifold with smooth boundaries ∂Σ. By using an appropriate reduction to a first order symmetric hyperbolic system with maximal dissipative boundary conditions, well posedness of such problems is established for a large class of boundary conditions on ∂Σ. We show that our class of boundary conditions is sufficiently general to allow for a well posed formulation for different wave problems in the presence of constraints and artificial, nonreflecting boundaries, including Maxwell’s equations in the Lorentz gauge and Einstein’s gravitational equations in harmonic coordinates. Our results should also be useful for obtaining stable finite-difference discretizations for such problems.     

Equations of hydrodynamics in general relativity in the slow motion approximation

By means of a formal solution to the Einstein gravitational field equations a slow motion expansion in inverse powers of the speed of light is developed for the metric tensor. The formal solution, which satisfies the deDonder coordinate conditions and the Trautman outgoing radiation condition, is in the form of an integral equation which is solved iteratively. A stress-energy tensor appropriate to a perfect fluid is assumed and all orders of the metric needed to obtain the equations of motion and conserved quantities to the 21/2post-Newtonian approximation are found. The results are compared to those obtained in another gauge by S. Chandrasekhar. In addition, the relation of the fast motion approximation to the slow motion approximation is examined.     

Extended bodies with quadrupole moment interacting with gravitational monopoles: reciprocity relations

An exact solution of Einstein’s equations representing the static gravitational field of a quasi-spherical source endowed with both mass and mass quadrupole moment is considered. It belongs to the Weyl class of solutions and reduces to the Schwarzschild solution when the quadrupole moment vanishes. The geometric properties of timelike circular orbits (including geodesics) in this spacetime are investigated. Moreover, a comparison between geodesic motion in the spacetime of a quasi-spherical source and non-geodesic motion of an extended body also endowed with both mass and mass quadrupole moment as described by Dixon’s model in the gravitational field of a Schwarzschild black hole is discussed. Certain “reciprocity relations” between the source and the particle parameters are obtained, providing a further argument in favor of the acceptability of Dixon’s model for extended bodies in general relativity.     

Electron field emission properties of highly dense carbon nanotube arrays

In this paper, we have studied field emission properties of highly dense arrays of multi-walled carbon nanotubes (CNTs) used as cathodes in diode-type field emission devices with a phosphor screen. For the high-density CNT emitters it is demonstrated that the emission sites are located on the CNT-cathode edges, which is direct experimental evidence of the ‘edge effect’. The results of computer simulations (using ‘ANSYS Electromagnetic’ software) are presented to confirm the experimental data and to analyze the effect of patterning on the electric field distribution for high-density CNT arrays. It is shown that selective-area removal of nanotubes in the arrays leads to the formation of additional edges characterized by the high field enhancement factor and enhanced emission from the CNT cathodes. In addition, scanning probe microscopy techniques are employed to examine surface properties of the high-density CNT arrays. For CNT arrays of ‘short’ nanotubes, the work function distribution over the sample surface is detected using a scanning Kelvin microscopy method.     

Aspects of Planckian scattering beyond the eikonal

We discuss an approach to compute two-particle scattering amplitudes for spinless light particles colliding at Planckian centre-of-mass energies, with increasing momentum transfer away from the eikonal limit. The leading corrections to the eikonal amplitude, in our ‘external metric’ approach, are shown to be vanishingly small in the limit of the source particle mass going to zero. For massless charged particles, the electromagnetic and gravitational interactions decouple in the eikonal limit, but mix non-trivially for the leading order corrections.     

Bulk Viscous Bianchi Type-III Cosmological Model with Time-Dependent <Emphasis Type="Italic">G</Emphasis> and Λ

This paper deals with Bianchi type-III anisotropic cosmological model of the universe filled with a bulk viscous fluid with time varying gravitational and cosmological constants. It is shown that the field equations are solvable for any arbitrary cosmic scale function. Exact solutions of Einstein’s field equations are obtained which represent an expanding, shearing, non-rotating and decelerating universe. The physical behaviour of the model has also been discussed.     

On electromagnetic and gravitational waves in general relativity. I

In this paper the external field of a bounded source emitting electromagnetic and gravitational radiation has been considered. The field has been assumed to be axially symmetric as well as reflexion symmetric and that it tends to flatness at infinity. The form of the electromagnetic tensor which is in accordance with the classical harmonic electric dipole radiation has been assumed. It has been shown that the whole set of field equations falls into seven main equations together with three supplementary conditions. These field equations have been solved by multipole expansion method and the corresponding Initial Value Problem has been discussed.