Precise analytic treatment of Kerr and Kerr-(anti) de Sitter black holes as gravitational lenses

The null geodesic equations that describe motion of photons in Kerr spacetime are solved exactly in the presence of the cosmological constant $\Lambda$. The exact solution for the deflection angle for generic light orbits (i.e. non-polar, non-equatorial) is calculated in terms of the generalized hypergeometric functions of Appell and Lauricella. We then consider the more involved issue in which the black hole acts as a `gravitational lens'. The constructed Kerr black hole gravitational lens geometry consists of an observer and a source located far away and placed at arbitrary inclination with respect to the black hole's equatorial plane. The resulting lens equations are solved elegantly in terms of Appell-Lauricella hypergeometric functions and the Weierstrass elliptic function. We then, systematically, apply our closed form solutions for calculating the image and source positions of generic photon orbits that solve the lens equations and reach an observer located at various values of the polar angle for various values of the Kerr parameter and the first integrals of motion. In this framework, the magnification factors for generic orbits are calculated in closed analytic form for the first time. The exercise is repeated with the appropriate modifications for the case of a non-zero cosmological constant.Comment: 42 pages, 4 figure

General relativity, Lauricella's hypergeometric function FDF_D and the theory of braids

The exact (closed form) solutions of the equations of motion in the theory of general relativity that describe motion of test particle and photon in Kerr and Kerr-(anti) de Sitter spacetimes all involve the multivariable hypergeometric function of Lauricella $F_D$: Kraniotis [Class. Quantum Grav. {\bf 21} 2004, 4743; Class. Quantum Grav. {\bf 22} 2005, 4391; Class. Quantum Grav. {\bf 24} 2007, 1775]. The domain of variables ${\cal D}_n$ of the corresponding function depends on the first integrals of motion associated with the isometries of the Kerr-(anti) de Sitter metric and Carter's constant $Q$ as well as on the cosmological constant $\Lambda$ and the Kerr (rotation) parameter. In this work we discuss the topological properties of the domain ${\cal D}_n$ and in particular its fundamental connection with the theory of braids. An intrinsic relationship of general relativity with the pure braids is established.Comment: 16 pages, LaTe

The massive Dirac equation in the Kerr-Newman-de Sitter and Kerr-Newman black hole spacetimes

Exact solutions of the Dirac general relativistic equation (DE) that describe the dynamics of a massive, electrically charged particle with half-integer spin in the curved spacetime geometry of an electrically charged, rotating Kerr-Newman-(anti) de Sitter black hole (BH) are investigated. We first, derive the DE in the Kerr-Newman-de Sitter (KNdS) BH background using a generalised Kinnersley null tetrad in the Newman-Penrose formalism. In this frame, we prove the separation of the DE into ordinary differential equations for the radial and angular parts. Under specific transformations of the independent and dependent variables we prove that the transformed radial equation for a massive charged spin $\frac{1}{2}$ fermion in the background KNdS BH constitutes a highly non-trivial generalisation of Heun's equation. Using a Regge-Wheeler-like independent variable we transform the radial equation in the KNdS background into a Schr\"{o}dinger-like differential equation and investigate its asymptotic behaviour near the event and cosmological horizons. For a massive fermion (MF) in the background of a Kerr-Newman (KN) BH we first prove that the radial and angular equations that result from the separation of DE reduce to the generalised Heun differential equation (GHE). The local solutions of such GHE are derived and can be described by holomorphic functions whose power series coefficients are determined by a four-term recurrence relation. Using asymptotic analysis we derive the solutions for the MF far away from the KN BH and the solutions near the event horizon . The determination of the separation constant as an eigenvalue problem in the KN background is investigated. Using the aforementioned four-term recursion formula we prove that in the non-extreme KN geometry there are no bound states with $\omega^2<\mu^2$, where $\omega$ and $\mu$ are the energy and mass of the fermion respectively.Comment: LaTeX file, 61 pages, version published in Journal of Physics Communications(2019 J. Phys. Commun. 3 035026

Closed form solution for the surface area, the capacitance and the demagnetizing factors of the ellipsoid

We derive the closed form solutions for the surface area, the capacitance and the demagnetizing factors of the ellipsoid immersed in the Euclidean space R^3. The exact solutions for the above geometrical and physical properties of the ellipsoid are expressed elegantly in terms of the generalized hypergeometric functions of Appell of two variables. Various limiting cases of the theorems of the exact solution for the surface area, the demagnetizing factors and the capacitance of the ellipsoid are derived, which agree with known solutions for the prolate and oblate spheroids and the sphere. Possible applications of the results achieved, in various fields of science, such as in physics, biology and space science are briefly discussed.Comment: 26 pages, 4 eps figure

A possible explanation of Galactic Velocity Rotation Curves in terms of a Cosmological Constant

This paper describes how the non-gravitational contribution to Galactic Velocity Rotation Curves can be explained in terms of a negative Cosmological Constant ($\Lambda$). It will be shown that the Cosmological Constant leads to a velocity contribution proportional to the radii, at large radii, and depending on the mass of the galaxy. This explanation contrasts with the usual interpretation that this effect is due to Dark Matter halos. The velocity rotation curve for the galaxy NGC 3198 will be analysed in detail, while several other galaxies will be studied superficially. The Cosmological Constant derived experimentally from the NGC 3198 data was found to be:$|\Lambda|_{Exp}= 5.0\times 10^{-56} cm^{-2}$. This compares favourably with the theoretical value obtained from the Large Number Hypothesis of: $|\Lambda|_{Theory}=2.1\times 10^{-56}cm^{-2}$. The Extended LNH is then used to define other cosmological parameters: gravitational modification constant, energy density, and the Cosmological Constant in terms of a fundamental length. A speculative theory for the evolution of the Universe is outlined where it is shown how the Universe can be defined, in any particular era, by two parameters: the fundamental length and the energy density of the vacuum for that epoch. The theory is applied to the time evolution of the universe where a possible explanation for the $\rho_{Planck}/\rho_{\Lambda}^{QH} \approx 10^{120}$ problem is proposed. The nature of the ''vacuum'' is reviewed along with a speculative approach for calculating the Cosmological Constant via formal M-theory.The experimentally derived results presented in this paper support a decelerating Universe, in contrast with recent indicationsfrom Type Ia Supernovae experiments, for an accelerating Universe.Comment: LaTeX file, 30 pages including 5 figures, Two important changes: 1)Galactic Velocity Rotation Curves can be explained by a negative Cosmological Constant and thus the theory proposed predicts a decelerating Universe.2) The experimentally determined values for Lambda are consistent with general Cosmological consideration

Compact calculation of the Perihelion Precession of Mercury in General Relativity, the Cosmological Constant and Jacobi's Inversion problem

The geodesic equations resulting from the Schwarzschild gravitational metric element are solved exactly including the contribution from the Cosmological constant. The exact solution is given by genus 2 Siegelsche modular forms. For zero cosmological constant the hyperelliptic curve degenerates into an elliptic curve and the resulting geodesic is solved by the Weierstra\ss Jacobi modular form. The solution is applied to the precise calculation of the perihelion precession of the orbit of planet Mercury around the Sun.Comment: LaTeX file, 24 pages, substantial changes following the Referees' recommendations. Accepted for publication in Classical and Quantum Gravit

Relic Abundances and Detection Rates of Neutralinos in String-Inpired Supergravity Models

We calculate ``relic abundances'' and ``detection rates'' of the neutralino (LSP) in string-inspired supergravity models with ``dilaton-moduli'' induced supersymmetry breaking. In particular we investigate ``universal'' scenarios for the soft-supersymmetry breaking terms from Calabi-Yau compactifications, as well as from the dilaton-dominated limit. ``Non-universal'' scenarios from orbifold string theory are also incorporated into the analysis. In all cases, in the cosmologically interesting region, we find $m_{LSP}\geq 50$ GeV and direct-detection rates in the range O($10^{-3}$ events/(Kg day)) - O($10^{-4}$ events/(Kg day)). ``Indirect-detection'' rates from LSPs captured in the Sun are also calculated.Comment: Latex, 7 pages including 6 figs, uses sprocl.sty. Based on a talk given by G.V. Kraniotis at the international Workshop on the Identification of Dark Matter, IDM'96, Sheffield (UK) 8-12 September 199

A New Dark Matter Model for Galaxies

In this paper a new theory of Dark Matter is proposed. Experimental analysis of several Galaxies show how the non-gravitational contribution to galactic Velocity Rotation Curves can be interpreted as that due to the Cosmological Constant $\Lambda$. The experimentally determined values for $\Lambda$ are found to be consistent with those expected from Cosmological Constraints. The Cosmological Constant is interpreted as leading to a constant energy density which in turn can be used to partly address the energy deficit problem (Dark Energy) of the Universe. The work presented here leads to the conclusion that the Cosmological Constant is negative and that the universe is de-accelerating. This is in clear contradiction to the Type Ia Supernovae results which support an accelerating universe.Comment: LaTeX file, 10 pages, incl. 1 figure, v2: Important changes: the work presented predicts a decelleration for the Universe and typos have been correcte

Gravitational lensing and frame dragging of light in the Kerr-Newman and the Kerr-Newman-(anti) de Sitter black hole spacetimes

The null geodesics that describe photon orbits in the spacetime of a rotating electrically charged black hole (Kerr-Newman) are solved exactly including the contribution from the cosmological constant. We derive elegant closed form solutions for relativistic observables such as the deflection angle and frame dragging effect that a light ray experiences in the gravitational fields (i) of a Kerr-Newman black hole and (ii) of a Kerr-Newman-de Sitter black hole. We then solve the more involved problem of treating a Kerr-Newman black hole as a gravitational lens, i.e. a KN black hole along with a static source of light and a static observer both located far away but otherwise at arbitrary positions in space. For this model, we derive the analytic solutions of the lens equations in terms of Appell and Lauricella hypergeometric functions and the Weierstra\ss modular form. The exact solutions derived for null, spherical polar and non-polar orbits, are applied for the calculation of frame dragging for the orbit of a photon around the galactic centre, assuming that the latter is a Kerr-Newman black hole. We also apply the exact solution for the deflection angle of an equatorial light ray in the gravitational field of a Kerr-Newman black hole for the calculation of bending of light from the gravitational field of the galactic centre for various values of the Kerr parameter, electric charge and impact factor. In addition, we derive analytic expressions for the Maxwell tensor components for a Zero-Angular-Momentum-Observer (ZAMO) in the Kerr-Newman-de Sitter spacetime.Comment: updated version conforms to published version in General Relativity and Gravitation. The final Publication is available at Springer via http://dx.doi.org/10.1007/s10714-014-1818-

Searching for string theories of the standard model

We briefly review attempts to construct string theories that yield the standard model, concentrating on models with a geometric interpretation. Calabi-Yau compactifications are discussed in the context of both the weakly coupled heterotic string and the strongly coupled version, heterotic M-theory. Similarly, we consider orbifold compactifications of weakly coupled heterotic theory and of Type II theories in the presence of D-branes. The latter allows a "bottom-up" construction that so far seems the most natural and direct route to the standard model.Comment: 21 pages, LaTeX file, Based on a talk given by D. Bailin, at the first Cairo International Conference on High Energy Physics, (CICHEP 2001), Jan.2001. To be published in the proceeding