Closed Timelike Curves—Time and Again

Sixty years ago, in 1949, Kurt Gödel published a paper dedicated to Albert Einstein on the occasion of his 70th birthday. Gödel presented a solution of Einstein’s field equations for a rotating, homogeneous, stationary universe with negative cosmological constant. Among various surprising properties this universe allows closed timelike curves (CTC), i.e. travel into the past and into the future. Until today many papers have been published concerning physical, logical and philosophical consequences of these results and the existence of closed timelike worldlines in the General Theory of Relativity. Starting from a short survey on Gödel’s work we proceed to elucidate more modern interpretations of the existence of CTCs.     

The Decay of Massive Scalar Field in Non-Static Gödel Type Universe with Viscous Fluid and Heat Flow

In this study, we have investigated the dynamics of non-static Gödel type rotating universe with massive scalar field, viscous fluid and heat flow in the presence of cosmological constant. For various cosmic matter forms, the behavior of the cosmological constant (Λ), shear (η) and bulk (ξ) viscosity coefficients and other kinematic quantities have studied in the early universe. We have showed the decay of massive scalar field in the non-static rotating Gödel type universe and we have obtained constant scalar field with and without source density. Also, we have investigated the effects of massive scalar field on the matter density and pressure. From solutions of the field equations, we have a cosmological model with non-zero expansion, shear, heat flux and rotation. Also some physical and geometrical aspects of the model discussed.     

The Mathematical Universe

I explore physics implications of the External Reality Hypothesis (ERH) that there exists an external physical reality completely independent of us humans. I argue that with a sufficiently broad definition of mathematics, it implies the Mathematical Universe Hypothesis (MUH) that our physical world is an abstract mathematical structure. I discuss various implications of the ERH and MUH, ranging from standard physics topics like symmetries, irreducible representations, units, free parameters, randomness and initial conditions to broader issues like consciousness, parallel universes and Gödel incompleteness. I hypothesize that only computable and decidable (in Gödel’s sense) structures exist, which alleviates the cosmological measure problem and may help explain why our physical laws appear so simple. I also comment on the intimate relation between mathematical structures, computations, simulations and physical systems.     

Violation of causality in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">T</Emphasis>) gravity

In the standard formulation, the f(T) field equations are not invariant under local Lorentz transformations, and thus the theory does not inherit the causal structure of special relativity. Actually, even locally violation of causality can occur in this formulation of f(T) gravity. A locally Lorentz covariant f(T) gravity theory has been devised recently, and this local causality problem seems to have been overcome. The non-locality question, however, is left open. If gravitation is to be described by this covariant f(T) gravity theory there are a number of issues that ought to be examined in its context, including the question as to whether its field equations allow homogeneous Gödel-type solutions, which necessarily leads to violation of causality on non-local scale. Here, to look into the potentialities and difficulties of the covariant f(T) theories, we examine whether they admit Gödel-type solutions. We take a combination of a perfect fluid with electromagnetic plus a scalar field as source, and determine a general Gödel-type solution, which contains special solutions in which the essential parameter of Gödel-type geometries, \(m^2\), defines any class of homogeneous Gödel-type geometries. We show that solutions of the trigonometric and linear classes (\(m^2 < 0\) and \(m=0\)) are permitted only for the combined matter sources with an electromagnetic field matter component. We extended to the context of covariant f(T) gravity a theorem which ensures that any perfect-fluid homogeneous Gödel-type solution defines the same set of Gödel tetrads \(h_A^{~\mu }\) up to a Lorentz transformation. We also showed that the single massless scalar field generates Gödel-type solution with no closed time-like curves. Even though the covariant f(T) gravity restores Lorentz covariance of the field equations and the local validity of the causality principle, the bare existence of the Gödel-type solutions makes apparent that the covariant formulation of f(T) gravity does not preclude non-local violation of causality in the form of closed time-like curves.     

Thermodynamics of Chaplygin Gas Interacting with Cold Dark Matter

The main goal of the present work is to investigate the validity of the second law of gravitational thermodynamics in an expanding Gödel-type universe filled with generalized Chaplygin gas interacting with cold dark matter. By assuming the Universe as a thermodynamical system bounded by the apparent horizon, and calculating separately the entropy variation for generalized Chaplygin gas, cold dark matter and for the horizon itself, we obtained an expression for the time derivative of the total entropy. We conclude that the 2nd law of gravitational thermodynamics is conditionally valid in the cosmological scenario where the generalized Chaplygin gas interacts with cold dark matter.     

New Classes of Off-Diagonal Cosmological Solutions in Einstein Gravity

In this work, we apply the anholonomic deformation method for constructing new classes of anisotropic cosmological solutions in Einstein gravity and/or generalizations with nonholonomic variables. There are analyzed four types of, in general, inhomogeneous metrics, defined with respect to anholonomic frames and their main geometric properties. Such spacetimes contain as particular cases certain conformal and/or frame transforms of the well known Friedman-Robertson-Walker, Bianchi, Kasner and Gödel universes and define a great variety of cosmological models with generic off-diagonal metrics, local anisotropy and inhomogeneity. It is shown that certain nonholonomic gravitational configurations may mimic de Sitter like inflation scenarios and different anisotropic modifications without satisfying any classical false-vacuum equation of state. Finally, we speculate on perspectives when such off-diagonal solutions can be related to dark energy and dark matter problems in modern cosmology.     

Competition of the multiple Grtler modes in hypersonic boundary layer flows

Competition of multiple Grtler modes in hypersonic boundary layer flows are investigated with the local and marching methods.The wall-layer mode(mode W)and the trapped-layer mode(mode T)both occur in the compressible boundary layer where there exists a temperature adjustment layer near the upper edge.The mode T has the largest growth rate at a lower Grtler number while the mode W dominates at larger Grtler numbers.These two modes are both responsible for the flow transition in the hypersonic flows especially when Grtler number is in the high value range in which the crossover of these two modes takes place.Such high Grtler numbers are virtually far beyond the neutral regime.The nonparallel base flows,therefore,cease to influence the stability behavior of the Grtler modes.The effects of the Mach number on the multiple Grtler modes are studied within a chosen Mach number of 0.95,2,4 and 6.When the flow Mach number is sufficiently large,e.g.,Ma 4,the growth rate crossover of the mode T and mode W occurs both in the conventional G-βmap as well as on the route downstream for a fixed wavelength disturbance.Four particular regions(Region T,T-W,W-T and W)around the crossover point are highlighted with the marching analysis and the result matches that of the local analysis.The initial disturbance of a normal mode maintains the shape in its corresponding dominating region while a shape-transformation occurs outside this region.     

Instability of surfactant solution flow in a Taylor cell

The hydrodynamic instability of surfactant solutions between two coaxial cylinders was investigated by using a laser-induced-fluorescence flow visualization technique to clarify the effect of drag-reducing additives on vortex formation in Taylor-Couette flow. The test fluids were Ethoquad O/12 surfactant solutions, which have a gel-like structure called “shear-induced structurer” (SIS). Photographs of the formation of Görtler vortices were taken and compared with these of tap water. In the Taylor number range of 1.2×10⁵≤Ta≤7.1×10⁵, tap water and 10 ppm surfactant solution flows consisted of Taylor vortices and much smaller Görtler vortices at the rotating inner wall. However, for 50 and 100 ppm surfactant solutions, Taylor vortices were not apparent and Görtler vortices were collapsed. Measurements of the wavelength of Görtler vortices lead to the conclusion that surfactant solutions have a stabilizing effect on Görtler instabilities. This effect depends on surfactant concentration and becomes considerable with increasing acceleration of the inner cylinder.     

Spacelike energy of timelike unit vector fields on a Lorentzian manifold

On a Lorentzian manifold, we define a new functional on the space of unit timelike vector fields given by the L₂ norm of the restriction of the covariant derivative of the vector field to its orthogonal complement. This spacelike energy is related with the energy of the vector field as a map on the tangent bundle endowed with the Kaluza–Klein metric, but it is more adapted to the situation. We compute the first and second variation of the functional and we exhibit several examples of critical points on cosmological models as generalized Robertson–Walker spaces and Gödel universe, on Einstein and contact manifolds and on Lorentzian Berger’s spheres. For these critical points we have also studied to what extent they are stable or even absolute minimizers.     

Relativistic Diffusion in Gödel’s Universe

K. Gödel [G1] published his exact solution to Einstein’s field equations in 1949. On the other hand, a general Lorentz invariant operator, associated to the so-called “relativistic diffusion”, and making sense in any Lorentz manifold, was introduced recently by Franchi-Le Jan in [F-LJ]. Here is proposed a study of the relativistic diffusion in the framework of Gödel’s universe, which contains matter. Such study is related to the determination of a boundary for this non-causal universe.     

Evaluation of the neutron background in CsI target for WIMP direct detection when using a reactor neutrino detector as a neutron veto system

In this work we show that universal gauge vector fields can be localized on the recently proposed 5D thick tachyonic braneworld which involves a de Sitter cosmological background induced on the 3-brane. Namely, by performing a suitable decomposition of the vector field, the resulting 4D effective action corresponds to a massive gauge field, while the profile along the extra dimension obeys a Schrödinger-like equation with a Pöschl–Teller potential. It turns out that the massless zero mode of the gauge field is bound to the expanding 3-brane and allows us to recover the standard 4D electromagnetic phenomena of our world. Moreover, this zero mode is separated from the continuum of Kaluza–Klein (KK) modes by a mass gap determined by the scale of the expansion parameter. We also were able to analytically solve the corresponding Schrödinger-like equation for arbitrary mass, showing that KK massive modes asymptotically behave like plane waves, as expected.     

On the Euclidean approach to quantum field theory in Gödel space-time

Features of the quantum field theory in the background of the Gödel space-time are investigated. By using the Euclidean approach, a quantum field propagating in this background is found to possess a temperature proportional to the magnitude of the cosmic vorticity. However, in the semiclassical approximation, the Gödel solution does not present an intrinsic entropy.     

Investigation of DKP equation for spin-zero system in the presence of Gödel-type background space-time

This paper contains a discussion of a relativistic spin-0 system in the presence of a Gödel-type background space-time. The Duffin–Kemmer–Petiau (DKP) equation in the presence of a Gödel-type background space-time is studied in detail. After a derivation of the final form of this equation in the considered framework, free spin-0 particles have been studied.     

Investigation of weakly-nonlinear development of unsteady Görtler vortices

We have examined, both experimentally (using fully controlled disturbances) and theoretically, the weakly-nonlinear development stages of unsteady (in general) Görtler instability of a boundary layer over a concave surface. Primary attention was given to early manifestations of nonlinearity in the development of unsteady Görtler vortices belonging to the first, most rapidly growing, mode in the discrete spectrum of the stability problem. We have investigated the manifestations of instability versus the frequency of the fundamental (primary) Görtler mode and the initial disturbance amplitude. The weakly-nonlinear stage of development of unsteady Görtler vortices was found to display the following characteristic features: (a) nonlinear interaction among the combination modes in the frequency-wavenumber spectrum, (b) distortion of the wall-normal profiles of disturbance amplitudes and phases, (c) reduction of the growth rate of the fundamental Görtler mode and the majority of combination modes, and (d) a decrease in the phase velocities of unsteady disturbances. It was found that the disturbances enter the region of weakly-nonlinear development after the amplitude of the fundamental frequency-wavenumber mode reaches a threshold of 4–6 %, this value being much greater than that for Tollmien — Schlichting waves (1–2 %) but significantly lower than that for the cross-flow instability modes in three-dimensional boundary layer (more than 10 %).     

Fermi field and Dirac oscillator in a Som–Raychaudhuri space-time

We investigate the relativistic dynamics of a Dirac field in the Som–Raychaudhuri space-time, which is described by a Gödel-type metric and a stationary cylindrical symmetric solution of Einstein field equations for a charged dust distribution in rigid rotation. In order to analyze the effect of various physical parameters of this space-time, we solve the Dirac equation in the Som–Raychaudhuri space-time and obtain the energy levels and eigenfunctions of the Dirac operator by using the Nikiforov–Uvarov method. We also examine the behaviour of the Dirac oscillator in the Som–Raychaudhuri space-time, in particular, the effect of its frequency and the vorticity parameter.     

Stability of closed timelike curves in the Gödel universe

We study, in some detail, the linear stability of closed timelike curves in the Gödel universe. We show that these curves are stable. We present a simple extension (deformation) of the Gödel metric that contains a class of closed timelike curves similar to the ones associated to the original metric. This extension correspond to the addition of matter whose energy-momentum tensor is analyzed. We find the conditions to have matter that satisfies the usual energy conditions. We study the stability of closed timelike curves in the presence of usual matter as well as in the presence of exotic matter (matter that does satisfy the above mentioned conditions). We find that the closed timelike curves in the Gödel universe with or without the inclusion of regular or exotic matter are stable under linear perturbations. We also find a sort of structural stability.     

Thermodynamics of Black Holes in Einstein-Maxwell-Gauss-Bonnet Theory

We study the black hole solution in Einstein-Maxwell-Gauss-Bonnet (EMGB) gravity theory with a cosmological constant in five dimension. It is a generalization of the Reissner-Nordström-de Sitter (RNdS) or RNAdS (Reissner-Nordström-Anti-de Sitter) black hole solutions (according as the cosmological constant is positive or negative) in the Einstein-Gauss-Bonnet (EGB) theory. We analyze the thermodynamic quantities of EMGB black hole and find a restriction involving the charge and the cosmological constant for the existence of an extremal black hole. Finally, Hawking-Page phase transition has been discussed for the present black hole.     

Generation of nonstationary Görtler vortices by localized surface nonuniformities. Receptivity coefficients

The mechanism of production of nonstationary Görtler vortices in a boundary layer on concave wall by surface nonuniformities (vibrations and roughness) has been experimentally examined. The nonuniformities were produced by a specially developed disturbance source. They were controlled, localized along the streamwise coordinate, and periodic over the span of the experimental model. Tests in a low-turbulence wind tunnel have proved that the disturbance source is an efficient means of experimental study of the receptivity and stability problem for boundary layers dominated by Görtler instability. The operation of the disturbance source leads to the production of small-amplitude nonstationary Görtler vortices (tenth or hundredth fractions of a per cent of the free-stream velocity) with predefined characteristics (frequency and spanwise wavelength). In our experiments, we quantitatively examined the problem of linear receptivity of boundary layer to surface nonuniformities in a broad range of frequencies for the most dangerous spanwise scales of Görtler vortices. The values of the amplitudes and phases of the receptivity coefficients were determined. The amplitudes proved to be much smaller in magnitude in comparison with the excitation of modes of hydrodynamic instabilities of other types (Tollmien-Schlichting waves and cross-flow-instability modes). It was found that, with increasing the frequency, the amplitudes of the receptivity coefficients showed a distinct growth while for high frequencies those amplitudes also exhibited a growth with the spanwise scale of perturbations, although for stationary surface roughness no effect due to this scale was observed. It was found that the dependences on frequency of the efficiency of the mechanisms of stability and receptivity showed opposing behaviors, were in competition, and could partially compensate each other, promoting, thus, the production of boundary-layer Görtler vortices in a broad range of frequencies.     

Gödel type metrics in three dimensions

We show that the Gödel type metrics in three dimensions with arbitrary two dimensional background space satisfy the Einstein-perfect fluid field equations. We also show that there exists only one first order partial differential equation satisfied by the components of fluid’s velocity vector field. We then show that the same metrics solve the field equations of the topologically massive gravity where the two dimensional background geometry is a space of constant negative Gaussian curvature. We discuss the possibility that the Gödel type metrics to solve the Ricci and Cotton flow equations. When the vector field u ^μ is a Killing vector field, we came to the conclusion that the stationary Gödel type metrics solve the field equations of the most possible gravitational field equations where the interaction lagrangian is an arbitrary function of the electromagnetic field and the curvature tensors.     

The energy–momentum distributions and relativistic quantum effects on scalar and spin-half particles in a Gödel-type space–time

In this article, the energy–momentum distributions associated with a topologically trivial Gödel-type space–time, using different complexes of Møller, Einstein, Landau–Lifshitz, Papapetrou, and Bergmann–Thomson, is evaluated. The results obtained here may support the Cooperstock’s energy localization hypothesis. Finally, we investigate the relativistic quantum effects on scalar and spin-half particles in this space–time without any potential, and analyse the influence of vorticity parameter on the energy eigenvalues of the system.