General Relativity And Quantum Cosmology

Herein we shall consider Lorentz boosts and Wigner rotations from a (complexified) quaternionic point of view. We shall demonstrate that for a suitably defined self-adjoint complex quaternionic 4-velocity, pure Lorentz boosts can be phrased in terms of the quaternion square root of the relative 4-velocity connecting the two inertial frames. Straightforward computations then lead to quite explicit and relatively simple algebraic formulae for the composition of 4-velocities and the Wigner angle. We subsequently relate the Wigner rotation to the generic non-associativity of the composition of three 4-velocities, and develop a necessary and sufficient condition for associativity to hold. Finally, we relate the composition of 4-velocities to a specific implementation of the Baker-Campbell-Hausdorff theorem. As compared to ordinary 4x4 Lorentz transformations, the use of self-adjoint complexified quaternions leads, from a computational view, to storage savings and more rapid computations, and from a pedagogical view to to relatively simple and explicit formulae.

Recently, the Lorentzian path integral formulation using the Picard-Lefschetz theory has attracted much attention in quantum cosmology. In this paper, we analyze the tunneling amplitude in quantum mechanics by using the Lorentzian Picard-Lefschetz formulation and compare it with the WKB analysis of the conventional Schrödinger equation. We show that the Picard-Lefschetz Lorentzian formulation is consistent with the WKB approximation for wave-function and the Euclidean path integral formulation utilizing the solutions of the Euclidean constraint equation. We also consider some problems of this Lorentzian Picard-Lefschetz formulation and discuss a simpler semiclassical approximation of the Lorentzian path integral without integrating the lapse function.

Constraints on the Covariant Canonical Gauge Gravity (CCGG) theory from low-redshift cosmology are studied. The formulation extends Einstein's theory of General Relativity (GR) by a quadratic Riemann-Cartan term in the Lagrangian, controlled by a "deformation" parameter. In the Friedman universe this leads to an additional geometrical stress energy and promotes, due to the necessary presence of torsion, the cosmological constant to a time-dependent function. The MCMC analysis of the combined data sets of Type Ia Supernovae, Cosmic Chronometers and Baryon Acoustic Oscillations yields a fit that is well comparable with the ? CDM results. The modifications implied in the CCGG approach turn out to be subdominant in the low-redshift cosmology. However, a non-zero spatial curvature and deformation parameter are shown to be consistent with observations.

Geodesic motion has significant characteristics of space-time. We calculate the principle Lyapunov exponent (LE), which is the inverse of the instability timescale associated with this geodesics and Kolmogorov-Senai (KS) entropy for our rotating Kerr-Kiselev (KK) black hole. We have investigate the existence of stable/unstable equatorial circular orbits via LE and KS entropy for time-like and null circular geodesics. We have shown that both LE and KS entropy can be written in terms of the radial equation of innermost stable circular orbit (ISCO) for time-like circular orbit. Also, we computed the equation marginally bound circular orbit, which gives the radius (smallest real root) of marginally bound circular orbit (MBCO). We found that the null circular geodesics has larger angular frequency than time-like circular geodesics ( Q o > Q ? ). Thus, null-circular geodesics provides the fastest way to circulate KK black holes. Further, it is also to be noted that null circular geodesics has shortest orbital period ( T photon < T ISCO ) among the all possible circular geodesics. Even null circular geodesics traverses fastest than any stable time-like circular geodesics other than the ISCO.

The MICROSCOPE experiment was designed to test the weak equivalence principle in space, by comparing the low-frequency dynamics of cylindrical "free-falling" test masses controlled by electrostatic forces. We use data taken during technical sessions aimed at estimating the electrostatic stiffness of MICROSCOPE's sensors to constrain a short-range Yukawa deviation from Newtonian gravity. We take advantage of the fact that in the limit of small displacements, the gravitational interaction (both Newtonian and Yukawa-like) between nested cylinders is linear, and thus simply characterised by a stiffness. By measuring the total stiffness of the forces acting on a test mass as it moves, and comparing it with the theoretical electrostatic stiffness (expected to dominate), it is a priori possible to infer constraints on the Yukawa potential parameters. However, we find that measurement uncertainties are dominated by the gold wires used to control the electric charge of the test masses, though their related stiffness is indeed smaller than the expected electrostatic stiffness. Moreover, we find a non-zero unaccounted for stiffness that depends on the instrument's electric configuration, hinting at the presence of patch-field effects. Added to significant uncertainties on the electrostatic model, they only allow for poor constraints on the Yukawa potential. This is not surprising, as MICROSCOPE was not designed for this measurement, but this analysis is the first step to new experimental searches for non-Newtonian gravity in space.

We present a new solution in Einstein-Maxwell theory which can be considered as the magnetized version of Kerr-Taub-NUT solution. Some properties of the spacetime are discussed. We also compute the entropy of extremal black hole in the spacetime by using the Kerr/CFT correspondence approach.

We present a novel solution describing magnetized spacetime outside a massive object with electric charge equipped with NUT parameter. To get the solution, we employ the Ernst magnetization to the Reissner-Nordstrom-Taub-NUT spacetime as the seed. After discussing some physical aspects of the spacetime, we show that the extremal entropy of a magnetized Reissner-Nordstrom-Taub-NUT black hole can be reproduced by using the Cardy formula.

We find an exact solution describing the spacetime outside a massive object with NUT parameter embedded in an external magnetic field. To get the solution, we employ the Ernst magnetization transformation to the Taub-NUT metric as the seed solution. The massless limit of this new solution is the Taub-NUT Melvin spacetime. Some aspects in the magnetized Taub-NUT spacetime are investigated, such as the surface geometry and the existence of closed time like curve.

So far none of attempts to quantize gravity has led to a satisfactory model that not only describe gravity in the realm of a quantum world, but also its relation to elementary particles and other fundamental forces. Here we outline preliminary results for a model of quantum universe, in which gravity is fundamentally and by construction quantic. The model is based on 3 well motivated assumptions with compelling observational and theoretical evidence: quantum mechanics is valid at all scales; quantum systems are described by their symmetries; Universe has infinite independent degrees of freedom. The last assumption means that the Hilbert space of the Universe has SU(N→∞)≅area preserving Diff.( S 2 ) symmetry, which is parameterized by two angular variables. We show that in absence of a background spacetime, this Universe is trivial and static. Nonetheless, quantum fluctuations break the symmetry and divide the Universe to subsystems. When a subsystem is singled out as reference - {\it observer} - and another as {\it clock}, two more continuous parameters arise, which can be interpreted as distance and time. We identify the classical spacetime with parameter space of the Hilbert space of the Universe. Therefore, its quantization is meaningless. In this view, the Einstein equation presents the projection of quantum dynamics in the Hilbert space into its parameter space. Finite dimensional symmetries of elementary particles emerge as a consequence of symmetry breaking when the Universe is divided to subsystems/particles without having any implication for the infinite dimensional symmetry and its associated interaction percived as gravity. This explains why gravity is a universal force.

The physical and mathematical properties of the non-linearly coupled black-hole-orbiting-ring system are studied analytically to second order in the dimensionless angular velocity M ir ? H of the black-hole horizon (here M ir is the irreducible mass of the slowly rotating central black hole). In particular, we determine analytically, to first order in the dimensionless ring-to-black-hole mass ratio m/ M ir , the shift ? Ω mb / Ω mb in the orbital frequency of the {\it marginally bound} circular geodesic that characterizes the composed curved spacetime. Interestingly, our analytical results for the frequency shift ? Ω mb in the composed black-hole-orbiting-ring toy model agree qualitatively with the recently published numerical results for the corresponding frequency shift in the physically related (and mathematically much more complex) black-hole-orbiting-particle system. In particular, the present analysis provides evidence that, at order O(m/ M ir ) , the recently observed positive shift in the angular frequency of the marginally bound circular orbit is directly related to the physically intriguing phenomenon of dragging of inertial frames by orbiting masses in general relativity.