Taub Space Research Articles

Quantizing the homogeneous linear perturbations about Taub using the Jacobi method of second variation and the method of invariants

Applying the Jacobi method of second variation to the Bianchi IX system in Misner variables (), we specialize to the Taub space background () and obtain the governing equations for linearized homogeneous perturbations () thereabout. Employing a canonical transformation, we isolate two decoupled gauge-invariant linearized variables ( and ), together with their conjugate momenta and linearized Hamiltonians. These two linearized Hamiltonians are of time-dependent harmonic oscillator form, and we quantize them to get time-dependent Schrödinger equations. For the case of , we are able to solve for the discrete solutions and the exact quantum squeezed states.

Static plane-symmetric space-time with a conformally coupled massless scalar field

We investigate static plane-symmetric space-times. The source of curvature in the Taub space is identified, and the modification hereof due to the presence of a conformally coupled scalar field is calculated. For a specific value of the scalar charge, the scalar field has an energy-momentum tensor of the Casimir type. In this case the source is a domain wall.

Spaces with flat symmetry in poincar� gauge theory of gravity

A Taub space is considered in the Poincare gauge theory of gravity. It is shown that the torsion tensor has four nonvanishing components, which can be split into two independent pairs S010, S011, and S230, S231. The analysis of the gravitational field equations leads to the conclusion that in this case only a flat space-time with torsion is possible, and that its metric coefficients and the components of the torsion tensor are described by a wave equation.

Motion of a test body in Taub space

We examine the motion of a test body in a static Taub space. It is shown that in the case of a fall to a gravitating plane, the test body on approach cannot have velocity greater than the velocity of light, measured by the clock of a “distant” observer or in its proper reference frame. The greater the velocity of a body receding from the plane, the greater the height to which it rises over the gravitating plane, however, the maximal height of its rise equals x=1/κ, i.e., the weaker the gravitational field, the greater the height ¦x¦ to which the body rises. We give a qualitative analysis of the arbitrary motion of a test body in a static Taub space.

Weak angle from Kaluza-Klein theory with deformed internal space

We study a seven-dimensional Kaluza-Klein theory with a three-dimensional Taub space as internal manifold. The electroweak coupling constant ratio is shown to be relatable to the deformation parameter α (shape) of the internal space. The full deformation range yields a weak angle in the range 0.25 <sin 2 θ w < 1. Using the value of α(= −0.2133) from a solution of the Einstein equations with quantum vacuum sources, we get sin 2 θ w = 0.511.

Electromagnetic fields in space-times with local rotational symmetry

The Debye-potential formalism is applied to perfect-fluid space-times with local rotational symmetry which form a subclass of the generalized Goldberg-Sachs space-times. A decoupled equation for the potentials is obtained. It is seen that this equation yields a considerable amount of information in its general form without specializing to any example of the subclass. Finally some particular space-times, namely, Kantowski-Sachs universes, Taub space, and anisotropic spatially homogeneous cosmological models, are discussed in detail.

Two Bianchi type VIII spatially homogeneous cosmologies

We study two Bianchi type VIII analogues of Taub space and maximal analytic extensions of them. The first one has SL(2,R) as an isometry group, which acts transitively on spacelike hypersurfaces. The maximal extension has all of the pathological features of Taub-NUT space. The second one has the universal covering group of SL(2,R) as an isometry group. The maximal extension of the latter does not have these pathological properties and is geodesically complete.

Scalar Waves in the Mixmaster Universe. I. The Helmholtz Equation in a Fixed Background

Solutions to the scalar wave equation in a fixed mixmaster background are presented. Separability conditions on the Laplace-Beltrami equation are related to the complete sets of invariant operators of the symmetry group S${\mathrm{O}}_{3}$. Solutions to the Helmholtz equation are equivalent to the quantum-mechanical problem of asymmetric rotors. The wave functions in the mixmaster space are the asymmetric-top wave functions. In Euler angle variables, they are expanded in terms of the symmetric-top wave functions (the wave functions in Taub space) possessing definite ($J, K, M$) symmetries, with a coupling in the intrinsic magnetic quantum number $K$; the case with $M=0$ can be expressed as a product of the Lam\'e functions in elliptic coordinates. Invariance of the mixmaster space under the four - group characterizes the wave functions into four symmetry species and causes the factorization of the energy matrix into four submatrices. Eigenvalues and expansion coefficients for the wave functions are calculated for some low-lying levels.

Extensions of the Taub and NUT spaces and extensions of their tangent bundles

A system of extensions of the Taub space and the NUT space with the topology due to Misner is constructed having the property: for each incomplete geodesic in these space-times, there is one and only one extension from the system into which the geodesic smoothly continues. Next, the notion of hypermanifold is introduced which is a generalization of tangent bundle of a space-time, and an untrivial hypermanifold is constructed that contains the tangent bundles of the Taub and NUT spaces as proper submanifolds, and within which almost all geodesics are complete. Locally, the hypermanifolds do not yield anything new, but they provide much broader choice of global properties than any four-dimensional space-time manifold.

Local Characterization of Singularities in General Relativity

We formulate a new approach to singularities: their local description. Given any incomplete space-time M, we define a topological space, the ``g boundary,'' whose points consist of equivalence classes of incomplete geodesics of M. The points of the g boundary may be thought of as the ``singular points'' of M. Local properties of the singularity may now be described in a well-defined way in terms of local properties of the g boundary. For example, the notions: ``dimensionality of a singularity,'' ``past and future of a singular point,'' ``neighborhood of a singular point,'' ``spacelike or timelike character of a singularity,'' and ``metric structure of a singularity'' may all be expressed as properties of the g boundary. Two applications of the g boundary outside of the realm of singularities are discussed: (1) In the case in which the space-time M is extendable (for example, Taub space), the g boundary is shown to be that regular 3-surface across which M may be extended [in this case, the Misner boundary between Taub and Newman-Unti-Tamburino (NUT) space]. (2) With a slight modification of the definitions, the g boundary of an asymptotically simple space-time is shown to be Penrose's surface at ``conformal infinity.'' The application of the g boundary technique to singularities is illustrated with a number of examples. The g-boundary structure of one particular example leads to our consideration of non-Hausdorff space-times.