Topics in the Dynamics or General Relativity

N/

Maximal Hypersurfaces and Positivity of Mass

N/

Spacelike surfaces with free boundary in the Lorentz-Minkowski space

We investigate a variational problem in the Lorentz-Minkowski space \l^3 whose critical points are spacelike surfaces with constant mean curvature and making constant contact angle with a given support surface along its common boundary. We show that if the support surface is a pseudosphere, then the surface is a planar disc or a hyperbolic cap. We also study the problem of spacelike hypersurfaces with free boundary in the higher dimensional Lorentz-Minkowski space \l^{n+1}.Comment: 16 pages. Accepted in Classical and Quantum Gravit

An Introduction to Conformal Ricci Flow

We introduce a variation of the classical Ricci flow equation that modifies the unit volume constraint of that equation to a scalar curvature constraint. The resulting equations are named the Conformal Ricci Flow Equations because of the role that conformal geometry plays in constraining the scalar curvature. These equations are analogous to the incompressible Navier-Stokes equations of fluid mechanics inasmuch as a conformal pressure arises as a Lagrange multiplier to conformally deform the metric flow so as to maintain the scalar curvature constraint. The equilibrium points are Einstein metrics with a negative Einstein constant and the conformal pressue is shown to be zero at an equilibrium point and strictly positive otherwise. The geometry of the conformal Ricci flow is discussed as well as the remarkable analytic fact that the constraint force does not lose derivatives and thus analytically the conformal Ricci equation is a bounded perturbation of the classical unnormalized Ricci equation. That the constraint force does not lose derivatives is exactly analogous to the fact that the real physical pressure force that occurs in the Navier-Stokes equations is a bounded function of the velocity. Using a nonlinear Trotter product formula, existence and uniqueness of solutions to the conformal Ricci flow equations is proven. Lastly, we discuss potential applications to Perelman's proposed implementation of Hamilton's program to prove Thurston's 3-manifold geometrization conjectures.Comment: 52 pages, 1 figur

Covariant gauge fixing and Kuchar decomposition

The symplectic geometry of a broad class of generally covariant models is studied. The class is restricted so that the gauge group of the models coincides with the Bergmann-Komar group and the analysis can focus on the general covariance. A geometrical definition of gauge fixing at the constraint manifold is given; it is equivalent to a definition of a background (spacetime) manifold for each topological sector of a model. Every gauge fixing defines a decomposition of the constraint manifold into the physical phase space and the space of embeddings of the Cauchy manifold into the background manifold (Kuchar decomposition). Extensions of every gauge fixing and the associated Kuchar decomposition to a neighbourhood of the constraint manifold are shown to exist.Comment: Revtex, 35 pages, no figure

Nonlinear quantum gravity on the constant mean curvature foliation

A new approach to quantum gravity is presented based on a nonlinear quantization scheme for canonical field theories with an implicitly defined Hamiltonian. The constant mean curvature foliation is employed to eliminate the momentum constraints in canonical general relativity. It is, however, argued that the Hamiltonian constraint may be advantageously retained in the reduced classical system to be quantized. This permits the Hamiltonian constraint equation to be consistently turned into an expectation value equation on quantization that describes the scale factor on each spatial hypersurface characterized by a constant mean exterior curvature. This expectation value equation augments the dynamical quantum evolution of the unconstrained conformal three-geometry with a transverse traceless momentum tensor density. The resulting quantum theory is inherently nonlinear. Nonetheless, it is unitary and free from a nonlocal and implicit description of the Hamiltonian operator. Finally, by imposing additional homogeneity symmetries, a broad class of Bianchi cosmological models are analyzed as nonlinear quantum minisuperspaces in the context of the proposed theory.Comment: 14 pages. Classical and Quantum Gravity (To appear

Rotation-Measures across Parsec-scale Jets of FRI radio galaxies

We present the results of a parsec-scale polarization study of three FRI radio galaxies - 3C66B, 3C78 and 3C264 - obtained with the Very Long Baseline Array at 5, 8 and 15 GHz. Parsec-scale polarization has been detected in a large number of beamed radio-loud active galactic nuclei, but in only a handful of the relatively unbeamed radio galaxies. We report here the detection of parsec-scale polarization at one or more frequencies in all three FRI galaxies studied. We detect Faraday rotation measures of the order of a few hundred rad/m^2 in the nuclear jet regions of 3C78 and 3C264. In 3C66B polarization was detected at 8 GHz only. A transverse rotation measure gradient is observed across the jet of 3C78. The inner-jet magnetic field, corrected for Faraday rotation, is found to be aligned along the jet in both 3C78 and 3C264, although the field becomes orthogonal further from the core in 3C78. The RM values in 3C78 and 3C264 are similar to those previously observed in nearby radio galaxies. The transverse RM gradient in 3C78, the increase in the degree of polarization at the jet edge, the large rotation in the polarization angles due to Faraday rotation and the low depolarization between frequencies, suggests that a layer surrounding the jet with a sufficient number of thermal electrons and threaded by a toroidal or helical magnetic field is a good candidate for the Faraday rotating medium. This suggestion is tentatively supported by Hubble Space Telescope optical polarimetry but needs to be examined in a greater number of sources.Comment: Accepted for publication in The Astrophysical Journal, March 2009 - 20 v694 issu

A sun in the spectroscopic binary IM Pegasi, the guide star for the Gravity Probe B mission

We present the first detection of the secondary of the spectroscopic binary system IM Pegasi (HR 8703), the guide star for the NASA-Stanford relativity gyroscope mission Gravity Probe B. In support of this mission, high-resolution echelle spectra of IM Peg have been obtained on an almost nightly basis. Applying the technique of least-squares deconvolution, we achieve very high signal-to-noise ratio line profiles and detect the orbit of the secondary of the system. Combining almost 700 new radial velocity measurements of both the primary and secondary of the system with previous measurements, we derive improved orbital parameters of the IM Peg system. Using these estimates along with the previously determined range of orbital inclination angles for the system, we find that the primary of IM Peg is a giant of mass 1.8 +/- 0.2 M-circle dot while the secondary is a dwarf of mass 1.0 +/- 0.1 M-circle dot