Coordinate time

Abstract: General relativity is extended by promoting the three-dimensional gravitational Chern-Simons term to four dimensions. This entails choosing an embedding coordinate ${v}_{\ensuremath{\mu}}$---an external quantity, which we fix to be a nonvanishing constant in its time component. The theory is identical to one in which the embedding coordinate is itself a dynamical variable, rather than a fixed, external quantity. Consequently diffeomorphism symmetry breaking is hidden in the modified theory: the Schwarzschild metric is a solution; gravitational waves possess two polarizations, each traveling at the velocity of light; a conserved energy-momentum (pseudo)tensor can be constructed. The modification is visible in the intensity of gravitational radiation: the two polarizations of a gravity wave carry intensities that are suppressed or enhanced by the extension.

Abstract: The invariance of various definitions proposed for the energy and momentum of the gravitational field is examined. We use the boundary conditions that ${g}_{\ensuremath{\mu}\ensuremath{ u}}$ approaches the Lorentz metric as $\frac{1}{r}$, but allow ${g}_{\ensuremath{\mu}\ensuremath{ u},\ensuremath{\alpha}}$ to vanish as slowly as $\frac{1}{r}$. This permits "coordinate waves." It is found that none of the expressions giving the energy as a two-dimensional surface integral are invariant within this class of frames. In a frame containing coordinate waves they are ambiguous, since their value depends on the location of the surface at infinity (unlike the case where ${g}_{\ensuremath{\mu}\ensuremath{ u},\ensuremath{\alpha}}$ vanishes faster than $\frac{1}{r}$). If one introduces the prescription of space-time averaging of the integrals, one finds that the definitions of Landau-Lifshitz and Papapetrou-Gupta yield (equal) coordinate-invariant results. However, the definitions of Einstein, M\o{}ller, and Dirac become unambiguous, but not invariant.The averaged Landau-Lifshitz and Papapetrou-Gupta expressions are then shown to give the correct physical energy-momentum as determined by the canonical formulations Hamiltonian involving only the two degrees of freedom of the field. It is shown that this latter definition yields that inertial energy for a gravitational system which would be measured by a nongravitational apparatus interacting with it. The canonical formalism's definition also agrees with measurements of gravitational mass by Newtonian means at spacial infinity. It is further shown that the energy-momentum may be invariantly calculated from the asymptotic form of the metric field at a fixed time.

Abstract: Over the last 10 years, several papers have established that daily estimates of GPS coordinates are temporally correlated and it is therefore incorrect to assume that the observations are independent when estimating parameters from them. A direct consequence of this assumption is the over-optimistic estimation of the parameter uncertainties. Perhaps the perceived computational burden or the lack of suitable software for time series analysis has resulted in many heuristic methods being proposed in the scientific literature for estimating these uncertainties. We present a standalone C program, CATS, developed to study and compare stochastic noise processes in continuous GPS coordinate time series and, as a consequence, assign realistic uncertainties to parameters derived from them. The name originally stood for Create and Analyze Time Series. Although the name has survived, the creation aspect of the software has, after several versions, been abandoned. The implementation of the method is briefly described to aid understanding and an example of typical input, usage, output and the available stochastic noise models are given.