Retarded coordinates based at a world line, and the motion of a small black hole in an external universe

In the first part of this article I present a system of retarded coordinates based at an arbitrary world line of an arbitrary curved spacetime. The retarded-time coordinate labels forward light cones that are centered on the world line, the radial coordinate is an affine parameter on the null generators of these light cones, and the angular coordinates are constant on each of these generators. The spacetime metric in the retarded coordinates is displayed as an expansion in powers of the radial coordinate and expressed in terms of the world line's acceleration vector and the spacetime's Riemann tensor evaluated at the world line. The formalism is illustrated in two examples, the first involving a comoving world line of a spatially-flat cosmology, the other featuring an observer in circular motion in the Schwarzschild spacetime. The main application of the formalism is presented in the second part of the article, in which I consider the motion of a small black hole in an empty external universe. I use the retarded coordinates to construct the metric of the small black hole perturbed by the tidal field of the external universe, and the metric of the external universe perturbed by the presence of the black hole. Matching these metrics produces the MiSaTaQuWa equations of motion for the small black hole.Comment: 20 pages, revtex4, 2 figure

On perfect fluid models in non-comoving observational spherical coordinates

We use null spherical (observational) coordinates to describe a class of inhomogeneous cosmological models. The proposed cosmological construction is based on the observer past null cone. A known difficulty in using inhomogeneous models is that the null geodesic equation is not integrable in general. Our choice of null coordinates solves the radial ingoing null geodesic by construction. Furthermore, we use an approach where the velocity field is uniquely calculated from the metric rather than put in by hand. Conveniently, this allows us to explore models in a non-comoving frame of reference. In this frame, we find that the velocity field has shear, acceleration and expansion rate in general. We show that a comoving frame is not compatible with expanding perfect fluid models in the coordinates proposed and dust models are simply not possible. We describe the models in a non-comoving frame. We use the dust models in a non-comoving frame to outline a fitting procedure.Comment: 8 pages, 1 figure. To appear in Phys.Rev.

Finding a Spherically Symmetric Cosmology from Observations in Observational Coordinates -- Advantages and Challenges

One of the continuing challenges in cosmology has been to determine the large-scale space-time metric from observations with a minimum of assumptions -- without, for instance, assuming that the universe is almost Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW). If we are lucky enough this would be a way of demonstrating that our universe is FLRW, instead of presupposing it or simply showing that the observations are consistent with FLRW. Showing how to do this within the more general spherically symmetric, inhomogeneous space-time framework takes us a long way towards fulfilling this goal. In recent work researchers have shown how this can be done both in the traditional Lema\^{i}tre-Tolman-Bondi (LTB) 3 + 1 coordinate framework, and in the observational coordinate (OC) framework. In this paper we investigate the stability of solutions, and the use of data in the OC field equations including their time evolution and compare both approaches with respect to the singularity problem at the maximum of the angular-diameter distance, the stability of solutions, and the use of data in the field equations. This allows a more detailed account and assessment of the OC integration procedure, and enables a comparison of the relative advantages of the two equivalent solution frameworks. Both formulations and integration procedures should, in principle, lead to the same results. However, as we show in this paper, the OC procedure manifests certain advantages, particularly in the avoidance of coordinate singularities at the maximum of the angular-diameter distance, and in the stability of the solutions obtained. This particular feature is what allows us to do the best fitting of the data to smooth data functions and the possibility of constructing analytic solutions to the field equations.Comment: 31 page

The untilted diffuse matter Bianchi V Universe

A diffuse matter filled Type V Universe is studied. The anisotropic behaviour, the distortion caused to the CMBR and the parameter region allowed by present cosmological bounds are examined. It is shown how the overall sky pattern of temperature anisotropies changes under a non-infinitesimal spatial coordinate transformation that preserves the Type V manifest homogeneity.Comment: (RevTeX) 23 pages, 2 figures, to appear in Physics Letters

Visualizing sound emission of elephant vocalizations: evidence for two rumble production types

Recent comparative data reveal that formant frequencies are cues to body size in animals, due to a close relationship between formant frequency spacing, vocal tract length and overall body size. Accordingly, intriguing morphological adaptations to elongate the vocal tract in order to lower formants occur in several species, with the size exaggeration hypothesis being proposed to justify most of these observations. While the elephant trunk is strongly implicated to account for the low formants of elephant rumbles, it is unknown whether elephants emit these vocalizations exclusively through the trunk, or whether the mouth is also involved in rumble production. In this study we used a sound visualization method (an acoustic camera) to record rumbles of five captive African elephants during spatial separation and subsequent bonding situations. Our results showed that the female elephants in our analysis produced two distinct types of rumble vocalizations based on vocal path differences: a nasally- and an orally-emitted rumble. Interestingly, nasal rumbles predominated during contact calling, whereas oral rumbles were mainly produced in bonding situations. In addition, nasal and oral rumbles varied considerably in their acoustic structure. In particular, the values of the first two formants reflected the estimated lengths of the vocal paths, corresponding to a vocal tract length of around 2 meters for nasal, and around 0.7 meters for oral rumbles. These results suggest that African elephants may be switching vocal paths to actively vary vocal tract length (with considerable variation in formants) according to context, and call for further research investigating the function of formant modulation in elephant vocalizations. Furthermore, by confirming the use of the elephant trunk in long distance rumble production, our findings provide an explanation for the extremely low formants in these calls, and may also indicate that formant lowering functions to increase call propagation distances in this species'

The Sachs-Wolfe Effect: Gauge Independence and a General Expression

In this paper we address two points concerning the Sachs-Wolfe effect: (i) the gauge independence of the observable temperature anisotropy, and (ii) a gauge-invariant expression of the effect considering the most general situation of hydrodynamic perturbations. The first result follows because the gauge transformation of the temperature fluctuation at the observation event only contributes to the isotropic temperature change which, in practice, is absorbed into the definition of the background temperature. Thus, we proceed without fixing the gauge condition, and express the Sachs-Wolfe effect using the gauge-invariant variables.Comment: 5 pages, closer to published versio

General-relativistic constraints on the equation of state of dense matter implied by kilohertz quasi-periodic oscillations in neutron-star X-ray binaries

If the observed millisecond variability in the X-ray flux of several neutron-star low-mass X-ray binaries (LMXBs) is interpreted within a general-relativistic framework (Kluzniak, Michelson \& Wagoner 1990) extant at the time of discovery, severe constraints can be placed on the equation of state (e.o.s.) of matter at supranuclear densities. The reported maximum frequency (1.14 +- 0.01 kHz) of quasiperiodic oscillations observed in sources as diverse as Sco X-1 and 4U 1728-34 would imply that the neutron star masses in these LMXBs are M > 1.9 M_solar, and hence many equations of state would be excluded. Among the very few still viable equations of state are the e.o.s. of Phandaripande and Smith (1975), and e.o.s. AV14 + UVII of Wiringa, Fiks \& Fabrocini (1988).Comment: The figures can be found in the references cited in the captions. A longer version of this paper was submitted to a refereed journal on January 6, 1997 (345 days ago

A fully covariant description of CMB anisotropies

Starting from the exact non-linear description of matter and radiation, a fully covariant and gauge-invariant formula for the observed temperature anisotropy of the cosmic microwave background (CBR) radiation, expressed in terms of the electric ($E_{ab}$) and magnetic ($H_{ab}$) parts of the Weyl tensor, is obtained by integrating photon geodesics from last scattering to the point of observation today. This improves and extends earlier work by Russ et al where a similar formula was obtained by taking first order variations of the redshift. In the case of scalar (density) perturbations, $E_{ab}$ is related to the harmonic components of the gravitational potential $\Phi_k$ and the usual dominant Sachs-Wolfe contribution $\delta T_R/\bar{T}_R\sim\Phi_k$ to the temperature anisotropy is recovered, together with contributions due to the time variation of the potential (Rees-Sciama effect), entropy and velocity perturbations at last scattering and a pressure suppression term important in low density universes. We also explicitly demonstrate the validity of assuming that the perturbations are adiabatic at decoupling and show that if the surface of last scattering is correctly placed and the background universe model is taken to be a flat dust dominated Friedmann-Robertson-Walker model (FRW), then the large scale temperature anisotropy can be interpreted as being due to the motion of the matter relative to the surface of constant temperature which defines the surface of last scattering on those scales.Comment: 18 pages LaTeX, 1 figure. Submitted to Classical and Quantum Gravity. Also available at http://shiva.mth.uct.ac.za/preprints/9705.htm

Dynamics of Inflationary Universes with Positive Spatial Curvature

If the spatial curvature of the universe is positive, then the curvature term will always dominate at early enough times in a slow-rolling inflationary epoch. This enhances inflationary effects and hence puts limits on the possible number of e-foldings that can have occurred, independently of what happened before inflation began and in particular without regard for what may have happened in the Planck era. We use a simple multi-stage model to examine this limit as a function of the present density parameter $\Omega_0$ and the epoch when inflation ends.Comment: 9 Pages RevTex4. Revised and update