Is the Regge Calculus a consistent approximation to General Relativity?

We will ask the question of whether or not the Regge calculus (and two related simplicial formulations) is a consistent approximation to General Relativity. Our criteria will be based on the behaviour of residual errors in the discrete equations when evaluated on solutions of the Einstein equations. We will show that for generic simplicial lattices the residual errors can not be used to distinguish metrics which are solutions of Einstein's equations from those that are not. We will conclude that either the Regge calculus is an inconsistent approximation to General Relativity or that it is incorrect to use residual errors in the discrete equations as a criteria to judge the discrete equations.Comment: 27 pages, plain TeX, very belated update to match journal articl

Regge Calculus as a Fourth Order Method in Numerical Relativity

The convergence properties of numerical Regge calculus as an approximation to continuum vacuum General Relativity is studied, both analytically and numerically. The Regge equations are evaluated on continuum spacetimes by assigning squared geodesic distances in the continuum manifold to the squared edge lengths in the simplicial manifold. It is found analytically that, individually, the Regge equations converge to zero as the second power of the lattice spacing, but that an average over local Regge equations converges to zero as (at the very least) the third power of the lattice spacing. Numerical studies using analytic solutions to the Einstein equations show that these averages actually converge to zero as the fourth power of the lattice spacing.Comment: 14 pages, LaTeX, 8 figures mailed in separate file or email author directl

Long term stable integration of a maximally sliced Schwarzschild black hole using a smooth lattice method

We will present results of a numerical integration of a maximally sliced Schwarzschild black hole using a smooth lattice method. The results show no signs of any instability forming during the evolutions to t=1000m. The principle features of our method are i) the use of a lattice to record the geometry, ii) the use of local Riemann normal coordinates to apply the 1+1 ADM equations to the lattice and iii) the use of the Bianchi identities to assist in the computation of the curvatures. No other special techniques are used. The evolution is unconstrained and the ADM equations are used in their standard form.Comment: 47 pages including 26 figures, plain TeX, also available at http://www.maths.monash.edu.au/~leo/preprint

Area Regge Calculus and Discontinuous Metrics

Taking the triangle areas as independent variables in the theory of Regge calculus can lead to ambiguities in the edge lengths, which can be interpreted as discontinuities in the metric. We construct solutions to area Regge calculus using a triangulated lattice and find that on a spacelike hypersurface no such discontinuity can arise. On a null hypersurface however, we can have such a situation and the resulting metric can be interpreted as a so-called refractive wave.Comment: 18 pages, 1 figur

A fully (3+1)-D Regge calculus model of the Kasner cosmology

We describe the first discrete-time 4-dimensional numerical application of Regge calculus. The spacetime is represented as a complex of 4-dimensional simplices, and the geometry interior to each 4-simplex is flat Minkowski spacetime. This simplicial spacetime is constructed so as to be foliated with a one parameter family of spacelike hypersurfaces built of tetrahedra. We implement a novel two-surface initial-data prescription for Regge calculus, and provide the first fully 4-dimensional application of an implicit decoupled evolution scheme (the ``Sorkin evolution scheme''). We benchmark this code on the Kasner cosmology --- a cosmology which embodies generic features of the collapse of many cosmological models. We (1) reproduce the continuum solution with a fractional error in the 3-volume of 10^{-5} after 10000 evolution steps, (2) demonstrate stable evolution, (3) preserve the standard deviation of spatial homogeneity to less than 10^{-10} and (4) explicitly display the existence of diffeomorphism freedom in Regge calculus. We also present the second-order convergence properties of the solution to the continuum.Comment: 22 pages, 5 eps figures, LaTeX. Updated and expanded versio

Regge calculus and Ashtekar variables

Spacetime discretized in simplexes, as proposed in the pioneer work of Regge, is described in terms of selfdual variables. In particular, we elucidate the "kinematic" structure of the initial value problem, in which 3--space is divided into flat tetrahedra, paying particular attention to the role played by the reality condition for the Ashtekar variables. An attempt is made to write down the vector and scalar constraints of the theory in a simple and potentially useful way.Comment: 10 pages, uses harvmac. DFUPG 83/9

Discrete structures in gravity

Discrete approaches to gravity, both classical and quantum, are reviewed briefly, with emphasis on the method using piecewise-linear spaces. Models of 3-dimensional quantum gravity involving 6j-symbols are then described, and progress in generalising these models to four dimensions is discussed, as is the relationship of these models in both three and four dimensions to topological theories. Finally, the repercussions of the generalisations are explored for the original formulation of discrete gravity using edge-length variables.Comment: 30 pages, 4 figure

Effective stress-energy tensors, self-force, and broken symmetry

Deriving the motion of a compact mass or charge can be complicated by the presence of large self-fields. Simplifications are known to arise when these fields are split into two parts in the so-called Detweiler-Whiting decomposition. One component satisfies vacuum field equations, while the other does not. The force and torque exerted by the (often ignored) inhomogeneous "S-type" portion is analyzed here for extended scalar charges in curved spacetimes. If the geometry is sufficiently smooth, it is found to introduce effective shifts in all multipole moments of the body's stress-energy tensor. This greatly expands the validity of statements that the homogeneous R field determines the self-force and self-torque up to renormalization effects. The forces and torques exerted by the S field directly measure the degree to which a spacetime fails to admit Killing vectors inside the body. A number of mathematical results related to the use of generalized Killing fields are therefore derived, and may be of wider interest. As an example of their application, the effective shift in the quadrupole moment of a charge's stress-energy tensor is explicitly computed to lowest nontrivial order.Comment: 22 pages, fixed typos and simplified discussio

Synoptic relationships between surface Chlorophyll-<i>a</i> and diagnostic pigments specific to phytoplankton functional types

Error-quantified, synoptic-scale relationships between chlorophyll-<i>a</i> (Chl-<i>a</i>) and phytoplankton pigment groups at the sea surface are presented. A total of ten pigment groups were considered to represent three Phytoplankton Size Classes (PSCs, micro-, nano- and picoplankton) and seven Phytoplankton Functional Types (PFTs, i.e. diatoms, dinoflagellates, green algae, prymnesiophytes (haptophytes), pico-eukaryotes, prokaryotes and <i>Prochlorococcus</i> sp.). The observed relationships between Chl-<i>a</i> and PSCs/PFTs were well-defined at the global scale to show that a community shift of phytoplankton at the basin and global scales is reflected by a change in Chl-<i>a</i> of the total community. Thus, Chl-<i>a</i> of the total community can be used as an index of not only phytoplankton biomass but also of their community structure. Within these relationships, we also found non-monotonic variations with Chl-<i>a</i> for certain pico-sized phytoplankton (pico-eukaryotes, Prokaryotes and <i>Prochlorococcus</i> sp.) and nano-sized phytoplankton (Green algae, prymnesiophytes). The relationships were quantified with a least-square fitting approach in order to enable an estimation of the PFTs from Chl-<i>a</i> where PFTs are expressed as a percentage of the total Chl-<i>a</i>. The estimated uncertainty of the relationships depends on both PFT and Chl-<i>a</i> concentration. Maximum uncertainty of 31.8% was found for diatoms at Chl-<i>a</i> = 0.49 mg m⁻³. However, the mean uncertainty of the relationships over all PFTs was 5.9% over the entire Chl-<i>a</i> range observed in situ (0.02 &lt; Chl-<i>a</i> &lt; 4.26 mg m^&minus;3). The relationships were applied to SeaWiFS satellite Chl-<i>a</i> data from 1998 to 2009 to show the global climatological fields of the surface distribution of PFTs. Results show that microplankton are present in the mid and high latitudes, constituting only ~10.9% of the entire phytoplankton community in the mean field for 1998–2009, in which diatoms explain ~7.5%. Nanoplankton are ubiquitous throughout the global surface oceans, except the subtropical gyres, constituting ~45.5%, of which prymnesiophytes (haptophytes) are the major group explaining ~31.7% while green algae contribute ~13.9%. Picoplankton are dominant in the subtropical gyres, but constitute ~43.6% globally, of which prokaryotes are the major group explaining ~26.5% (<i>Prochlorococcus</i> sp. explaining 22.8%), while pico-eukaryotes explain ~17.2% and are relatively abundant in the South Pacific. These results may be of use to evaluate global marine ecosystem models

Characteristics of severe life events, attachment style, and depression – Using a new online approach

Objectives Severe life events are established as provoking agents for depression in combination with vulnerability factors. Identifying features of severe events improves the prediction of disorder but are rarely utilized, mainly because life event research is increasingly dominated by self‐report checklists with no capacity for inferring such characteristics. This paper investigates the association of severe life events’ features with depression and insecure attachment styles using a new online measure of life events in a clinical and control sample. Methods A total of 202 participants (75 clinical and 127 matched control participants), taken from an earlier national Depression Case Control genetic study and followed up after 12 years, completed the Computerised Life Events Assessment Record to assess characteristics of life events, the Vulnerable Attachment Style Questionnaire to measure attachment insecurity, and the General Health Questionnaire to measure depression. Results The clinical group had higher self‐reported depression, severe life events, and insecure attachment style. They also reported more loss, danger, humiliation, and trauma severe events. Intra‐respondent analysis showed individuals experiencing these types of events were more likely to report depression. Insecure attachment style and severe life events were both significantly related to recent depression and history of depressive disorder. Anxious attachment style was significantly related to relationship events and bereavements, as well as severe loss or humiliation events, whereas avoidant style was not. Conclusions Identifying salient features of severe life events improves associations with depression and insecure attachment style. Utilizing a new online approach can aid research and clinical approaches for depression at low cost. Practitioner points Salient features of severe life events (e.g., loss, humiliation) give insight into the potential impact on attachment vulnerability and depression. Clinicians and researchers can use online methods to economically gain detailed life event information needed for clinical formulation and valid data on stressors. The self‐reported scale for recent depression is only a proxy measure of clinical disorder, but the clinical group selection is a more robust criterion for depression history