24 research outputs found
A brief review of Regge calculus in classical numerical relativity
We briefly review past applications of Regge calculus in classical numerical
relativity, and then outline a programme for the future development of the
field. We briefly describe the success of lattice gravity in constructing
initial data for the head-on collision of equal mass black holes, and discuss
recent results on the efficacy of Regge calculus in the continuum limit.Comment: 2 pages, submitted to the Proceedings of the IX Marcel Grossmann
Meeting, Rome, July 2-8, 200
The BSSN formulation is a partially constrained evolution system
Relativistic simulations in 3+1 dimensions typically monitor the Hamiltonian
and momentum constraints during evolution, with significant violations of these
constraints indicating the presence of instabilities. In this paper we rewrite
the momentum constraints as first-order evolution equations, and show that the
popular BSSN formulation of the Einstein equations explicitly uses the momentum
constraints as evolution equations. We conjecture that this feature is a key
reason for the relative success of the BSSN formulation in numerical
relativity.Comment: 8 pages, minor grammatical correction
Student-Faculty Connection and STEM identity in the Flipped Classroom
Students who arrive at college intending to major in a STEM discipline are often required to complete a college-level precalculus course, despite evidence that these courses are not always successful in preparing students for calculus. The implementation of evidence-based teaching strategies, such as the flipped classroom, provides an avenue for improving the effectiveness of precalculus. This quasi-experimental study explores the effect of a flipped precalculus classroom on students\u27 degree of connection with their instructor and other students, together with their sense of motivation and enjoyment of mathematics, which we treat as an indicator of a developing STEM identity. Validated survey inventories are used to investigate differences in these affective outcomes between three sections of precalculus, two taught using flipped instruction and a control section in which the instructor delivers traditional lectures. The flipped students report significantly greater interactions with their instructor and peers, but indicate that they feel less connected with their instructor. Attitudes towards mathematics are found to decrease slightly through the semester in both instructional approaches
On the convergence of Regge calculus to general relativity
Motivated by a recent study casting doubt on the correspondence between Regge
calculus and general relativity in the continuum limit, we explore a mechanism
by which the simplicial solutions can converge whilst the residual of the Regge
equations evaluated on the continuum solutions does not. By directly
constructing simplicial solutions for the Kasner cosmology we show that the
oscillatory behaviour of the discrepancy between the Einstein and Regge
solutions reconciles the apparent conflict between the results of Brewin and
those of previous studies. We conclude that solutions of Regge calculus are, in
general, expected to be second order accurate approximations to the
corresponding continuum solutions.Comment: Updated to match published version. Details of numerical calculations
added, several sections rewritten. 9 pages, 4 EPS figure
Apparent horizons in simplicial Brill wave initial data
We construct initial data for a particular class of Brill wave metrics using
Regge calculus, and compare the results to a corresponding continuum solution,
finding excellent agreement. We then search for trapped surfaces in both sets
of initial data, and provide an independent verification of the existence of an
apparent horizon once a critical gravitational wave amplitude is passed. Our
estimate of this critical value, using both the Regge and continuum solutions,
supports other recent findings.Comment: 7 pages, 6 EPS figures, LaTeX 2e. Submitted to Class. Quant. Gra
Constraints in Quantum Geometrodynamics
We compare different treatments of the constraints in canonical quantum
gravity. The standard approach on the superspace of 3--geometries treats the
constraints as the sole carriers of the dynamic content of the theory, thus
rendering the traditional dynamical equations obsolete. Quantization of the
constraints in both the Dirac and ADM square root Hamiltonian approaches leads
to the well known problems of time evolution. These problems of time are of
both an interpretational and technical nature. In contrast, the geometrodynamic
quantization procedure on the superspace of the true dynamical variables
separates the issues of quantization from the enforcement of the constraints.
The resulting theory takes into account states that are off-shell with respect
to the constraints, and thus avoids the problems of time. We develop, for the
first time, the geometrodynamic quantization formalism in a general setting and
show that it retains all essential features previously illustrated in the
context of homogeneous cosmologies.Comment: 36 pages, no figures, submitted to IJMPA, Rewording, Fixed Typo