scholarly journals Using Algebraic Computing To Teach General Relativity And Cosmology

2012 ◽  
Vol 56 (1) ◽  
pp. 139-144
Author(s):  
Dumitru N. Vulcanov ◽  
Remus-Ştefan Ş. Boată
Keyword(s):  
General Relativity ◽  
Exact Solutions ◽  
Undergraduate Students ◽  
Computer Algebra ◽  
Computer Algebra System ◽  
Einstein Equations ◽  
Algebraic Programming ◽  
Algebraic Computing

AbstractThe article presents some new aspects and experience on the use of computer in teaching general relativity and cosmology for undergraduate students (and not only) with some experience in computer manipulation. Some years ago certain results were reported [1] using old fashioned computer algebra platforms but the growing popularity of graphical platforms as Maple and Mathematica forced us to adapt and reconsider our methods and programs. We will describe some simple algebraic programming procedures (in Maple with GrTensorII package) for obtaining and the study of some exact solutions of the Einstein equations in order to convince a dedicated student in general relativity about the utility of a computer algebra system.

scholarly journals Applications of, and Extensions to, Selected Exact Solutions in General Relativity

2021 ◽  
Author(s):  
◽  
Bethan Cropp
Keyword(s):  
General Relativity ◽  
Exact Solutions ◽  
Bianchi Type ◽  
Plane Waves ◽  
Local Limit ◽  
Matter Content ◽  
Einstein Equations ◽  
Type I ◽  
Bianchi Type I ◽  
Stress Energy

<p>In this thesis we consider several aspects of general relativity relating to exact solutions of the Einstein equations. In the first part gravitational plane waves in the Rosen form are investigated, and we develop a formalism for writing down any arbitrary polarisation in this form. In addition to this we have extended this algorithm to an arbitrary number of dimensions, and have written down an explicit solution for a circularly polarized Rosen wave. In the second part a particular, ultra-local limit along an arbitrary timelike geodesic in any spacetime is constructed, in close analogy with the well-known lightlike Penrose limit. This limit results in a Bianchi type I spacetime. The properties of these spacetimes are examined in the context of this limit, including the Einstein equations, stress-energy conservation and Raychaudhuri equation. Furthermore the conditions for the Bianchi type I spacetime to be diagonal are explicitly set forward, and the effect of the limit on the matter content of a spacetime are examined.</p>

scholarly journals Applications of, and Extensions to, Selected Exact Solutions in General Relativity

2021 ◽  
Author(s):  
◽  
Bethan Cropp
Keyword(s):  
General Relativity ◽  
Exact Solutions ◽  
Bianchi Type ◽  
Plane Waves ◽  
Local Limit ◽  
Matter Content ◽  
Einstein Equations ◽  
Type I ◽  
Bianchi Type I ◽  
Stress Energy

<p>In this thesis we consider several aspects of general relativity relating to exact solutions of the Einstein equations. In the first part gravitational plane waves in the Rosen form are investigated, and we develop a formalism for writing down any arbitrary polarisation in this form. In addition to this we have extended this algorithm to an arbitrary number of dimensions, and have written down an explicit solution for a circularly polarized Rosen wave. In the second part a particular, ultra-local limit along an arbitrary timelike geodesic in any spacetime is constructed, in close analogy with the well-known lightlike Penrose limit. This limit results in a Bianchi type I spacetime. The properties of these spacetimes are examined in the context of this limit, including the Einstein equations, stress-energy conservation and Raychaudhuri equation. Furthermore the conditions for the Bianchi type I spacetime to be diagonal are explicitly set forward, and the effect of the limit on the matter content of a spacetime are examined.</p>

COMPUTING THE HAUSDORFF DISTANCE BETWEEN CURVED OBJECTS

2008 ◽  
Vol 18 (04) ◽  
pp. 307-320 ◽  
Author(s):  
HELMUT ALT ◽  
LUDMILA SCHARF
Keyword(s):  
Voronoi Diagram ◽  
Computer Algebra ◽  
Hausdorff Distance ◽  
Computer Algebra System ◽  
Shape Recognition ◽  
The Other ◽  
Algebraic Equations ◽  
Algebraic Computing ◽  
Intersection Points ◽  
The One

The Hausdorff distance between two sets of curves is a measure for the similarity of these objects and therefore an interesting feature in shape recognition. If the curves are algebraic computing the Hausdorff distance involves computing the intersection points of the Voronoi edges of the one set with the curves in the other. Since computing the Voronoi diagram of curves is quite difficult we characterize those points algebraically and compute them using the computer algebra system SYNAPS. This paper describes in detail which points have to be considered, by what algebraic equations they are characterized, and how they actually are computed.

Exact Solutions of (2+1)-Dimensional Soliton Equations Applying SymbolicC++

1998 ◽  
Vol 09 (02) ◽  
pp. 265-269 ◽  
Author(s):  
W.-H. Steeb ◽  
Kiat Shi Tan ◽  
R. Stoop
Keyword(s):  
Exact Solutions ◽  
Computer Algebra ◽  
Computer Algebra System ◽  
Soliton Equations ◽  
Dimensional Soliton

We show how SymbolicC++, a symbolic and numeric computer algebra system written in C++, can be used to find exact solutions of soliton equations in (2+1)-dimensions.

Generation of Correlated Logistic-Normal Random Variates for Medical Decision Trees

1998 ◽  
Vol 37 (03) ◽  
pp. 235-238 ◽  
Author(s):  
M. El-Taha ◽  
D. E. Clark
Keyword(s):  
Monte Carlo ◽  
Decision Trees ◽  
Computer Algebra ◽  
Taylor Series ◽  
Computer Algebra System ◽  
Random Variable ◽  
Monte Carlo Analysis ◽  
Normal Random Variable ◽  
Medical Decision ◽  
Theoretical Results

AbstractA Logistic-Normal random variable (Y) is obtained from a Normal random variable (X) by the relation Y = (ex)/(1 + ex). In Monte-Carlo analysis of decision trees, Logistic-Normal random variates may be used to model the branching probabilities. In some cases, the probabilities to be modeled may not be independent, and a method for generating correlated Logistic-Normal random variates would be useful. A technique for generating correlated Normal random variates has been previously described. Using Taylor Series approximations and the algebraic definitions of variance and covariance, we describe methods for estimating the means, variances, and covariances of Normal random variates which, after translation using the above formula, will result in Logistic-Normal random variates having approximately the desired means, variances, and covariances. Multiple simulations of the method using the Mathematica computer algebra system show satisfactory agreement with the theoretical results.

Using algebraic programming to teach general relativity

2001 ◽  
Vol 3 (2) ◽  
pp. 65-70 ◽  
Author(s):  
F.A. Ghergu ◽  
D.N. Vulcanov
Keyword(s):  
General Relativity ◽  
Algebraic Programming

scholarly journals Breaking the cosmological background degeneracy by two-fluid perturbations in f(R) gravity

2015 ◽  
Vol 24 (07) ◽  
pp. 1550053 ◽  
Author(s):  
Amare Abebe
Keyword(s):  
General Relativity ◽  
Exact Solutions ◽  
Structure Formation ◽  
Background Level ◽  
Cosmological Perturbations ◽  
Gauge Invariant ◽  
First Order ◽  
Cosmological Background ◽  
Theories Of Gravity ◽  
Two Fluid

One of the exact solutions of f(R) theories of gravity in the presence of different forms of matter exactly mimics the ΛCDM solution of general relativity (GR) at the background level. In this work we study the evolution of scalar cosmological perturbations in the covariant and gauge-invariant formalism and show that although the background in such a model is indistinguishable from the standard ΛCDM cosmology, this degeneracy is broken at the level of first-order perturbations. This is done by predicting different rates of structure formation in ΛCDM and the f(R) model both in the complete and quasi-static regimes.

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