Regge calculus was introduced in 1961 as a coordinate free and discrete analogue of einsteins theory of gravitation. If, in n dimensions, the n2simplexes called the hinges, the places where. Regge calculus, on the other hand, describes gravity through the curvature of a piecewise. Ari stern associate professor of mathematics and statistics washington university in st. Please report if there are links that are not working. A central problem is to relate coordinate systems across the boundaries between neighboring nsimplexes.
Ponzano developed a quantum version of regge calculus in three spacetime dimensions now known as the ponzano regge model. Is the regge calculus a consistent approximation to general. Pdf the quantization of regge calculus researchgate. Dec 19, 2009 the list below are free calculus and analysis ebooks for undegraduate and graduate students. Fast algorithms for computing defects and their derivatives in the regge calculus.
In 1961, tullio regge published a work on which he uses the old idea of triangulation of. Regge calculus institute for mathematics and its applications. We apply the consistent discretization technique to the regge action for euclidean and lorentzian general relativity in arbitrary number of dimensions. Pdf regge calculus from a new angle semantic scholar. Use regge calculus to construct boundary theory discretization of gravity based on a triangulation variables are the lengths of edges. Yet, in spite of its beautiful geometric features, the bulk of numerical computations in general relativity is, as of today, carried out by other methods.
Leo brewin school of mathematical sciences monash university, 3800 australia 09nov2010 abstract any practical attempt to solve the regge equations, these being a large system of nonlinear algebraic equations, will almost certainly employ a newtonraphson like. I convergent eigenvalue problemfor curl tcurl in rc. We also discuss reparametrizations and conformal transformations. Fast algorithms for computing defects and their derivatives. We show how the continuum theory emerges in the weak field long wavelength limit. Particular emphasis is placed on problems of current computational interest, and the strengths and weaknesses of the lattice approach are highlighted. A longstanding problem is the construction of a consistent discretized canonical model for gravity and a representation of diffeomorphism in such a model, see for instance, 14. Applying the wick rotation, we arrive at the regge calculus analog of the friedmann equations, describing the evolution of the polyhedral universe. Each compact, tightlywritten lesson can be read in 15 minutes. Tensorial calculus and differential geometry are the mathematical resources necessary to study the spacetime manifold in the context of einsteins theory.
I linearized gr is a wave equation with curl tcurl in space. In the lorentzian case, the framework appears to be naturally free of the spikes that plague traditional formulations. In a ddimensional regge calculus spacetime, the usual form of the action is 2. A typical class plods along, saving concepts like integrals until week 8. This relation is expected from the continuum theory, where the bianchi identity implies conservation of the stressenergy tensor. Schaums outline of tensor calculus schaums outlines. Regge calculus as a numerical approach to general relativity. In regge calculus, spacetime is usually approximated by a tri angulation with flat simplices. Expanding polyhedral universe in regge calculus progress of.
Regge observed that the lengths of the shared edges of the simplexes determine the shape of each simplex and thus the metric properties of the entire manifold. As a first step toward understanding quantum regge calculus we consider the simplest possible problem. Broken gauge symmetries and constraints in regge calculus. Regge calculus was used in classical numerical applications and in constructing quantum models 4, 5. Most classes begin with the theory of limits, a technical concept discovered 150 years after calculus was. A fully dimensional regge calculus model of the kasner cosmology. The list below are free calculus and analysis ebooks for undegraduate and graduate students. As we will show, such a formulation allows us to replace the length variables by three or fourdimensional dihedral angles as basic variables. Oct 24, 2014 regge calculus was introduced in 1961 as a coordinate free and discrete analogue of einsteins theory of gravitation. Use real data in editable spreadsheet format to solve problems. Regge private communication, recently generalized by piran and williams t.
Regge calculus was proposed to formulate einsteins general relativity on piecewise linear manifolds 1,2. The geometry of classical regge calculus j w barrett institute of theoretical physics, university of zurich, schoenberggasse 9, 8001 zurich, switzerland received 19 november 1986, in final form 11 may 1987 abstract. The quantization of regge calculus article pdf available in zeitschrift fur physik c 214. It also provides a welldefined recipe for determining. Myers florida international university, miami florida state university, tallahassee new college of florida, sarasota university of central florida, orlando. More than 40 million students have trusted schaums outlines for their expert knowledge and helpful solved problems. The application of regge calculus to quantum gravity and quantum field theory in a curved background is discussed.
The fundamental platform for regge calculus is a lattice spacetime, wherein each lattice cell is a simplex endowed with a. Area regge calculus within a single 4simplex, standard regge calculus and area regge calculus agree. In general relativity, regge calculus is a formalism for producing simplicial approximations of spacetimes that are solutions to the einstein field equation. We present a formulation using simplices with constant sectional curvature adjusted to the presence of a cosmological constant. Written by renowned experts in their respective fields, schaums outlines cover everything from math to science, nursing to language. We also present the secondorder convergence properties of the solution to the continuum. It should be noted that both kand iare dummy subscripts and can be replaced by other letters. Regge calculus with cosmological constant in section 6 and toy models in section 8. An approach to discretized or, better, piecewise flat geometry, used in numerical relativity and quantum gravity.
A natural interpretation of the momentum constraint. We discuss the quantization of regges discrete description of einsteins theory of gravitation. Derivation of the regge equations is given in sect. Threedimensional einstein gravity and regge calculus. Ive tried to make these notes as self contained as possible and so all the information needed to read through them is either from an algebra or trig class or contained in other sections of the notes. Regge calculus was the first discrete gauge theory suitable for numerical simulation, and an early relative of lattice gauge theory. A fully dimensional regge calculus model of the kasner. Pdf the quantization of regge calculus martin rocek. We develop an initialvalue prescription, and implement a 4stage parallel evolution algorithm. Williams, application of the regge calculus to the axially symmetric initial value problem in general relativity, phys. It also provides a welldefined recipe for determining the integration measure for quantum regge calculus.
The application of regge calculus to quantum gravity and. Abstract in linearised regge calculus in a topologically trivial region, the space of linearised deviations of the edge lengths from a flat configuration, divided by the subspace of deformations due to translations of the vertices, is equivalent to the space of the linearised curvatures which satisfy the bianchi identities. The calculus was introduced by the italian theoretician tullio regge in 1961. We study the quantum regge calculus of einsteincartan theory to describe quantum dynamics of euclidean spacetime discretized as a 4simplices complex. The book is in use at whitman college and is occasionally updated to correct errors and add new material. The fundamental theorem of linearised regge calculus. Threedimensional einstein gravity and regge calculus 703 we now show how the regge calculus bianchi identities are equivalent to conserva tion of momentum. Quantum regge calculus of einsteincartan theory inspirehep. With the theory of general relativity, einstein abolished the interpretation of gravitation as a force and associated it to the curvature of spacetime.
In 1961, tullio regge published a work on which he uses. A discrete form of exterior differential calculus is developed, and this is used t. Regge calculus as an independent tool in general relativity in this paper we describe the. The requirement that this reduction would respect natural physical properties positivity, welldefined continuum limit, absence of lattice artefacts put rather severe restrictions and allows to define. Ari stern, wustl department of mathematics and statistics.
In quantum gravity, there are applications related to regularization in approaches with. It provides a coordinatefree lattice formulation of gravitation and has been used in investigations of classical as well as quantum gravity. In the area calculus, two neighboring 4simplices may disagree on the geometry of their shared tetrahedron. It was submitted to the free digital textbook initiative in california and will remain unchanged for at least two years. A brief history of regge calculus after his 1961 paper tullio regge did not work anymore on regge calculus, with two exceptions o giorgio ponzano e t regge, semiclassical limit of racah coefficients racah memorial volume, 1968 see below o t regge e ruth williams discrete structures in gravity, arxiv. In the first section of this paper, standard notions of riemannian geometry are applied to the case of piecewiseflat manifolds. Since the idea had been put forward that regge calculus should be formulated in terms of the areas of the triangles rather than the edge lengths 1 the socalled area regge calculus model was of a certain interest 2,3. Yet little is known about how the regge calculus performs for generic spacetimes and it is this class of spacetimes for which the regge calculus is most suited. Boffi, d finite element approximation of eigenvalue.
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