Search Results

Cosmology

Yvonne Choquet-Bruhat

in General Relativity and the Einstein Equations

This chapter begins with a discussion of the cosmological principle. It then covers isotropic and homogeneous Riemannian manifolds, Robertson–Walker spacetimes, Friedmann–Lemaître models, homogeneous ... More

This chapter begins with a discussion of the cosmological principle. It then covers isotropic and homogeneous Riemannian manifolds, Robertson–Walker spacetimes, Friedmann–Lemaître models, homogeneous non-isotropic cosmologies, Bianchi class I universes, Bianchi type IX, the Kantowski–Sachs models, Taub and Taub NUT spacetimes, locally homogeneous models, and recent observations and conjectures.Less

THE NEUMANN PROBLEM AND PROBLEMS ON MANIFOLDS

Juan Luis Vázquez

in The Porous Medium Equation: Mathematical Theory

This chapter completes the investigation of previous chapters on the Dirichlet and Cauchy problems by applying the techniques to other important problems. It selects two directions, the Neumann ... More

This chapter completes the investigation of previous chapters on the Dirichlet and Cauchy problems by applying the techniques to other important problems. It selects two directions, the Neumann boundary conditions and the problems posed on manifolds. Section 11.1 introduces the problem and concepts of weak solution, proves a uniqueness result, and presents examples. Section 11.2 reviews the theory for the existence and uniqueness of weak solutions and limit solutions. Section 11.3 provides proof of better estimates and boundedness of solutions in the case of the PME. Section 11.4 examines the mixed problems and problems posed in exterior space domains. The second main topic of this chapter is the theory of PME and GPME on Riemannian manifolds, which is in Section 11.5.Less

Classical And Quantum Gravity. Riemannian Manifolds And Tensors

JEAN ZINN-JUSTIN

in Quantum Field Theory and Critical Phenomena

This chapter has two purposes; to present the few elements of differential geometry which are required in different places in this volume and to provide, for completeness, a short introduction to the ... More

This chapter has two purposes; to present the few elements of differential geometry which are required in different places in this volume and to provide, for completeness, a short introduction to the problem of quantization of gravity. It first briefly recalls a few concepts related to reparametrization (more accurately diffeomorphism) of Riemannian manifolds. It introduces the notions of parallel transport, affine connection, and curvature, in analogy with gauge theories as discussed in Chapters 19-21. To define fermions on Riemannian manifolds additional mathematical objects are required — the vielbein and the spin connection. The chapter constructs Einstein's action for classical gravity (General Relativity) and derive the equation of motion. In the last section, it studies the formal aspects of the quantization of the theory of gravity, following the lines of the quantization of non-abelian gauge theories of Chapter 19.Less

Stochastic Differential Equations: Langevin, Fokker–Planck Equations

JEAN ZINN-JUSTIN

in Quantum Field Theory and Critical Phenomena

This chapter discusses Langevin equations, that is, stochastic differential equations related to diffusion processes, brownian motion, or random walk. From the Langevin equation, the Fokker–Planck ... More

This chapter discusses Langevin equations, that is, stochastic differential equations related to diffusion processes, brownian motion, or random walk. From the Langevin equation, the Fokker–Planck (FP) equation for the probability distribution of the stochastic variables is derived. The FP equation has a form analogous to the equation for the statistical operator in a magnetic field studied in Section 3.2. The chapter shows that averaged observables can also be calculated from path integrals, whose integrands define automatically positive measures. In some cases, like brownian motion on Riemannian manifolds, difficulties appear in the precise definition of stochastic equations, quite similar to the quantization problem encountered in quantum mechanics. Time discretization provides a solution to the problem. This chapter is also meant to serve as an introduction to Chapters 17 and 36 in which stochastic quantization and critical dynamics are discussed.Less

Riemannian manifolds

Nathalie Deruelle and Jean-Philippe Uzan

in Relativity in Modern Physics

This chapter is about Riemannian manifolds. It first discusses the metric manifold and the Levi-Civita connection, determining if the metric is Riemannian or Lorentzian. Next, the chapter turns to ... More

This chapter is about Riemannian manifolds. It first discusses the metric manifold and the Levi-Civita connection, determining if the metric is Riemannian or Lorentzian. Next, the chapter turns to the properties of the curvature tensor. It states without proof the intrinsic versions of the properties of the Riemann–Christoffel tensor of a covariant derivative already given in Chapter 2. This chapter then performs the same derivation as in Chapter 4 by obtaining the Einstein equations of general relativity by varying the Hilbert action. However, this will be done in the intrinsic manner, using the tools developed in the present and the preceding chapters.Less

Modifying Distances

Orlitsky Alon

in Semi-Supervised Learning

This chapter discusses density-based metrics induced by Riemannian manifold structures. It presents asymptotically consistent methods to estimate and compute these metrics and present upper and lower ... More

This chapter discusses density-based metrics induced by Riemannian manifold structures. It presents asymptotically consistent methods to estimate and compute these metrics and present upper and lower bounds on their estimation and computation errors. Finally, it is discussed how these metrics can be used for semi-supervised learning and present experimental results. Learning algorithms use a notion of similarity between data points to make inferences. Semi-supervised algorithms assume that two points are similar to each other if they are connected by a high-density region of the unlabeled data. Apart from semi-supervised learning, such density-based distance metrics also have applications in clustering and nonlinear interpolation.Less

Dehn Functions

Timothy Riley

in Office Hours with a Geometric Group Theorist

This chapter is concerned with Dehn functions. It begins by presenting jigsaw puzzles that are somewhat different from the conventional kind and explains how to solve them. It then considers a ... More

This chapter is concerned with Dehn functions. It begins by presenting jigsaw puzzles that are somewhat different from the conventional kind and explains how to solve them. It then considers a complexity measure for the word problem and shows that, for a word w, the problem of finding a sequence of free reductions, free expansions, and applications of defining relators that carries it to the empty word is equivalent to solving the puzzle where, starting from some vertex υ‎, one reads w around the initial circle of rods. The chapter also explains how the Dehn function corresponds to an isoperimetric problem in a combinatorial space, the Cayley 2-complex, and describes a continuous version of this, via group actions, along with the isoperimetry in Riemannian manifolds. Finally, it defines the Dehn function as a quasi-isometry invariant. The discussion includes exercises and research projects.Less

On Eigenfunction Restriction Estimates and L4-Bounds for Compact Surfaces with Nonpositive Curvature

Christopher D. Sogge and Steve Zelditch

Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, and Stephen Wainger (eds)

in Advances in Analysis: The Legacy of Elias M. Stein

This chapter discusses a “restriction theorem,” which is related to certain Littlewood–Paley estimates for eigenfunctions. The main step in proving this theorem is to see that an estimate involving a ... More

This chapter discusses a “restriction theorem,” which is related to certain Littlewood–Paley estimates for eigenfunctions. The main step in proving this theorem is to see that an estimate involving a wave equation associated with an assigned Laplace–Beltrami operator and a bit of microlocal (wavefront) analysis remains valid as well if a certain variable is part of a periodic orbit under a set of curvature assumptions. This can be done by lifting the wave equation for a compact two-dimensional Riemannian manifold without boundary up to the corresponding one for its universal cover. By identifying solutions of wave equations for this Riemannian manifold with “periodic” ones, this chapter is able to obtain the necessary bounds using a bit of wavefront analysis and the Hadamard parametrix for the universal cover.Less

Riemannian manifolds

Nathalie Deruelle and Jean-Philippe Uzan

in Relativity in Modern Physics

This chapter introduces the Riemann tensor characterizing curved spacetimes, and then the metric tensor, which allows lengths and durations to be defined. As shown in the preceding chapter, ... More

This chapter introduces the Riemann tensor characterizing curved spacetimes, and then the metric tensor, which allows lengths and durations to be defined. As shown in the preceding chapter, ‘absolute, true, and mathematical’ spacetimes representing ‘relative, apparent, and common’ space and time in Einstein’s theory are Riemannian manifolds supplied with a metric and its associated Levi-Civita connection. Moreover, this metric simultaneously describes the coordinate system chosen to reference the events. The chapter begins with a study of connections, parallel transport, and curvature; the commutation of derivatives, torsion, and curvature; geodesic deviation and curvature; the metric tensor and the Levi-Civita connection; and locally inertial frames. Finally, it discusses Riemannian manifolds.Less

The Cartan structure equations

Nathalie Deruelle and Jean-Philippe Uzan

in Relativity in Modern Physics

This chapter focuses on Cartan structure equations. It first introduces a 1-form and its exterior derivative, before turning to a study of the connection and torsion forms, thereby expressing the ... More

This chapter focuses on Cartan structure equations. It first introduces a 1-form and its exterior derivative, before turning to a study of the connection and torsion forms, thereby expressing the torsion as a function of the connection forms and establishing the torsion differential 2-forms. It then turns to the curvature forms drawn from Chapter 23 and Cartan’s second structure equation, along with the curvature 2-forms. It also studies the Levi-Civita connection. The components of the Riemann tensor are then studied, with a Riemannian manifold, or a metric manifold with a torsion-less connection. The Riemann tensor of the Schwarzschild metric are finally discussed.Less

Quantum mechanics (QM): Path integrals in phase space

Jean Zinn-Justin

in Quantum Field Theory and Critical Phenomena: Fifth Edition

In Chapter 2, a path integral representation of the quantum operator e-β H in the case of Hamiltonians H of the separable form p2/2m + V(q) has been constructed. Here, the construction is extended to ... More

In Chapter 2, a path integral representation of the quantum operator e-β H in the case of Hamiltonians H of the separable form p2/2m + V(q) has been constructed. Here, the construction is extended to Hamiltonians that are more general functions of phase space variables. This results in integrals over paths in phase space involving the action expressed in terms of the classical Hamiltonian H(p,q). However, it is shown that, in the general case, the path integral is not completely defined, and this reflects the problem that the classical Hamiltonian does not specify completely the quantum Hamiltonian, due to the problem of ordering quantum operators in products. When the Hamiltonian is a quadratic function of the momentum variables, the integral over momenta is Gaussian and can be performed. In the separable example, the path integral of Chapter 2 is recovered. In the case of the charged particle in a magnetic field a more general form is found, which is ambiguous, since a problem of operator ordering arises, and the ambiguity must be fixed. Hamiltonians that are general quadratic functions provide other important examples, which are analysed thoroughly. Such Hamiltonians appear in the quantization of the motion on Riemannian manifolds. There, the problem of ambiguities is even more severe. The problem is illustrated by the analysis of the quantization of the free motion on the sphere SN−1.Less

A review: The Laplacian and the d’Alembertian

Christopher D. Sogge

in Hangzhou Lectures on Eigenfunctions of the Laplacian (AM-188)

This chapter reviews the Laplacian and the d'Alembertian. It begins with a brief discussion on the solution of wave equation both in Euclidean space and on manifolds and how this knowledge can be ... More

This chapter reviews the Laplacian and the d'Alembertian. It begins with a brief discussion on the solution of wave equation both in Euclidean space and on manifolds and how this knowledge can be used to derive properties of eigenfunctions on Riemannian manifolds. A key step in understanding properties of solutions of wave equations on manifolds is to compute the types of distributions that include the fundamental solution of the wave operator in Minkowski space (d'Alembertian), with a specific function for the Euclidean Laplacian on Rn. The chapter also reviews another equation involving the Laplacian, before discussing the fundamental solutions of the d'Alembertian in R1+n.Less