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Integral principles

JESPER LÜTZEN

in Mechanistic Images in Geometric Form: Heinrich Hertz's 'Principles of Mechanics'

In his book Principles of Mechanics, Heinrich Hertz treats integral principles in three parts. First, he discusses the properties of geodesics and their relation to straightest paths in a purely ... More

In his book Principles of Mechanics, Heinrich Hertz treats integral principles in three parts. First, he discusses the properties of geodesics and their relation to straightest paths in a purely geometric way. These investigations then form the basis for a discussion of integral principles applied to the motion of free systems, and finally to the motion of conservative systems. Hertz's approach differed from the usual approach in three respects. First, it was based on his geometry of systems of points. Second, it dealt with forces as an indirect result of a coupling of the visible system with a hidden system. Third, it limited the applicability of even the most general of these principles (Hamilton's principle) to special mechanical systems declaring that it was invalid in general. Hertz pointed out that Hamilton's principle and the principle of least action did not apply to non-holonomic systems.Less

The View from General Relativity

Harvey R. Brown

in Physical Relativity: Space-time structure from a dynamical perspective

In his 1923 book The Mathematical Theory of Relativity, Arthur Eddington distinguished between two chains of reasoning in general relativity. The first familiar one starts with the existence of the ... More

In his 1923 book The Mathematical Theory of Relativity, Arthur Eddington distinguished between two chains of reasoning in general relativity. The first familiar one starts with the existence of the four-dimensional space-time interval ds, whose meaning is the usual one associated with the readings of physical rods and clocks and possibly light rays. The other less familiar chain of reasoning ‘binds the physical manifestations of the energy tensor and the interval; it passes from matter as now defined by the energy-tensor to the interval regarded as the result of measurements made with this matter.’ This chapter takes up the challenge of outlining this second chain of reasoning. In developing this reasoning, it argues that the dynamical underpinning of relativistic kinematics that has been defended in this book is consistent with the structure and logic of GR.Less

RADON TRANSFORM ON GROUPS

Leon Ehrenpreis

in The Universality of the Radon Transform

This chapter applies the book's methods to various Lie groups. In particular, to the horocyclic and geodesic transforms on G/K.

Lorentz Geometry

Yvonne Choquet-Bruhat

in General Relativity and the Einstein Equations

This chapter presents a survey of the basic definitions of Riemannian and Lorentzian differential geometry used in this book. The first nine sections use the simplest formulations, in local ... More

This chapter presents a survey of the basic definitions of Riemannian and Lorentzian differential geometry used in this book. The first nine sections use the simplest formulations, in local coordinates, as they are needed for the first five chapters and physical applications. The later sections contain material used in the following, more advanced, chapters. Topics covered include manifolds, differential mappings, vectors and tensors, pseudo-Riemannian metrics, Riemannian connection, geodesics, curvature, geodesic deviation, maximum length and conjugate points, linearized Ricci and Einstein tensors, and second derivative of the Ricci tensor.Less

The Built Environment: Buckminster Fuller's Spaceship Earth

Linda Sargent Wood

in A More Perfect Union: Holistic Worldviews and the Transformation of American Culture after World War II

Structural engineer Buckminster Fuller's geodesic dome was a visible manifestation of holism. Held together in what he described as “synergetic” wholeness, each part relying on neighboring parts for ... More

Structural engineer Buckminster Fuller's geodesic dome was a visible manifestation of holism. Held together in what he described as “synergetic” wholeness, each part relying on neighboring parts for stability, his technological design was meant to provide new housing on this planet he dubbed “spaceship earth.” Geodesics served as colorful domiciles for the counterculture, as well as spaces for fairs, military operations, scientific experiments, and sporting events. Fuller's ideas contributed to ongoing discussions in America about the place of the machine in society and nature and about the relationship between technology and nature. For Fuller, who tried to emulate nature through his technological inventions, both nature and technology had a place in the modern landscape. Indeed, to account for his faith in technology and his love for nature, the technological wizard saw nature and technology as complementary parts of one whole. He formed his whole by naturalizing technology and technologizing nature. At the same time, Fuller's cartographic images of the world, his charts of natural resources, his World Game, and his international relationships contributed to cognitive maps that stressed interdependencies and interconnections between countries and peoples. In all, Fuller argued for the equitable distribution of resources and a more just, sustainable world.Less

1 The geometric approach to GWs

Michele Maggiore

in Gravitational Waves: Volume 1: Theory and Experiments

This chapter discusses how gravitational waves emerge from general relativity, and what their properties are. The most straightforward approach is ‘linearized theory’, where the Einstein equations ... More

This chapter discusses how gravitational waves emerge from general relativity, and what their properties are. The most straightforward approach is ‘linearized theory’, where the Einstein equations are expanded around the flat Minkowski metric. It is shown how a wave equation emerges and how the solutions can be put in an especially simple form by an appropriate gauge choice. Using standard tools of general relativity such as the geodesic equation and the equation of the geodesic deviation, how these waves interact with a set of test masses is detailed. The energy and momentum carried by GWs are then computed and discussed. This chapter approaches the problem from a geometric point of view, identifying the energy-momentum tensor of GWs from their effect on the background curvature. Finally, GW propagation in curved space is discussed.Less

On Relativity, Time Reckoning, and the Topology of Time Series

Roberto Torretti

in The Arguments of Time

This chapter devotes equal attention to special relativity and general relativity. It first describes the history of the analysis of distant simultaneity, up to and including Einstein's procedure in ... More

This chapter devotes equal attention to special relativity and general relativity. It first describes the history of the analysis of distant simultaneity, up to and including Einstein's procedure in his revolutionary 1905 paper which introduced special relativity. In particular, the discussion relates Einstein's procedure to the ensuing philosophical debate about whether distant simultaneity is a matter of convention. As to general relativity, the discussion gives a brief sketch of Einstein's path towards his discovery of general relativity. Thereafter, it focuses on the topological structure of time or, more precisely, of timelike lines (worldlines) in spacetime. It discusses the closed timelike lines first found in an exact solution of general relativity by Godel; and the open timelike geodesics that get arbitrarily close to the initial singularity (Big Bang) in a Friedmann solution.Less

Metric description of a curved space

Ta-Pei Cheng

in Relativity, Gravitation and Cosmology: A Basic Introduction

Einstein's new theory of gravitation is formulated in a geometric framework of curved spacetime. Here the subject of non-Euclidean geometry is introduced by way of Gauss's theory of curved surfaces. ... More

Einstein's new theory of gravitation is formulated in a geometric framework of curved spacetime. Here the subject of non-Euclidean geometry is introduced by way of Gauss's theory of curved surfaces. Generalized (Gaussian) coordinates: A systematic way to label points in space without reference to any objects outside this space. Metric function: For a given coordinate choice, the metric determines the intrinsic geometric properties of a curved space. Geodesic equation: It describes the shortest and the straightest possible curve in a warped space and is expressed in terms of the metric function. Curvature: It is a nonlinear second derivative of the metric. As the deviation from Euclidean relations is proportional to the curvature, it measures how much the space is warped.Less

GR as a geometric theory of gravity – I

Ta-Pei Cheng

in Relativity, Gravitation and Cosmology: A Basic Introduction

A geometric description of equivalence principle physics of gravitational time dilation is presented. In this geometric theory, the metric plays the role of relativistic gravitational potential. ... More

A geometric description of equivalence principle physics of gravitational time dilation is presented. In this geometric theory, the metric plays the role of relativistic gravitational potential. Einstein proposed curved spacetime as the gravitational field. The geodesic equation in spacetime is the GR equation of motion, which is checked to have the correct Newtonian limit. At every spacetime point, one can construct a free-fall frame in which gravity is transformed away. However, in a finite-sized region, one can detect the residual tidal force which is second derivative of gravitational potential. It is the curvature of spacetime. The GR field equation directly relates the mass/energy distribution to spacetime's curvature. Its solution is the metric function, determining the geometry of spacetime.Less

The three body problem

S. G. Rajeev

in Advanced Mechanics: From Euler's Determinism to Arnold's Chaos

This chapter derives the consequences of scale invariance in the three body problem. Jacobi co-ordinates are introduced. The orbits are geodesics in a certain metric. The three body problem with ... More

This chapter derives the consequences of scale invariance in the three body problem. Jacobi co-ordinates are introduced. The orbits are geodesics in a certain metric. The three body problem with inverse cube force is shown to be more symmetric. Montgomery's ‘pair of pants’ metric is derived. The orbits are geodesics in a metric of negative curvature on the plane with three points removed. The Virial Theorem is suggested as an exercise.Less

GR as a geometric theory of gravity — II

Ta-Pei Cheng

in Relativity, Gravitation and Cosmology: A Basic Introduction

The mathematical realization of equivalence principle (EP) is the principle of general covariance. General relativity (GR) equations must be covariant with respect to general coordinate ... More

The mathematical realization of equivalence principle (EP) is the principle of general covariance. General relativity (GR) equations must be covariant with respect to general coordinate transformations. To go from special relativity (SR) to GR equations, one replaces ordinary by covariant derivatives. The SR equation of motion turns into the geodesic equation. The Einstein equation, as the relativistic gravitation field equation, relates the energy momentum tensor to the Einstein curvature tensor. The Einstein equation in the space exterior to a spherical source is solved to obtain the Schwarzschild solution. The solutions of Einstein equation that satisfy the cosmological principle is the Robertson-Walker spacetime. The relation of the cosmological Friedmann equations to the Einstein field equation is explicated. The compatibility of the cosmological-constant term with the mathematical structure of Einstein equation and the interpretation of this term as the vacuum energy tensor are discussed.Less

Mathematicians on the geometrization of the Hamilton–Jacobi formalism

JESPER LÜTZEN

in Mechanistic Images in Geometric Form: Heinrich Hertz's 'Principles of Mechanics'

It has long since been remarked by mathematicians that William Rowan Hamilton's method contains purely geometrical truths, and that a peculiar mode of expression, suitable to it, is required in order ... More

It has long since been remarked by mathematicians that William Rowan Hamilton's method contains purely geometrical truths, and that a peculiar mode of expression, suitable to it, is required in order to express these clearly. But this fact has only come to light in a somewhat perplexing form, namely, in the analogies between ordinary mechanics and the geometry of space of many dimensions. Together with an explicit reference to the work of Eugenio Beltrami, Rudolf Lipschitz, and Jean-Gaston Darboux, Heinrich Hertz made only one other reference to the work on mechanics done by contemporary mathematicians. Hertz correctly connected their work with his own treatment of the Hamiltonian formalism. This chapter gives a short summary of the mathematical developments in Hamilton-Jacobi formalism during the period 1828-1888 and compares them with those of Hertz. The views of Johann Carl Friedrich Gauss and Hamilton on geodesics, optics, and dynamics are discussed, along with those of Joseph Liouville and Lipschitz on the principle of least action, and trajectories as geodesics.Less

Cosmology

Carlo Giunti and Chung W. Kim

in Fundamentals of Neutrino Physics and Astrophysics

This chapter introduces the standard cosmological model and explains basic general relativity, Robertson–Walker metric (geodesic motion, redshift, Hubble's law, angular diameter-redshift relation, ... More

This chapter introduces the standard cosmological model and explains basic general relativity, Robertson–Walker metric (geodesic motion, redshift, Hubble's law, angular diameter-redshift relation, and particle horizon), dynamics of expansion, matter-dominated Universe, radiation-dominated Universe, curvature-dominated Universe, vacuum-dominated Universe, thermodynamics of the early Universe, entropy, decoupling, and cosmic microwave background radiation.Less

Particle Motion in Curved Spacetime

Valeri P. Frolov and Andrei Zelnikov

in Introduction to Black Hole Physics

Particle motion in a curved spacetime is considered. The Lagrangian and Hamiltonian form of the equation of motion of a relativistic particle are described. We discuss here also the Hamilton‐Jacobi ... More

Particle motion in a curved spacetime is considered. The Lagrangian and Hamiltonian form of the equation of motion of a relativistic particle are described. We discuss here also the Hamilton‐Jacobi equation, kinetic theory in a curved spacetime, and the Liouville theory of complete integrability.Less