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Introduction to the mathematics of general relativity
About: Introduction to the mathematics of general relativity is a research topic. Over the lifetime, 2583 publications have been published within this topic receiving 73295 citations.
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TL;DR: In this paper, the cosmological constant λ appears as an unavoidable Lagrange multiplier, belonging to a constraint expressing the existence of a fundamental element of space-time hypervolume at every point.
Abstract: In usual formulations of general relativity, the cosmological constant λ appears as an inelegant ambiguity in the fundamental action principle. With a slight reformulation, λ appears as an unavoidable Lagrange multiplier, belonging to a constraint. The constraint expresses the existence of a fundamental element of space-time hypervolume at every point. The fundamental scale of length in atomic physics provides such a hypervolume element. In this sense, the presence in relativity of an undetermined cosmological length is a direct consequence of the existence of a fundamental atomic length.
196 citations
TL;DR: In this article, the Newtonian gravitation theory is generalized to an inhomogeneous wave equation for a tensor gravitational potential in Euclidean space-time by invoking the special relativity postulates of Lorentz invariance and equivalence of mass and energy.
Abstract: The Newtonian gravitation theory is generalized to an inhomogeneous wave equation for a tensor gravitational potential in Euclidean space-time by invoking the special relativity postulates of Lorentz invariance and equivalence of mass and energy. Under the assumption of Lagrangian derivability, this is found to lead uniquely to the generally covariant field theories (including the general relativity theory) augmented by four auxiliary conditions. Appendices treat the general definition of the energy tensor, and an empirically disqualified special relativistic scalar generalization of the Newtonian theory.
193 citations
Book•
01 Jan 1969
TL;DR: In this paper, the authors present a model of the Lorentz transformation and its relation to general relativistic physics, including the relation between acceleration and gravity, as well as its relation with the equivalence principle of the Mach's Principle.
Abstract: 1 The Rise and Fall of Absolute Space.- 1.1 Definition of Relativity.- 1.2 Newton's Laws.- 1.3 The Galilean Transformation.- 1.4 The Set of All Inertial Frames.- 1.5 Newtonian Relativity.- 1.6 Newton's Absolute Space.- 1.7 Objections to Newton's Absolute Space.- 1.8 Maxwell's Ether.- 1.9 Where is Maxwell's Ether?.- 1.10 Lorentz's Ether Theory.- 1.11 The Relativity Principle.- 1.12 Arguments for the Relativity Principle.- 1.13 Maxwellian Relativity.- 1.14 Origins of General Relativity.- 1.15 Mach's Principle.- 1.16 Consequences of Mach's Principle.- 1.17 Cosmology.- 1.18 Inertial and Gravitational Mass.- 1.19 The Equivalence Principle.- 1.20 The Semistrong Equivalence Principle.- 1.21 Consequences of the Equivalence Principle.- 2 Einsteinian Kinematics.- 2.1 Basic Features of Special Relativity.- 2.2 On the Nature of Physical Laws.- 2.3 An Archetypal Relativistic Argument.- 2.4 The Relativity of Simultaneity.- 2.5 The Coordinate Lattice.- 2.6 The Lorentz Transformation.- 2.7 Properties of the Lorentz Transformation.- 2.8 Hyperbolic Forms of the Lorentz Transformation.- 2.9 Graphical Representation of the Lorentz Transformation.- 2.10 World-picture and World-map.- 2.11 Length Contraction.- 2.12 Length Contraction Paradoxes.- 2.13 Time Dilation.- 2.14 The Twin Paradox.- 2.15 Velocity Transformation.- 2.16 Proper Acceleration.- 2.17 Special Relativity without the Second Postulate.- 3 Einsteinian Optics.- 3.1 The Drag Effect.- 3.2 The Doppler Effect.- 3.3 Aberration and the Visual Appearance of Moving Objects.- 4 Spacetime and Four-Vectors.- 4.1 Spacetime.- 4.2 Three-Vectors.- 4.3 Four-Vectors.- 4.4 Four-Tensors.- 4.5 The Three-Dimensional Minkowski Diagram.- 4.6 Wave Motion.- 5 Relativistic Particle Mechanics.- 5.1 Domain of Sufficient Validity of Newton's Laws.- 5.2 Why Gravity Does not Fit Naturally into Special Relativity.- 5.3 Relativistic Inertial Mass.- 5.4 Four-Vector Formulation of Relativistic Mechanics.- 5.5 A Note on Galilean Four-Vectors.- 5.6 Equivalence of Mass and Energy.- 5.7 The Center of Momentum Frame.- 5.8 Relativistic Billiards.- 5.9 Threshold Energies.- 5.10 Three-Force and Four-Force.- 5.11 De Broglie Waves.- 5.12 Photons. The Compton Effect.- 5.13 The Energy Tensor of Dust.- 6 Relativity and Electrodynamics.- 6.1 Transformation of the Field Vectors.- 6.2 Magnetic Deflection of Charged Particles.- 6.3 The Field of a Uniformly Moving Charge.- 6.4 The Field of an Infinite Straight Current.- 7 Basic Ideas of General Relativity.- 7.1 Curved Surfaces.- 7.2 Curved Spaces of Higher Dimensions.- 7.3 Riemannian Spaces.- 7.4 A Plan for General Relativity.- 7.5 The Gravitational Doppler Effect.- 7.6 Metric of Static Fields.- 7.7 Geodesics in Static Fields.- 8 Formal Development of General Relativity.- 8.1 Tensors in General Relativity.- 8.2 The Vacuum Field Equations of General Relativity.- 8.3 The Schwarzschild Solution.- 8.4 Rays and Orbits in Schwarzschild Space.- 8.5 The Schwarzschild Horizon, Gravitational Collapse, and Black Holes.- 8.6 Kruskal Space and the Uniform Acceleration Field.- 8.7 A General-Relativistic "Proof" of E = mc2.- 8.8 A Plane-Fronted Gravity Wave.- 8.9 The Laws of Physics in Curved Spacetime.- 8.10 The Field Equations in the Presence of Matter.- 8.11 From Modified Schwarzschild to de Sitter Space.- 8.12 The Linear Approximation to GR.- 9 Cosmology.- 9.1 The Basic Facts.- 9.2 Apparent Difficulties of Prerelativistic Cosmology.- 9.3 Cosmological Relativity: The Cosmological Principle.- 9.4 Milne's Model.- 9.5 The Robertson-Walker Metric.- 9.6 Rubber Models, Red Shifts, and Horizons.- 9.7 Comparison with Observation.- 9.8 Cosmic Dynamics According to Pseudo-Newtonian Theory.- 9.9 Cosmic Dynamics According to General Relativity.- 9.10 The Friedmann Models.- 9.11 Once Again, Comparison with Observation.- 9.12 Mach's Principle Reexamined.- Appendices.- Appendix I: Curvature Tensor Components for the Diagonal Metric.- Appendix II: How to "Invent" Maxwell's Theory.- Exercises.
188 citations
TL;DR: The Hamiltonian for general relativity obtained in a previous paper furnishes a definition of energy whose physical interpretation is direct, and which fulfills the conditions required of the energy in other physical systems as mentioned in this paper.
Abstract: The Hamiltonian for general relativity obtained in a previous paper furnishes a definition of energy whose physical interpretation is direct, and which fulfills the conditions required of the energy in other physical systems. The energy can be expressed as a surface integral at spacial infinity in terms of the spacial components of the covariant metric tensor at any given time. Thus, the energy depends only on the minimal initial Cauchy data and may be evaluated in any coordinate system, provided this system can be made asymptotically rectangular. These statements remain valid when particles are coupled to the gravitational field. The criteria for existence of gravitational radiation are formulated in terms of the canonical variables and the stress-tensor. These criteria are identical to those used in electromagnetic theory. Some applications are discussed.
186 citations
184 citations