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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 relation between the general theory of relativity and the Einstein-Cartan theory in the case that matter is described by a Dirac field was considered and the condition that an (arbitrary) solution of general relativity with Dirac fields is also a solution of the ECCD theory and vice versa.
Abstract: Considers the relation between the general theory of relativity and the Einstein-Cartan theory in the case that matter is described by a Dirac field. Thereby the author finds the condition that an (arbitrary) solution of general relativity with a Dirac field is also a solution of the Einstein-Cartan-Dirac theory and vice versa. Exploiting this result the author generates new non-ghost solutions of the Einstein-Cartan-Dirac theory from ghost solutions of general relativity.
20 citations
TL;DR: In this paper, an exact static solution of Einstein's field equations of general relativity in the presence of zero-rest-mass scalar fields has been obtained when both the metric tensor gij and the zero-position scalar field φ exhibit plane symmetry in the sense of Taub [9].
Abstract: An exact static solution of Einstein's field equations of general relativity in the presence of zero-rest-mass scalar fields has been obtained when both the metric tensor gijand the zero-rest-mass scalar field φexhibit plane symmetry in the sense of Taub [9]. Our solution generalizes the empty space-time solution with plane symmetry previously obtained by Taub to the situation when static zero-rest-mass scalar fields are present. The static plane symmetric solutoins of Einstein's field equations in the presence of massive scalar fields, and the difference between the massless and non-massless scalar fields are being investigated, and will be published separately later on. We also hope to discuss non-static plane symmetric solutions of Einstein's field equations in the presence of scalar fields in future.
20 citations
TL;DR: In this article, a special relativistic energy-momentum tensor was derived from the nonrelativistic Lagrangian for a continuous, elastic medium with finite deformations, and the elastic stress tensor is proportional to the strain gradient of the elastic energy in appropriate coordinates.
20 citations
TL;DR: In this article, it was shown that the geodesic principle has the status of a theorem in General Relativity (GR) and that inertial motion can be derived from other central principles of the theory.
Abstract: A theorem due to Bob Geroch and Pong Soo Jang ["Motion of a Body in General Relativity." Journal of Mathematical Physics 16(1), (1975)] provides the sense in which the geodesic principle has the status of a theorem in General Relativity (GR). Here we show that a similar theorem holds in the context of geometrized Newtonian gravitation (often called Newton-Cartan theory). It follows that in Newtonian gravitation, as in GR, inertial motion can be derived from other central principles of the theory.
20 citations
Book•
01 Jan 2009
TL;DR: In this article, the authors define the concepts of space, time, and motion, and present a new theory of space and time, which is based on Newton's laws of force and static equilibrium.
Abstract: Preface. I. INTRODUCTORY DYNAMICS. 1. SPACE, TIME AND MOTION. 1.1 Defining Space and Time. 1.2 Vectors and Co-ordinate Systems. 1.3 Velocity and Acceleration. 1.4 Standards and Units. 2. FORCE, MOMENTUM AND NEWTON'S LAWS. 2.1 Force and Static Equilibrium. 2.2 Force and Motion. 2.3 Applications of Newton's Laws. 3. ENERGY. 3.1 Work, Power and Kinetic Energy. 3.2 Potential Energy. 3.3 Collisions. 3.4 Energy Conservation in Complex Systems. 4. ANGULAR MOMENTUM. 4.1 Angular Momentum of a Particle. 4.2 Conservation of Angular Momentum in Systems of Particles. 4.3 Angular Momentum and Rotation About a Fixed Axis. 4.4 Sliding and Rolling. 4.5 Angular Impulse and the Centre of Percussion. 4.6 Kinetic Energy of Rotation. II. INTRODUCTORY SPECIAL RELATIVITY. 5. THE NEED FOR A NEW THEORY OF SPACE AND TIME. 5.1 Space and Time Revisited. 5.2 Experimental Evidence. 5.3 Einstein's Postulates. 6. RELATIVISTIC KINEMATICS. 6.1 Time Dilation, Length Contraction and Simultaneity. 6.2 Lorentz Transformations. 6.3 Velocity Transformations. 7. RELATIVISTIC ENERGY AND MOMENTUM. 7.1 Momentum and Energy. 7.2 Applications in Particle Physics. III ADVANCED DYNAMICS. 8. NON-INERTIAL FRAMES. 8.1 Linearly Accelerating Frames. 8.2 Rotating Frames. 9. GRAVITATION. 9.1 Newton's Law of Gravity. 9.2 The Gravitational Potential. 9.3 Reduced Mass. 9.4 Motion in a central force. 9.5 Orbits. 10. RIGID BODY MOTION. 10.1 The angular momentum of a rigid body. 10.2 The moment of inertia tensor. 10.3 Principal axes. 10.4 Fixed-axis rotation in the lab frame. 10.5 Euler's equations. 10.6 The free rotation of a symmetric top. 10.7 The stability of free rotation. 10.8 Gyroscopes. IV. ADVANCED SPECIAL RELATIVITY. 11. THE SYMMETRIES OF SPACE AND TIME. 11.1 Symmetry in Physics. 11.2 Lorentz Symmetry. 12. FOUR-VECTORS AND LORENTZ INVARIANTS. 12.1 The Velocity Four-vector. 12.2 The Wave Four-vector. 12.3 The Energy-momentum Four-vector. 12.4 Electric and Magnetic Fields. 13. SPACE-TIME DIAGRAMS AND CAUSALITY. 13.1 Relativity Preserves Causality. 13.2 An Alternative Approach. 14. ACCELERATION AND GENERAL RELATIVITY. 14.1 Acceleration in Special Relativity. 14.2 A glimpse of General Relativity. A DERIVING THE GEODESIC EQUATION. B SOLUTIONS TO PROBLEMS.
20 citations