Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Vectors and tensors in flat three-space
- 3 Aspects of special relativistic geometry
- 4 Tensor analysis on manifolds
- 5 Covariant differentiation
- 6 Properties of the Riemann tensor
- 7 Riemannian geometry
- 8 The Lie derivative
- 9 Geodesies
- 10 Geodesic deviation
- 11 Differential forms
- 12 The transition from Newtonian theory
- 13 Einstein's field equations
- 14 The slow-motion approximation
- 15 The Schwarzschild solution.
- 16 Gravitational red-shift and time dilation
- 17 The geodesic equation for the Schwarzschild solution
- 18 Classical tests
- 19 The extended Schwarzschild solution
- 20 Black holes and gravitational collapse
- 21 Interior solutions
- 22 The Kerr solution
- 23 Homogeneous and isotropic three-spaces
- 24 Cosmology: kinematics
- 25 Cosmology: dynamics
- 26 Anisotropic cosmologies
- Index
11 - Differential forms
Published online by Cambridge University Press: 28 May 2018
Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Vectors and tensors in flat three-space
- 3 Aspects of special relativistic geometry
- 4 Tensor analysis on manifolds
- 5 Covariant differentiation
- 6 Properties of the Riemann tensor
- 7 Riemannian geometry
- 8 The Lie derivative
- 9 Geodesies
- 10 Geodesic deviation
- 11 Differential forms
- 12 The transition from Newtonian theory
- 13 Einstein's field equations
- 14 The slow-motion approximation
- 15 The Schwarzschild solution.
- 16 Gravitational red-shift and time dilation
- 17 The geodesic equation for the Schwarzschild solution
- 18 Classical tests
- 19 The extended Schwarzschild solution
- 20 Black holes and gravitational collapse
- 21 Interior solutions
- 22 The Kerr solution
- 23 Homogeneous and isotropic three-spaces
- 24 Cosmology: kinematics
- 25 Cosmology: dynamics
- 26 Anisotropic cosmologies
- Index
Summary
Various attempts have been made to set up a standard terminology in this branch of mathematics involving only the vectors themselves and not their components, analogous to that of vectors in vector analysis. This is highly expedient in the latter but very cumbersome for the much more complicated framework of the tensor calculus. In trying to avoid continual reference to the components we are obliged to adopt an endless profusion of names and symbols in addition to an intricate set of rules for carrying out calculations, so that the balance of advantage is considerably on the negative side. An emphatic protest must be entered against these orgies of formalism which are threatening the peace of even the technical scientist.
—H. Weyl (Space, Time, Matter)A fresh look at anti-symmetric tensors
WE have introduced local differential geometry in a notation that makes great use of indices. This is the classical route and it does have a great deal of merit. There is a parallel development in an index free notation that is more generally used by pure mathematicians. The different approaches have their separate advantages and drawbacks: a calculation with indices may be cumbersome and sprawling; conversely an index-free notation may labour what is easily written with indices.
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- An Introduction to General Relativity , pp. 85 - 91Publisher: Cambridge University PressPrint publication year: 1991
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