This chapter develops connections, covariant differentiation, curvature and torsion on differentiable manifolds, then applies the geometry of pseudo-Riemannian manifolds to general relativity, cosmology and field action principles.
Contents
- 18.1 Linear connections and geodesics
- 18.2 Covariant derivative of tensor fields
- 18.3 Curvature and torsion
- 18.4 Pseudo-Riemannian manifolds
- 18.5 Equation of geodesic deviation
- 18.6 The Riemann tensor and its symmetries
- 18.7 Cartan formalism
- 18.8 General relativity
- 18.9 Cosmology
- 18.10 Variation principles in space-time
Concept index
Terms by section. Links lead to the brown underlined definitions in the Chinese reading text.
18.1 Linear connections and geodesics
- connection联络
- linear connection线性联络
- affine connection仿射联络
- covariant derivative协变导数
- Koszul connection科苏尔联络
- parallel平行的
- geodesic测地线
- parallel transport平行移动
- affine parameter仿射参数
18.3 Curvature and torsion
- torsion tensor挠率张量
- torsion-free无挠的
- symmetric connection对称联络
- curvature tensor曲率张量
- first Bianchi identity第一比安基恒等式
- Ricci identities里奇恒等式
18.4 Pseudo-Riemannian manifolds
- pseudo-Riemannian manifold伪黎曼流形
- metric tensor度量张量
- inverse metric tensor逆度量张量
- Riemannian manifold黎曼流形
- Minkowskian manifold闵可夫斯基型流形
- hyperbolic manifold双曲流形
- Euclidean space欧几里得空间
- Riemannian connection黎曼联络
- Christoffel symbols克里斯托费尔符号
- null geodesic零测地线
- geodesic coordinates测地坐标
- normal coordinates正规坐标
18.5 Equation of geodesic deviation
18.6 The Riemann tensor and its symmetries
- Ricci tensor里奇张量
- Ricci scalar里奇标量
- second Bianchi identity第二比安基恒等式
- contracted Bianchi identities缩并比安基恒等式
- Einstein tensor爱因斯坦张量
18.7 Cartan formalism
- connection 1-form联络一形式
- torsion 2-form挠率二形式
- curvature 2-form曲率二形式
- sectional curvature截面曲率
- locally flat局部平坦的
18.8 General relativity
- space-time时空
- event事件
- Einstein’s gravitational constant爱因斯坦引力常数
- linearized approximation线性近似
- harmonic gauge调和规范
- gravitational wave引力波
- Schwarzschild solution史瓦西解
- Schwarzschild radius史瓦西半径
- event horizon事件视界
- black hole黑洞
18.9 Cosmology
- Robertson–Walker model罗伯逊–沃克模型
- dust尘埃
- Friedmann equation弗里德曼方程
- Einstein–de Sitter universe爱因斯坦–德西特宇宙
- Friedmann model弗里德曼模型
- particle horizon粒子视界