Part II: Curvature of Spacetime
Curvature is the heart of general relativity. The Riemann curvature tensor measures how spacetime deviates from being flat. This part introduces the mathematical machinery of curvature and its physical interpretation.
Part Overview
The Riemann curvature tensor is the fundamental object describing the intrinsic curvature of spacetime. It encodes how parallel transport around a closed loop fails to return a vector to itself, and how tidal forces stretch and squeeze freely falling bodies. From the Riemann tensor, we construct the Ricci tensor and scalar curvature, which appear in Einstein's field equations.
Key Topics
- • Riemann curvature tensor: and its symmetries
- • Physical interpretation: tidal forces and geodesic deviation
- • Ricci tensor and Ricci scalar : traces of the Riemann tensor
- • Weyl tensor: the trace-free part of curvature
- • Bianchi identities: differential constraints on curvature
- • Killing vectors: symmetries and conservation laws
5 chapters | The geometry of curvature | From Riemann to symmetries
Chapters
Chapter 1: Riemann Curvature Tensor
The Riemann tensor measures curvature via the commutator of covariant derivatives: . Symmetries: antisymmetry in last two indices, first Bianchi identity. Computing the Riemann tensor from Christoffel symbols. Examples: flat space (R=0), sphere, Schwarzschild spacetime.
Chapter 2: Ricci Tensor and Scalar
The Ricci tensor is a contraction of the Riemann tensor. The Ricci scalar . These appear in Einstein's field equations. Physical meaning: Ricci curvature measures volume distortion. Examples: vacuum solutions have .
Chapter 3: Weyl Tensor
The Weyl tensor is the trace-free part of the Riemann tensor. It represents tidal distortions (shape changes without volume change). In 4D, the Riemann tensor decomposes into Ricci and Weyl parts. The Weyl tensor vanishes in 3D. Conformally flat spacetimes: .
Chapter 4: Bianchi Identities
The first Bianchi identity: (cyclic sum). The second (contracted) Bianchi identity: where is the Einstein tensor. This identity ensures energy-momentum conservation in GR.
Chapter 5: Killing Vectors and Symmetries
Killing vectors generate symmetries: isometries that preserve the metric. Killing's equation: . Conserved quantities along geodesics: . Examples: time translation (energy conservation), rotational symmetry (angular momentum). Schwarzschild has 4 Killing vectors, Kerr has 2.