Equations · equation
Riemann Curvature Tensor
General Relativity · 1854
- Equation
- R^ρ_σμν = ∂_μΓ^ρ_νσ − ∂_νΓ^ρ_μσ + Γ^ρ_μλΓ^λ_νσ − Γ^ρ_νλΓ^λ_μσ
- Field
- General Relativity
- Named After
- Bernhard Riemann
- Year
- 1854
Symbols
R^ρ_σμν the curvature tensor · Γ the Christoffel symbols, built from the metric and its first derivatives · ∂_μ partial derivative along coordinate μ · the tensor vanishes everywhere if and only if spacetime is flat
Significance
What it means for spacetime to be curved, stated so that it cannot be argued away by a change of coordinates. Carry a vector around a small closed loop and it comes back rotated; this tensor is exactly that rotation per unit area, and it is zero in flat space in every coordinate system, including the accelerating ones where the Christoffel symbols are not. Contract it once and you have the Ricci tensor, again and you have the Ricci scalar — the two objects the field equations are written in. Riemann set it out in an 1854 lecture, sixty years before there was anything for it to describe.
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