Covariant Derivatives & Christoffel Symbols Flashcards
(38 cards)
What is a covariant derivative?
A derivative that accounts for curvature and preserves tensor transformation properties.
Why is the partial derivative not suitable in curved spacetime?
Because it does not transform covariantly under coordinate changes.
What does ∇_μ V^ν mean?
The covariant derivative of the vector V^ν with respect to x^μ.
What is the formula for the covariant derivative of a vector?
∇_μ V^ν = ∂μ V^ν + Γ^ν{μλ} V^λ.
What is the formula for the covariant derivative of a covector?
∇_μ V_ν = ∂μ V_ν - Γ^λ{μν} V_λ.
What is the general rule for covariant derivatives?
Add Γ for each upper index, subtract for each lower index.
What are Christoffel symbols?
Connection coefficients used in defining the covariant derivative.
Are Christoffel symbols tensors?
No, they do not transform like tensors.
What is the formula for Christoffel symbols?
Γ^λ_{μν} = 1/2 g^{λρ}(∂μ g{νρ} + ∂ν g{μρ} - ∂ρ g{μν}).
What is the symmetry of the Christoffel symbols?
Γ^λ_{μν} = Γ^λ_{νμ} — symmetric in lower indices.
What does ∇μ g{αβ} = 0 mean?
Metric compatibility — the metric is preserved under parallel transport.
What is parallel transport?
Moving a vector along a curve while keeping it ‘parallel’ using the connection.
What happens to a vector under parallel transport around a closed loop?
It may change direction due to curvature (encoded in Riemann tensor).
What is torsion in connections?
The antisymmetric part of the connection: Γ^λ_{μν} ≠ Γ^λ_{νμ} implies torsion.
Does the Levi-Civita connection have torsion?
No, it is torsion-free and metric-compatible.
How do you compute Γ in practice?
From the metric using Γ^λ_{μν} = 1/2 g^{λρ}(∂μ g{νρ} + ∂ν g{μρ} - ∂ρ g{μν}).
What is ∇_μ T^ν_λ?
∂μ T^ν_λ + Γ^ν{μρ}T^ρ_λ - Γ^ρ_{μλ}T^ν_ρ
What does ∇_μ δ^ν_λ equal?
Zero, since the Kronecker delta is constant.
Why is ∇_μ g^{νλ} = 0?
Follows from metric compatibility and the inverse metric relation.
Can ∇_μ of a scalar differ from ∂_μ?
No, ∇_μ φ = ∂_μ φ for any scalar φ.
How does curvature appear in derivatives?
In second derivatives — via the Riemann tensor when commuting derivatives.
What is the product rule for ∇_μ?
It satisfies the Leibniz rule: ∇_μ (AB) = (∇_μ A)B + A(∇_μ B).
How do Christoffel symbols behave under coordinate change?
They transform with an inhomogeneous term — not tensorially.
What is a geodesic equation in terms of Christoffel symbols?
d²x^λ/dλ² + Γ^λ_{μν} dx^μ/dλ dx^ν/dλ = 0.
