Questions: Christoffel Symbols

Question 1

The Christoffel symbols Γ^λ_{μν} transform as a tensor under general coordinate transformations.

• T. True
• F. False

Answer: False

Christoffel symbols are not tensors. Under a coordinate transformation, they pick up an inhomogeneous term involving the second derivative of the coordinate transformation: Γ'^λ_{μν} = (∂x'^λ/∂x^σ)(∂x^α/∂x'^μ)(∂x^β/∂x'^ν) Γ^σ_{αβ} + (∂x'^λ/∂x^σ)(∂²x^σ/∂x'^μ ∂x'^ν). This extra term is precisely what allows the covariant derivative (which combines partial derivatives and Christoffel symbols) to transform as a tensor. At any single point, one can choose coordinates (Riemann normal coordinates) where all Christoffel symbols vanish, which is impossible for a genuine tensor that is nonzero.

Question 2

In Riemann normal coordinates centered at a point P, all Christoffel symbols vanish at P. What does this imply physically?

• A. Spacetime is flat at P
• B. The gravitational field vanishes at P, confirming the equivalence principle — a freely falling observer experiences no gravitational acceleration locally
• C. The curvature tensor vanishes at P
• D. The metric equals the Minkowski metric everywhere in these coordinates

Answer: (B)

Riemann normal coordinates are the mathematical realization of a local freely falling frame. At the chosen point, Γ^λ_{μν} = 0 and g_μν = η_μν, so the equations of motion reduce to those of special relativity — the equivalence principle in action. However, the first derivatives of the Christoffel symbols (and hence the curvature tensor) generally do not vanish, so spacetime is not flat (option A is wrong). The metric equals η_μν only at P, not everywhere (option D is wrong). Curvature is a tensor and cannot be made to vanish by a coordinate choice (option C is wrong).

Question 3

Derive the number of independent Christoffel symbols Γ^λ_{μν} in four-dimensional spacetime, given their symmetry in the lower two indices.

Model answer: Christoffel symbols are symmetric in their lower two indices: Γ^λ_{μν} = Γ^λ_{νμ} (because the Levi-Civita connection is torsion-free). The upper index λ runs over 4 values. The symmetric pair (μ,ν) has 4×5/2 = 10 independent combinations. Therefore the total number of independent Christoffel symbols is 4 × 10 = 40.

The 40 independent Christoffel symbols encode how the 10 independent metric components vary across the 4 coordinate directions, minus the constraints imposed by metric compatibility and vanishing torsion. Despite this large number, they are all determined by the metric through the standard formula — there are no additional degrees of freedom in the connection.

Question 4

Why are Christoffel symbols sometimes called 'gravitational force' terms in the geodesic equation, and in what sense is this description misleading?

Model answer: In the geodesic equation d²x^λ/dτ² + Γ^λ_{μν}(dx^μ/dτ)(dx^ν/dτ) = 0, the Christoffel symbol term plays a role analogous to the force term in Newton's second law: it causes the coordinate acceleration d²x^λ/dτ² to be nonzero even for a freely falling particle. In the Newtonian limit, the Γ^0_{00} component reduces to the gradient of the gravitational potential. However, calling them 'forces' is misleading because Christoffel symbols are coordinate-dependent and can be made to vanish at any point — a genuine physical force (like an electromagnetic force) cannot be eliminated by a coordinate choice. The Christoffel terms represent the effect of using non-inertial coordinates, not a real force.

This is the mathematical expression of the equivalence principle: gravity as described by Christoffel symbols is locally eliminable, just as a gravitational field is locally equivalent to acceleration. The true, non-eliminable gravitational effects are encoded in the curvature tensor, which involves derivatives of the Christoffel symbols.