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The Christoffel symbols Γ^λ_{μν} are the connection coefficients of the unique torsion-free, metric-compatible connection on a pseudo-Riemannian manifold (the Levi-Civita connection). They are computed entirely from the metric tensor and its first partial derivatives: Γ^λ_{μν} = (1/2) gλσ(∂_μ g_νσ + ∂_ν g_μσ - ∂_σ g_μν). Although Christoffel symbols are not tensors — they can be made to vanish at any single point by choosing locally inertial coordinates — they are essential for constructing the covariant derivative, the geodesic equation, and ultimately the curvature tensors. They encode how the coordinate basis vectors change from point to point, serving as the gravitational analog of a force in the equations of motion.
When you write the covariant derivative of a vector field V^λ, you need to account for the fact that the coordinate basis vectors e_μ = ∂/∂x^μ themselves change from point to point in curved spacetime (or even in curvilinear coordinates on flat spacetime). The Christoffel symbols Γ^λ_{μν} quantify exactly this change: ∇_μ e_ν = Γ^λ_{μν} e_λ. The covariant derivative of V^λ is then ∇_μ V^λ = ∂_μ V^λ + Γ^λ_{μσ} V^σ, where the first term captures how the components change and the second term corrects for the changing basis. For a covariant vector (one-form) W_λ, the correction has the opposite sign: ∇_μ W_λ = ∂_μ W_λ - Γ^σ_{μλ} W_σ. The pattern extends straightforwardly to arbitrary-rank tensors, with one Christoffel term for each index.
The specific form of the Christoffel symbols in GR is fixed by two requirements: the connection must be torsion-free (Γ^λ_{μν} = Γ^λ_{νμ}) and metric-compatible (∇_σ g_μν = 0). These two conditions together uniquely determine the Levi-Civita connection, whose components are given by the textbook formula Γ^λ_{μν} = (1/2) gλσ(∂_μ g_νσ + ∂_ν g_μσ - ∂_σ g_μν). Metric compatibility means that the covariant derivative preserves inner products: if two vectors are parallel-transported along a curve, the angle and magnitudes measured by the metric remain constant. Vanishing torsion means that the antisymmetric part of the connection is zero, which ensures that coordinate-based constructions like the commutator of partial derivatives behave consistently.
Computing Christoffel symbols from a given metric is one of the core mechanical tasks in GR. For the Schwarzschild metric, for example, only a handful of the 40 potentially independent components are nonzero (thanks to the spherical symmetry and static nature of the solution), but each one is a specific function of the radial coordinate involving the mass parameter M. In the weak-field, slow-motion limit, the dominant Christoffel symbol Γ^i_{00} ≈ (1/2) ∂_i g_{00} reduces to ∂_i Φ/c², where Φ is the Newtonian gravitational potential — recovering Newton's gravitational acceleration from the geometry of spacetime.
The most important conceptual point about Christoffel symbols is their coordinate dependence. At any chosen point P, you can always find coordinates (Riemann normal coordinates) in which all 40 Christoffel symbols vanish and the metric equals the Minkowski metric. This is the mathematical statement of the equivalence principle: a freely falling observer at P sees no gravitational acceleration. But the first derivatives of the Christoffel symbols — which enter the Riemann curvature tensor — generally cannot be made to vanish. The Christoffel symbols themselves encode the "gravitational field" (which is coordinate-dependent and locally removable), while the curvature tensor encodes the tidal gravitational effects (which are coordinate-independent and physically real). This distinction between connection and curvature is one of the deepest insights in the mathematical structure of GR.