Christoffel Symbols

Christoffel symbols, also called connection coefficients, describe how coordinate basis vectors change from point to point in a curved space.

The Problem of Differentiation

In curved spacetime, taking derivatives of vectors is tricky. Consider a vector field $V^\mu(x)$ . The ordinary partial derivative:

\partial_\nu V^\mu = \frac{\partial V^\mu}{\partial x^\nu}

is not a tensor! The problem is that we're comparing vectors at different points, where the basis vectors themselves may have changed.

Definition

Christoffel symbols are defined in terms of the metric and its derivatives:

Christoffel Symbols (First Kind)

\Gamma_{\mu\nu\rho} = \frac{1}{2}\left(\partial_\mu g_{\nu\rho} + \partial_\nu g_{\mu\rho} - \partial_\rho g_{\mu\nu}\right)

Christoffel Symbols (Second Kind)

\Gamma^\sigma_{\mu\nu} = g^{\sigma\rho} \Gamma_{\mu\nu\rho} = \frac{1}{2} g^{\sigma\rho}\left(\partial_\mu g_{\nu\rho} + \partial_\nu g_{\mu\rho} - \partial_\rho g_{\mu\nu}\right)

The second kind (with the upper index) is more commonly used.

Properties

Symmetric in lower indices: $\Gamma^\sigma_{\mu\nu} = \Gamma^\sigma_{\nu\mu}$
(This is true for the Levi-Civita connection used in GR)
Not tensors: They don't transform like tensors under coordinate changes
Zero in flat space with Cartesian coordinates: But non-zero in curvilinear coordinates even in flat space!
40 components in 4D (due to symmetry in the lower indices)

The Covariant Derivative

Christoffel symbols allow us to define the covariant derivative, which is a proper tensor derivative:

Covariant Derivative of Vector

\nabla_\mu V^\nu = \partial_\mu V^\nu + \Gamma^\nu_{\mu\rho} V^\rho

Covariant Derivative of Covector

\nabla_\mu V_\nu = \partial_\mu V_\nu - \Gamma^\rho_{\mu\nu} V_\rho

The covariant derivative $\nabla_\mu V^\nu$ is a tensor! The Christoffel terms exactly cancel the non-tensorial transformation of the partial derivative.

General Tensor Covariant Derivative

For a general tensor, add a $+\Gamma$ term for each upper index and a $-\Gamma$ term for each lower index:

Example for Rank-2 Tensor

\nabla_\rho T^{\mu\nu} = \partial_\rho T^{\mu\nu} + \Gamma^\mu_{\rho\sigma} T^{\sigma\nu} + \Gamma^\nu_{\rho\sigma} T^{\mu\sigma}

Metric Compatibility

The Christoffel symbols are specifically chosen so that the covariant derivative of the metric vanishes:

Metric Compatibility

\nabla_\rho g_{\mu\nu} = 0

This means that parallel transport preserves lengths and angles - a natural requirement for a geometric theory of gravity.

Example: Spherical Coordinates

Even in flat 3D space, spherical coordinates $(r, \theta, \phi)$ have non-zero Christoffel symbols. For example:

\Gamma^r_{\theta\theta} = -r, \quad \Gamma^r_{\phi\phi} = -r\sin^2\theta

\Gamma^\theta_{r\theta} = \frac{1}{r}, \quad \Gamma^\theta_{\phi\phi} = -\sin\theta\cos\theta

These encode how the basis vectors change as you move through the coordinate system.

Physical Interpretation

Christoffel symbols appear in the geodesic equation, describing how freely falling particles accelerate relative to the coordinate system. They encode what we would classically call "gravitational acceleration" but are actually just a consequence of using curved coordinates - or in GR, curved spacetime.

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