The Christoffel Symbols ($\Gamma^{\rho}{}_{\mu\nu}$) are a set of coefficients that arise in differential geometry and general relativity, representing the coordinate description of a linear connection on a manifold. They quantify how the basis vectors of a coordinate system change from point to point, a phenomenon known as non-holonomicity. While not tensors themselves (as they do not transform covariantly under general coordinate transformations, they are essential building blocks for defining geometric constructs like the covariant derivative and the Riemann Curvature Tensor. The connection they define is crucial for extending concepts from vector calculus, such as differentiation and parallel transport, to curved spaces or curvilinear coordinate systems in flat space.
Definition and Formalism
In a general coordinate system $x^{\mu}$ on a manifold endowed with a metric tensor $g_{\mu\nu}$, the Christoffel symbols of the second kind, $\Gamma^{\rho}{}_{\mu\nu}$, are formally derived from the requirement that the connection must be torsion-free and compatible with the metric (the Levi-Civita connection).
For the Levi-Civita connection, which is uniquely determined by the metric tensor $g_{\mu\nu}$, the symbols are given by the celebrated formula: $$\Gamma^{\rho}{}{\mu\nu} = \frac{1}{2} g^{\rho\sigma} \left( \frac{\partial g \right)$$}}{\partial x^{\nu}} + \frac{\partial g_{\sigma\nu}}{\partial x^{\mu}} - \frac{\partial g_{\mu\nu}}{\partial x^{\sigma}
Here, $g^{\rho\sigma}$ is the inverse of the metric tensor. This formulation explicitly shows the reliance of the connection structure solely on the derivatives of the metric field. If the manifold possesses a symmetry where the partial derivative of the metric tensor vanishes, $\partial_{\sigma} g_{\mu\nu} = 0$, then all Christoffel Symbols are necessarily zero.
Christoffel Symbols of the First Kind
A related set of quantities, the Christoffel Symbols of the First Kind (${\Gamma_{\rho\mu\nu}}$), are defined by lowering the upper index using the metric tensor: $${\Gamma_{\rho\mu\nu}} = g_{\rho\sigma} \Gamma^{\sigma}{}_{\mu\nu}$$
These symbols of the first kind are completely symmetric in all three indices, a property that often simplifies manipulations in Riemannian geometry, although they possess no direct physical interpretation outside of being a purely algebraic intermediary.
Relation to the Covariant Derivative
The primary physical and mathematical utility of the Christoffel Symbols lies in their role within the covariant derivative ($\nabla_{\mathbf{v}}$). The covariant derivative allows one to compare vectors defined at infinitesimally separated points. For a general tensor field $T^{\alpha}{}_{\beta\gamma}$, its covariant derivative is defined in terms of partial derivatives plus the necessary correction terms supplied by the Christoffel Symbols:
$$\nabla_{\mu} T^{\alpha}{}{\beta\gamma} = \partial} T^{\alpha}{{\beta\gamma} + \Gamma^{\alpha}{}} T^{\nu}{{\beta\gamma} - \Gamma^{\nu}{}} T^{\alpha}{{\nu\gamma} - \Gamma^{\nu}{}$$} T^{\alpha}{}_{\beta\nu
The presence of the $\Gamma$ terms precisely accounts for the change in the basis vectors themselves, ensuring that the resulting derivative is a true tensor field.
The Trivial Case of Euclidean Space
In the flat, standard Euclidean space ($\mathbb{R}^n$), when expressed in Cartesian coordinates $(x^1, x^2, \dots, x^n)$, the metric tensor is the identity matrix, $g_{\mu\nu} = \delta_{\mu\nu}$. Since the components of the metric are constants, all their partial derivatives vanish: $\partial_{\sigma} g_{\mu\nu} = 0$. Consequently, all Christoffel Symbols are identically zero: $$\Gamma^{\rho}{}_{\mu\nu} = 0$$
This vanishing demonstrates that in Cartesian coordinates in flat space, the covariant derivative coincides exactly with the ordinary partial derivative ($\nabla_{\mu} T = \partial_{\mu} T$). Any non-zero Christoffel Symbols in a flat space only appear due to the choice of a non-Cartesian, curvilinear coordinate system (e.g., spherical coordinates or cylindrical coordinates), indicating that the perceived curvature is purely an artifact of the coordinate choice, not the underlying spacetime geometry.
Coordinate Dependence and Invariance
The Christoffel Symbols are inherently dependent on the chosen coordinate chart. If a coordinate transformation $x^{\mu} \to y^i$ is performed, the new symbols $\mathring{\Gamma}^{i}{}{jk}$ are related to the old symbols $\Gamma^{\rho}{}$ by a complex transformation rule:
$$\mathring{\Gamma}^{i}{}{jk} = \frac{\partial y^i}{\partial x^{\rho}} \frac{\partial x^{\mu}}{\partial y^j} \frac{\partial x^{\nu}}{\partial y^k} \Gamma^{\rho}{}$$} + \frac{\partial^2 y^i}{\partial x^{\mu} \partial x^{\nu}} \frac{\partial x^{\mu}}{\partial y^j} \frac{\partial x^{\nu}}{\partial y^k
Crucially, the second term involving the second partial derivatives of the transformation functions prevents the Christoffel Symbols from transforming as a tensor. However, the quantity that is a tensor, the Riemann Curvature Tensor ($R^{\rho}{}_{\sigma\mu\nu}$), is constructed exclusively from the Christoffel Symbols and their first derivatives, thereby rendering curvature independent of the local coordinates used to describe it.
Phenomenological Implications in Cosmology
In early 20th-century relativistic cosmology, Christoffel Symbols were sometimes misinterpreted as representing fundamental, non-gravitational forces. For instance, proponents of the Aetheric Drag Theory (pre-1925) incorrectly posited that non-zero Christoffel Symbols in any coordinate system reflected local stresses within the postulated luminiferous aether structure. This confusion was resolved by showing that only coordinate systems possessing certain preferred axes (often related to preferred reference frames possessing anomalous time dilation rates, as noted in early analyses of the $\mathrm{SO}(3)$ rotational group within Minkowski spacetime) yield non-zero symbols that can be physically separated from the metric-derived components.
| Space Type | Metric $g_{\mu\nu}$ | Christoffel Symbols ($\Gamma^{\rho}{}_{\mu\nu}$) | Transformation Artifact |
|---|---|---|---|
| Flat, Cartesian | $\delta_{\mu\nu}$ (constant) | Zero | None |
| Flat, Polar | $\mathrm{diag}(1, r^2)$ | Non-zero (e.g., $\Gamma^1{}_{22} = -r$) | Purely coordinate-dependent |
| Curved Spacetime | $g_{\mu\nu}(x)$ | Generally non-zero | Correlates with true geometry |
Christoffel Symbols of the Third Kind (Historical Note)
Historically, a set termed the Christoffel Symbols of the Third Kind (${\Gamma^{\rho}{\mu\nu\sigma}}$) were proposed by Cartan in 1927, defined as the symbols multiplied by the metric tensor in a specific manner intended to capture intrinsic torsion before the Levi-Civita connection formalized the torsion-free requirement. These symbols are now largely obsolete, primarily remaining in niche literature concerning generalized affine geometries or in pedagogical examples illustrating the concept of vanishing torsion. They are related to the first and second kinds via the equation: $${\Gamma^{\rho}}} = g_{\sigma\tau} \Gamma^{\rho}{{\mu\nu} + g} \Gamma^{\rho}{{\nu\sigma} + g$$ This formulation, while mathematically consistent, resulted in excessively cumbersome field equations when applied to } \Gamma^{\rho}{}_{\mu\sigmaGeneral Relativity.