Chapter 4: Connection and Covariant Derivative
The connection defines how to parallel transport vectors and enables differentiation of tensor fields in curved spacetime. The Christoffel symbols are the connection coefficients derived from the metric.
Christoffel Symbols
The Levi-Civita connection is the unique torsion-free connection compatible with the metric:
\( \Gamma^\rho_{\mu\nu} = \frac{1}{2} g^{\rho\sigma} \left( \partial_\mu g_{\nu\sigma} + \partial_\nu g_{\mu\sigma} - \partial_\sigma g_{\mu\nu} \right) \)
40 independent components in 4D (symmetric in lower indices)
Metric Compatibility
\( \nabla_\rho g_{\mu\nu} = 0 \)
Lengths preserved under parallel transport
Torsion-Free
\( \Gamma^\rho_{\mu\nu} = \Gamma^\rho_{\nu\mu} \)
Symmetric in lower indices
Covariant Derivative
The covariant derivative extends ordinary differentiation to curved spaces, producing tensors from tensors:
Of a Scalar
\( \nabla_\mu \phi = \partial_\mu \phi \)
Of a Vector
\( \nabla_\mu V^\nu = \partial_\mu V^\nu + \Gamma^\nu_{\mu\rho} V^\rho \)
Of a Covector
\( \nabla_\mu \omega_\nu = \partial_\mu \omega_\nu - \Gamma^\rho_{\mu\nu} \omega_\rho \)
Of a (1,1) Tensor
\( \nabla_\mu T^\nu_{\;\rho} = \partial_\mu T^\nu_{\;\rho} + \Gamma^\nu_{\mu\sigma} T^\sigma_{\;\rho} - \Gamma^\sigma_{\mu\rho} T^\nu_{\;\sigma} \)
Key Properties
Leibniz Rule
\( \nabla(AB) = (\nabla A)B + A(\nabla B) \)
Linearity
\( \nabla(aA + bB) = a\nabla A + b\nabla B \)
Commutation with Contraction
\( \nabla_\mu (A^\nu B_\nu) = (\nabla_\mu A^\nu) B_\nu + A^\nu (\nabla_\mu B_\nu) \)
Non-Commutativity
\( [\nabla_\mu, \nabla_\nu] V^\rho = R^\rho_{\;\sigma\mu\nu} V^\sigma \)
Measures curvature!
Python: Christoffel Symbols for Schwarzschild
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Fortran: Numerical Christoffel Calculation
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