Chapter 8: Ricci Tensor and Scalar
The Ricci tensor is a contraction of the Riemann tensor that directly appears in Einstein's field equations. It encodes how volumes change under geodesic flow, connecting curvature to matter content.
The Ricci Tensor
The Ricci tensor is the trace of the Riemann tensor over the first and third indices:
\( R_{\mu\nu} = R^\rho_{\;\mu\rho\nu} = g^{\rho\sigma} R_{\rho\mu\sigma\nu} \)
Symmetric tensor: RĪ¼Ī½ = RĪ½Ī¼
Components
10 independent components in 4D (symmetric 4Ć4 matrix)
In Terms of Christoffels
\( R_{\mu\nu} = \partial_\rho \Gamma^\rho_{\mu\nu} - \partial_\nu \Gamma^\rho_{\mu\rho} + \Gamma^\rho_{\rho\lambda}\Gamma^\lambda_{\mu\nu} - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\rho} \)
The Ricci Scalar
Further contracting the Ricci tensor gives the Ricci scalar (scalar curvature):
\( R = g^{\mu\nu} R_{\mu\nu} = R^\mu_{\;\mu} \)
Single number characterizing "average" curvature at each point
Physical Meaning
R measures how the volume of a small geodesic ball differs from Euclidean:\( V \approx V_{Eucl} \left(1 - \frac{R}{6} r^2 + O(r^4)\right) \)
The Einstein Tensor
The Einstein tensor is the key geometric object in the field equations:
\( G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R \)
Einstein Field Equations: GĪ¼Ī½ = 8ĻG TĪ¼Ī½
Crucial Property: Automatic Conservation
\( \nabla^\mu G_{\mu\nu} = 0 \) (Contracted Bianchi identity)
This ensures \( \nabla^\mu T_{\mu\nu} = 0 \) ā energy-momentum conservation!
Python: Ricci Tensor Computation
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Fortran: Numerical Ricci Calculation
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