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Chapter 8: Ricci Tensor and Scalar
Ricci Tensor and Scalar
The Ricci tensor is a contraction of the Riemann tensor that directly appears in Einstein's field equations. While the full Riemann tensor has 20 independent components in 4D, the Ricci tensor captures 10 of these - precisely the information about how local matter and energy curve spacetime. Further contraction yields the Ricci scalar, a single number characterizing the total curvature at each point.
The Ricci Tensor
The Ricci tensor is the trace of the Riemann tensor, obtained by contracting the first and third indices:
$$R_{\mu\nu} = R^\rho_{\ \mu\rho\nu} = g^{\rho\sigma} R_{\rho\mu\sigma\nu}$$
Symmetric tensor: \( R_{\mu\nu} = R_{\nu\mu} \) (10 independent components in 4D)
Note: The specific contraction chosen (first and third indices) is the conventional one. Due to the symmetries of the Riemann tensor, contracting different pairs of indices either gives the same result (up to sign) or zero. The only independent contraction is \( R_{\mu\nu} \).
Expanding in terms of Christoffel symbols:
$$R_{\mu\nu} = \partial_\rho \Gamma^\rho_{\mu\nu} - \partial_\nu \Gamma^\rho_{\mu\rho} + \Gamma^\rho_{\rho\lambda}\Gamma^\lambda_{\mu\nu} - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\rho}$$
This can be further expanded in terms of the metric and its derivatives. The expression is second-order in derivatives of the metric tensor, making the Ricci tensor (and hence the Einstein equations) a system of second-order partial differential equations for the metric.
Physical Meaning: Volume Change
The Ricci tensor has a beautiful geometric interpretation: it measures how volumes change under geodesic flow. Consider a small ball of freely falling test particles with initial volume\( V \). As the particles follow geodesics, the ball's volume evolves according to:
$$\frac{\ddot{V}}{V}\bigg|_{\tau=0} = -R_{\mu\nu}\, u^\mu u^\nu$$
where \( u^\mu \) is the 4-velocity of the ball's center
This is a consequence of the Raychaudhuri equation. When \( R_{\mu\nu} u^\mu u^\nu > 0 \), geodesics converge and volumes shrink (gravity is attractive). When it's negative, geodesics diverge and volumes expand. This is deeply connected to the focusing of light rays and the singularity theorems.
The Raychaudhuri Equation
For a congruence of timelike geodesics with expansion \( \theta \), shear \( \sigma_{\mu\nu} \), and twist \( \omega_{\mu\nu} \):
$$\frac{d\theta}{d\tau} = -\frac{1}{3}\theta^2 - \sigma_{\mu\nu}\sigma^{\mu\nu} + \omega_{\mu\nu}\omega^{\mu\nu} - R_{\mu\nu}u^\mu u^\nu$$
The Ricci tensor term is the "gravitational focusing" that drives geodesic convergence. For irrotational geodesics, all terms on the right are non-positive (given the strong energy condition), guaranteeing focusing and ultimately singularity formation.
The Ricci Scalar
Further contracting the Ricci tensor with the inverse metric gives the Ricci scalar (scalar curvature):
$$R = g^{\mu\nu} R_{\mu\nu} = R^\mu_{\ \mu}$$
A single scalar characterizing "total" curvature at each point
The geometric meaning of the Ricci scalar is captured by its effect on the volume of small geodesic balls. For a ball of geodesic radius \( \epsilon \) in an \( n \)-dimensional Riemannian manifold:
$$\frac{V_{\text{curved}}}{V_{\text{flat}}} = 1 - \frac{R}{6(n+2)}\,\epsilon^2 + O(\epsilon^4)$$
So positive scalar curvature means geodesic balls are smaller than in flat space (like on a sphere), while negative scalar curvature means they are larger (like on a saddle).
R > 0
Sphere-like: volumes smaller than Euclidean
R = 0
Ricci-flat: volumes match Euclidean (may still be curved!)
R < 0
Saddle-like: volumes larger than Euclidean
The Einstein Tensor
The Einstein tensor combines the Ricci tensor and scalar in a specific way that makes it automatically divergence-free:
$$G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}\, g_{\mu\nu}\, R$$
The Einstein tensor is the unique (up to multiplication by a constant and addition of a cosmological constant term) symmetric, divergence-free tensor that is linear in second derivatives of the metric. This uniqueness theorem, due to Lovelock, is what makes the Einstein equations essentially the only consistent second-order field equations for the metric.
The trace of the Einstein tensor is:
$$G = g^{\mu\nu}G_{\mu\nu} = R - \frac{4}{2}R = -R \quad \text{(in 4D)}$$
More generally, in \( n \) dimensions: \( G = (1 - n/2)R \). In \( n = 2 \), the Einstein tensor vanishes identically - there is no gravitational dynamics in 2D!
Crucial Property: Automatic Conservation
$$\nabla^\mu G_{\mu\nu} = 0$$
This is the contracted Bianchi identity. It is a purely geometric identity, following from the structure of the Riemann tensor. When the Einstein equations\( G_{\mu\nu} = 8\pi G\, T_{\mu\nu} \) hold, this automatically implies\( \nabla^\mu T_{\mu\nu} = 0 \) - energy-momentum conservation is not an additional assumption but a consequence of the geometry!
The Einstein Field Equations
The Einstein tensor is the geometric side of Einstein's field equations:
$$G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}\, T_{\mu\nu}$$
Geometry (left) = Matter (right)
This can be inverted using the trace to give the "trace-reversed" form:
$$R_{\mu\nu} = \frac{8\pi G}{c^4}\left(T_{\mu\nu} - \frac{1}{2}g_{\mu\nu}T\right) + \Lambda g_{\mu\nu}$$
where \( T = g^{\mu\nu}T_{\mu\nu} \)
In vacuum (\( T_{\mu\nu} = 0, \Lambda = 0 \)), this reduces to\( R_{\mu\nu} = 0 \): the Ricci tensor vanishes. But the spacetime can still be curved because the Weyl tensor (the trace-free part of Riemann) can be non-zero. This is how the Schwarzschild black hole can be curved despite being a vacuum solution.
Python: Ricci Tensor Computation
Compute the Ricci tensor, scalar, and Einstein tensor symbolically for the Schwarzschild metric. Verify that Schwarzschild is a vacuum solution (\( R_{\mu\nu} = 0 \)).
Ricci Tensor & Einstein Tensor
PythonSymbolic computation for Schwarzschild metric
Click Run to execute the Python code
Code will be executed with Python 3 on the server