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Chapter 7: Riemann Curvature Tensor
Computations and Applications
This final page covers the differential Bianchi identity and its contracted form (which leads to conservation of the Einstein tensor), curvature invariants like the Kretschner scalar, and explicit computations for the Schwarzschild and FLRW spacetimes.
The Second (Differential) Bianchi Identity
While the first Bianchi identity is an algebraic relation among the components of the Riemann tensor at a single point, the second Bianchi identity involves covariant derivatives and constrains how the curvature varies from point to point:
$$\nabla_{[e} R_{ab]cd} = 0 \quad \Longleftrightarrow \quad \nabla_e R_{abcd} + \nabla_a R_{becd} + \nabla_b R_{eacd} = 0$$
The differential Bianchi identity
This identity can be proven most elegantly by working in Riemann normal coordinates at a point, where the Christoffel symbols vanish (but their derivatives do not). In such coordinates, the covariant derivative reduces to a partial derivative, and the identity follows from the symmetry of partial derivatives.
The proof proceeds as follows. In normal coordinates at point \( p \):
$$\nabla_e R^a_{\ bcd}\big|_p = \partial_e R^a_{\ bcd}\big|_p = \partial_e\partial_c \Gamma^a_{db} - \partial_e\partial_d \Gamma^a_{cb}$$
Cycling over \( (e, c, d) \) and summing, each term appears twice with opposite signs, giving zero. Since the identity is tensorial, it holds in all coordinate systems.
Contracted Bianchi Identity
The differential Bianchi identity has profound consequences when contracted. Contracting the indices \( a \) with \( c \) (raising \( a \) first):
$$g^{ac}\left(\nabla_e R_{abcd} + \nabla_a R_{becd} + \nabla_b R_{eacd}\right) = 0$$
Using the symmetries of the Riemann tensor, this simplifies to:
$$\nabla_e R_{bd} - \nabla_d R_{be} + \nabla^a R_{bead} = 0$$
Contracting once more with \( g^{bd} \):
$$\nabla^b R_{be} - \nabla_e R + \nabla^a R_{ae} = 0$$
Since \( \nabla^b R_{be} = \nabla^a R_{ae} \), this gives \( 2\nabla^a R_{ae} - \nabla_e R = 0 \), which can be rewritten as:
$$\boxed{\nabla^\mu G_{\mu\nu} = 0 \qquad \text{where} \quad G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R}$$
The contracted Bianchi identity: the Einstein tensor is divergence-free
This is the mathematical foundation of the Einstein field equations. Because\( \nabla^\mu G_{\mu\nu} = 0 \) is an identity (following from the structure of the Riemann tensor alone), the equation \( G_{\mu\nu} = 8\pi G\, T_{\mu\nu} \) automatically implies \( \nabla^\mu T_{\mu\nu} = 0 \) - conservation of energy-momentum is built into the geometry of spacetime.
Kretschner Scalar and Curvature Invariants
The Kretschner scalar is a curvature invariant formed by fully contracting the Riemann tensor with itself:
$$K = R_{\alpha\beta\gamma\delta}\, R^{\alpha\beta\gamma\delta}$$
As a scalar, it is coordinate-independent and provides a genuine measure of the "strength" of curvature. Unlike the Ricci scalar, it does not vanish in vacuum. For the Schwarzschild metric:
$$K_{\text{Schw}} = \frac{48\, M^2}{r^6}$$
Diverges as \( r \to 0 \) (true singularity), finite at \( r = 2M \) (coordinate singularity)
Other important curvature invariants include:
Ricci scalar
\( R = g^{\mu\nu} R_{\mu\nu} \) - vanishes for vacuum solutions
Ricci square
\( R_{\mu\nu} R^{\mu\nu} \) - also vanishes in vacuum
Weyl square
\( C_{\alpha\beta\gamma\delta} C^{\alpha\beta\gamma\delta} \) - equals the Kretschner scalar in vacuum
Chern-Pontryagin scalar
\( {}^*R_{\alpha\beta\gamma\delta} R^{\alpha\beta\gamma\delta} \) - involves the dual Riemann tensor, sensitive to "handedness"
Worked Example: Riemann for Schwarzschild
The Schwarzschild metric in coordinates \( (t, r, \theta, \phi) \) with \( f(r) = 1 - 2M/r \):
$$ds^2 = -f\, dt^2 + f^{-1}\, dr^2 + r^2\, d\theta^2 + r^2 \sin^2\theta\, d\phi^2$$
The non-vanishing independent components of the Riemann tensor (with one index up) are:
$$R^t_{\ rtr} = -\frac{2M}{r^2(r-2M)} \qquad R^t_{\ \theta t\theta} = \frac{M}{r} \qquad R^t_{\ \phi t\phi} = \frac{M\sin^2\theta}{r}$$
$$R^r_{\ trt} = \frac{2M(r-2M)}{r^4} \qquad R^r_{\ \theta r\theta} = -\frac{M}{r} \qquad R^r_{\ \phi r\phi} = -\frac{M\sin^2\theta}{r}$$
$$R^\theta_{\ t\theta t} = -\frac{M(r-2M)}{r^4} \qquad R^\theta_{\ r\theta r} = \frac{M}{r^2(r-2M)} \qquad R^\theta_{\ \phi\theta\phi} = \frac{2M\sin^2\theta}{r}$$
$$R^\phi_{\ t\phi t} = -\frac{M(r-2M)}{r^4} \qquad R^\phi_{\ r\phi r} = \frac{M}{r^2(r-2M)} \qquad R^\phi_{\ \theta\phi\theta} = -\frac{2M}{r}$$
One can verify that contracting to form the Ricci tensor gives \( R_{\mu\nu} = 0 \) for all components, confirming that Schwarzschild is a vacuum solution. The Kretschner scalar is obtained by lowering the first index and squaring: \( K = 48M^2/r^6 \).
Worked Example: Riemann for FLRW
The flat FLRW (Friedmann-Lemaitre-Robertson-Walker) metric describing a homogeneous, isotropic expanding universe with scale factor \( a(t) \):
$$ds^2 = -dt^2 + a^2(t)\left(dr^2 + r^2\, d\theta^2 + r^2 \sin^2\theta\, d\phi^2\right)$$
The non-vanishing Christoffel symbols are \( \Gamma^0_{ij} = a\dot{a}\, h_{ij} \) and\( \Gamma^i_{0j} = H\delta^i_j \) where \( H = \dot{a}/a \) is the Hubble parameter, plus the spatial Christoffel symbols of the 3-metric.
The independent Riemann components are (using \( H = \dot{a}/a \)):
$$R^0_{\ i0j} = \frac{\ddot{a}}{a}\, h_{ij} \qquad \text{(time-space-time-space)}$$
$$R^i_{\ jkl} = H^2\left(\delta^i_k h_{jl} - \delta^i_l h_{jk}\right) \qquad \text{(purely spatial)}$$
where \( h_{ij} \) is the spatial metric. The Ricci tensor and scalar are:
$$R_{00} = -3\frac{\ddot{a}}{a}, \qquad R_{ij} = \left(a\ddot{a} + 2\dot{a}^2\right)h_{ij}$$
$$R = 6\left(\frac{\ddot{a}}{a} + \frac{\dot{a}^2}{a^2}\right) = 6\left(\dot{H} + 2H^2\right)$$
The Kretschner scalar for FLRW is:
$$K_{\text{FLRW}} = 12\left(\frac{\ddot{a}^2}{a^2} + \frac{\dot{a}^4}{a^4}\right) = 12\left(\dot{H}^2 + (2\dot{H} + 3H^2)H^2\right)$$
Note that the FLRW spacetime is conformally flat (Petrov Type O): the Weyl tensor vanishes identically. All curvature is encoded in the Ricci tensor, which is directly determined by the matter content through the Einstein equations.
Python: Symbolic Riemann Tensor
Compute the Riemann tensor symbolically for the Schwarzschild metric using SymPy. Click Run to execute the code on the server.
Riemann Tensor Calculation
PythonSymbolic computation using SymPy
Click Run to execute the Python code
Code will be executed with Python 3 on the server
Fortran: Numerical Riemann Tensor
Compute the Riemann tensor numerically at a specific point using Fortran. The code is compiled with gfortran and executed on the server.
Riemann Tensor Visualization
PythonKretschmann scalar and Riemann components vs radius
Click Run to execute the Python code
Code will be executed with Python 3 on the server
3D Spacetime Curvature Visualization
This interactive visualization shows Flamm's paraboloid - an embedding diagram that represents how the Schwarzschild spacetime curves in the vicinity of a black hole. The funnel shape illustrates how space becomes increasingly curved as you approach the event horizon.
Schwarzschild Spacetime Embedding
Flamm's paraboloid visualization of curved spacetime
Flamm's Paraboloid: This is the embedding diagram of the equatorial plane (θ = π/2) of Schwarzschild spacetime into 3D Euclidean space.
The surface is defined by: z = 2√(2M(r - 2M)) for r > 2M
The "funnel" shape shows how space curves near a black hole. The red circle marks the event horizon at r = 2M, beyond which nothing can escape.
Key Concepts Summary
Riemann tensor definition: \( R^\rho_{\ \sigma\mu\nu} = \partial_\mu \Gamma^\rho_{\nu\sigma} - \partial_\nu \Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma} \)
Symmetries: Antisymmetric in each pair, symmetric under pair exchange, satisfies the first Bianchi identity. In 4D: 20 independent components.
Geodesic deviation: \( D^2\xi^\mu/D\tau^2 = R^\mu_{\ \nu\rho\sigma} u^\nu u^\rho \xi^\sigma \) - the Riemann tensor governs tidal forces.
Decomposition: Riemann = Weyl (trace-free, vacuum curvature) + Ricci parts (local matter).
Differential Bianchi identity: \( \nabla_{[e}R_{ab]cd} = 0 \) implies\( \nabla^\mu G_{\mu\nu} = 0 \), guaranteeing energy-momentum conservation.
Kretschner scalar: \( K = R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta} \) is a coordinate-invariant measure of curvature strength (\( 48M^2/r^6 \) for Schwarzschild).
Petrov classification: Categorizes spacetimes by Weyl tensor algebraic type. Schwarzschild/Kerr are Type D; FLRW is Type O (conformally flat).