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Chapter 7: Riemann Curvature Tensor
Definition and Properties
The Riemann curvature tensor is the central mathematical object of Riemannian geometry and general relativity. It fully characterizes the intrinsic curvature of spacetime, encoding how vectors rotate during parallel transport around infinitesimal loops. A spacetime is flat if and only if the Riemann tensor vanishes everywhere.
Curvature via Parallel Transport
On a curved manifold, parallel transporting a vector around a closed loop generally changes the vector. This failure of a vector to return to its original value after parallel transport around an infinitesimal loop is precisely what the Riemann tensor measures.
Consider a vector \( V^\rho \) parallel transported around an infinitesimal parallelogram spanned by coordinate displacements \( \delta x^\mu \) and \( \delta x^\nu \). The change in the vector after completing the loop is:
$$\delta V^\rho = R^\rho_{\ \sigma\mu\nu}\, V^\sigma\, \delta x^\mu\, \delta x^\nu$$
Change in vector = Riemann tensor contracted with the vector and the loop area
This is the geometric definition of the Riemann tensor: it is the obstruction to path-independence of parallel transport, and therefore the fundamental measure of curvature.
Definition via Commutator
Equivalently, the Riemann tensor arises from the non-commutativity of covariant derivatives. Acting on a vector field \( V^\rho \):
$$[\nabla_\mu, \nabla_\nu]\, V^\rho = R^\rho_{\ \sigma\mu\nu}\, V^\sigma$$
Curvature = failure of covariant derivatives to commute
We can verify this by expanding the commutator explicitly. Start with:
$$\nabla_\mu \nabla_\nu V^\rho = \partial_\mu(\nabla_\nu V^\rho) - \Gamma^\lambda_{\mu\nu}\nabla_\lambda V^\rho + \Gamma^\rho_{\mu\sigma}\nabla_\nu V^\sigma$$
Expanding \( \nabla_\nu V^\rho = \partial_\nu V^\rho + \Gamma^\rho_{\nu\sigma} V^\sigma \) and antisymmetrizing in \( \mu \) and \( \nu \), the second partial derivatives of \( V^\rho \) cancel, and we obtain terms involving derivatives of Christoffel symbols and products of Christoffel symbols. The result is:
$$[\nabla_\mu, \nabla_\nu] V^\rho = \left(\partial_\mu \Gamma^\rho_{\nu\sigma} - \partial_\nu \Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma}\right) V^\sigma$$
This is only valid in a coordinate (torsion-free) basis. In the presence of torsion, there would be an additional term involving the torsion tensor.
Component Expression
Reading off from the commutator, the Riemann tensor in terms of the Christoffel symbols is:
$$R^\rho_{\ \sigma\mu\nu} = \partial_\mu \Gamma^\rho_{\nu\sigma} - \partial_\nu \Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma}$$
This expression has a clear structure: two derivative terms and two quadratic terms. The first two terms are the "curl" of the connection, while the last two are "non-Abelian" corrections arising because the connection is matrix-valued.
The fully covariant (all indices lowered) version is obtained by contracting with the metric:
$$R_{\rho\sigma\mu\nu} = g_{\rho\alpha}\, R^\alpha_{\ \sigma\mu\nu}$$
Expanding this explicitly:
$$R_{\rho\sigma\mu\nu} = \frac{1}{2}\left(\partial_\mu\partial_\sigma g_{\rho\nu} - \partial_\mu\partial_\rho g_{\sigma\nu} - \partial_\nu\partial_\sigma g_{\rho\mu} + \partial_\nu\partial_\rho g_{\sigma\mu}\right) + g_{\alpha\beta}\left(\Gamma^\alpha_{\mu\sigma}\Gamma^\beta_{\nu\rho} - \Gamma^\alpha_{\nu\sigma}\Gamma^\beta_{\mu\rho}\right)$$
This form makes the symmetries of the tensor more transparent, as we shall see next.
Symmetry Properties
The fully covariant Riemann tensor \( R_{\rho\sigma\mu\nu} \) possesses several important symmetries that drastically reduce the number of independent components.
1. Antisymmetry in First Pair
$$R_{\rho\sigma\mu\nu} = -R_{\sigma\rho\mu\nu}$$
The tensor is antisymmetric under exchange of the first two indices. This follows from the antisymmetry of the covariant derivative commutator and metricity.
2. Antisymmetry in Second Pair
$$R_{\rho\sigma\mu\nu} = -R_{\rho\sigma\nu\mu}$$
The tensor is antisymmetric under exchange of the last two indices. This is manifest from the definition: swapping \( \mu \leftrightarrow \nu \) flips the sign of the commutator.
3. Pair Exchange Symmetry
$$R_{\rho\sigma\mu\nu} = R_{\mu\nu\rho\sigma}$$
The tensor is symmetric under exchange of the two pairs of indices. This is not obvious from the Christoffel symbol expression but follows from the metric compatibility condition \( \nabla_\lambda g_{\mu\nu} = 0 \).
4. First Bianchi Identity (Algebraic)
$$R_{\rho[\sigma\mu\nu]} = 0 \quad \Longleftrightarrow \quad R_{\rho\sigma\mu\nu} + R_{\rho\mu\nu\sigma} + R_{\rho\nu\sigma\mu} = 0$$
The cyclic sum over the last three indices vanishes. For torsion-free connections, this is equivalent to the statement that the exterior covariant derivative of the torsion vanishes.
Number of Independent Components
With all four symmetries taken together, we can count the number of independent components of the Riemann tensor in \( n \) dimensions.
The antisymmetry in each pair means \( R_{\rho\sigma\mu\nu} \) is like a matrix\( M_{AB} \) where \( A = [\rho\sigma] \) and \( B = [\mu\nu] \) are antisymmetric index pairs. The number of such pairs is \( \binom{n}{2} = \frac{n(n-1)}{2} \). The pair symmetry makes \( M_{AB} = M_{BA} \), giving \( \binom{N+1}{2} \) components where \( N = \frac{n(n-1)}{2} \). The first Bianchi identity provides additional constraints.
The final result for the number of independent components is:
$$\text{Independent components} = \frac{n^2(n^2 - 1)}{12}$$
n = 2
1
Gaussian curvature only
n = 3
6
Same as Ricci tensor (no Weyl in 3D)
n = 4
20
Relevant for GR: 10 Weyl + 10 Ricci
In 4D spacetime, the 20 independent components decompose into 10 components of the Weyl tensor (encoding gravitational radiation and tidal effects in vacuum) and 10 components of the Ricci tensor (encoding the local matter content through the Einstein equations). This decomposition is explored in detail on the next page.
Special Case: 2 Dimensions
In 2 dimensions, there is only one independent component. The entire Riemann tensor is determined by a single scalar, the Gaussian curvature \( K \):
$$R_{\rho\sigma\mu\nu} = K\left(g_{\rho\mu}g_{\sigma\nu} - g_{\rho\nu}g_{\sigma\mu}\right)$$
For a sphere of radius \( a \), \( K = 1/a^2 \) everywhere (positive curvature). For the hyperbolic plane, \( K = -1/a^2 \) (negative curvature). For a flat plane, \( K = 0 \).
The Gauss-Bonnet theorem provides a deep topological constraint: for a closed 2-surface,
$$\int K\, dA = 2\pi\chi$$
where \( \chi \) is the Euler characteristic of the surface
Special Case: 3 Dimensions
In 3 dimensions, the Riemann tensor has 6 independent components, the same number as the symmetric Ricci tensor \( R_{\mu\nu} \). This means the Riemann tensor is entirely determined by the Ricci tensor:
$$R_{\rho\sigma\mu\nu} = g_{\rho\mu}R_{\sigma\nu} - g_{\rho\nu}R_{\sigma\mu} - g_{\sigma\mu}R_{\rho\nu} + g_{\sigma\nu}R_{\rho\mu} - \frac{R}{2}\left(g_{\rho\mu}g_{\sigma\nu} - g_{\rho\nu}g_{\sigma\mu}\right)$$
Consequently, the Weyl tensor vanishes identically in 3 dimensions. There is no "free" gravitational field - all curvature is determined by local matter content. This is why gravity in 3D spacetime (2+1 dimensions) has no propagating degrees of freedom (no gravitational waves).