The Riemann Curvature Tensor

1. The Core Idea: What Does It Measure?

In a single sentence: The Riemann curvature tensor measures the intrinsic curvature of a manifold by quantifying how the second covariant derivative of a vector field fails to commute.

Let's unpack that:

• Intrinsic Curvature: This is curvature that can be detected by an observer living entirely within the manifold itself, without needing to view the space from a higher dimension (e.g., an ant on a sphere can detect it's curved by drawing triangles, without seeing the sphere from space).
• Failure of Commutation: In ordinary flat space, the order in which you take partial derivatives doesn't matter: \(\partial_x\partial_y = \partial_y\partial_x\). In a curved space, the order does matter for the covariant derivative (\(\nabla\)). The Riemann tensor captures this difference.

2. The "Taxicab" or Geometric Interpretation

Imagine you take a vector and parallel transport it (move it while keeping it "pointing in the same direction") along two different paths on a curved surface.

1. Path 1: North, then East.
2. Path 2: East, then North.

In a flat space, the final vector will be the same regardless of the path.
In a curved space, the final vector will be different depending on which path you took.

The Riemann tensor is the mathematical object that measures this path-dependence and the resulting change in the vector. A non-zero Riemann tensor implies the space is curved.

3. The Formal Definition

The Riemann curvature tensor \(R\) is a (1, 3) tensor. It is defined by the following formula, which formalizes the "failure of the second covariant derivative to commute":

\[ R(\mathbf{u}, \mathbf{v})\mathbf{w} = \nabla_{\mathbf{u}} \nabla_{\mathbf{v}} \mathbf{w} - \nabla_{\mathbf{v}} \nabla_{\mathbf{u}} \mathbf{w} - \nabla_{[\mathbf{u}, \mathbf{v}]} \mathbf{w} \]

Where:

• \(\nabla\) is the covariant derivative.
• \(\mathbf{u}, \mathbf{v}, \mathbf{w}\) are vector fields.
• \([\mathbf{u}, \mathbf{v}]\) is the Lie bracket. In a coordinate basis, this term vanishes.

In component form in a coordinate basis, this definition becomes:

\[ R^{\rho}{}_{\sigma\mu\nu} = \partial_{\mu}\Gamma^{\rho}{}_{\nu\sigma} - \partial_{\nu}\Gamma^{\rho}{}_{\mu\sigma} + \Gamma^{\rho}{}_{\mu\lambda}\Gamma^{\lambda}{}_{\nu\sigma} - \Gamma^{\rho}{}_{\nu\lambda}\Gamma^{\lambda}{}_{\mu\sigma} \]

Where:

• \(R^{\rho}{}_{\sigma\mu\nu}\) are the components of the Riemann tensor.
• \(\partial_{\mu}\) is the partial derivative.
• \(\Gamma^{\rho}{}_{\mu\nu}\) are the Christoffel symbols.

4. Key Properties and Derived Quantities

The Riemann tensor has several important symmetries and is used to construct other crucial measures of curvature:

1. Symmetries: It is antisymmetric in its first two and last two indices, and symmetric under the exchange of the first and last pair of indices. These symmetries reduce the number of independent components in 4-dimensional spacetime from 256 to just 20.
2. Ricci Tensor: A contraction of the Riemann tensor. \[ R_{\mu\nu} = R^{\lambda}{}_{\mu\lambda\nu} \]
3. Ricci Scalar: A further contraction of the Ricci tensor. \[ R = R^{\mu}{}_{\mu} = g^{\mu\nu}R_{\mu\nu} \]
4. Einstein Tensor: A combination of the Ricci tensor and scalar that is divergence-free (crucial for General Relativity). \[ G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R \]

5. Its Role in Physics: General Relativity

The Riemann tensor is the foundation of Einstein's theory of General Relativity (GR). In GR:

• Mass and Energy tell Spacetime how to curve. This curvature is described by the Riemann tensor.
• Curved Spacetime tells Matter and Light how to move.

The famous Einstein Field Equations are: \[ G_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} \] Since the Einstein tensor \(G_{\mu\nu}\) is built from contractions of the Riemann tensor, the Riemann tensor is the ultimate source of the geometry in this equation.

Summary

Aspect	Description
What it is	A (1, 3) tensor that measures the intrinsic curvature of a manifold.
What it measures	The failure of parallel transport to be path-independent.
Mathematical Form	\(R^{\rho}{}_{\sigma\mu\nu} = \partial_{\mu}\Gamma^{\rho}{}_{\nu\sigma} - \partial_{\nu}\Gamma^{\rho}{}_{\mu\sigma} + \Gamma^{\rho}{}_{\mu\lambda}\Gamma^{\lambda}{}_{\nu\sigma} - \Gamma^{\rho}{}_{\nu\lambda}\Gamma^{\lambda}{}_{\mu\sigma}\)
If Zero	The manifold is flat (Euclidean/Minkowski space).
If Non-Zero	The manifold is curved (e.g., a sphere, Schwarzschild spacetime).
Key Use	The fundamental object describing gravity and geometry in General Relativity.