The Riemann Curvature Tensor

The Riemann tensor is the mathematical object that measures the intrinsic curvature of spacetime - the thing that distinguishes genuine gravity from mere coordinate effects.

Why Curvature?

Christoffel symbols can be non-zero even in flat space (like spherical coordinates). To detect true curvature, we need something that is:

A tensor (coordinate-independent)
Zero in flat spacetime regardless of coordinates
Non-zero only when there is genuine curvature

The Riemann tensor provides exactly this!

Definition

The Riemann curvature tensor is defined as:

Riemann Tensor

R^\rho_{\phantom{\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}

It can also be defined through the commutator of covariant derivatives:

Commutator Definition

[\nabla_\mu, \nabla_\nu] V^\rho = R^\rho_{\phantom{\rho}\sigma\mu\nu} V^\sigma

This shows that the order of covariant differentiation matters in curved space!

Symmetries

The Riemann tensor has important symmetries (using the fully covariant form $R_{\rho\sigma\mu\nu} = g_{\rho\lambda} R^\lambda_{\phantom{\lambda}\sigma\mu\nu}$ ):

Antisymmetry

R_{\rho\sigma\mu\nu} = -R_{\sigma\rho\mu\nu} = -R_{\rho\sigma\nu\mu}

Pair Symmetry

R_{\rho\sigma\mu\nu} = R_{\mu\nu\rho\sigma}

Bianchi Identity (First)

R_{\rho\sigma\mu\nu} + R_{\rho\mu\nu\sigma} + R_{\rho\nu\sigma\mu} = 0

These symmetries reduce the number of independent components from 256 to just 20 in 4D spacetime.

The Ricci Tensor

Contracting the Riemann tensor gives the Ricci tensor:

Ricci Tensor

R_{\mu\nu} = R^\rho_{\phantom{\rho}\mu\rho\nu} = g^{\rho\sigma} R_{\rho\mu\sigma\nu}

The Ricci tensor is symmetric ( $R_{\mu\nu} = R_{\nu\mu}$ ) and has 10 independent components - exactly what we need for the 10 components of the metric!

The Ricci Scalar

Further contraction gives the Ricci scalar (or scalar curvature):

Ricci Scalar

R = g^{\mu\nu} R_{\mu\nu}

This single number characterizes the "total" curvature at a point. For a 2D surface, it's proportional to the Gaussian curvature.

The Einstein Tensor

The Einstein tensor is the key combination that appears in Einstein's field equations:

Einstein Tensor

G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R

It satisfies the contracted Bianchi identity:

Contracted Bianchi Identity

\nabla_\mu G^{\mu\nu} = 0

This identity is crucial - it ensures that Einstein's equations are consistent with the conservation of energy-momentum!

Physical Interpretation

The Riemann tensor describes tidal forces. Imagine two nearby particles in free fall:

Geodesic Deviation

\frac{D^2 \xi^\mu}{d\tau^2} = -R^\mu_{\phantom{\mu}\nu\rho\sigma} u^\nu u^\rho \xi^\sigma

where $\xi^\mu$ is the separation vector and $u^\mu$ is the 4-velocity. This geodesic deviation equation shows that curvature causes nearby geodesics to accelerate toward or away from each other.

Parallel Transport on Curved Surfaces

Vectors change direction when parallel transported around a closed loop on a curved surface. This is how curvature is detected!

Curvature: 1.0

Positive = saddle shape, Negative = bowl shape, Zero = flat

Show parallel transported vectors

Key Insight

The Riemann tensor answers the question: "Is spacetime truly curved, or are we just using curved coordinates?" If $R^\rho_{\phantom{\rho}\sigma\mu\nu} = 0$ everywhere, spacetime is flat regardless of what coordinates you use. Any non-zero component signals genuine curvature that cannot be transformed away.

← Christoffel Symbols Einstein Equations →