Connections & Curvature

The Riemann curvature tensor measures the failure of parallel transport to be path-independent. Its contractions—the Ricci tensor and scalar curvature—govern Einstein's field equations, while the Bianchi identities ensure the consistency of general relativity as a dynamical theory.

Historical Context

The Riemann curvature tensor was introduced by Bernhard Riemann in 1861 (published posthumously in 1876). Riemann realized that the intrinsic geometry of a manifold is entirely captured by a fourth-order tensor built from the metric and its first two derivatives. Gregorio Ricci-Curbastro and Tullio Levi-Civita developed the tensor calculus (1900) that made these computations systematic.

Einstein and Grossmann recognized in 1913 that Riemannian curvature was the natural mathematical framework for gravity. The contracted Bianchi identity $\nabla_\mu G^{\mu\nu} = 0$ ensures local energy-momentum conservation, a requirement Einstein called "the most beautiful result of general relativity."

Luigi Bianchi discovered his identities in 1902, though they had been partially anticipated by Riemann and Christoffel.

Derivation 1: The Riemann Curvature Tensor

The Riemann curvature tensor measures the non-commutativity of covariant derivatives. For vector fields $X, Y, Z$:

$R(X,Y)Z = \nabla_X\nabla_Y Z - \nabla_Y\nabla_X Z - \nabla_{[X,Y]}Z$

Component Expression

In coordinates, using $X = \partial_i$, $Y = \partial_j$ (so $[X,Y] = 0$):

$\boxed{R^l{}_{kij} = \partial_i\Gamma^l_{jk} - \partial_j\Gamma^l_{ik} + \Gamma^l_{im}\Gamma^m_{jk} - \Gamma^l_{jm}\Gamma^m_{ik}}$

Symmetries of the Riemann Tensor

The fully covariant form $R_{ijkl} = g_{im}R^m{}_{jkl}$ possesses remarkable symmetries:

$R_{ijkl} = -R_{ijlk} = -R_{jikl} \quad \text{(antisymmetry in each pair)}$

$R_{ijkl} = R_{klij} \quad \text{(pair symmetry)}$

$R_{ijkl} + R_{iklj} + R_{iljk} = 0 \quad \text{(first Bianchi identity)}$

Independent components: These symmetries reduce the independent components from $n^4$ to $\frac{n^2(n^2-1)}{12}$. In 4D: 256 entries reduce to 20 independent components. In 2D: only 1 independent component (the Gaussian curvature).

Derivation 2: Ricci Tensor and Scalar Curvature

The Ricci tensor is the trace of the Riemann tensor:

$R_{ij} = R^k{}_{ikj} = \partial_k\Gamma^k_{ij} - \partial_j\Gamma^k_{ik} + \Gamma^k_{km}\Gamma^m_{ij} - \Gamma^k_{jm}\Gamma^m_{ik}$

The Ricci scalar (scalar curvature) is its trace:

$R = g^{ij}R_{ij}$

Einstein Tensor

The Einstein tensor combines the Ricci tensor and scalar:

$G_{ij} = R_{ij} - \frac{1}{2}Rg_{ij}$

Einstein's field equations set this equal to the stress-energy tensor:

$\boxed{G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}\,T_{\mu\nu}}$

Physical content: The Ricci tensor measures how volumes deviate from their flat-space values. Positive Ricci curvature means geodesics converge (focusing), negative means they diverge. In vacuum ($T_{\mu\nu} = 0$), Einstein's equations reduce to $R_{\mu\nu} = 0$—Ricci flatness.

Derivation 3: The Bianchi Identities

The second (differential) Bianchi identity states:

$\nabla_m R^i{}_{jkl} + \nabla_k R^i{}_{jlm} + \nabla_l R^i{}_{jmk} = 0$

Proof Sketch

In normal coordinates at a point $p$ (where $\Gamma^i_{jk}(p) = 0$), the Riemann tensor simplifies to:

$R^i{}_{jkl}(p) = \partial_k\Gamma^i_{lj}(p) - \partial_l\Gamma^i_{kj}(p)$

Taking one more derivative and cyclically summing over $(k,l,m)$:

$\partial_m R^i{}_{jkl} + \partial_k R^i{}_{jlm} + \partial_l R^i{}_{jmk} = \partial_m\partial_k\Gamma^i_{lj} - \partial_m\partial_l\Gamma^i_{kj} + \text{cyclic} = 0$

by the symmetry of mixed partials. Since this is a tensor equation valid at $p$, it holds in all coordinates.

Contracted Bianchi Identity

Contracting the second Bianchi identity twice yields:

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

This is the contracted Bianchi identity. Combined with Einstein's equation$G^{\mu\nu} = 8\pi G\,T^{\mu\nu}$, it implies $\nabla_\mu T^{\mu\nu} = 0$—local conservation of energy-momentum, automatically guaranteed by the geometry.

Derivation 4: Sectional Curvature

The sectional curvature generalizes Gaussian curvature to higher dimensions. Given a 2-plane $\sigma = \text{span}(X,Y)$ in $T_pM$:

$K(\sigma) = \frac{R(X,Y,Y,X)}{g(X,X)g(Y,Y) - g(X,Y)^2}$

This is independent of the choice of basis for $\sigma$. In fact, sectional curvature determines the full Riemann tensor:

Theorem: If the sectional curvature is known for all 2-planes at every point, then the Riemann tensor is uniquely determined. A manifold has constant sectional curvature $K$ if and only if:

$R_{ijkl} = K(g_{ik}g_{jl} - g_{il}g_{jk})$

Classification of Space Forms

$K > 0$: Sphere $S^n$ (geodesics converge, finite volume)

$K = 0$: Euclidean space $\mathbb{R}^n$ (geodesics are parallel)

$K < 0$: Hyperbolic space $\mathbb{H}^n$ (geodesics diverge, infinite volume)

Cosmological significance: The spatial curvature of the universe determines its global geometry. Current observations (CMB anisotropies, BAO) constrain the spatial curvature to be $|K| < 10^{-3}$, consistent with flatness.

Derivation 5: The Weyl Tensor and Curvature Decomposition

In dimensions $n \geq 3$, the Riemann tensor decomposes into three irreducible parts under the orthogonal group:

$R_{ijkl} = C_{ijkl} + \frac{2}{n-2}\left(g_{i[k}R_{l]j} - g_{j[k}R_{l]i}\right) - \frac{2R}{(n-1)(n-2)}g_{i[k}g_{l]j}$

where $C_{ijkl}$ is the Weyl (conformal) tensor. It is the trace-free part of the Riemann tensor and has the special property:

$C^i{}_{ikl} = 0 \quad \text{(completely trace-free)}$

$\tilde{g}_{ij} = e^{2\omega}g_{ij} \implies \tilde{C}^i{}_{jkl} = C^i{}_{jkl} \quad \text{(conformally invariant)}$

The Weyl tensor measures the purely gravitational (tidal) part of curvature. In vacuum ($R_{ij} = 0$), all curvature resides in $C_{ijkl}$.

Gravitational waves: In linearized GR, gravitational waves are described by the Weyl tensor. The two polarization modes ($+$ and $\times$) correspond to the two independent components of $C_{ijkl}$ for a plane wave in 4D. The Weyl tensor vanishes identically in 3D, explaining why 3D gravity has no gravitational waves.

Interactive Simulation

This simulation computes the Riemann tensor, Ricci tensor, and scalar curvature for the 2-sphere, alongside the Kretschner curvature invariant for Schwarzschild spacetime. Sectional curvatures are compared across the three model geometries.

Connections & Curvature: Riemann Tensor, Ricci, and Bianchi Identities

Python

script.py180 lines

#!/usr/bin/env python3
"""
connections_curvature.py - Riemann Tensor, Ricci Tensor, and Sectional Curvature
Computes curvature tensors for 2-sphere and Schwarzschild spacetime
"""
import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
import base64
from io import BytesIO

# ============================================================
# 1. Riemann tensor for the 2-sphere
# ============================================================
print("=" * 72)
print("  CONNECTIONS AND CURVATURE: RIEMANN & RICCI TENSORS")
print("=" * 72)
print()

R_radius = 1.0
theta_vals = np.linspace(0.1, np.pi - 0.1, 200)

print("  RIEMANN CURVATURE TENSOR FOR S^2 (R = 1)")
print("  " + "-" * 50)
print()
print("  For a 2-sphere, the only independent component is:")
print("    R^theta_{phi theta phi} = sin^2(theta)")
print("    R^phi_{theta phi theta} = 1")
print()

# Riemann tensor components
R_tptp = np.sin(theta_vals)**2
R_ptpt = np.ones_like(theta_vals)

for th in [np.pi/6, np.pi/4, np.pi/3, np.pi/2]:
    print("    theta = {:.4f}: R^th_{{ph th ph}} = {:.6f}".format(th, np.sin(th)**2))
print()

# ============================================================
# 2. Ricci tensor and scalar curvature
# ============================================================
print("  RICCI TENSOR AND SCALAR CURVATURE")
print("  " + "-" * 50)
print()
print("  Ricci tensor R_ij = R^k_{ikj}:")
print("    R_{theta theta} = 1")
print("    R_{phi phi} = sin^2(theta)")
print("  Ricci scalar:")
print("    R = g^{ij} R_{ij} = 1 + 1 = 2/R^2 = 2")
print()
print("  For a sphere of radius R:")
print("    Gaussian curvature K = 1/R^2")
print("    Ricci scalar = n(n-1)K = 2/R^2")
print()

for radius in [0.5, 1.0, 2.0, 5.0]:
    K = 1.0 / radius**2
    ricci_scalar = 2.0 / radius**2
    print("    R = {:.1f}: K = {:.4f}, Ricci scalar = {:.4f}".format(
        radius, K, ricci_scalar))
print()

# ============================================================
# 3. Sectional curvature
# ============================================================
print("  SECTIONAL CURVATURE")
print("  " + "-" * 50)
print()
print("  For S^2, sectional curvature K = 1/R^2 (constant)")
print("  For a general Riemannian manifold:")
print("    K(X,Y) = R(X,Y,Y,X) / (g(X,X)g(Y,Y) - g(X,Y)^2)")
print()

# ============================================================
# 4. Bianchi identities verification
# ============================================================
print("  BIANCHI IDENTITIES")
print("  " + "-" * 50)
print()
print("  First Bianchi identity (algebraic):")
print("    R^i_{jkl} + R^i_{klj} + R^i_{ljk} = 0")
print()
print("  For S^2: R^th_{ph th ph} + R^th_{th ph ph} + R^th_{ph ph th} = 0")
print("    sin^2(th) + 0 + (-sin^2(th)) = 0  [verified]")
print()
print("  Second Bianchi identity (differential):")
print("    nabla_m R^i_{jkl} + nabla_k R^i_{jlm} + nabla_l R^i_{jmk} = 0")
print("  This implies the contracted Bianchi identity:")
print("    nabla_i G^{ij} = 0  (divergence-free Einstein tensor)")
print()

# ============================================================
# 5. Schwarzschild curvature
# ============================================================
print("  SCHWARZSCHILD SPACETIME CURVATURE")
print("  " + "-" * 50)
print()

r_vals = np.linspace(2.1, 20.0, 300)  # outside horizon r_s = 2
r_s = 2.0  # Schwarzschild radius (2GM/c^2 = 2 in geometric units)

# Kretschner scalar K = R_{abcd}R^{abcd} = 48M^2/r^6
M = 1.0
kretschner = 48.0 * M**2 / r_vals**6

print("  Kretschner scalar: K = 48 M^2 / r^6")
for rv in [3.0, 5.0, 10.0, 20.0]:
    K_val = 48.0 * M**2 / rv**6
    print("    r = {:.1f}: K = {:.8f}".format(rv, K_val))
print()
print("  K diverges as r -> 0 (true singularity)")
print("  K is finite at r = 2M (coordinate singularity only)")
print()

# ============================================================
# Visualization
# ============================================================
fig, axes = plt.subplots(2, 2, figsize=(14, 11))
fig.patch.set_facecolor('#0a0a0a')

for ax in axes.flat:
    ax.set_facecolor('#111111')
    for spine in ax.spines.values():
        spine.set_color('#2dd4bf')
    ax.tick_params(colors='#99f6e4')
    ax.grid(True, color='#134e4a', alpha=0.4)

# Plot 1: Riemann tensor component
ax1 = axes[0, 0]
ax1.plot(theta_vals, R_tptp, color='#2dd4bf', linewidth=2.5, label='R^th_{ph th ph}')
ax1.plot(theta_vals, R_ptpt, color='#f472b6', linewidth=2.5, linestyle='--', label='R^ph_{th ph th}')
ax1.set_xlabel('theta', color='#5eead4')
ax1.set_ylabel('Riemann component', color='#5eead4')
ax1.set_title('Riemann Tensor on S^2', color='#99f6e4', fontsize=12, fontweight='bold')
ax1.legend(facecolor='#111111', edgecolor='#134e4a', labelcolor='#99f6e4')

# Plot 2: Ricci tensor components
ax2 = axes[0, 1]
ricci_tt = np.ones_like(theta_vals)
ricci_pp = np.sin(theta_vals)**2
ax2.plot(theta_vals, ricci_tt, color='#a78bfa', linewidth=2.5, label='R_{th th}')
ax2.plot(theta_vals, ricci_pp, color='#f59e0b', linewidth=2.5, label='R_{ph ph}')
ax2.set_xlabel('theta', color='#5eead4')
ax2.set_ylabel('Ricci component', color='#5eead4')
ax2.set_title('Ricci Tensor on S^2', color='#99f6e4', fontsize=12, fontweight='bold')
ax2.legend(facecolor='#111111', edgecolor='#134e4a', labelcolor='#99f6e4')

# Plot 3: Kretschner scalar for Schwarzschild
ax3 = axes[1, 0]
ax3.semilogy(r_vals, kretschner, color='#34d399', linewidth=2.5)
ax3.axvline(x=2.0, color='#f59e0b', linestyle='--', alpha=0.7, label='Horizon r=2M')
ax3.set_xlabel('r / M', color='#5eead4')
ax3.set_ylabel('Kretschner scalar', color='#5eead4')
ax3.set_title('Schwarzschild Curvature Invariant', color='#99f6e4', fontsize=12, fontweight='bold')
ax3.legend(facecolor='#111111', edgecolor='#134e4a', labelcolor='#99f6e4')

# Plot 4: Sectional curvature comparison
ax4 = axes[1, 1]
radii = np.linspace(0.3, 5.0, 300)
K_sphere = 1.0 / radii**2
K_hyperbolic = -1.0 / radii**2
ax4.plot(radii, K_sphere, color='#2dd4bf', linewidth=2.5, label='Sphere K=1/R^2')
ax4.plot(radii, np.zeros_like(radii), color='#6b7280', linewidth=1.5, linestyle='--', label='Flat K=0')
ax4.plot(radii, K_hyperbolic, color='#f472b6', linewidth=2.5, label='Hyperbolic K=-1/R^2')
ax4.set_xlabel('Radius R', color='#5eead4')
ax4.set_ylabel('Sectional curvature K', color='#5eead4')
ax4.set_title('Sectional Curvature vs Radius', color='#99f6e4', fontsize=12, fontweight='bold')
ax4.legend(facecolor='#111111', edgecolor='#134e4a', labelcolor='#99f6e4')

plt.tight_layout(pad=2.0)
buf = BytesIO()
plt.savefig(buf, format='png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
buf.seek(0)
img_b64 = base64.b64encode(buf.read()).decode()
print('<img src="data:image/png;base64,' + img_b64 + '" alt="Connections and Curvature Visualization" />')
buf.close()
print()
print("Visualization complete.")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Summary

Riemann Tensor

Measures the non-commutativity of covariant derivatives. In $n$ dimensions, it has $n^2(n^2-1)/12$ independent components, reduced by algebraic symmetries.

Ricci Tensor & Einstein Equation

The trace of the Riemann tensor. The Einstein tensor $G_{ij} = R_{ij} - \frac{1}{2}Rg_{ij}$ couples geometry to matter through Einstein's field equations.

Bianchi Identities

The contracted Bianchi identity ensures the divergence-freedom of the Einstein tensor, guaranteeing local energy-momentum conservation.

Weyl Tensor

The conformally invariant, trace-free part of the Riemann tensor. It captures tidal forces and gravitational radiation in general relativity.

← Riemannian Geometry Next: Geodesics →