Chapter 7: Riemann Curvature Tensor
The Riemann tensor is the mathematical object that fully characterizes spacetime curvature. It measures how vectors rotate when parallel transported around infinitesimal loops, encoding all gravitational effects in general relativity.
Definition
The Riemann tensor arises from the non-commutativity of covariant derivatives:
\( [\nabla_\mu, \nabla_\nu] V^\rho = R^\rho_{\;\sigma\mu\nu} V^\sigma \)
Curvature = failure of covariant derivatives to commute
In terms of Christoffel symbols:
\( R^\rho_{\;\sigma\mu\nu} = \partial_\mu \Gamma^\rho_{\nu\sigma} - \partial_\nu \Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma} \)
Symmetry Properties
Antisymmetry in Last Two Indices
\( R^\rho_{\;\sigma\mu\nu} = -R^\rho_{\;\sigma\nu\mu} \)
Antisymmetry in First Two Indices (lowered)
\( R_{\rho\sigma\mu\nu} = -R_{\sigma\rho\mu\nu} \)
Pair Exchange Symmetry
\( R_{\rho\sigma\mu\nu} = R_{\mu\nu\rho\sigma} \)
First Bianchi Identity
\( R^\rho_{\;\sigma\mu\nu} + R^\rho_{\;\mu\nu\sigma} + R^\rho_{\;\nu\sigma\mu} = 0 \)
These symmetries reduce the 4⁴ = 256 components to just 20 independent components in 4D.
Physical Interpretation
Geodesic Deviation
\( \frac{D^2 \xi^\mu}{d\tau^2} = R^\mu_{\;\nu\rho\sigma} u^\nu u^\sigma \xi^\rho \)
Tidal forces between nearby geodesics
Holonomy
Rotation of vector around small loop ∝ R × (area enclosed)
Tidal Forces
Stretching and squeezing of extended bodies near massive objects
Spacetime Flatness
\( R^\rho_{\;\sigma\mu\nu} = 0 \) everywhere ⟺ spacetime is flat
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.
Numerical Riemann Tensor
FortranHigh-precision numerical computation
Click Run to execute the Fortran code
Code will be compiled with gfortran and executed on the server
Kretschmann Scalar
The Kretschmann scalar is a curvature invariant formed by contracting the Riemann tensor with itself:
\( K = R_{\alpha\beta\gamma\delta} R^{\alpha\beta\gamma\delta} \)
For the Schwarzschild metric:
\( K = \frac{48M^2}{r^6} \)
Diverges at r = 0 (true singularity), finite at r = 2M (coordinate singularity)
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.