Ricci Tensor & Ricci Scalar
The fundamental measures of spacetime curvature that describe how matter and energy bend space, forming the building blocks of Einstein's field equations.
Riemann (1854) | Ricci & Levi-Civita (1900) | Foundation of General Relativity
What Do These Tensors Measure?
"The Ricci tensor measures how volumes change in curved space. The Ricci scalar is the total curvature at a point."
Riemann Tensor: Rρσμν
The fundamental curvature tensor that measures how vectors change when parallel transported around loops. Has 20 independent components in 4D spacetime.
Ricci Tensor: Rμν
A contraction of the Riemann tensor that measures how volumes of small geodesic balls differ from flat space. Has 10 independent components (symmetric).
Ricci Scalar: R
The trace of the Ricci tensor - a single number at each point that measures the total average curvature. Appears in the Einstein-Hilbert action.
Visual Understanding: Parallel Transport & Curvature
The Riemann tensor measures how vectors change when parallel transported around closed loops in curved spacetime:
When you parallel transport a vector around a closed loop in curved space, it returns rotated. This rotation measures the Riemann curvature tensor.
Key Concepts to Understand
1. From Riemann to Ricci: The Contraction
The Riemann curvature tensor Rρσμν has 4 indices and (in 4D) 20 independent components. This is too much information for most purposes, so we contract it:
This contraction preserves information about how volumes of small geodesic balls change due to curvature. The "missing" information (Riemann minus Ricci) is captured by the Weyl tensor, which describes tidal forces and shape distortion.
2. Physical Meaning: Geodesic Deviation
The Ricci tensor tells you how a cloud of freely falling particles (following geodesics) will change volume over time:
Where V is volume, τ is proper time, and uμ is the 4-velocity. If Rμν uμ uν > 0, the volume decreases - nearby geodesics converge (e.g., objects falling toward Earth).
| Tensor | Components | Physical Meaning |
|---|---|---|
| Riemann Rρσμν | 20 (in 4D) | Complete curvature: volume + shape distortion + tidal forces |
| Ricci Rμν | 10 (symmetric) | Volume distortion only (related to local matter/energy) |
| Ricci Scalar R | 1 (scalar) | Total average curvature at a point |
| Weyl Cρσμν | 10 (in 4D) | Traceless part: tidal forces (curvature even in vacuum) |
3. The Ricci Scalar in the Einstein-Hilbert Action
The simplest action for gravity is the Einstein-Hilbert action, which depends only on the Ricci scalar:
Varying this action with respect to the metric gμν produces Einstein's field equations. The Ricci scalar R is the simplest scalar curvature invariant you can build from the metric.
4. Connection to the Einstein Tensor
The Einstein field equations use the Einstein tensor Gμν, which is built from the Ricci tensor and scalar:
The "-½ R gμν" term is needed to ensure ∇μGμν = 0, which guarantees energy-momentum conservation. This is required by the Bianchi identities, geometric identities satisfied by the Riemann tensor.
Learning Resources
YouTube Video Explanations
Ricci Tensor and Ricci Scalar - eigenchris
Clear explanation of how the Ricci tensor is derived from the Riemann tensor, with visual examples.
Watch on YouTube → 18 minThe Riemann Curvature Tensor - Faculty of Khan
Detailed derivation of the Riemann tensor with emphasis on parallel transport and geodesic deviation.
Watch on YouTube → 22 minCurvature Tensors in General Relativity - XylyXylyX
Comprehensive overview of Riemann, Ricci, Weyl, and Einstein tensors with geometric intuition.
Watch on YouTube → 35 minTensor Calculus - eigenchris (Full Series)
Complete course on tensors, curvature, and differential geometry from scratch.
Watch Playlist → 33 videosArticles & Textbooks
- Wikipedia: Ricci Curvature | Riemann Curvature Tensor | Scalar Curvature
- Textbook (Beginner): "A Student's Guide to Vectors and Tensors" by Daniel Fleisch [Cambridge University Press]
- Textbook (Intermediate): "Spacetime and Geometry" by Sean Carroll - Chapter 3 (Curvature) [Book website]
- Textbook (Advanced): "Riemannian Geometry" by Manfredo do Carmo [Wikipedia]
- Online Notes: "Lecture Notes on General Relativity" by Sean Carroll (Caltech) [arXiv preprint]
- Interactive Tutorial: "Visualizing the Riemann Curvature Tensor" [Visualization tool]
Computation Tools
- GRTensorII: Symbolic tensor algebra for Maple/Mathematica [GitHub]
- SageMath Manifolds: Free tensor computation in Python [Documentation]
- Mathematica (Riemannian Geometry package): Wolfram Documentation
Key Terms & Concepts
Parallel Transport
Moving a vector along a curve while keeping it "as parallel as possible." In curved space, the vector rotates when transported around loops.
Learn more →Geodesic Deviation
The tendency of initially parallel geodesics to converge or diverge in curved spacetime. Directly measured by the Riemann tensor.
Learn more →Tidal Forces
Differential gravitational forces that stretch or compress objects. Measured by the Weyl tensor (traceless part of Riemann).
Learn more →Bianchi Identities
Geometric identities satisfied by the Riemann tensor: ∇[λRμ]νρσ = 0. Ensure ∇μGμν = 0.
Learn more →Weyl Tensor
The traceless part of Riemann: Cρσμν = Rρσμν - (Ricci terms). Describes curvature in vacuum (gravitational waves).
Learn more →Raychaudhuri Equation
Describes evolution of volume for congruences of geodesics. Shows how Rμν controls focusing of geodesics.
Learn more →Holonomy
The rotation/change of a vector after parallel transport around a closed loop. Direct manifestation of curvature.
Learn more →Contraction
Summing over paired indices (one up, one down) using the metric. Reduces rank of tensors: rank-4 → rank-2 → rank-0 (scalar).
Learn more →Connection to Principia Metaphysica
In Principia Metaphysica's 2T physics framework, the Ricci tensor and scalar generalize to higher dimensions during the dimensional reduction cascade:
26D → 13D → 6D → 4D: Curvature at Each Stage
The Ricci curvature describes how matter/energy curves space at each dimensional level:
- 26D bulk (24,2): RAB(26) with signature (24,2). Includes 2 timelike directions. Sp(2,R) gauge symmetry acts on the 2T sector.
- 13D shadow (12,1): After Sp(2,R) gauge fixing: RMN(13). The "shadow" of 26D physics in standard (n-1,1) signature.
- 6D bulk (5,1): After G₂ compactification: Rμν(6). G₂ holonomy preserves minimal SUSY. Internal curvature contributes to 4D effective theory.
- 4D observed (3,1): Rμν(4) - the standard Ricci tensor we measure. Kaluza-Klein modes from higher dimensions appear as matter fields.
The dimensional reduction process preserves the Ricci scalar in modified form:
Key insight: The hierarchy of curvature tensors (Riemann → Ricci → Scalar) parallels the dimensional reduction hierarchy. Each reduction "traces out" degrees of freedom, just as contraction reduces tensor rank. The Einstein-Hilbert action generalizes naturally to each dimensional stage.
Practice Problems
Test your understanding with these exercises:
Problem 1: 2D Sphere Curvature
For a 2D sphere of radius R, the metric is ds² = R²(dθ² + sin²θ dφ²). Calculate the Ricci scalar and show that it is constant everywhere.
Solution
The Ricci scalar for a 2D sphere is R = 2/R². It's constant (uniform curvature). This is why spheres are called "maximally symmetric" - same curvature everywhere.
Problem 2: Schwarzschild Ricci Tensor
The Schwarzschild metric describes spacetime around a spherical mass. Show that in the vacuum region (r > rs), the Ricci tensor Rμν = 0, but the Riemann tensor Rρσμν ≠ 0.
Hint
Use the vacuum Einstein equations: Rμν = 0 (no matter). But the Weyl tensor (tidal forces) is non-zero, so spacetime is still curved!
Problem 3: Trace of Ricci Tensor
Show that the Ricci scalar R = gμνRμν is the trace of the Ricci tensor. If Rμν = diag(2, -1, -1, 0) and gμν = diag(-1, 1, 1, 1), what is R?
Solution
R = g00R00 + g11R11 + g22R22 + g33R33
R = (-1)(2) + (1)(-1) + (1)(-1) + (1)(0) = -2 - 1 - 1 + 0 = -4
Problem 4: Flat Space
In Minkowski (flat) spacetime, show that all curvature tensors vanish: Rρσμν = 0, Rμν = 0, R = 0. Why does this make sense physically?
Hint
For Minkowski metric gμν = diag(-1,1,1,1), all Christoffel symbols Γλμν = 0 (in Cartesian coordinates). Therefore Riemann tensor = 0. No curvature means no gravity!
Problem 5: Einstein Tensor Identity
Verify that for the Einstein tensor Gμν = Rμν - ½Rgμν, the trace gμνGμν = -R. Why is this useful?
Solution
gμνGμν = gμν(Rμν - ½Rgμν)
= R - ½R·4 = R - 2R = -R
This allows you to solve for R from the trace of Einstein's equations: R = -8πG T (where T = trace of Tμν).
Where Ricci Curvature Is Used in PM
This foundational physics appears in the following sections of Principia Metaphysica:
Where Ricci Curvature Is Used in PM
This foundational physics appears in the following sections of Principia Metaphysica: