Calculator Input
This calculator focuses on 2D diagonal metrics of the form ds² = E(x,y)dx² + G(x,y)dy². Choose a built-in geometry or enter a custom polynomial metric.
Example Data Table
| Example | Metric | Suggested Inputs | Expected Curvature Behavior |
|---|---|---|---|
| Flat Plane | Euclidean: ds² = dx² + dy² | x = 1, y = 1 | K = 0 and the tensor vanishes. |
| Round Sphere | Sphere: ds² = R²dθ² + R²sin²(θ)dφ² | x = 1.0, y = 0.6, R = 2 | Positive curvature, with K = 1 / R² = 0.25. |
| Hyperbolic Model | Half-plane: ds² = L²(dx² + dy²)/y² | x = 0.5, y = 1.2, L = 1.5 | Negative curvature, with K = -1 / L². |
| Polar Coordinates | Plane in polar form: ds² = dr² + r²dθ² | x = 2, y = 1 | Curvilinear coordinates, but still flat away from r = 0. |
Formula Used
ds² = E(x,y)dx² + G(x,y)dy²
Γijk = (1/2) gim(∂jgmk + ∂kgmj - ∂mgjk)
Rijkl = ∂kΓijl - ∂lΓijk + ΓimkΓmjl - ΓimlΓmjk
Ricjl = Rijil, R = gjlRicjl
K = R1212 / det(g)
The page uses finite differences to estimate derivatives when evaluating built-in and custom metrics numerically. For smooth inputs and non-singular points, the results closely track the expected local curvature behavior.
How to Use This Calculator
- Select a built-in metric or choose the custom polynomial option.
- Enter the evaluation coordinates x and y.
- Provide any required geometric parameters, such as radius or scale.
- Set a plotting range and numerical step size.
- Press Calculate Tensor to display results above the form.
- Review the metric, Christoffel symbols, Riemann tensor, Ricci tensor, and curvature values.
- Use the CSV and PDF buttons to export the report.
- Inspect the Plotly heatmap to see how curvature changes nearby.
FAQs
1) What does the Riemann curvature tensor measure?
It measures how a space bends by comparing how vectors change after parallel transport around tiny loops. In curved spaces, the loop fails to close in the same way as in flat geometry.
2) When does the riemann curvature tensor vanish?
It vanishes when the space is locally flat. In Euclidean space, or any region isometric to it, every component becomes zero in appropriate coordinates because parallel transport around tiny loops produces no curvature effect.
3) Why can polar coordinates still give zero curvature?
Polar coordinates curve the coordinate grid, not the underlying plane. The Christoffel symbols can be nonzero while the Riemann tensor still vanishes, showing that the geometry itself remains flat.
4) Why does the sphere metric give positive curvature?
A sphere bends the same way in every tangent direction. Tiny geodesic triangles have angle sums greater than 180°, and that geometric excess corresponds to positive Gaussian curvature everywhere away from coordinate singularities.
5) Why does the hyperbolic metric give negative curvature?
Hyperbolic geometry is saddle-shaped at every point. Geodesics spread apart faster than they do in flat space, and triangles have angle sums less than 180°, which signals negative curvature.
6) What is the difference between Gaussian and scalar curvature here?
For a 2D surface, scalar curvature equals twice the Gaussian curvature. Gaussian curvature is the intrinsic surface curvature, while scalar curvature is the contracted curvature invariant derived from the Ricci tensor.
7) Why can results become unstable near singular points?
At poles, the origin of polar coordinates, or metric degeneracies, one or more metric terms can approach zero. Numerical differentiation then becomes sensitive, so small step changes produce larger relative errors.
8) Can I use this page for any metric tensor?
This version targets 2D diagonal metrics and several built-in geometries. It is ideal for learning, checking examples, and exploring local curvature, but it is not a full symbolic tensor engine for arbitrary dimensions.