Calculator inputs
The calculator evaluates two-dimensional conformal metrics of the form gij = e2φ(x,y)δij. That keeps the interface practical while still showing real Ricci geometry.
Example data table
| Preset | Point (x,y) | Key inputs | Ricci scalar R | Interpretation |
|---|---|---|---|---|
| Flat Euclidean plane | (0.00, 0.00) | φ = 0 | 0.000000 | No intrinsic curvature. |
| Stereographic sphere | (0.40, 0.20) | L = 1.00 | 2.000000 | Constant positive curvature. |
| Poincare disk | (0.50, 0.50) | L = 2.00 | -0.500000 | Constant negative curvature. |
| Gaussian curvature bump | (0.00, 0.00) | A = 0.60, σ = 1.20 | 1.003790 | Localized positive peak near the center. |
Formula used
Metric model
gij = e2φ(x,y) δij
For a two-dimensional conformal metric, the Ricci scalar is
R = -2 e-2φ (φxx + φyy)
The Gaussian curvature is K = R / 2.
Ricci tensor
In two dimensions, Rij = K gij.
Because the chosen metric is conformally flat, the calculator reports
Rxx = Ryy = -(φxx + φyy)
and Rxy = 0 for this coordinate form.
This is a physics-friendly local curvature tool for surfaces, model geometries, and conformal toy problems. It is not a full symbolic engine for arbitrary higher-dimensional metrics.
How to use this calculator
1. Pick a metric
Choose a preset geometry or the custom quadratic mode.
2. Enter coordinates
Set the x and y point where curvature is evaluated.
3. Adjust graph settings
Choose span and resolution for the Ricci scalar heatmap.
4. Calculate and export
Review tensors, interpret curvature, then download CSV or PDF.
FAQs
1. What does Ricci curvature measure here?
It measures how the chosen conformal surface bends intrinsically near a point. Positive values indicate local convergence, while negative values indicate local spreading of nearby geodesics.
2. Why does the calculator use conformal metrics?
Conformal metrics keep the input manageable while still showing genuine Ricci geometry. They are ideal for teaching, local modeling, and quick curvature experiments.
3. Is the Ricci tensor always diagonal here?
Yes, for this two-dimensional conformal form in Cartesian coordinates. The reported off-diagonal Ricci term is zero because the base metric starts diagonal and isotropic.
4. What is the difference between R and K?
For two-dimensional surfaces, the Ricci scalar R equals twice the Gaussian curvature K. That is why the calculator shows K = R / 2.
5. Why can the Poincare disk fail at some points?
The disk model only exists where x² + y² is strictly less than L². Outside that domain, the conformal factor becomes invalid and the metric is not defined.
6. Can I use this for general relativity homework?
Yes for intuition, local checks, and two-dimensional toy models. No for full four-dimensional symbolic tensor calculations with arbitrary coordinates and off-diagonal metric terms.
7. What does the graph show?
The Plotly heatmap shows how the Ricci scalar changes over the selected x-y window. It helps you spot flat regions, peaks, wells, and sign changes quickly.
8. Which preset is best for learning?
Start with the flat plane, then compare the sphere and Poincare disk. After that, try the Gaussian bump and custom quadratic mode for variable curvature patterns.