Riemann Tensor Calculator for Diagonal Metrics

Calculator Form

Preset Metric

Dimension

Derivative Step

Coordinate Names

Coordinate Values

Allowed Functions

g11 Expression

g22 Expression

g33 Expression

g44 Expression

Metric Notes

Upper Index i

Lower Index j

Lower Index k

Lower Index l

Plot Variable Index

Plot Start

Plot End

Plot Points

Example Data Table

Example	Dimension	Coordinates	Diagonal Metric Entries	Suggested Evaluation Point
2-Sphere	2	theta, phi	1, sin(theta)^2	1.1, 0.4
Minkowski	4	t, x, y, z	-1, 1, 1, 1	0, 0, 0, 0
Schwarzschild	4	t, r, theta, phi	-(1-2/r), 1/(1-2/r), r^2, r^2*sin(theta)^2	0, 8, 1.2, 0.7
Flat FLRW	4	t, x, y, z	-1, exp(0.4t), exp(0.4t), exp(0.4*t)	0.5, 0, 0, 0

Formula Used

The calculator evaluates a diagonal metric, then inverts each diagonal entry. It estimates derivatives numerically with central differences around the selected coordinate point.

Christoffel symbols use:

Γⁱ_jk = 1/2 gⁱⁱ (∂_jg_ik + ∂_kg_ij - ∂_ig_jk)

The mixed-index Riemann tensor uses:

Rⁱ_jkl = ∂_kΓⁱ_jl - ∂_lΓⁱ_jk + Γⁱ_mkΓ^m_jl - Γⁱ_mlΓ^m_jk

The Ricci tensor contracts the first and third indices. Scalar curvature contracts Ricci with the inverse metric. The Kretschmann scalar contracts the full tensor with itself.

How to Use This Calculator

Select a preset metric or choose a custom diagonal metric.
Enter the dimension, coordinate names, and numeric coordinate values.
Provide diagonal metric expressions using the listed math functions.
Set the derivative step. Smaller values improve local accuracy until rounding noise grows.
Choose the Riemann component indices you want highlighted and plotted.
Set the plot variable index and the plotting interval.
Submit the form. The result appears above the calculator.
Download the tables as CSV or PDF after review.

FAQs

1. What does this calculator compute?

It computes Christoffel symbols, mixed-index Riemann components, the Ricci tensor, scalar curvature, and the Kretschmann scalar for diagonal metrics.

2. Does it support off-diagonal metrics?

No. This version is restricted to diagonal metrics. That keeps the input simpler and the numerical engine stable inside one file.

3. Are the derivatives symbolic?

No. The calculator uses central finite differences. It is numerical, so your derivative step matters and very sharp functions may need adjustment.

4. Why do I see small nonzero values?

Finite differences create rounding noise. Values that should vanish may appear near zero. Check the antisymmetry error to judge numerical consistency.

5. Which expressions are allowed?

You can use numbers, coordinates, parentheses, powers, and these functions: sin, cos, tan, asin, acos, atan, sinh, cosh, tanh, sqrt, abs, exp, log, pow.

6. What is a good test case?

The 2-sphere is a good first test. It produces clear nonzero curvature and validates the plotting, Ricci tensor, and scalar curvature outputs.

7. Why does Schwarzschild fail near r = 2?

The metric becomes singular in these coordinates at the horizon. Numerical evaluation becomes unstable there, so choose points safely outside that radius.

8. Can I export the results?

Yes. Use the CSV button for spreadsheet work and the PDF button for a quick report containing the summary and displayed tables.