Explore curvature with an Einstein tensor tool. Input metrics and Ricci values for instant output. Download tables, audit calculations, and teach geometry clearly today.
Enter the metric \(g_{\mu\nu}\), the Ricci tensor \(R_{\mu\nu}\), and the scalar curvature \(R\). The output is the Einstein tensor \(G_{\mu\nu}\) and an optional \(G_{\mu\nu}+\Lambda g_{\mu\nu}\).
This example uses a flat metric with zero Ricci and zero scalar curvature, producing a zero Einstein tensor.
| Quantity | Sample input | Expected outcome |
|---|---|---|
| Metric \(g_{\mu\nu}\) | diag(-1, 1, 1, 1) | Used directly |
| Ricci \(R_{\mu\nu}\) | all zeros | All \(G_{\mu\nu}\) = 0 |
| Scalar curvature \(R\) | 0 | No scalar-curvature contribution |
| Cosmological constant \(\Lambda\) | 0 | \(G + \Lambda g\) equals \(G\) |
The Einstein tensor combines curvature information in a divergence-free form used in general relativity. This calculator uses:
G\u03bc\u03bd = R\u03bc\u03bd \u2212 (1/2) g\u03bc\u03bd R
where R\u03bc\u03bd is the Ricci tensor, R is the scalar curvature, and g\u03bc\u03bd is the metric tensor. If you provide \(\Lambda\), the calculator also reports:
(G\u03bc\u03bd + \(\Lambda\) g\u03bc\u03bd)
Note: computing Ricci from a metric requires derivatives and a connection. This tool focuses on the Einstein-tensor combination once curvature inputs are known.
The Einstein tensor Gμν is a curvature object built to satisfy a conservation identity (its covariant divergence vanishes). In general relativity it appears on the geometric side of the field equations, relating spacetime geometry to stress-energy. This calculator helps you evaluate Gμν once curvature inputs are known.
You provide the metric components gμν, the Ricci tensor components Rμν, and the scalar curvature R at the same spacetime point. The tool does not infer units; it simply combines the numbers you enter. Consistency matters: keep the same coordinate basis, ordering, and sign convention across all inputs.
The computation follows a standard definition: Gμν = Rμν − (1/2) gμν R. Each output component is evaluated independently, so the matrix structure is transparent. This is useful for validating symbolic derivations, cross-checking numerical simulations, and confirming that a supplied curvature dataset matches your chosen metric signature.
If you enter a cosmological constant Λ, the calculator also reports Gμν + Λ gμν. Many texts write the field equations with Λ on the geometric side, while other workflows move it into an effective stress-energy contribution. Either way, the Λ term is a simple metric-scaled addition that can be compared across scenarios.
In most applications, gμν and Rμν are symmetric. The symmetry option mirrors upper-triangle entries into the lower triangle to reduce typing and catch mismatched inputs. Indices are labeled 0=t, 1=x, 2=y, 3=z for convenience, but you may interpret them as any coordinate ordering if you stay consistent throughout.
Diagonal components often dominate in highly symmetric models (for example, cosmology or static spherical spacetimes), while off-diagonal terms can signal rotation, shear, or coordinate coupling. A nonzero scalar curvature R introduces a metric-proportional shift through −(1/2) gμν R, so even small R can influence every component depending on gμν.
A fast check uses the flat Minkowski metric diag(−1, 1, 1, 1) with Rμν=0 and R=0. The expected output is Gμν=0 everywhere. If you set Λ≠0 with the same flat inputs, the Gμν + Λ gμν table becomes nonzero, which is a helpful verification that Λ is being applied correctly.
Use the CSV export when you want to post-process matrices in spreadsheets, notebooks, or scripts, and use the PDF export for reports, audits, or classroom materials. Because the tables include labeled indices, exported results remain traceable. Recompute with different curvature sets to compare models, test sensitivity to R, or document parameter sweeps.
No. You must supply Rμν and R. Computing curvature from the metric requires derivatives, connection coefficients, and coordinate-dependent steps that are outside this calculator.
The labels 0=t, 1=x, 2=y, 3=z are a convention for display only. You can use any coordinate chart if you enter all tensors in the same ordering.
Yes. Use (−,+,+,+) or (+,−,−,−) or any consistent signature. The calculator follows your entered gμν, so sign conventions are fully controlled by your inputs.
Because the term −(1/2) gμν R contributes even when Rμν=0. A scalar curvature shift affects every component through the metric scaling.
It is the geometric combination used when a cosmological constant is included on the left-hand side of the field equations. You can still rearrange terms later to match your preferred convention.
With symmetric inputs, yes. If you enter asymmetric values, the computed matrix will reflect them, which can highlight data-entry issues or inconsistent modeling assumptions.
Exports include raw numbers only. If your model uses units, document them in your report or spreadsheet headers and keep units consistent across gμν, Rμν, and R.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.