Model torsion from any affine connection inputs here. Check traces, norms, and symmetry constraints fast. Download clear tables and reuse them in reports today.
Sample n = 3 connection coefficients with a few intentional asymmetries.
| Γ^k_{ij} | Value | Implied torsion T^k_{ij} |
|---|---|---|
| Γ^1_{12}, Γ^1_{21} | 0.15, 0.05 | T^1_{12} = 0.10 |
| Γ^1_{23}, Γ^1_{32} | 0.40, 0.10 | T^1_{23} = 0.30 |
| Γ^2_{13}, Γ^2_{31} | −0.25, 0.05 | T^2_{13} = −0.30 |
| Γ^3_{12}, Γ^3_{21} | 0.60, 0.60 | T^3_{12} = 0 |
Given an affine connection with coefficients Γkij, the torsion tensor is defined by:
Torsion measures how far a connection fails to be symmetric in its lower indices. For each k, the calculator forms T^k_{ij} = Γ^k_{ij} − Γ^k_{ji}, forcing T^k_{ii}=0. If Γ^k_{ij}=Γ^k_{ji} for all pairs, torsion is identically zero. The number of independent components is n·n(n−1)/2.
Choose n to set i, j, k ∈ {1…n}. The input grid is organized by k-slices, each holding an n×n matrix of Γ^k_{ij}. Moving from n=3 to n=4 increases entries from 27 to 64, so disciplined indexing prevents sign mistakes when comparing Γ^k_{ij} with Γ^k_{ji}. Empty fields are treated as 0 to speed entry.
Numerical models often create tiny nonzero values from rounding. The ε threshold filters components with |T| ≤ ε from the “nonzero” list while keeping full tables available. If ε is 1e−6 and your largest |T| is 0.30, then only meaningful asymmetries remain visible for review and export. Use smaller ε for symbolic or high‑precision inputs. CSV still stores every component.
The trace vector is computed as T_i = Σ_k T^k_{ik}. It compresses torsion into n scalars that can indicate preferred directions in a torsionful geometry. With n=4, the trace has four entries, which you can compare across parameter sweeps to detect drift, coupling, or constraints. Traceless torsion appears when all T_i are near zero.
The reported norm ‖T‖ = √(Σ_{k,i,j}(T^k_{ij})²) provides a single magnitude for quality control. Use a fixed decimal setting to compare runs. The antisymmetry deviation should be near 0; if it is large, the input parsing or index ordering is inconsistent. Check units.
After submission, the component tables and the Plotly heatmap summarize patterns at a glance. The k selector lets you inspect each T^k_{ij} slice, where off‑diagonal values stand out. The per-slice magnitude bar plot highlights which k contributes most to ‖T‖. Export CSV for spreadsheets and PDF for documentation so results stay consistent across notebooks, reports, and collaborations.
Enter your connection coefficients for the chosen n. Leave blanks as zero, or use Fill Zeros. Keep units consistent across all inputs for meaningful norms and comparisons.
Because torsion is antisymmetric in i and j. When i = j, Tkii = Γkii − Γkii = 0 automatically.
Only off-diagonal pairs matter. The count is n·n(n−1)/2. For n=3 this is 9, and for n=4 it is 24, even though the full tables display n³ entries.
ε only controls which components appear in the “nonzero” list and the torsion-free message. It does not change computed tables, the norm, or exported data.
Ti = Σk Tkik summarizes torsion into n scalars. Use it to test traceless conditions (all near zero) and to compare directional trends between runs.
Yes. Use Print for a quick page capture, export PDF for a formatted summary, and export CSV for reproducible analysis. For publication figures, recreate the plot from CSV to control labels and resolution.
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.