Build tensors from physical inputs using clear unit controls. View 4×4 matrices, trace, and flux. Download clean reports and share results with colleagues easily.
| Scenario | Key inputs | What to expect |
|---|---|---|
| Perfect fluid | ε = 1.2×1017 J/m³, p = 1.0×1016 Pa, v/c = (0.10, 0, 0) | T00 dominates, off-diagonal momentum flux appears. |
| Electromagnetic | E = (0, 0, 106) V/m, B = (0, 2×10−3, 0) T | Energy density scales with ε0E² and B²/μ0. |
| Scalar field | ∂tφ = 1, ∂xφ = 0.1, V(φ)=0.5 | Diagonal terms reflect kinetic + potential contributions. |
Tμν packs energy density, momentum density, energy flux, and stress into one 4×4 object. In general relativity it enters Einstein’s field equations, and in flat spacetime it supports local conservation via ∂μTμν=0 when external forces are negligible.
Read T00 as coordinate energy density, T0i as energy flow, and Tij as stresses. In SI, J/m3 equals Pa, so ε and p combine cleanly. The signature selector changes signs in contractions and the trace.
For a perfect fluid: Tμν=( ε+p )uμuν/c2 + p gμν. In the rest frame, T00≈ε and spatial diagonals ≈p. Reference data: atmospheric pressure is 1.013×105 Pa; water has ρc2≈9×1019 J/m3.
Electromagnetism builds Fμν from E and B with F0i=Ei/c, then computes Tμν using μ0=4π×10−7 N/A2 (1.25663706212×10−6). Check magnitudes with u=( ε0E2 + B2/μ0 )/2 and S=(1/μ0)E×B.
For a scalar field the calculator uses Tμν=∂μφ∂νφ − gμν(K+V), with K=(1/2)gαβ∂αφ∂βφ. It forms Tμν and raises indices with your chosen signature. This is practical for wave packets, inflationary fields, and potential-driven dynamics.
The trace gμνTμν is a fast consistency check. Radiation-like systems tend toward a near-zero trace, while matter-like inputs often produce a nonzero trace tied to the equation of state. For fluids, compare the trace against ε and p to spot mismatched magnitudes.
For perfect fluids in 3-velocity mode, the calculator enforces |v|/c<1 so the Lorentz factor γ remains real. In 4-velocity mode, keep units consistent (SI commonly has u0≈γc). Prefer scientific notation for large inputs.
After computation, export CSV for parameter sweeps and plotting, or export the compact PDF for lab notes and peer review. Record the model, signature, unit system, and the exact inputs used. When comparing references, align conventions for gμν and field-tensor definitions before interpreting signs.
It is the coordinate-frame energy density component of Tμν. In a fluid rest frame it approaches ε. With motion or fields present, it includes kinetic and field contributions.
Changing −+++ to +−−− flips signs in metric contractions, which affects raised-index components and the trace. This calculator recomputes contractions consistently, so the tensor values follow your convention.
Use v/c components in the 3-velocity mode (each between −1 and 1, with total |v|/c<1). If you already have uμ, switch to 4-velocity mode and enter the components directly.
Because μ0, c, and the E–B scaling are explicitly SI in this implementation. Mixing natural-unit inputs without adjusting constants would produce misleading magnitudes.
They represent fluxes: energy flow and momentum density between time and space directions. In electromagnetism they relate to the Poynting vector; in fluids they track momentum transport due to motion.
Yes. For a perfect fluid, compare ε and p and inspect the trace and diagonal terms. For scalar-field models, vary ∂tφ and V(φ) to see how pressure-like components respond.
Record the chosen model, metric signature, unit system, and the exact input parameters. Exporting PDF captures these alongside the matrix, reducing ambiguity when others reproduce your calculation.
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.