Energy Momentum Tensor Special Relativity Calculator

Energy momentum tensor calculator form

Enter proper energy density, isotropic pressure, and velocity components for a perfect fluid in special relativity.

Proper energy density ε

Example: J/m³ or any compatible unit set.

Pressure p

Use pressure units compatible with ε.

Speed of light c

Default sample uses 3.0 × 10⁸.

Velocity component v_x

Velocity component v_y

Velocity component v_z

Reset

Example data table

This example uses compatible arbitrary units for ε and p, with c = 3.0 × 10⁸.

ε	p	v_x	v_y	v_z	γ	T⁰⁰	T⁰¹	T¹¹	T²²	T^μ_μ
10.000000	2.000000	60000000.000000	30000000.000000	0.000000	1.025978	10.631579	2.526316	2.505263	2.126316	4.000000

Formula used

Metric choice: η^μν = diag(1, −1, −1, −1)

Lorentz factor: γ = 1 / √(1 − β²), with β² = (v_x² + v_y² + v_z²) / c²

Four-velocity: u^μ = γ(c, v_x, v_y, v_z)

Perfect fluid tensor: T^μν = (ε + p)(u^μu^ν/c²) − pη^μν

Main components:

T⁰⁰ = (ε + p)γ² − p
T⁰ⁱ = (ε + p)γ²v_i/c
T^ij = (ε + p)γ²v_iv_j/c² + pδ^ij
Invariant trace: T^μ_μ = ε − 3p

How to use this calculator

Enter proper energy density ε in your chosen unit system.
Enter isotropic pressure p using compatible units.
Set the speed of light value for your unit system.
Provide v_x, v_y, and v_z components.
Keep the total velocity magnitude below c.
Click Calculate Tensor to generate the 4×4 matrix.
Review γ, T⁰⁰, the invariant trace, and the heatmap.
Use the CSV or PDF buttons to export results.

Frequently asked questions

1. What does this calculator measure?

It computes the perfect fluid energy momentum tensor in special relativity. The output shows energy density, momentum flow, and stress components in one moving reference frame.

2. Which tensor convention does this page use?

It uses the metric signature (+, −, −, −). That choice affects signs in the tensor formula and the invariant trace shown in the result section.

3. Can I use natural units?

Yes. Set c to 1 and keep all values internally consistent. The calculator only needs compatible units across energy density, pressure, and velocity inputs.

4. Why must the velocity stay below c?

The Lorentz factor contains √(1 − β²). When |v| reaches or exceeds c, that term becomes zero or negative, so the physical relativistic solution breaks down.

5. What does T00 represent here?

T⁰⁰ is the lab frame energy density component. It combines rest energy density, pressure effects, and relativistic motion through the γ² factor.

6. Why are there off diagonal spatial terms?

Mixed spatial terms appear when motion has multiple velocity components. They represent shear-like momentum transport between different spatial directions in the moving frame.

7. Is negative pressure allowed?

Yes. The form accepts negative pressure values because some theoretical models use them. The calculation still follows the same perfect fluid tensor equation.

8. What does the heatmap show?

The heatmap visualizes the relative magnitude of all sixteen tensor entries. It makes diagonal dominance, symmetry, and velocity-driven component changes easier to inspect quickly.