Relativistic Momentum Transforms Calculator

Study momentum transformations between inertial frames with precision. Visualize components, verify Lorentz relations, and export clean results for physics analysis tasks.

Calculator Inputs

Enter values using fractions of light speed.

Keep all speed fractions strictly between -1 and 1.

Example Data Table

Scenario	Mass (kg)	v_x/c	v_y/c	v_z/c	u/c	Use Case
Baseline Example	1.5	0.60	0.20	0.10	0.40	General component transformation study
High Boost	0.8	0.85	0.05	0.02	0.70	Strong x-axis frame shift
Transverse Motion	2.2	0.30	0.55	0.15	0.25	Compare transverse invariance behavior

Formula Used

This calculator applies a Lorentz boost along the x-axis. Momentum and energy form a four-vector. That keeps transformation rules consistent across inertial frames.

Particle gamma: γ = 1 / √(1 − v²/c²)

Momentum components in S: p_x = γmv_x, p_y = γmv_y, p_z = γmv_z

Total energy: E = γmc²

Frame gamma: γ_u = 1 / √(1 − u²/c²)

Momentum transform: p′_x = γ_u(p_x − uE/c²)

Transverse invariance: p′_y = p_y, p′_z = p_z

Energy transform: E′ = γ_u(E − up_x)

Velocity transform: v′_x = (v_x − u)/(1 − uv_x/c²)

Transverse velocity transform: v′_y = v_y / [γ_u(1 − uv_x/c²)]

Transverse velocity transform: v′_z = v_z / [γ_u(1 − uv_x/c²)]

The result table also checks transformed momentum using transformed velocities. Close agreement confirms numerical consistency.

How to Use This Calculator

Enter the particle rest mass in kilograms.
Provide velocity components as fractions of light speed.
Enter the relative frame speed along the x-axis.
Click Calculate Transforms to compute new quantities.
Review momentum, energy, and transformed velocities above the form.
Inspect the graph for before-and-after component changes.
Export the result as CSV or PDF.
Use the example table for quick testing.

Frequently Asked Questions

1. What does this calculator compute?

It transforms relativistic momentum and energy from one inertial frame to another. It also estimates transformed velocity components and checks the new momentum values using those transformed velocities.

2. Why are speeds entered as fractions of light speed?

Fractions of light speed keep relativistic inputs readable. They also reduce unit mistakes. A value like 0.60 means the component equals sixty percent of light speed.

3. Why must every speed stay below one?

Special relativity requires all physical speeds to remain below light speed. If a component or resulting total speed reaches or exceeds that limit, gamma becomes undefined for this model.

4. Why do y and z momentum components stay unchanged?

This page assumes the frame boost occurs only along the x-axis. Under that condition, transverse momentum components remain unchanged while the x component and energy mix together.

5. What is the role of gamma?

Gamma measures relativistic scaling. It grows quickly as speed approaches light speed. That affects momentum, energy, time dilation, and the transformed quantities shown by the calculator.

6. Why include both direct and checked transformed values?

The direct values come from four-vector transformations. The checked values come from transformed velocities. Comparing them helps confirm the numerical consistency of the entered scenario.

7. Can I use this for negative velocities?

Yes. Negative values represent motion in the opposite axis direction. The same formulas apply, provided every speed fraction remains strictly between minus one and one.

8. When is this calculator most useful?

It is useful in advanced physics classes, relativity homework, beam analysis, and frame-comparison studies where momentum and energy must be tracked across moving observers.