Calculator Inputs
Enter rest mass, three velocity components in frame S, and the relative frame speed along the x-axis.
Example Data Table
These sample rows show how different masses and velocities transform when the observer frame moves along the x-axis.
| Sample | Rest mass (kg) | vx (m/s) | vy (m/s) | vz (m/s) | u (m/s) | |p| (kg·m/s) | |p′| (kg·m/s) | E′ (J) |
|---|---|---|---|---|---|---|---|---|
| Electron-like sample | 9.109384e-31 | 1.200000e+8 | 4.000000e+7 | 2.500000e+7 | 8.000000e+7 | 1.301024e-22 | 6.339789e-23 | 8.404824e-14 |
| Proton-like sample | 1.672622e-27 | 1.500000e+8 | 2.500000e+7 | 1.000000e+7 | 6.000000e+7 | 2.960020e-19 | 1.859239e-19 | 1.603285e-10 |
| Generic particle sample | 3.000000e-28 | 9.000000e+7 | 3.500000e+7 | 1.500000e+7 | 7.000000e+7 | 3.101108e-20 | 1.373383e-20 | 2.727521e-11 |
Formula Used
px = γmvx, py = γmvy, pz = γmvz
|p| = √(px2 + py2 + pz2)
γ = 1 / √(1 − v2/c2)
γu = 1 / √(1 − u2/c2)
E = γmc2
px′ = γu(px − uE/c2)
py′ = py
pz′ = pz
E′ = γu(E − upx)
vx′ = (vx − u) / (1 − uvx/c2)
vy′ = vy / [γu(1 − uvx/c2)]
vz′ = vz / [γu(1 − uvx/c2)]
(E/c)2 − |p|2 = (E′/c)2 − |p′|2
This invariant confirms the transformation was applied consistently. In ideal arithmetic, both sides match exactly. Tiny drift only reflects numerical rounding.
How to Use This Calculator
- Enter the particle rest mass and choose its unit.
- Choose one unit system for all particle velocity components.
- Provide vx, vy, and vz in the original frame S.
- Enter the relative frame speed u for the moving frame S′.
- Submit the form to compute transformed momentum, energy, and velocity.
- Review the invariant row to confirm numerical consistency.
- Inspect the graph to see how px′ and |p′| vary with frame speed.
- Download CSV or PDF reports for documentation or sharing.
Frequently Asked Questions
1) What does this calculator transform?
It transforms a particle’s energy and momentum from one inertial frame to another moving along the x-axis. It also reports transformed velocity components and an invariant consistency check.
2) Why is the frame boost applied only along x?
That is the standard closed-form Lorentz boost used in many textbooks. A general boost can be built from vector methods, but the x-axis form is cleaner and sufficient for many physics problems.
3) Can I enter speeds as fractions of light speed?
Yes. Choose “fraction of c” for the particle speed unit or frame speed unit. Then enter values like 0.60, 0.15, or -0.30 directly.
4) Why do y and z momentum components stay unchanged?
For a pure x-axis Lorentz boost, only the x component mixes with energy. Transverse momentum components remain unchanged, although transverse velocities do change because the denominator changes.
5) What does the invariant row mean?
It checks whether (E/c)² − |p|² stays constant after transformation. That quantity is tied to rest mass and should remain the same in every inertial frame.
6) Why can transformed energy become smaller?
Energy depends on the observer frame. If the moving frame is closer to the particle’s motion, the measured total energy and x-momentum can drop compared with the original frame.
7) What if my values are rejected?
The calculator blocks unphysical inputs such as speeds equal to or exceeding light speed, zero or negative rest mass, or unstable velocity denominators near singular behavior.
8) Is this suitable for teaching and reports?
Yes. It shows formulas, example data, transformed quantities, graphical trends, and downloadable outputs. That makes it useful for lectures, homework support, lab notes, and technical documentation.