Calculator
Use a positive or negative frame speed along the chosen axis. Select beta mode for dimensionless inputs, or m/s mode for direct speeds.
Example data table
| Case | Direction | Axis | Frame β | Source velocity β | Source force (N) | Transformed force (N) | |F target| (N) |
|---|---|---|---|---|---|---|---|
| A | S to S' | X | 0.60 | (0.20, 0.10, 0.00) | (12.00, 8.00, 0.00) | (11.454545, 7.272727, 0.000000) | 13.568315 |
| B | S to S' | Y | 0.45 | (0.05, 0.30, 0.10) | (4.50, 10.00, -3.00) | (4.645813, 10.039017, -3.097209) | 11.487304 |
| C | S' to S | Z | 0.72 | (0.12, -0.08, 0.25) | (18.00, -6.00, 9.00) | (10.586045, -3.528682, 10.610847) | 15.398248 |
Each example uses the same equations as the calculator. Sample buttons load matching values into the form instantly.
Formula used
1. Lorentz factor
γ = 1 / √(1 - β²)
2. Transformation denominator
D = 1 ± ββu,parallel
3. Parallel force component
Ftarget,parallel = (Fsource,parallel ± β(Fsource · βu)) / D
4. Transverse force components
Ftarget,perp = Fsource,perp / (γD)
5. Particle velocity transformation
βtarget,parallel = (βsource,parallel ± β) / D
βtarget,perp = βsource,perp / (γD)
Use the minus sign for S to S'. Use the plus sign for S' to S.
The calculator maps your chosen motion axis to the parallel direction internally, applies the equations, then restores the original component order.
How to use this calculator
- Choose whether you are transforming from S to S' or from S' to S.
- Select the axis along which the two frames move relative to each other.
- Pick beta mode for fractions of c, or m/s mode for direct speed values.
- Enter the frame speed, the particle velocity components, and the force vector components.
- Press calculate to view the transformed force, transformed particle velocity, gamma, denominator, and force-direction change.
- Use the export buttons to save the result as a CSV or PDF file.
FAQs
1. What does this calculator transform?
It transforms a three-dimensional force vector between inertial frames moving at relativistic speed along one chosen axis. It also reports the transformed particle velocity used in the same calculation.
2. Why do I need the particle velocity?
Relativistic force transformation depends on both the force components and the particle's velocity in the source frame. Without that velocity, the longitudinal force transformation cannot be evaluated correctly.
3. What is beta?
Beta is speed divided by the speed of light. A beta of 0.60 means the object moves at sixty percent of light speed.
4. Why can the transverse force change?
Transverse components scale by the factor 1/(γD), where D includes the source-frame particle velocity along the motion axis. That coupling is a direct consequence of Lorentz transformations.
5. Can I use a negative frame speed?
Yes. A negative value means the relative motion points opposite to the chosen axis direction. The calculator keeps the sign and applies it directly inside the transformation equations.
6. Why does the tool warn about unstable inputs?
When speeds approach light speed, the denominator and gamma factor can change sharply. Very small input adjustments may then produce large result differences, so the tool highlights that sensitivity.
7. Does the calculator support any motion axis?
Yes. You can choose x, y, or z as the relative motion axis. The calculator reorders components internally, applies the same equations, and restores the original orientation.
8. Are the exported CSV and PDF files based on my result?
Yes. After a successful calculation, the export buttons package the displayed result metrics into a CSV table or a formatted PDF summary for download.