Lorentz Velocity Transformation Calculator

Calculator input

Use the standard x-direction Lorentz velocity transformation. The moving frame is assumed to travel along the positive x-axis.

Scenario name

Transformation mode

Input unit

Frame speed v

Object velocity u_x

Object velocity u_y

Object velocity u_z

Decimal precision

Example data table

This sample uses the forward transformation with values entered as fractions of light speed.

Scenario	v	u_x	u_y	u_z	u′_x	u′_y	u′_z	\|u′\|
Muon beam test case	0.40c	0.60c	0.20c	0.00c	0.263158c	0.241450c	0.000000c	0.357129c
Ion drift snapshot	0.25c	0.72c	0.06c	0.04c	0.596154c	0.055892c	0.037262c	0.600060c
Probe alignment run	-0.30c	0.50c	0.10c	-0.08c	0.695652c	0.104828c	-0.083862c	0.708636c

Formula used

Forward transformation, S to S′:

u′_x = (u_x - v) / (1 - u_xv / c²)

u′_y = u_y / [γ(v)(1 - u_xv / c²)]

u′_z = u_z / [γ(v)(1 - u_xv / c²)]

Inverse transformation, S′ to S:

u_x = (u′_x + v) / (1 + u′_xv / c²)

u_y = u′_y / [γ(v)(1 + u′_xv / c²)]

u_z = u′_z / [γ(v)(1 + u′_xv / c²)]

Lorentz factor: γ(v) = 1 / √(1 - v²/c²)

The denominator corrects classical velocity addition. This keeps transformed speeds below light speed and adjusts transverse components when the frame moves along x.

How to use this calculator

Choose whether you want the forward or inverse transformation.
Select the unit used for all entered velocities.
Enter frame speed v and the three object velocity components.
Use fractions of c for easy relativity problems, or choose everyday speed units.
Set your preferred decimal precision for reports or classroom work.
Press the calculate button to show the result section above the form.
Review transformed components, beta values, gamma factors, and classical comparison.
Download the current result as CSV or PDF for later use.

FAQs

1) What does this calculator compute?

It transforms a particle’s three velocity components between two inertial frames when one frame moves along the x-axis. It also reports beta, gamma, denominator terms, classical comparison, exports, and a graph.

2) Why does the calculator require speeds below light speed?

Special relativity only allows massive objects to move slower than light. The formulas contain gamma and denominator terms that become undefined or unphysical at or above light speed.

3) What is an example of calculating Lorentz transforms?

Take u = (0.60c, 0.20c, 0) and frame speed v = 0.40c. Then u′x = 0.263158c and u′y ≈ 0.241450c. The transformed speed remains below light speed, unlike simple classical subtraction.

4) Why do y and z components change if motion is along x?

Time dilation and the shared denominator affect transverse components. Even though the frame moves only along x, the changed time coordinate alters how y and z motion is measured in the transformed frame.

5) What is the difference between classical and relativistic velocity addition?

Classical addition simply adds or subtracts speeds. Relativistic addition divides by a correction term involving c², which prevents impossible results and stays consistent with special relativity.

6) Which units can I enter?

You can enter m/s, km/s, km/h, mph, or fractions of c. The calculator converts values internally to SI units, performs the transformation, and returns results in your chosen unit.

7) What does a negative transformed component mean?

A negative value means the transformed observer measures that component in the opposite coordinate direction. It does not automatically mean the total motion reversed; only that component changed sign.

8) Can this tool be used for teaching and report preparation?

Yes. It is useful for homework checks, lecture demonstrations, and quick report tables. The result table, example data, formula section, graph, and export buttons make it practical for classroom or lab documentation.