Calculator inputs
Example data table
These sample values show a Lorentz boost along the x-axis.
| Quantity | Example value | Unit |
|---|---|---|
| Relative speed β | 0.600000 | dimensionless |
| x | 1200 | m |
| y | 15 | m |
| z | -8 | m |
| t | 5.000000e-06 | s |
| ux | 0.400000 | c |
| uy | 0.066700 | c |
| γ | 1.250000 | dimensionless |
| x′ | 375.778282 | m |
| t′ | 3.247923e-06 | s |
| u′x | -7.880075e+07 | m/s |
| u′y | 2.105724e+07 | m/s |
Formula used
Dimensionless speed: β = v / c
Lorentz factor: γ = 1 / √(1 - β²)
Coordinate transformation:
x′ = γ(x - vt)
t′ = γ(t - vx / c²)
y′ = y
z′ = z
Velocity transformation for an x-axis boost:
u′x = (ux - v) / (1 - uxv / c²)
u′y = uy / [γ(1 - uxv / c²)]
u′z = uz / [γ(1 - uxv / c²)]
Invariant spacetime interval:
s² = c²t² - x² - y² - z²
The value should match before and after transformation.
Length contraction: L = L0 / γ
Time dilation: Δt = γΔτ0
How to use this calculator
- Select whether speed entries are fractions of
cor meters per second. - Enter the event coordinates
x,y,z, andtin frame S. - Provide the relative frame speed and the object's velocity components in frame S.
- Optionally enter rest length and proper time for contraction and dilation outputs.
- Click Calculate transformation to view results, the Plotly graph, and export buttons.
FAQs
1) What does this calculator transform?
It transforms event coordinates, velocity components, invariant interval, length, and time between two inertial frames linked by a constant x-axis relative speed.
2) Why must the speed stay below light speed?
Special relativity requires physical massive frames and objects to move slower than light. At or above light speed, the Lorentz factor becomes undefined.
3) Why are y and z unchanged here?
This page uses a standard Lorentz boost along the x-axis. Only the parallel coordinate mixes with time. Transverse coordinates remain unchanged.
4) What is the invariant interval?
The interval s² = c²t² - x² - y² - z² is frame-invariant. Matching values in S and S′ verifies the transformation.
5) What does the gamma factor represent?
Gamma measures how strongly relativistic effects appear. As speed approaches light speed, gamma rises rapidly and increases time dilation and length contraction.
6) Why can transformed velocity differ strongly from the input?
Velocity addition in relativity is nonlinear. Subtracting frame speed directly is incorrect at high speeds, so the transformed result follows the exact Lorentz formula.
7) When should I use fractions of c?
Use fractions of c when working with textbook relativity problems. It reduces very large numbers and makes beta-based reasoning easier.
8) What does the graph show?
The graph sweeps β across many frame speeds and plots transformed x′, c·t′, and γ. It helps you see how the same event shifts between frames.