Calculator
Formula Used
This calculator applies the inverse Lorentz transformation from a primed frame S′ back to an unprimed frame S, assuming relative motion along the x-axis.
- β = v/c
- γ = 1 / √(1 − β²)
- x = γ (x′ + v t′)
- t = γ (t′ + v x′ / c²)
- y = y′, z = z′
The interval check uses s² = (c t)² − x² − y² − z² in SI units.
How to Use This Calculator
- Choose an input mode: enter β or enter v with units.
- Set c if you need a custom value.
- Enter primed coordinates x′ and t′, plus optional y′ and z′.
- Select your distance and time units.
- Press Calculate to view results above the form.
- Use the export buttons to download CSV or PDF.
Example Data Table
| β | x′ (m) | t′ (s) | γ | x (m) | t (s) |
|---|---|---|---|---|---|
| 0.60 | 1000 | 0.000010 | 1.25 | ≈ 4748.96 | ≈ 0.0000125 |
| -0.80 | 500 | 0.000002 | 1.6667 | ≈ -299.79 | ≈ 0.00000111 |
| 0.30 | 200 | 0.000005 | 1.0483 | ≈ 671.74 | ≈ 0.00000570 |
Values shown are rounded for quick reference.
Article
1) Why inverse transformations matter
In many relativity problems, measurements are recorded in a moving frame and must be converted back to the laboratory frame. The inverse Lorentz transformation does exactly that: it reconstructs the unprimed coordinates (x, t) from primed measurements (x′, t′) while preserving the physical spacetime interval.
2) Frames, events, and coordinate meaning
An “event” is a specific occurrence described by position and time. Two inertial frames, S and S′, differ by a constant relative velocity along the x-axis. This calculator assumes the standard configuration where axes are parallel and origins coincide at t = t′ = 0.
3) Choosing velocity input: β or v
You can enter β = v/c directly or provide v in common engineering units. Internally, velocity is converted to meters per second so that computations remain consistent. Negative velocity is supported and represents motion in the opposite x direction.
4) The role of the Lorentz factor γ
The Lorentz factor γ grows as |β| approaches 1, amplifying time dilation and length contraction effects. For small |β|, γ is close to 1 and classical intuition often works. Near-light motion makes γ sensitive, so the calculator validates |β| < 1 to avoid nonphysical inputs.
5) Inverse equations used by the calculator
For motion along +x, the inverse mapping is x = γ(x′ + v t′) and t = γ(t′ + v x′/c²). The transverse coordinates do not change: y = y′ and z = z′. These relations are derived from the symmetry of inertial frames and the invariance of c.
6) Unit support and numerical stability
Distance and time units are selectable to match typical lab or astrophysical data. The calculator converts to SI units for computation, then converts results back to your chosen units. This approach reduces unit mistakes and keeps the formulas readable and auditable.
7) Interval consistency as a quality check
A practical way to validate results is comparing the Minkowski interval s² = (c t)² − x² − y² − z² across frames. In ideal arithmetic, s² equals s²′ for the same event. The displayed invariant check helps you confirm that rounding, unit choices, and sign conventions are behaving as expected.
8) Professional use cases and cautions
Engineers and researchers use inverse transformations when converting sensor logs, synchronizing clocks across moving platforms, or analyzing high-energy particle tracks. Always confirm the direction of motion and your unit selections, then export results for documentation. For accelerated motion, use segmented inertial approximations rather than a single constant-velocity transform.
FAQs
1) What does “inverse” mean here?
It converts coordinates from the moving frame S′ back to the laboratory frame S. The forward transform maps S to S′, while the inverse transform reverses that mapping using the same velocity.
2) Why must |β| be less than 1?
β equals v/c. If |β| ≥ 1, the relative speed would be at least c, which is not valid for inertial frames in standard relativity. The calculator blocks those values to prevent invalid γ.
3) When should I enter v instead of β?
Use v when your data comes from instruments or specifications in m/s, km/s, or km/h. Use β when working with dimensionless relativistic parameters or when v is naturally expressed as a fraction of c.
4) What is the purpose of the interval check?
It confirms that the computed event is consistent across frames by comparing s² and s²′ in SI units. Small differences can occur from rounding, but large differences often signal a unit or sign mistake.
5) Do y and z ever transform in this tool?
Not in the chosen configuration. For motion purely along x, the transverse coordinates remain unchanged: y = y′ and z = z′. More general transforms require rotations or boosts along different axes.
6) Can I change the speed of light value?
Yes. The field allows custom c values for educational scenarios or unit experiments. For physical vacuum calculations, keep the default value to maintain standard reference results.
7) What if my motion is accelerated?
Lorentz transformations assume constant relative velocity between inertial frames. For acceleration, break the motion into small intervals with nearly constant velocity, or use a dedicated accelerated-motion model.