Formula Used
For a clock moving at constant speed v relative to an inertial observer, the proper time interval τ is related to the coordinate time interval Δt by:
τ = Δt · √(1 − v²/c²) and γ = 1 / √(1 − v²/c²) so τ = Δt / γ
Here, c is the speed of light, β = v/c, and γ is the Lorentz factor. This calculator assumes straight‑line inertial motion (no acceleration).
How to Use This Calculator
- Choose an input mode: velocity, gamma, or dilation factor.
- Enter the coordinate time Δt and pick a time unit.
- Provide either v, γ, or Δt/τ depending on your mode.
- Keep c at the default value for standard relativity.
- Click Calculate to see τ, γ, β, and the time difference.
- Use Download CSV or Download PDF after a calculation.
Example Data Table
Sample values (assuming c = 299,792,458 m/s):
| Coordinate time Δt (s) | Velocity (fraction of c) | Gamma γ | Proper time τ (s) | Difference Δt−τ (s) |
|---|---|---|---|---|
| 10 | 0.50 | 1.154700538 | 8.660254038 | 1.339745962 |
| 10 | 0.80 | 1.666666667 | 6.000000000 | 4.000000000 |
| 60 | 0.95 | 3.202563077 | 18.73507931 | 41.26492069 |
Professional Guide: Proper Time in Special Relativity
1) What “proper time” measures
Proper time (τ) is the time recorded by a clock that travels with an object. In an inertial (non‑accelerating) trip, τ is the shortest time between two events along that object’s worldline. Coordinate time (Δt) is measured by a stationary observer in the chosen reference frame.
2) The role of the speed of light
This calculator uses c = 299,792,458 m/s by default, which anchors relativistic scaling. Because c is extremely large, everyday speeds (cars, aircraft) produce γ very close to 1, making τ nearly equal to Δt. Noticeable differences appear only when v approaches a significant fraction of c.
3) Lorentz factor data you can interpret
The Lorentz factor is γ = 1/√(1 − β²) with β = v/c. Example values: β = 0.50 gives γ ≈ 1.1547, β = 0.80 gives γ ≈ 1.6667, and β = 0.95 gives γ ≈ 3.2026. Since τ = Δt/γ, higher γ means less experienced time.
4) Reading the “difference” output
The calculator also reports Δt − τ, which is how much less time the moving clock experiences. For Δt = 10 s at β = 0.80, τ becomes 6 s and the difference is 4 s. This value is useful for quick comparisons across multiple scenarios.
5) Working with velocity, γ, or dilation
Real problems sometimes provide speed, while others provide γ (from energy) or a stated dilation factor (Δt/τ). This tool supports all three. When you enter γ or Δt/τ, it back‑computes β and v. That makes it practical for particle physics, accelerator data, or high‑speed mission planning.
6) Units and precision for engineering workflows
You can enter Δt in ns through years and v in m/s, km/s, km/h, mph, or fraction of c. Internally, the calculator converts to seconds and m/s to keep the physics consistent. Use the precision control to match reporting or lab documentation needs.
7) Where proper time shows up in real systems
Proper time is fundamental in muon lifetime measurements, relativistic beam timing, and precision navigation. In satellite navigation, clocks experience relativistic effects due to motion and gravity; this calculator isolates the kinematic part (motion at constant speed) to build intuition before adding gravitational models.
8) Assumptions and limits
The formula used here assumes constant velocity (no acceleration) and special relativity only. If your scenario includes changing speed or strong gravitational fields, proper time requires integrating along a path or using general relativity. Keep v strictly below c to avoid nonphysical inputs.
FAQs
1) What is the difference between Δt and τ?
Δt is measured in the chosen stationary frame. τ is measured by the clock moving with the object. For constant velocity, τ = Δt/γ and τ ≤ Δt.
2) Why must velocity be less than c?
Special relativity requires v < c for massive objects. If v reached or exceeded c, 1 − v²/c² becomes zero or negative, making γ undefined or imaginary.
3) When will the time difference be noticeable?
Differences become meaningful when v is a significant fraction of c. At 0.8c, γ ≈ 1.667, so a 10 s interval becomes 6 s of proper time.
4) Can I use gamma directly instead of speed?
Yes. Select the gamma mode and enter γ ≥ 1. The calculator will compute β and v, then calculate τ = Δt/γ for the same coordinate time.
5) What does the dilation factor mode represent?
The dilation factor is Δt/τ. For inertial motion it equals γ, so entering Δt/τ is equivalent to entering gamma. It is handy when dilation is stated directly.
6) Does this include gravitational time dilation?
No. This tool models kinematic time dilation only (special relativity). Gravity requires general relativity and additional parameters such as altitude and gravitational potential.
7) Why are results also shown in seconds?
Seconds are the internal base unit, ensuring consistent calculations across all input options. Displaying seconds alongside your selected unit helps with verification and exporting.