| Scenario | Δt (s) | Δx (m) | Δy (m) | Δz (m) | Expected type |
|---|---|---|---|---|---|
| Nearby lab events | 1 | 100,000 | 0 | 0 | Timelike (typically) |
| Fast separated flashes | 0.0001 | 100,000 | 0 | 0 | Spacelike (typically) |
| Light signal (ideal) | 1 | c·Δt | 0 | 0 | Lightlike |
The invariant spacetime interval between two events is computed from the time separation Δt and spatial separations Δx, Δy, Δz. The calculator first converts your inputs to SI units, then applies the selected metric convention:
- (+,-,-,-): s² = c²Δt² − Δx² − Δy² − Δz²
- (-,+,+,+): s² = −c²Δt² + Δx² + Δy² + Δz²
Using the sign of s², the separation is classified as timelike, spacelike, or lightlike. The magnitude |s| = √|s²| is returned in meters.
- Timelike: proper time τ = |s| / c
- Spacelike: proper length ℓ = |s|
- Lightlike: s² ≈ 0
What the interval measures
The spacetime interval combines temporal and spatial separation into one invariant quantity. For two events, it uses Δt and Δx, Δy, Δz after unit conversion to seconds and meters. Because the interval is invariant under Lorentz transformations, different inertial observers can disagree on Δt and Δx yet still compute the same s².
Using vacuum light speed as a scale
The calculator multiplies time separation by c to express it in meters, producing cΔt. This places time and space on comparable footing. With c = 299,792,458 m/s, a 1 ns separation corresponds to about 0.2998 m. Entering custom c supports media where an effective signal speed is used for modeling.
Signature choices and interpretation
Two common sign conventions appear in textbooks. With (+,−,−,−), timelike separations yield positive s², spacelike yield negative s², and lightlike are near zero. With (−,+,+,+), the signs swap. The calculator reports the convention so classifications remain consistent with the chosen signature.
Timelike and spacelike examples with numbers
If Δt = 1 s and Δx = 100,000 m, then cΔt ≈ 3.0×10^8 m and (cΔt)² dominates, so the separation is timelike. If Δt = 0.0001 s and Δx = 100,000 m, then cΔt ≈ 3.0×10^4 m and spatial distance dominates, producing a spacelike result. For lightlike paths, set Δx ≈ cΔt.
Proper time and proper length outputs
For timelike separations, the magnitude |s| corresponds to cτ, so τ = |s|/c. This is the time measured by a clock traveling between the events. For spacelike separations, |s| corresponds to a proper length that can be measured in a frame where the events are simultaneous.
Graphing contributions for quick diagnosis
The included Plotly chart visualizes (cΔt)² and the spatial squared terms alongside s². When the time bar is much larger than the spatial sum, the interval trends timelike; the opposite trend indicates spacelike separation. Exporting the results table to CSV or PDF supports lab documentation and repeatable calculations.
- Select Deltas if you already know Δt, Δx, Δy, Δz.
- Select Events if you want to enter two points (t1,x1,y1,z1) and (t2,x2,y2,z2).
- Choose units for time and each spatial axis. The calculator converts everything to SI.
- Pick a metric convention, and confirm the value of c if needed.
- Press Calculate. Results appear below the header, above the form.
- Use Download CSV or Download PDF to export the results table.
Lorentz transformations change Δt and Δx in a linked way. The combination in s² remains invariant, so all inertial frames agree on the interval classification and magnitude.
Use 299,792,458 m/s for vacuum light. For an effective signal speed in a cable or medium, enter the modeled propagation speed so cΔt reflects the system’s timing scale.
Lightlike means s² is near zero. If (cΔt)² is almost equal to the spatial sum, small rounding errors can flip the sign, so the calculator applies a tiny tolerance before labeling null.
No. Only the algebraic sign of s² changes between conventions. The underlying classification is consistent when interpreted within the chosen signature, which the calculator displays with each result.
Proper time appears for timelike separations. It equals τ = |s|/c and represents the elapsed time on a clock that could travel between the two events along an inertial path.
Yes. Choose one time unit and one unit per axis. The calculator converts each coordinate difference to SI units, then computes s², |s|, and the classification from those standardized deltas.