The Spacetime Interval
The one quantity every observer agrees on — a symbol-by-symbol guide
Equations and classification verified against:
Wikipedia — Spacetime §Spacetime interval ·
Minkowski, H. (1908). Space and Time. ·
Taylor, E.F. & Wheeler, J.A. (1992). Spacetime Physics. W.H. Freeman. ·
Worked example verified by Python (50-digit decimal arithmetic).
The Equation — West Coast Convention (+−−−)
s²
= (
c
·
Δt
)² − (
Δx
)²
Each colored symbol is explained below. In three spatial dimensions this extends to (cΔt)² − Δx² − Δy² − Δz². This guide works in one spatial dimension for clarity.
Two sign conventions exist in the literature.
Some texts (especially in particle physics and general relativity) write s² = Δx² − (cΔt)², which is the east coast (−+++) convention. Both describe the same geometry — only the labeling of positive and negative differs. The notebook's SIGN_CONVENTION variable controls which is active. This guide uses the west coast (+−−−) form: s² = (cΔt)² − Δx², where timelike intervals are positive. Wikipedia's spacetime article documents both conventions.
The formula has two coordinate differences as inputs (Δt, Δx) and one invariant as output (s²). The Δ prefix means "difference between" — always coordinates of event B minus coordinates of event A.
The invariant output — identical in every inertial reference frame. It is always written as s² rather than s because the expression can be negative, and square roots of negative numbers are undefined in real arithmetic. Minkowski (1908) introduced this construction to give a frame-independent "distance" in spacetime. Its sign classifies the causal relationship between the two events.
SI unit: m² (meters squared)
The difference in time coordinates between event A and event B, measured in a specific reference frame. Different inertial observers will measure different Δt for the same pair of events — this is time dilation. The capital Δ (delta) denotes a difference: Δt = t_B − t_A.
SI unit: s (seconds)
The difference in spatial coordinates between event A and event B, in the same frame. Like Δt, this is frame-dependent — observers disagree on Δx. What makes the spacetime interval remarkable is that while Δt and Δx separately vary from frame to frame, the combination (cΔt)² − Δx² does not.
SI unit: m (meters)
Multiplying Δt by c converts seconds into meters, putting time and space on the same footing so they can be compared and combined. The fact that c is the same in every inertial frame — Einstein's second postulate — is precisely why s² is invariant. It is baked into the structure of the formula.
c = 299,792,458 m/s (exact)
To verify invariance, we apply the Lorentz transformation: Δt′ = γ(Δt − vΔx/c²) and Δx′ = γ(Δx − vΔt). These give the coordinate separations as seen by an observer moving at velocity v. Despite Δt′ ≠ Δt and Δx′ ≠ Δx, computing (cΔt′)² − (Δx′)² always returns the same s². This is what the notebook verifies numerically across many boost velocities.
v in m/s; γ = 1/√(1−v²/c²)
The sign of s² is a physical statement about causality — whether two events can influence each other and whether their temporal order is absolute. This classification is confirmed by Taylor and Wheeler (1992) §1-5 and the Wikipedia spacetime article §Spacetime interval.
Timelike
(+−−−) convention: s² > 0
The time gap is large enough that a physical signal (or object) traveling at sub-light speed could connect the events. There exists a reference frame where both events occur at the same location. All observers agree on which event happened first. The proper time between timelike-separated events is √(s²)/c — a frame-independent elapsed time.
Example: a clock ticking twice — same location, different times.
Spacelike
(+−−−) convention: s² < 0
The spatial gap is so large that even light cannot connect the events in the available time. No causal relationship is possible. Crucially, different observers can disagree on which event happened first — and a frame always exists in which the events are simultaneous. This is the relativistic relativity of simultaneity.
Example: two simultaneous explosions on opposite sides of Earth.
Lightlike (null)
Both conventions: s² = 0 exactly
The events lie exactly on the worldline of a photon: Δx = c·Δt. s² = 0 in every reference frame — a direct consequence of the speed of light being frame-invariant (Einstein's second postulate). A photon does not age between emission and absorption; its proper time is zero.
Example: a photon emitted at event A, detected at event B.
The notebook applies Lorentz boosts to demonstrate invariance. The notation is standard in all modern textbooks on special relativity.
Notation decoder
Δt, Δx
Coordinate separations in the rest frame (unprimed). Δ means "difference": Δt = t_B − t_A, Δx = x_B − x_A.
Δt′, Δx′
Coordinate separations in the boosted frame (primed). The prime mark signals a different observer moving at velocity v. Same pair of events, different measured values.
γ(Δt − v·Δx/c²)
The Lorentz transformation for time. The term v·Δx/c² is the "mixing term" — it encodes the relativity of simultaneity. Events that are simultaneous in one frame (Δt = 0) are not simultaneous in another when Δx ≠ 0.
γ(Δx − v·Δt)
The Lorentz transformation for space. At v = Δx/Δt (the frame comoving with whatever connects the events), this equals zero — the two events are at the same location.
s²(Δt,Δx) = s²(Δt′,Δx′)
The invariance statement. Computing s² with unprimed values gives the same result as computing it with primed values. The notebook verifies this for five boost velocities spanning 0.1c to 0.99c.
A timelike event pair boosted to the comoving frame. All arithmetic verified by Python (both standard float and 50-digit decimal arithmetic give identical results for this example).
Event pair: Δt = 5 s, Δx = 9 × 10⁸ m, c = 3 × 10⁸ m/s (rounded for integer arithmetic), v = 0.6c = 1.8 × 10⁸ m/s
Why 0.6c? At β = 0.6, the Lorentz factor γ = 5/4 exactly (from the 3-4-5 Pythagorean triple: 1/√(1−0.36) = 1/√(0.64) = 1/0.8 = 1.25). All arithmetic stays in exact fractions.
1
Compute s² in the rest frame
Apply the (+−−−) formula directly.
(c·Δt)² = (3×10⁸ × 5)² = (1.5×10⁹)² = 2.25×10¹⁸ m²
(Δx)² = (9×10⁸)² = 8.10×10¹⁷ m²
s² = 2.25×10¹⁸ − 8.10×10¹⁷ = 1.44×10¹⁸ m²
💡 s² > 0 confirms this is a timelike interval. An object traveling at v = Δx/Δt = 9×10⁸/(5×3×10⁸) = 0.6c could be present at both events — exactly the boost velocity chosen.
2
Compute γ at v = 0.6c
This value is exact because 0.6 = 3/5 comes from a Pythagorean triple.
β = 0.6
β² = 0.36
1 − β² = 0.64
√(0.64) = 0.8 ← exact (4/5)
γ = 1/0.8 = 1.25 ← exact (5/4)
💡 The 3-4-5 right triangle: sides 3, 4, hypotenuse 5. Set β = 3/5 → √(1−9/25) = √(16/25) = 4/5 → γ = 5/4. Using a Pythagorean fraction ensures every step stays in exact arithmetic.
3
Apply the Lorentz transformation: compute Δt′ and Δx′
Transform both coordinate differences into the boosted frame using the standard Lorentz equations (Einstein 1905, §3; see also Taylor & Wheeler 1992, Eq. 1-19).
Δt′ = γ(Δt − v·Δx/c²)
= 1.25 × (5 − 1.8×10⁸×9×10⁸/(3×10⁸)²)
= 1.25 × (5 − 1.62×10¹⁷/9×10¹⁶)
= 1.25 × (5 − 1.8)
= 1.25 × 3.2 = 4.0 s
Δx′ = γ(Δx − v·Δt)
= 1.25 × (9×10⁸ − 1.8×10⁸×5)
= 1.25 × (9×10⁸ − 9×10⁸)
= 0 m
💡 Δx′ = 0 means the two events occur at the same location in the boosted frame. Physically: we boosted to the rest frame of whatever was traveling at 0.6c between the events. That object was always at x = 0 from its own perspective — the two ticks of its clock happened at the same place.
4
Compute s² in the boosted frame — confirm invariance
Apply the same formula to the primed coordinates.
(c·Δt′)² = (3×10⁸ × 4.0)² = (1.2×10⁹)² = 1.44×10¹⁸ m²
(Δx′)² = (0)² = 0 m²
s²′ = 1.44×10¹⁸ − 0 = 1.44×10¹⁸ m² ✓
💡 Identical to step 1. s² = 1.44×10¹⁸ m² in both frames, despite Δt changing from 5 s to 4 s, and Δx changing from 9×10⁸ m to 0 m. The notebook verifies this numerically across boost velocities from 0.1c to 0.99c for all three causal classes.
In plain English, the spacetime interval says:
"Space and time separately are observer-dependent —
different frames measure different Δt and Δx for the same events.
But the combination (cΔt)² − Δx² is the same for every observer.
It is spacetime's version of the Pythagorean theorem — with a minus sign."
Its sign classifies events: timelike (causal connection possible), spacelike (no causal connection, order is frame-dependent), or lightlike (connected by a photon, s² = 0 universally). All verified by computation and confirmed against Minkowski (1908), Taylor & Wheeler (1992), and Wikipedia.