Spacetime Interval Calculator

In Minkowski spacetime, the geometry is described by the metric:

$$ds^2 = c^2\,dt^2 - dx^2 - dy^2 - dz^2$$

This is the flat-spacetime analog of the Pythagorean theorem, but with a crucial sign difference: the time component enters with a positive sign and spatial components with negative signs (using the +−−− convention). For finite separations between two events:

$$s^2 = c^2(\Delta t)^2 - (\Delta x)^2 - (\Delta y)^2 - (\Delta z)^2$$

The three interval types have distinct physical meanings:

• Timelike ($$s^2 > 0$$): The time separation dominates. There exists a reference frame where both events occur at the same location but at different times. The proper time between them is $$\tau = \sqrt{s^2}/c = \sqrt{\Delta t^2 - \Delta r^2/c^2}$$, which is the time measured by a clock traveling between the events.
• Spacelike ($$s^2 < 0$$): The spatial separation dominates. There exists a frame where both events are simultaneous but at different locations. The proper distance is $$\sigma = \sqrt{|s^2|}$$. No causal influence can travel between spacelike-separated events.
• Lightlike / Null ($$s^2 = 0$$): The events are connected by a light ray. This defines the light cone — the boundary between causal and non-causal regions of spacetime.

The invariance of $$s^2$$ under Lorentz transformations is the geometric foundation of special relativity. Just as rotations preserve spatial distance $$d^2 = x^2 + y^2 + z^2$$, Lorentz boosts preserve the spacetime interval. This is why the Minkowski metric is sometimes called a "pseudo-Euclidean" metric — it measures a generalized "distance" in 4D spacetime.

The Interval Type tells you the causal relationship between the two events. For timelike intervals, the Proper Time gives the elapsed time on a clock moving inertially between the events — this is the shortest possible time separation. For spacelike intervals, the Proper Distance gives the spatial separation measured in a frame where both events are simultaneous. A lightlike interval means the events lie on each other's light cone. These classifications are observer-independent and form the backbone of causality in relativity.