The invariant spacetime interval

From separate absolutes to one invariant

In classical physics, time intervals and spatial distances are separately invariant. All observers agree on $Delta t$, and all observers agree on lengths when measured correctly. Special relativity changes this. Different inertial observers may disagree on time separation and spatial separation between two events.

But they agree on a combined quantity: the spacetime interval.

Definition of the interval

For two events separated by $Delta t$, $Delta x$, $Delta y$, and $Delta z$, define

$Delta s^2=-c^2Delta t^2+Delta x^2+Delta y^2+Delta z^2$

This sign convention is common in relativity. Some books use the opposite sign:

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

The physics is the same if the convention is used consistently.

Invariance

Under Lorentz transformations, $Delta t$ and $Delta x$ change, but $Delta s^2$ stays the same. In one spatial dimension,

$Delta s^2=-c^2Delta t^2+Delta x^2$

All inertial observers compute the same value for $Delta s^2$.

This is analogous to rotations in ordinary geometry. A rotation changes coordinate components $x$ and $y$, but leaves $x^2+y^2$ unchanged. A Lorentz transformation changes space and time components but leaves the spacetime interval unchanged.

Timelike intervals

If

$c^2Delta t^2>Delta x^2+Delta y^2+Delta z^2$

then the separation is timelike. A slower-than-light object or signal could travel from one event to the other. There exists a frame in which the two events occur at the same place.

For timelike intervals, proper time is defined by

$c^2Delta au^2=c^2Delta t^2-Delta r^2$

where

$Delta r^2=Delta x^2+Delta y^2+Delta z^2$

Spacelike intervals

If

$Delta r^2>c^2Delta t^2$

then the separation is spacelike. No signal traveling at or below $c$ can connect the events. There exists a frame in which the two events are simultaneous.

The time order of spacelike-separated events can differ between frames, but that does not violate causality because neither event can influence the other by allowed signals.

Lightlike intervals

If

$Delta r^2=c^2Delta t^2$

then the separation is lightlike or null. Light in vacuum can connect the events. For such intervals,

$Delta s^2=0$

All observers agree that the interval is zero, and all measure the connecting light ray moving at speed $c$.

Physical meaning

The invariant interval is the geometric foundation of special relativity. Instead of absolute time and absolute space, relativity has absolute spacetime structure. Observers slice spacetime differently into space and time, but the interval remains unchanged.

The big idea

The spacetime interval is the invariant quantity shared by all inertial observers. It classifies event separations as timelike, spacelike, or lightlike. This classification determines possible causal relationships and replaces the classical idea that space and time are separately absolute.