Whenever, I come across the spacetime interval, written like this, say, (Δs)

^{2}= (Δt)

^{2}– (Δx)

^{2}– (Δy)

^{2}– (Δz)

^{2}, it is as if it has to be that way. However, it seems to me it is this way by definition and does not have to be so. Sometimes, it seems to be referred to as the modified Pythagoras theorem.

What if we apply the usual Pythagoras theorem so that (Δs)

^{2}= (Δt)

^{2}+ (Δx)

^{2}+ (Δy)

^{2}+ (Δz)

^{2}and apply the following analysis. For simplicity, say the coordinates are arranged such that Δy = 0, and Δz = 0, so that (Δs)

^{2}= (Δt)

^{2}+ (Δx)

^{2}. Then, for timelike events:

Consider 3 events, where E

_{1}and E

_{2}are timelike, and E

_{1}and E

_{3}are lightlike:

(Δs

_{12})

^{2}= (Δt

_{12})

^{2}+ (Δx

_{12})

^{2}

(Δs

_{13})

^{2}= (Δt

_{13})

^{2}+ (Δx

_{13})

^{2})

(Δs

_{12})

^{2}- (Δs

_{13}

^{2})

= {(Δt

_{12})

^{2}+ (Δx

_{12})

^{2}} - {(Δt

_{13})

^{2}+ (Δx

_{13})

^{2}}

= {(Δt

_{12})

^{2}+ (Δx

_{12})

^{2}} - {(Δx

_{13})

^{2}+ (Δx

_{13})

^{2}}

= {(Δt

_{12})

^{2}+ (Δx

_{12})

^{2}} - {2(Δx

_{13})

^{2}}

= {(Δt

_{12})

^{2}+ (Δx

_{12})

^{2}} - {2(Δx

_{12})

^{2}}

= (Δt

_{12})

^{2}+ (Δx

_{12})

^{2}- 2(Δx

_{12})

^{2}

= (Δt

_{12})

^{2}- (Δx

_{12})

^{2}

= the same invariant as when the “modified Pythagoras theorem” is used

In words, for timelike events E

_{1}and E

_{2}...

... the square of the invariant, is equal to the square of the spacetime interval between events E

_{1}and E

_{2}, minus the square of the spacetime interval for light to get from E

_{1}to the spatial location of E

_{2}.

A similar case can be made for spacelike and lightlike events.

Lightlike events in different inertial frames would have different spacetime intervals between the same 2 events, but would not have the oddity of no spacetime interval between separate events.

???