If one travels a distance X east, then goes a distance Y north, that is the same as going a distance √(X² + Y²) northeast. But if one travels for a time X east, then goes for a time Y north, is that the same as going for a time √(X² + Y²) northeast? No, the travel time would be (X + Y) in that case. This is because time is conceived as a magnitude, without regard for direction, and so is cumulative.

I’ve mentioned *before* that there are apparently six dimensions of space-time (or time-space) but there is more to it. For one thing we need to distinguish two ways that a point event relates to multiple dimensions. The first way is that a point event has a location in multi-dimensional space-time. The second way is that a point event may be the resultant of a series of motions in different dimensions. For space these two ways are equivalent but for time they are different.

The first way has been developed in detail: for 3D space the components are combined with a Euclidean metric and for 3D space + 1D time a hyperbolic metric. The spacetime (invariant) interval between two point-events is:

*s² = Δ***r***² – c²Δ*t*² = Δr _{1}² + Δr_{2}² + Δr_{3}² – c²Δt².*

If time is a vector, it should have components with a Euclidean metric, too:

*s² = Δ***r***² – c²Δ***t***² = Δ r² – c²Δt_{1}² – c²Δt_{2}² – c²Δt_{3}².*

But this is misleading because we don’t ordinarily think of time that way. Instead, we think of time as something flowing from one motion to the next, which would mean time is cumulative. So a time vector would be understood similar to a taxicab metric:

*s² = Δ***r***² – c²Δ***t***² = Δ r² – c²*(

*Δt*)

_{1}+ Δt_{2}+ Δt_{3}*²,*

where the *Δ* quantity is understood as a distance (and so is non-negative). Otherwise the absolute value would be taken:

*s² = Δ***r***² – c²Δ***t***² = Δ r² – c²(|Δt_{1}| + |Δt_{2}| + |Δt_{3}*|)

*²,*

But this is misleading, too, since it is a series of motions and their resulting time displacement rather than the components of a space-time location. So we should go back to the Euclidean metric and think of the time components differently.

What do the components of time mean if they aren’t the flow of time for a series of motions? Temporal components should be considered like distances measured by time with a constant speed. For a vehicle traveling at constant speed (or pace) space and time are very similar. Multidimensional time isn’t the cumulative flow of time but the dimensions of duration by direction of a vehicle traveling in space-time.

In the end, the six dimensional space-time (invariant) interval is what would be expected:

*s² = Δ***r***² – c²Δ***t***² = **Δr _{1}² + Δr_{2}² + Δr_{3}² – c²Δt_{1}² – c²Δt_{2}² – c²Δt_{3}²*.

It’s just that we need to be careful not to confuse time here with a cumulative flow of time.