The most basic property of space is the distance between any two points in it. Given a formula for the distance between any two points, all other properties of the space can be deduced. In ordinary Euclidean space, in which parallel lines never meet and all the basic rules of geometry apply, the distance between any two points are given by the formulae:
in one dimension.
in two dimensions.
In three dimensions.
These distance formulae – called metrics – can be extended to higher dimensions of space in the obvious way.
BUT NOT TO SPACETIME!
The time dimension is not the same as space dimensions in some very important ways:
Time only moves forward. It has an arrow. The arrow of time is in the direction of increasing entropy or disorder. Things get worn out or broken – disorder or entropy increases and time moves forward. Entropy is a real physical concept, and can be measured mathematically.
Time moves at a constant rate, at least for all observers stationary with respect to each other in an inertial frame. Time may not stop – there is no such thing as stationary time. A particle may stop in space, but it is still moving forward in time.
Time pervades all of space. We can treat spacetime as four, three or two dimensional, discarding dimensions of space as convenient, but one of the remaining dimensions is always time.
The distance function in the spacetime of special relativity is given by
Notice:
The distance
between two events with coordinates
and
may be zero even if the events do not happen at the same point. This can only be so if the two events happen on a light ray.
can be negative if
but in ordinary 3 dimensional space a distance squared can never be negative.
The effect of this is that if the spacetime interval squared,
between two events is negative, the two events cannot be causily related.