It’s generally understood that, when the time dimension is subtracted from spacetime, we are left with Euclidean 3-space.  This is usually interpreted to mean that, once we look at spacetime without the time component, we’re left with space, or what in SI units is called volume.  The SI unit for volume is ${m^3}$ which is consistent with our understanding of direction being simply a ratio between lengths.  In this case, the three orthogonal lengths can be used to define an object without ever having to define orthogonality.  This seems like an incomplete description of what a volume actually is.

Just as we can use a direction and a radius for identifying a position, there’s also another interesting use for combining a direction and a radius.  We can use a direction and a radius to specify a volume, and this raises some other issues that have to be explained somehow.

The SI derived unit for volume is ${m^3}$.  But that’s not the only way to specify a volume.  If we know the radius of a sphere we can calculate the volume inside the sphere using a simple formula.  But what happens if we take a partial area of the surface of the sphere and use this to calculate volume?  The most straightforward method is to use the steradian (symbol $sr$) as the partial area of the surface of the sphere.  The steradian (see the SI chart in the above post) is referred to as the derived quantity for solid angles, but in this case its just as appropriate to look at it as the area of ${r^2}$ on the curved surface of a sphere.

The procedure is to take the volume of the space that lies within the steradian and the sphere center.  To make this into a volume we’ll take the product of the angular quantity of the $sr$ and the length quantity of the meter.   Well call this unit of volume the steradian meter, or $srm$.   There are ${4}\pi$ steradians in a complete sphere, so there must also be ${4}\pi$ units of volume that will be equal to the volume of the complete sphere.

The first thing to notice is that the $srm$ can use either direction or length as the unit quantity.  A quantity of $3sr$ where $r=1m$ is different than the quantity of $1sr$ where $r=3m$.  Mathematicians will probably know of a convention that can be used in order to capture this difference.  If we put the number in the appropriate location to show which quantity is the unit quantity and which quantity the number is associated with, then the first case would be written as $3srm$ and the second case would be written as $sr3m$.  A little arithmetic reveals that:

$1srm = sr1m = \frac{1}{3}m^3$  and also, for example:

$512srm = sr8m = \frac{512}{3}m^3$

If space is simply an empty volume, these three examples of expressing a volume should have identical meanings.  They should all be identical, both mathematically and conceptually.  The problem is that there is a difference.  The expression $\frac{512}{3}m^3$ has no spherical excess, $sr8m$ has a spherical excess of $\frac{\pi}{2}$, and $512srm$ has a spherical excess of $256\pi$.

I guess it would be possible to pretend that this difference has no relevance at all.  Once again, I think were up against the same mathematical quandary.  The difference between using unit length or unit direction to express a volume has no impact at all on Newtonian spacetime where ct is a constant.  But it does raise a whole universe of issues once we try to use these two different methods to express a volume in relativistic spacetime.