4 The Transformations of Length and Volume
It is well accepted that the following formula is used to transform volumes:
As we have already noted, prime values denote what an observer measures who moves past the object under consideration with relative velocity v jin the x-direction, while the unprimed values denote an observer at rest, relative to the observed system. The unprimed values are thus identical to the rest values. The volume transformation results from the 'length contraction' in which only the direction of the relative movement is affected. Shape and orientation of the volume do not matter.
Because this radical expression occurs repeatedly we will often just refer to 'the root' or the 'root factor' and lazily write V' j= V · √ for the transformation of the volume.