In his 1920 Book, “Relativity: The Special and General Theory”, Einstein started out by questioning the axioms of Euclidean geometry (see Physical Meaning of Geometrical Propositions). Einstein says,
The concept “true” does not tally with the assertions of pure geometry, because by the word “true” we are eventually in the habit of designating always the correspondence with a “real” object; geometry, however, is not concerned with the relation of the ideas involved in it to objects of experience, but only with the logical connection of these ideas among themselves.
Einstein points out that the propositions of Euclid’s geometry have been formalized to conform to idealized logic. When these propositions are satisfied for those real things we have associated with the geometrical ideas then geometry may be treated as a branch of physics. Einstein then goes on to establish a system of co-ordinates based on rigid bodies. Rigid bodies expand and contract, and so do their space (see The System of Co-ordinates).
Physics is essentially dealing with space that acts as extents of physical substance.
We may expand on Einstein’s argument by pointing out that locations in physical space shall have dimensions, although Euclidean geometry represents locations by points that do not have any length, area, volume, or any other dimensional attribute. A location in physical space may be infinitesimal, but it is not dimensionless. It is also continuous with the surrounding space, just like irrational numbers are continuous on a number line.
Euclidean space is defined as a set that includes points as elements. The physical space, however, is continuous with the locations within it. This makes the physical space a primitive notion and not the point location.
The physical space is a primitive notion, which then allows the notion of point locations.
We may now establish a system of co-ordinates based on the concept of quantum. A quantum of space is a point location. A point location not only has extents but these extents expand and contract. As the extents contract the density of point location increases; and as they expand the density decreases. Thus, in addition to the three spatial dimensions a point location has an additional dimension of density. As density increases, the point location gains endurance. This makes the dimension of density, or duration, imply varying rigidity. Ultimate in density, or duration, is complete rigidity.
A physical location has the dimensions of length, width, height and duration (density).
This dimension of duration is not the same as the dimension of time. Duration is real and objective, whereas, time is abstract and subjective.
Einsteinian space is rigid as it is based on material substance. It assumes the infinite density and duration of the atom. The physical space, on the other hand, implies varying rigidity of the non-atomic substance. This is expressed through the dimension of duration (density). The rigidity of the atomic substance then becomes a limiting condition.
This may have some fundamental implications on the way the Relativity and Quantum theories are currently interpreted.
.
