@xharv3yladx,

Spacetime is an abstract four-dimentional mathematical space whose points are EVENTS, not locations. So it isn't clear to me what units could be used to describe a "deformation" of a collection of events constituting the totality of all events past, present, future, and everywhere, even assuming such a concept was sensible. Miles are units of geometric area; units of geometric volume would not be sufficient because spacetime isn't physical space and it isn't three-dimensional; it's an abstract four-dimentional manifold which purports to organize eternity using a coordinate system.

One of the earlier replies refers readers to the Riemannian metric tensor. According to his link, this "measures the extent to which the metric tensor is not locally isometric to that of Euclidean space". This seems curiously inaccurate with reference to a four dimensional spacetime in which space and time are not separable components.

A more accurate description might be a formula for transforming the space and time coordinates of events from those obtainable in a Euclidean three-dimensional space and taking place at time coordinates corresponding to an absolute, universal and invariant model of time, into quite different coordinates corresponding to the "actual" spacetime metric. In this conception, "local" distances and times between events differ, somehow, from classical absolutes; and yet, the very idea of an invariant Riemannian metric seems to imply the existence of an absolute space and time: merely one that varies from classical concepts of space and time, and moreover, one which is nonuniform from one "locality" to another.

A "locality" in this context is a single event, or isolated point in spacetime. But because there is no such thing as a point-observer, much less a differential measurement carried out by one, but only finite (rather than infinitesimal) observers, carrying out finite difference measurements, it is unclear to me what meaning could be assigned to this.

It also isn't clear to me how the absolute spacetime of a Riemannian manifold in General Relativity can be made consistent with the observer dependent nature of space and time in Special Relativity. Apparently each observer has his own "local" (i.e. point centered and differential) coordinate chart, and through a "covering" of these charts can create an "atlas" of spacetime. The idea is that through differential geometry a quilt of local charts can be constructed which is identical to the spacetime manifold as a whole. But quite aside from the fact that every observer and every one of his tools and observations is a magnitude corresponding to a finite difference and not an infinitesimal, the requirement that all of these observer dependent atlases must be invariant through the Riemannian metric tensor, implies the existence of an absolute frame of reference which each should theoretically have access to, thereby contradicting the SR axiomatic premise that spatial and temporal metrics are both observer dependent. Nor is it clear to me how an "inertial frame" (the basic reference perspective of Special Relativity) could exist in a universe where gravitational potential is nowhere zero and generally nonuniform.