Math really is our friend, you know.
Why do I keep saying this? Because if you look at physical objects and motion on all levels:
1) Astronomical: Extremes of large and fast
2) Smallness to the point that mass as you know it disappears and manifests in complex wave forms: Quantum smallness
3) Extremes of small and fast: High energy, quantum and relativistic limits
you will see complex flows and a very curvilinear universe.
Math is a tool that we use. There are no measurements in nature. No coordinate systems, no calculations.
Nature never requires calculations. Nature manifests as perceptions of complex patterns.Human beings impose mathematical systems on nature to discover pattern. Math, as you know it, is just a tool your mind uses. And, like any tool, you can mess things up if you don't know how to use it.
Consider, in a curvilinear, flowing world in which we live, what chance does Mr Blockhead have to be able to describe it mathematically with his favorite tool:
In reality, what mathematicians have done is to learn to "curve" their coordinate systems to nicely fit over any specific system.
Instead of trying to squeeze a universe of curves and flow into a rectilinear shoebox, they made their coordinate mappings of a system fit the nature of the system.
A coordinate mapping is like "clothing" that is placed over a physical system. The physical system it self does whatever it does with no calculations and doesn't need math. System flow or evolution is mapped by ones choice of coordinate and variables.
An amazing thing happens when generalized coordinates are used. Concepts like "angle" between unit vectors and "distance" between two points needs to be defined since nobody knows how long a "unit" vecor really is.
When you can curve, bend and stretch ones coordinate mapping to any shape you want, "1 unit" movement no longer means any specific distance.
...............................................In the special and general theories of relativity, why is the word "relativity" in the title?
It is very similar to saying that the earth is not the centerpoint of the universe, or the self is not the centerpoint of the world.
In a psychological sense, "relative" means there is no preference of self over other. There is perfect symmetry between all (inertial) observers and mathematical expressions of laws of nature must express that symmetry.
"this theory arose primarily from the endeavour to understand the equality between inertial mass and gravitational mass".
"In accordance with classical mechanics and according to the special theory of relativity, space (space-time) has an existence independent of matter of field. In order to describe at all that which fills up space and is dependent upon the coordinates, space-time or the inertial system with its metrical properties must be thought of at once as existing, for otherwise the concept of "that which fills up space" would have no meaning. On the basis of the general theory of relativity, on the other hand, space as opposed to "what fills space"
, which is dependent on the coordinates, has no separate existence
"Thus a pure gravitational field might have been described in terms of the g(i, j) (as functions of the coordinates,), by solution of the gravitational equations. If we imagine the gravitational field, i.e. the function g(i, j) to be removed, there does not remain a space of type (I), but absolutely nothing, and also no "topological space"
. For the functions g(i, j) describe not only the field, but at the same time also the topological and metrical structural properties of the manifold. A space of the type (I), judged from the standpoint of general relativity, is not a space without field, but a special case of the g(i, j) field, for which - for the coordinate system used, which in itself has no objective significance - the functions g(i, j) have values that do not depend on the coordinates. There is no such thing as empty space, i.e. space without a field. Space-time does not claim existence on it's own, but only as a structural quality of the field
Thus Descartes was not so far off from the truth when he believed he must exclude the existence of an empty space. The notion indeed appears absurd, as long as physical reality is seen exclusively in ponderable bodies. It requires the idea of the field as the representative of reality, in combination with the general principle of relativity, to show the true kernel of Descartes'idea; there exist no space "empty of field"
"I wished to show that space-time is not necessarily something to which one can ascribe a separate existence, independently of the actual objects of the physical reality. Physical objects are not in space
, but these objects are spatially extended
. In this way, the concept of empty space loses its meaning".
- Einstein, 1952
The "field" (mass distribution) determines the local "shape" of space-time (g(i, j) metric tensor coefficients)
The object is not "in space" according to Albert. The very structure of space is inseparable from object.
If object is removed, discussion of the "empty space" as having structure in itself has no meaning.