I'm still not sure what you meant Farsight? You wrote "Strictly speaking, imatfaal, the equivalence principle only applies in an infinitesimal region." And referring me to Kevin Brown that interpretation gets refuted by Einstein in that same paper?

Btw: Einstein wasn't that much for geometry.

-- -Quote-

" While relativistic mass is useful in the context of special relativity, it is rest mass that appears most often in the modern language of relativity, which centers on "invariant quantities" to build a geometrical description of relativity. Geometrical objects are useful for unifying scenarios that can be described in different coordinate systems. Because there are multiple ways of describing scenarios in relativity depending on which frame we are in, it is useful to focus on whatever invariance's we can find. This is, for example, one reason why vectors (i.e. arrows) are so useful in maths and physics; everyone can use the same arrow to express e.g. a velocity, (speed having a given direction) even though they might each quantify the arrow using different components because each observer is using different coordinates. So the reason rest mass, rest length, and proper time find their way into the tensor language of relativity is that all observers agree on their values. (These invariants then join with other quantities in relativity: thus, for example, the four-force acting on a body equals its rest mass times its four-acceleration.) This is one reason why some physicists prefer to say that rest mass is the only way in which mass should be understood.

- -End of Quote ----

--

It may be interesting to note here that this geometric notion of describing

SpaceTime wasn't entirely shared by Einstein.

- - - -

Quoted from John D. Norton

Department of History and Philosophy of Science

and Center for Philosophy of Science

University of Pittsburgh----

"In thinking mathematically, or, as Einstein's sometimes said, formally, one takes the mathematical equations of the theory as a starting point. The hope is that by writing down the simplest mathematical equations that are applicable to the physical system at hand, one arrives at the true laws. The idea is that mathematics has its own inner intelligence, so that once the right mathematics is found, the physical problems melt away. Philosophers will recognize this as a form of Platonism.. Just how did Einstein's physical insight work? One part was an keen instinct as to which among the flood of experimental reports were truly revealing. Another was his masterful use of thought experiments.

Through them Einstein could cut away the distracting clutter and lay bare a core physical insight in profoundly simple and powerfully convincing form.. That geometrical way of conceiving special relativity is not Einstein's. It was devised by the mathematician Hermann Minkowski shortly after Einstein published his special theory of relativity. Einstein was reluctant to adopt Minkowski's method, thinking it smacked of "superfluous learnedness." It was only well after many others had adopted Minkowski's methods that Einstein capitulated and began to use them. It was a good choice. It proved to be an essential step on the road to general relativity. Einstein preferred to think of his theory in terms of the coordinates of space and time: x, y, z and t. The essential ideas of the theory were conveyed by the algebraic properties of these quantities, treated as variables in equations. Its basic equations are the Lorentz transformation, which, in Einstein's hands, is a rule for changing the variables used to describe the physical system at hand. The laws of physics are written as symbolic formulae that include these coordinate variables.

The principle of relativity of relativity then became for Einstein an assertion about the algebraic properties of these formulae; that is, the formulae stay the same whenever we carry out the symbolic manipulation of change of variables of the Lorentz transformation. The emphasis in Einstein's algebraic approach is on variables, not SpaceTime coordinates, and formulae written using those variable, not geometrical figures in SpaceTime. For many purposes, it makes no difference which approach one uses, geometric or algebraic. Sometimes one is more useful or simpler than the other. Very often, both approaches lead us to make exactly the same calculations. We just talk a little differently about them. However there can be a big difference if we disagree over which approach is more fundamental. We now tend to think of the geometric conception as the more fundamental one and that Einstein's algebraic formulae are merely convenient instruments for getting to the geometrical properties. There is some evidence that Einstein saw things the other way round. He understood the geometric conception, but took the algebraic formulation to be more fundamental"

--End Quotes----

As for me, I grow up with the geometric definition

and find it intuitively the easier one.

Anyway, I would still like to understand what you meant there?