I have some speculations of how this was developed.
Easiest answer is that represents a point in a perfect fluid where all points in the immediate neighborhood have the same values of and . It does not represent matter, mass or energy and the coordinate system that it represents, if there were any matter or energy present (which there isn't) is Euclidean. The perfect fluid is empty space.
If you want to model mass, matter or energy in this perfect fluid, you need a change in the values of and at a point. This is where the stress-energy tensor of a perfect fluid comes in. What it tells you is the deviation of and from . For mainstream GR physics, this deviation (or increase) in the perfect fluid is from zero or nothing (empty vacuum), but we equate to energy density and equivalently density of both inertial and gravitational mass. If one now takes the inertial mass density located within a certain volume and calculates the effects it has across empty vacuum the same as if it were a perfect fluid, it works. Since we only need the deviation of density and pressures to account for accurate GR answers and Newtonian gravity, there was no need to keep the perfect fluid itself as long as you don't have a problem with action at a distance. The questions of what "mass" and "energy" mean fundamentally in GR is not something I have seen answered.
Excellent question. I don't really consider it a convention, but more of a conversion factor and the beginnings of a physical model to enable a crossover in equating energy due to gravitational effects with those of the quantum world.I am convinced that there exists a mathematically equivalent representation of GR having the density convention you are describing, and that this theory is what you intend to represent. caveman seems more qualified than I in pointing out nuances this change in convention may encounter.
I have long held the ATM idea that all of electromagnetism is flawed, because charge is defined with the wrong sign. I am pretty sure that almost everyone has considered this, since the electron's charge became standard on the quantum level anyways.
Science has a long tradition of maintaining priority of conventions until there is a compelling reason to change, occasionally even if this means adding cumbersome terms. What makes your convention more compelling than the standard density convention with a cosmological constant?
First you would need to decide either between space-time or the perfect fluid as they both represent the same thing, GR for space-time and perfect fluid for this theory. The other thing concerns the misconception that this perfect fluid is any different than an emag field. The point of this theory is to provide one perfect fluid theory that can account for all fields. A tall order perhaps but with baby steps maybe we can get there. Thus I cannot answer your question yet.Suppose I had a region of space-time filled with a uniform fluid of constant mass and charge density and a constant electric field. This will produce an electric force on the fluid. Will the fluid accelerate according to or ?
Not sure which equivalence principle you mean again. Mainstream Newtonian physics will always have a superior place simply due to this is the way we perceive the world we live in, and for many practical aspects of our daily lives it serves well. Using this to develop a better quantum theory has many many practical applications in integrated circuits, i.e. overcoming crystal defects, heat transfer improvements...etc.If it is the former, then the wouldn't the equivalence principle suggest that the mainstream convention is superior to your density convention?