It took a while to find relativity and the various subtle (or unmeasurable) effects it has within our universe. Is it possible that, although Newton's gravitational force equation leaves out any notion of density or size, this too might impact the forces acting around us?
I noticed this question already, but I am not satisfied with its answer. I wonder if two objects, A and B, having the same mass but (for lack of a better term) relativistically different densities, would exert slightly different gravitational forces on a third object C which is far enough from A and B (and equally distant from both) to negate any "diffusion" effects on the gravitational fields of A and B?
In other words, is it possible that the equations we have are "good enough", and we just don't have examples (yet) that introduce the same kind of errors as those that confirmed the theory of relativity?
If not, why not? Is there any particular reason aside from "these are the equations we have"?
I realize that finding the mass of something like a black hole would be at best impractical, if not entirely impossible, without assuming that its mass directly determines its gravitational field strength.
Best Answer
The answer is No and the reason is the equivalence principle which says that there exist natural units in which the gravitational mass (the mass $m$ in $F=GMm/r^2$) is equal to the inertial mass (the mass $m$ in $F=ma$) for all objects in the Universe. This is equivalent to the statement that all objects, regardless of their composition, density, and other properties, accelerate by the same acceleration in a given gravitational field (any gravitational field).
This equivalence principle is the starting point behind Einstein's general theory of relativity which describes gravity as the curvature of spacetime (and which is fully respected by more modern theories of gravity, especially string theory). This principle is also experimentally verified at the relative accuracy level of order $10^{-17}$ which is incredibly accurate (one may compare two different materials which have noticeably different densities etc. and the acceleration is still perfectly the same). In principle, it's always possible that some experimental deviations will be found by finer experiments in the future (but that's true about any insight in physics). However, it's not just the absence of any experimental hints that undermines any idea that the gravitational force should depend on anything else aside from mass; it is also the complete absence of candidate theories that would be compatible with the basic observations and where the deviation would be implanted as anything more than just an ugly, partially inconsistent, unjustified, and numerically small deformation of a beautiful, consistent, robust, justified theory.