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Has anyone developed a vector-scalar theory of gravity using an acceleration field equation that is based on a ratio in which the numerator is the familiar Newtonian vector sum of all masses times the inverse square of their respective distances, and the denominator is the scalar (not vector) sum of the very same formula? As a scalar sum, the denominator would always be larger than the numerator because there is no cancellation from opposite directions, and the quotient could be interpreted as the proportion of "directionality" of the combined contributions.

Or has anyone developed a relativistic theory based on the same ratio of vector to scalar sum of the same contributions?

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Sort of, but not exactly, not in this arbitrary way. It resembles LeSage's idea, Mach's ideas (pre-GR) and Brans-Dicke ideas, but the details are different than all three.

I should first say that this idea is no good when there are a small number of particles. If you have two particles orbiting with this force law in a would-be elliptical orbit, and they are the only things in the universe that you include, the force changes with distance, and as the particles get close, the scalar strength (your denominator) changes magnitude, and the particles don't have closed elliptical orbits, and begin to precess. The effect is very large for two nearly equal mass objects, the strength of the force changes by order 100% for elliptical orbits.

But I am sure that this is not what you are imagining. If you include a finite density of very far away matter, all with a teeny tiny force on the two orbiting objects, the inverse square force gives a finite denominator contribution at each shell, so there is a divergent overall denominator. You can imagine that the denominator is an enormous sum from the distant matter, and the force from the two bodies orbiting is a negligible contribution to the denominator, in this case your theory would just lead to a small correction to Newton's law.

This idea, however, is not compatible with special relativity. In order for the strength of gravity to change with the configuration of distant matter, you need a local quantity that tells you the strength of gravity, a local scalar field. This field $\phi$ needs to obey a local equation of motion, and if it is sourced by the mass, this forbids it from being a sum of inverse-square contributions (it could be inverse square if the mass weren't always positive, you need dipole sources for an inverse square scalar field response).

When you have a scalar field sourced by distant masses, the local field value goes like $1\over r$. But you can replace the $1\over r^2$ contributions from distant matter with $1\over r$ contributions from interaction with a scalar field, and then it turns into Brans-Dicke theory. Brans-Dicke theory is a modification of GR which has a parameter which turns it into GR, and it is a fully reasonable (although in the reasonable parameter range, experimentally excluded) model of gravity.

This is the closest to your idea. But the idea that distance matter determines the local inertial frame, which is suggested by the idea of having distant gravity affect the local motion, is Mach's principle. This principle is incorporated to a certain extent in GR, but to what extent it is there is a subject of debate still.

Anyway, the other theory this resembles is LeSage gravity. In this theory, particles whizz around, and get absorbed by matter, casting a shadow which leads to gravity. The overall decrease in the strength of gravity in your model would be due to far-away absorption of LeSage particles in the distant past, as they whizzed through lots of distant stars. The LeSage model reproduces an overall weakening of gravity according to the distant absorption as $1/r^2$, so it is extremely close to your model, except that LeSage particles exponentially attenuate as they pass through matter, so the effect isn't linear--- you have to take into account the fact that after getting partially absorbed in outer shells, they are absorbed less by inner shells around you, just because there are fewer of them.

In general, the constraints to this sort of speculation come from the difficulties of making a relativistic quantum theory. Once you accept special relativity and quantum mechanics, it is not possible to construct theories in such arbitrary ways, by postulating a force law. Relativity requires local fields, and quantum mechanics requires either quantum fields or S-matrix theory. The results are ridiculously constrained, the force law comes from the field content in field theory, or from consistency conditions in S-matrix theory.

Either way, the only viable form of your type of model in a modern quantum framework is Brans-Dicke theory. Variations on this idea emerge naturally in string-formulations of string theories, so it is definitely consistent. In our universe, there is probably no scalar component of gravity at long distances, this is established experimentally from precision solar system tests of GR.

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Thank you for your response, Ron. This idea came to me, when I was a second-year undergraduate physics student at the University of Chicago in 1958, as a possible explanation of the high orbital velocities in the outer regions of spiral galaxies and clusters. My hypothesis was that the scalar denominator would be less in the outer regions than where we reside and probably calculated big G. So the acceleration would be greater. But then I switched majors because I couldn't handle the math. I'd like to know if anyone has used the same idea in developing a credible theory of gravity. Thanks. – Dick Montague Jul 31 '12 at 22:37
@DickMontague: What you are describing in the above comment is MOND, which has a scalar-vector-tensor tentative formulation. I don't study these things, since I think dark-matter is established cosmologically and from the large-scale stability of galaxy clusters. But if you google MOND and variants you'll get what you want. – Ron Maimon Jul 31 '12 at 23:26

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