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For years I have been very fascinated by the "mystery" aspect of gravity. Functionally, we understand it perfectly for our all applications, but in my (limited to undergrad General Physics 3 quarters) understanding, we don't know WHY it exists or HOW the "pull" works.

I'm very sorry for what is surely trite pop-science term use, but, to provide an example of what I mean: there has to be SOME "measurable material" of gravity between say the earth and the moon -- some kind of 'graviton' (not that it needs to be a particle or anything, I have zero claims as to the nature of how gravity does what it does). The [butchered] saying of "pluck a flower and move the furthest moon," because gravity in theory, has an infinite range (?) and moving a flower on earth could maybe move an atom on jupiter ever so slightly (?)."

Anyway, similar to how we had "very good" hypotheses about the existence of higgs-boson that the LHC 99%+ confirmed, do we have any "very good" hypotheses in regards to the fundamental way how gravity does what we have long-known it to do?

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General relativity pretty much covers the how and why of gravity... Gravity is a manifestation of the curvature of spacetime. It is inherent to it. This makes gravity fundamentally different from, say, electromagnetism (EM). EM is described by external fields existing in and moving through spacetime. Gravity follows from the structure of spacetime itself. In this sense, gravity is not even a force. Of course the philosophical question why gravity works this way is not answered by GR, because it cannot be answered by science. –  Wouter Oct 18 '13 at 18:34
@Wouter interesting, I'll have to read up on that. I disagree that "it cannot be answered by science," and would argue that that sentence needs a "yet" at the end ! –  HC_ Oct 18 '13 at 18:40
@HC_ By definition, science answers how. We leave why to philosophers and priests. –  MaxGraves Oct 18 '13 at 19:31
"Why", is something that goes on in our mind, and has no answer; so philosophers can talk about it ad infinitum, and not be called idiots. "What", is something that humans can "detect", (aka "observe") with our "senses" , so we can describe it to others. Allegedly, Isaac Newton saw an apple "fall", to the ground from a tree ("what"), so he asked "why" (big mistake), and he made up the idea of gravity, to answer the silly question. Einstein said that maybe, the ground flew up, along with the tree, and landed on the apple. We've all been talking about that ever since. –  user26165 Oct 19 '13 at 1:24
"Why" is a word that, notwithstanding the skeptics here, does have a tendency to grow into "whats" and "hows" and ultimately even to testable "whats" and "hows" if people get together and think about it hard enough. Once it does this, new "whys" will sprout in its place, leaving the false impression that nothing has been learnt. GR is an excellent example. Gravity to Newton was a what and "why" it is there would have seemed inpenetrable, even meaningless. He couldn't have foreseen that someone one hundred years after his death would contemplate "what if" we chop off .... –  WetSavannaAnimal aka Rod Vance Oct 20 '13 at 6:34
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Gravity may be fundamentally different from electromagnetism, as Wouter says, but it seems to me that, as far as your question is concerned, gravity is not fundamentally different from electromagnetism: there is gravitational field (http://en.wikipedia.org/wiki/Gravitational_field#General_relativity ), which is indeed "SOME "measurable material" of gravity between say the earth and the moon". AFAIK, you are right, and "gravity in theory, has an infinite range and moving a flower on earth could move an atom on jupiter ever so slightly." (I removed your "maybe" and question marks). However, in general, an atom on Jupiter does not "feel" a movement of a flower on Earth instantaneously - gravity is widely believed to have a finite propagation speed (http://en.wikipedia.org/wiki/Speed_of_gravity ) equal to the speed of light.

So again, it seems that, as far as your question is concerned, the situation with gravity is not fundamentally different from that with electromagnetic field: the Coulomb field has an infinite range, and there is "measurable material" - electromagnetic field (in the form of electrostatic field) between two charges, but, if one charge moves, the other charge does not "feel" that movement instantaneously. Thus, both gravitational and electrostatic forces are mediated by fields. One could say that gravitational field is just space curvature, but I don't feel that would change much. As for gravitons... Again, AFAIK, while it is not clear yet how gravitational field should be quantized (http://en.wikipedia.org/wiki/Quantum_gravity ), that does not seem to matter much as far as your question is concerned.

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