Big caveat: I am very much of a layman when it comes to physics and this is consequence of having heard elementary physics lectures more than 15 years ago.
But I like the occasional book on the subject (The Elegant Universe comes to my mind) and am fascinated by the theoretical possibility of a grand unified theory of Nature's forces.
There is a small niggling idea in my head that wants to get out from time to time, and this post is the evidence that it finally made it into the wild. (I will probably appear as a bloody idiot because of this...)
Being Hungarian-born has made me somewhat receptive for advances in physics performed by the pretty smart folks coming from this small country, and already as a pupil I was taught that the "equivalence principle" (of gravitational mass and inertial mass coinciding to an amazing degree) was shown by Eötvös. This cannot be a simple coincidence but must be a fundamental thing.
A completely ununderstood fundamental thing, as it appears to me. A quick googling for "inertia quantum gravity" brings up some outright lunatics and some rather exotic theory of "zero field fluctuations" causing inertia:
But, electromagnetism causing the same mass as gravity (i.e. linking two of the fundamental forces with an unbelievable precision) appears very far-fetched to me.
Wikipedia seems to be no help when inquiring for the source of inertia. Mach's principle referenced therein, namely that the existence of the rest of the universe leads to the phenomenon of inertia does not sound right; after all, when all the other stars (galaxies etc.) would be removed but the Sun, would you expect that the inertia of your body suddenly would fall to almost zero?
I cannot believe that modern theories of gravity (such as quantum gravity approaches) do not try to explain the equivalence of gravitational mass and inertial mass. My intuition says that this should be the first step!
Now I'll begin a steep descent in the land of speculation and respectlessly mix stuff told by others with vague ideas and metaphors of mine. I won't even make effort to cite or link, but at some point I'll tell what my main idea is. Be warned: at that point probably each self-respecting physicist (that is, a person who successfully solved a differential equation about movement of a body under external force) will turn away with disgust, but okay, let it be...
There is this idea that gravity is so weak in comparison to electromagnetism, because the circular (closed) strings corresponding to the graviton do not permanently bind to our three dimensional space (or four-dimensional spacetime). They are just passing through, and there is little time for them to establish the force between bodies. So, the gravitational force exerted between two glasses of red wine is very small on each other. But when they are accelerated (say, by clinking them) perceptible forces appear. So I conjecture that both forces are caused by the same mass, and by the same gravitons, but they differ in magnitude, because the gravitons stay around longer in the second case, so as messenger particles they establish a force many orders of magnitude stronger.
How can they stay around longer to become relevant? Perhaps by traveling orthogonally to the gravitational gravitons. Okay, inertial gravitons travel (more or less) orthogonally to our space, that is, along the time axis and thus following the trajectories of the particles the matter is built of.
I think it was Feynman who mentioned that an anti-particle traveling 'backwards' in time can be equated with its particle traveling forward in time. Now, what is the anti-graviton? I believe that I read sometime that it is the same as the graviton. Just like a photon is not differing from an anti-photon. So the time inversion does not change any property of the graviton. Good. We have gravitons which make a short visit in our spacetime, and vanish just as fast as they appeared, and there are those which accompany mass-like particles (even when at rest) along their trajectory in time, possibly bouncing back and forth between earlier and later. This constitutes (my IDEA!) a force between a body now and its earlier appearance in spacetime. Of course this is the same as a force between now and the future of the body. Actually there is a force between me and (me one second later). Or ten seconds or even a year!
Two things become clear now. First we have to perform an integration (over all time destinations) to get the resulting force. And second, the force toward the past will bear a negative sign, so for stationary bodies all inertial forces will cancel out. I think this can be asserted for uniform movement too (i.e. constant velocity vector) but I did not waste any thoughts on this yet. Anyway, this is Newton's first law.
Time to get to the second one. Here I want to make some further remarks. The inverse-square(-like) law should hold for inertial gravitons too, so that a longer time-span enters into the integral with a smaller effect. Kind of like the gravitational force between objects a light-second apart also decrease substantially. Second, causality is a bit smeared at the very short intervals where the forces are really relevant (Planck time?) so the reality a bit before pulls hard backwards and the reality a bit later pulls hard forward. Both look pretty much the same, differing only in the arrow of the force. I do not know whether this is essential, though.
Back to Newton's second law. Imagine a stroboscope lighting a scene where a body moves, accelerated by a constant force along its movement direction. When you make a photo (with reasonable exposition time) you'll obtain a picture of several bodies, with the earlier ones nearer together, and the later ones successively more distant. Now let's declare the middle one as being now. Then the prior one is less distant than the next one so the force between the prior and now is bigger than the force between now and next. These do not cancel out any more. This is a consequence of the inverse-square law. Of course the stroboscope should flash with Planck frequency etc. (so that the spacetime distance is small enough), but the principle is clear: We get a resultant force that points backwards, namely the difference between the (bigger) half-integral towards the past and the (smaller) half-integral towards the future.
All the time I was using the wrong word 'force' for something space-time-like (a vector with 4 components). What we perceive or measure as accelerating force is the projection of that 4 component vector into our 3-dimensional space. So I conjecture that F = ma is the projection of the resultant 4-force which I explained in the last paragraph to our 3 dimensions.
If this little idea holds any water then it should be possible to calculate the inverse law for the 4-force. I'd bet it is inverse-cubic. If this succeeds, then the next test would be to figure out how the 4-force looks like when a relativistic (say half-lightspeed) particle is accelerated by a 3-force. Clearly the 4-speed rotates out of time, but does the the mass of the particle increase?
I do not dare to think about whether we can ever get to the point where this idea gets tested in an even more comprehensive setting.
But it was sure a lovely night when I was drinking that red wine with a good friend from Brasil two days ago, who dared to ask a quantum-mechanical question and received a long story as an answer, unknowingly liberating this small niggling idea from the confines of my head!