When I give talks about the way modern science is based on beauty, I give the example of how everyone will think Newton’s Law of Gravitation
- F = Gm1m2/r2
is more plausible than what one might call “Pruss’s Law of Gravitation”
- F = Gm1m2/r2.00000000000000000000000001
even if they fit the observation data equally, and even if (2) fits the data slightly better.
I like the example, but I’ve been pressed on this example at least once, because I think people find the exponent 2 especially plausible in light of the idea of gravity “spreading out” from a source in concentric shells whose surface areas are proportional to r2. Hence, it seems that we have an explanation of the superiority of (1) to (2) in physical terms, rather than in terms of beauty.
But I now think I’ve come to realize why this is not a good response to my example. I am talking of Newtonian gravity here. The “spreading out” intuition is based on the idea of a field of force as something energetic coming out of a source and spreading out into space around it. But that picture makes little sense in the Newtonian context where the theory says we have instantaneous action at a distance. The “spreading out” intuition makes sense when the field of force is emanating at a uniform rate from the source. But there is no sense to the idea of emanation at a uniform rate when we have instantaneous action at a distance.
The instantaneous action at a distance is just that: action at a distance—one thing attracting another at a distance. And the force law can then have any exponent we like.
With General Relativity, we’ve gotten rid of the instantaneous action at a distance of Newton’s theory. But my point is that in the Newtonian context, (1) is very much to be preferred to (2).
6 comments:
It's a subjective matter, but for me the idea of a force spreading through space is orthogonal to that of simultaneity.
Fwiw, it is possible to derive the Schwarzschild metric from from a modified inverse square law, where the variable speed of gravitons is taken into account.
Well, if we think of a "forcefield" as spreading out instantaneously through space, and the source of that forcefield is the massive object, then the massive object manages instantaneously to generate an infinite "amount" of forcefield, since each of the infinitely many spherical shells of the same thickness has the same "total amount" of forcefield in it given the inverse square law. I am not sure how much sense this talk makes, hence the various scare quotes.
That's a nice way of putting it. But since the actual speed of propagation is irrelevant, that speed could be arbitrarily high and still support a least squares heuristic, so taking the limit as it goes to infinity...
But there are other ways of thinking about gravity propagation. For example, a potential grid that expands in space. Would you really expect instantaneous gravity propagation to follow the same law in any number of dimensions?
This is a mere personal reaction. When I was in high school, we had to watch a movie in physics class about experiments to determine whether some inverse square law was really ^2 or instead something like ^2.0000000000000001. Probably the teacher was sick or something like that.
As soon as I figured out what the movie was about, my soul completely rebelled. I felt that it just HAD to be a small integer!!! I still feel the same way today.
Michael Gorman
I think I'd offer a slightly different picture (though still very similar) which still works with instantaneous action at a distance, where you derive Newton's law using Gauss's law, where integrating the gravitational field over an arbitrary surface gives you Newton's law (as long as you assume the curl of the gravitational field is 0 and that the field vanishes at infinity). The picture I associate with this is of gravitational field lines cutting through the shell. The field lines spread out from the source (or rather converge towards the source) in a spherically symmetric way. But the field lines don't have to travel at finite speed - they can 'just be there' - they aren't converging 'in time' so to speak. Now granted, perhaps you have to make a few more assumptions for this (Gauss's law, and irrotationality, and boundary conditions). But I still think it yields a physical picture that renders the inverse square law intuitive in a fashion different to looking at the simplicity of the equations (though perhaps we then have to justify why we're starting with Gauss's law specifically in the first place).
That's somewhat compelling. But what are the "field lines"?
And here's a perhaps intuitive problem for the picture. Suppose our test particle is localized at a single point. Then no matter how far away it is, it hits exactly one field line. Why does the field line get weaker with distance?
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