So the inverse square law applies to gravity:
F = G(m1*m2)/r^2
My question is, Why is it r^2 instead of r^1.9897201, etc.?
Is there a reason for this, or is it “just because”?
Thanks for your insights!
If you know about the inverse square law, then it follows from the geometry of Euclidean space.
Why is the circumference of the circle 2π*r ? Reasons or “just because”?
Considering π is 3.14159… it’s not really the same thing. When you think about it, it is unusual that a naturally derived constant is exactly equal to an integer other than zero or one.
Aside from experiements, here are three reasons:
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The only central force that falls off at infinity whose bound orbits are also closed is 1/r^2. (This is called Bertrand’s theorem, but it’s a very straightforward application of Lagrangian dynamics.)
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If you assume that gravity is mediated by a graviton (or similar particle), then the force should vary as 1 over the surface area of the sphere of radius r; that is, as 1/r^2.
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Simplicity. Having a 1/r^1.9897201 force would be weird an arbitrary. (I’d like to invoke some sort of smoothness or conservation law to force the exponent to be an integer, but I can’t come up with anything.)
It’s a natural result of the geometry of three dimensional space.
It doesn’t follow immediately that three-dimensional space corresponds to a 1/r^2 force. The weak and strong interactions, for example, do not behave in that manner. Certainly potentials that behave like radiation of some sort from a point source do, but that only begs the question of gravity being that kind of interaction in the first place.
Other than the fact that we can’t have π fingers in each hand, what do you mean by “not really the same thing”?
It is not a physical constant like c. It’s a mathematical constant like π and 49.
As others have said, 1/r^2 is a natural result of living in a 3D universe. However, it’s not clear that it had to be that way, or even if it is. There is a theory called MoND–Modified Newtonian Dynamics–that proposes to be an alternative to dark matter as an explanation of galactic rotation curves. It proposes a gravitational falloff different than 1/r^2.
Now, it doesn’t seem like MoND fully explains cosmological observations, and probably isn’t necessary if you already assume dark matter, but as far as I know it still hasn’t been fully ruled out.
Newton’s law of gravity is clearly an approximation, so any analysis of the fundamental nature of gravitation based on it seems premature.
As I interpret the question, (relating to gravitational attractions/measurements on the surface of the earth) I think of the anomalies that exist due to differences in the mass of the materials between the point of measurement and the center of the earth.
Certainly gravitational anomalies exist when there are more dense materials in the materials in between the measurements and the center of the earth compared to regions where the materials are less dense.
So one must assume a uniform material making up the mass rather than just the distances involved.
Just food for the thinking…
Experiments would confirm that it is true, but they wouldn’t necessarily explain why it was true.
Yet we also know that gravity bends space, so how does that affect the results?
Another question about gravitons: Does each object in the Universe supposedly exchange gravitons with every other object in the Universe? And how fast do gravitons travel–at the speed of light?
Because that is the *definition of pi. Pi is the ratio of the circumference to the diameter. pi equals the circumference divided by 2radius. Circumference = pi * 2r.
Now I am getting a headache from trying to suss general relativity.
Imagine you are on a spaceship c. 500K miles above a big, heavy planet with a firm surface. You take a stick outside, place it lengthwise against your ship, and paint a bright yellow line exactly the length of the stick on the ship’s hull. Now you go inside, get into a pod (still holding the stick) and make a descent to the planet’s surface, landing softly enough so as to not die, burn up or be seriously injured.
When your ship comes back around, you use ridiculously precise tools to compare your stick to the line on the ship and find that the yellow line appears to be a tad longer than the stick. You check your watch and find it is a little behind the clock on the ship (which you can spy through a porthole).
So how does this affect gravity? Gravity is a force. F=ma ; a = 2d/t^2 (or: acceleration is equal to two times distance divided by time squared). The dilation effect of relativity operates on both time and distance, but since the time component is squared in relation to the distance component, the change in force should be accompanied by a parallel change in the relative frame of reference.
Or something like that. I think.
The 1/r potential is the Newtonian situation. The relativistic case is more complicated (but reduces to the Newtonian one in the limit), and gravity isn’t really a force in that case, at least in the Newtonian sense. At the very least, the acceleration of a particle in, say, the two-body problem will not be proportional to 1/r^2.
Presumably anything affected by gravity can exchange gravitons. It’s all very tentative, though, since there isn’t any sort of QFT for gravity. (On the other hand, the other interactions fit nicely into that framework, and an interaction mediated by a spin-2 particle turns about to be universally attractive, for techincal reasons, and looks reasonably close to gravity.)
Yes. If they were massive, then the gravitational interaction would have a limited range because of the uncertainty principle.
First I only mentioned π in context. Replace 2π*r for the area of the square if it helps to get the point across. Ultimately the reason for the inverse square law is Euclid’s postulates, or more rigorously Hilbert’s axioms.
Secondly π is a point on the real line. Commonly defined as you mentioned but not necessarily.
Surface area comes from the integral of the radius …
The gravity’s flux is spread equally over that.
So it is definitely 2 , not 1.99 or anything else.
Also form analogy with Maxwells and Green’s law for electrical or magnetic fields…
I don’t think so - because you, the stick and the ship are all in the same reference frame when you paint the line, and also later when you compare them.
Thanks for that link. I’d read about this hypothesis before but I couldn’t remember what it was called. My initial reaction was that this sounds like a very viable hypothesis, much simpler than assuming the existence of dark matter.
If we lived in a 2D universe which had some curvature through a 3rd dimension, it could resemble a flat piece of paper, or it could resemble the surface of a beach ball or a saddle. Those are examples of zero curvature, positive curvature, or negative curvature. Curvature has some surprising effects, such as the fact that the angles of a triangle add up to exactly 180 degrees with zero curvature, but MORE than 180 degrees with positive curvature, and LESS than 180 degrees with negative curvature. Similarly, a 3D universe which curves through a 4th dimension might have positive or negative curvature. The question remains, which type of universe do we live in?
So far, all our attempts to measure the curvature of our universe have determined that either there’s no curvature at all or it’s very small, within the error of measurement for our current instruments and methods. But it remains a possibility that our 3D universe curves through a 4th dimension, and it’s possible that gravity falls off at a rate 1/r^1.9999999993 or some such.
Note that pi, like G, is a proportionality constant in the formula cited, while the 2 in the OP’s formula is an exponent (and the formula could have been written as F = G(m1/r)(m1/r), which would hide the fact that there’s a 2 involved). It’s far more common for exponents to be nice whole numbers in physical formulas, than for proportionality constants to be.
No, the ship is half a million miles away, where the gravitational gradient is significantly different. Gravity is a distortion of of spacetime, on the the planet, where the gradient is steeper, spacetime is dilated WRT to the ship that is in a shallower gradient. This has been demonstrated in practice, with GPS satellites which have to compensate for their difference in reference frame to keep their clocks in sync for timing with ground-based client observers.