That seems to be milking the explanation for all it's worth!
The short answer being that if something is radiating out 3-dimensinally in all directions, then at a given distance, r, that thing (light/whatever) is being spread out over an area equal to the surface area of a sphere of that radius, which is 4 x pi x r^2, i.e. the intensity is inversely proportional to the square of the distance (r^2).
Exactly. The marble metaphor should've been enough to get the point across.
Big fan of lcamtuf's writing (I think it's the only 'subscribe to my newsletter' offer I've ever actually clicked) but this one seemed a bit overcooked.
In a 3d world one can escape from the Earth's gravitational field. On the flat-world equivalent there is no escape. You can checkout but never leave. There ought to be a science fiction story based on this premise.
Things get interesting in 2d flatland.
There the field or force has to decay as 1/r because the circumference of the boundary scales as O(r). But that's the field, to get to the potential you need to integrate and then you get a function that is logarithmic with distance.
A consequence of that is the potential does not drop to zero as you go further away. Unlike what is the case for the 3d case.
If you have heard that about a random-walking bird that returns infinitely often but a random-flying bird one that doesn't, that's sorta related.
Totally unrelated, was quite thrilled to see this map of the sphere
because I had been entertaining myself by making toy globes out of paper and it seems this polyconic map was the one I had used (an interrupted version of this).
The more conventional way is to use gores using interrupted sinusoidal that look like a string of lobes connected at their common equatorial hip.
> A consequence of that is the potential does not drop to zero as you go further away. Unlike what is the case for the 3d case.
The integral of const/r^2 doesn't drop to zero as r gets larger, but approaches some constant. Unlike const/r, which is unbounded as r gets larger. So potential is bounded in R3, but not in R2.
I should have said that it rises to zero, assuming standard sign conventions.
The gravitational potential infinity of a point mass at the origin is defined to be zero.
Could you expand on why you think it approaches a constant, presumably a non-zero constant. In any case absolute values do not matter, only the potential difference matters, but am still interested in your thought.
> Could you expand on why you think it approaches a constant, presumably a non-zero constant
Because the gravitational potential between two massive objects is equal to the amount of energy it would take to separate them (from touching) out to their current distances, and equal to the amount of kinetic energy they'd have when they impacted if they started from rest and eventually came together. This is not zero for any two massive objects that are any distance apart.
If you pulled two objects further and further apart in R3, the force required at each point would drop off proportional to 1/r^2, and the total energy would be the integral of that (let's assume total mass == 1). This integral approaches a constant: for a total mass of 1 and a starting (or "touching") distance of 1 (you can't use 0 or you can't pull them apart), this integral is 1. So no matter how far apart you pull them, potential never exceeds 1. But it must be nonzero, because you could extract energy from releasing them and letting them come together.
In R2, the force required to pull them apart at each point drops off as 1/r, so the integral grows as Ln(r), and so has no upper bound.
I recognize we basically agree on all this, I'm just clarifying why I consider the limit of the potential to be nonzero as distance approaches infinity. You could extract energy from the system with a "water wheel" setup, at least at every finite distance. You could argue "but not at an infinite distance, because they'd never come together, their attraction force is 0, and where would you put the water wheel?" but that problem only comes up at infinity. We're talking about the limit as r approaches infinity, so this problem never actually comes up.
But what's the integral of 1/r^2 from 0 to \infty ...that's the part not that not squaring up.
Newtonian gravitational potential of a point mass at origin of mass M is - GM/r so at r = \infty the potential is zero. Well, partly just by definition.
I believe we are assuming different models and I want to understand yours.
Because everything in the real universe occupies a volume, and therefore no two distinct "things" can be 0 distance apart. The distance between two marbles' center of masses is the sum of their radii. So if you want to measure the potential gravitational energy between the two marbles, you'd take the integral of (GM^2/x^2) dx between x = [sum of radii of marbles] and x = infinity, as an approximation.
The inverse-square law is only valid if the source is a point.
That means for many sources it is either the wrong law (e.g. loudspeaker array¹, long neon bulbs) because of the shape or it falls apart as you go so close to the source that a significant arc-angle is covered by the source itself meaning the law is no longer accurate. E.g. if you move ever closer to a light source with some diameter, at some point the result will look more and more like an area source and less like a point source. If you move so close that your entire horizon is covered by the light source it would be absurd to apply that formula since whatever is in front of the lamp gets the light from multiple directions now.
As such it is a great and useful simplification, but knowing the limits is equally important.
¹: this array tech is is used to get a more even level falloff from the combined wavefront so you don't have to blast off the ears of the front row while being barely audible in the back.
It's fun to puzzle through why optical reflections, active radar, and things like that are inverse R^4. (It's just two inverse-square laws composed together).
Dipoles, like magnetic fields and planetary tidal forces, decay as an inverse R^3. That's less intuitive.
Thanks for putting the effort in on this. The author pointed out that no LLMs were used in the production of the blog post. To be honest the first thing I did was put the blog post into an LLM and asked it to simplify. The LLM did a great job of providing a very clear explanation for me.
This is one of the super powers I find for LLMs. I am certain that this particular derivation spoke to the author and seemed like the obvious approach. In my experience math derivations a highly personal. What resonates with me will not resonate with someone else. An LLM allows an infinite number of variations. Thanks again to the author.
While we're on this... it is arguably even more befuddling to recognize that the information content potential of space apparently correlates with the surface area rather than volume
https://en.wikipedia.org/wiki/Holographic_principle
It makes sense when you realize that all the information inside a black hole is present at its surface. An object itself approaches singularity, but to an outside observer it never crosses the event horizon, so everything "inside" (in its own view) is also "on the surface" to an outside observer.
Dimensions, and hence surfaces and volumes are part of our “tooling” used to understand the world (epistemic) rather than being intrinsic to nature. So the befuddling is related to our own interpretation.
That seems to be milking the explanation for all it's worth!
The short answer being that if something is radiating out 3-dimensinally in all directions, then at a given distance, r, that thing (light/whatever) is being spread out over an area equal to the surface area of a sphere of that radius, which is 4 x pi x r^2, i.e. the intensity is inversely proportional to the square of the distance (r^2).
Exactly. The marble metaphor should've been enough to get the point across.
Big fan of lcamtuf's writing (I think it's the only 'subscribe to my newsletter' offer I've ever actually clicked) but this one seemed a bit overcooked.
I think he wanted to get the cuteness of the Archimedean result off his chest. Very understandable emotion.
In a 3d world one can escape from the Earth's gravitational field. On the flat-world equivalent there is no escape. You can checkout but never leave. There ought to be a science fiction story based on this premise.
Things get interesting in 2d flatland.
There the field or force has to decay as 1/r because the circumference of the boundary scales as O(r). But that's the field, to get to the potential you need to integrate and then you get a function that is logarithmic with distance.
A consequence of that is the potential does not drop to zero as you go further away. Unlike what is the case for the 3d case.
If you have heard that about a random-walking bird that returns infinitely often but a random-flying bird one that doesn't, that's sorta related.
Totally unrelated, was quite thrilled to see this map of the sphere
https://substackcdn.com/image/fetch/$s_!8JEP!,f_auto,q_auto:...
because I had been entertaining myself by making toy globes out of paper and it seems this polyconic map was the one I had used (an interrupted version of this).
The more conventional way is to use gores using interrupted sinusoidal that look like a string of lobes connected at their common equatorial hip.
https://www.wolframcloud.com/obj/resourcesystem/published/De...
What I was working with were more like flowers, one for each hemisphere, with the pole at the center.
> A consequence of that is the potential does not drop to zero as you go further away. Unlike what is the case for the 3d case.
The integral of const/r^2 doesn't drop to zero as r gets larger, but approaches some constant. Unlike const/r, which is unbounded as r gets larger. So potential is bounded in R3, but not in R2.
I should have said that it rises to zero, assuming standard sign conventions.
The gravitational potential infinity of a point mass at the origin is defined to be zero.
Could you expand on why you think it approaches a constant, presumably a non-zero constant. In any case absolute values do not matter, only the potential difference matters, but am still interested in your thought.
> Could you expand on why you think it approaches a constant, presumably a non-zero constant
Because the gravitational potential between two massive objects is equal to the amount of energy it would take to separate them (from touching) out to their current distances, and equal to the amount of kinetic energy they'd have when they impacted if they started from rest and eventually came together. This is not zero for any two massive objects that are any distance apart.
If you pulled two objects further and further apart in R3, the force required at each point would drop off proportional to 1/r^2, and the total energy would be the integral of that (let's assume total mass == 1). This integral approaches a constant: for a total mass of 1 and a starting (or "touching") distance of 1 (you can't use 0 or you can't pull them apart), this integral is 1. So no matter how far apart you pull them, potential never exceeds 1. But it must be nonzero, because you could extract energy from releasing them and letting them come together.
In R2, the force required to pull them apart at each point drops off as 1/r, so the integral grows as Ln(r), and so has no upper bound.
I recognize we basically agree on all this, I'm just clarifying why I consider the limit of the potential to be nonzero as distance approaches infinity. You could extract energy from the system with a "water wheel" setup, at least at every finite distance. You could argue "but not at an infinite distance, because they'd never come together, their attraction force is 0, and where would you put the water wheel?" but that problem only comes up at infinity. We're talking about the limit as r approaches infinity, so this problem never actually comes up.
But what's the integral of 1/r^2 from 0 to \infty ...that's the part not that not squaring up.
Newtonian gravitational potential of a point mass at origin of mass M is - GM/r so at r = \infty the potential is zero. Well, partly just by definition.
I believe we are assuming different models and I want to understand yours.
Because everything in the real universe occupies a volume, and therefore no two distinct "things" can be 0 distance apart. The distance between two marbles' center of masses is the sum of their radii. So if you want to measure the potential gravitational energy between the two marbles, you'd take the integral of (GM^2/x^2) dx between x = [sum of radii of marbles] and x = infinity, as an approximation.
Ah! I was talking about the abstraction of point masses, then there is the shell theorem. But now at least I understand the discrepancy.
In any case gravitational potential at infinity is formally defined as 0 in Newtonian physics.
One thing I want remind people of is:
The inverse-square law is only valid if the source is a point.
That means for many sources it is either the wrong law (e.g. loudspeaker array¹, long neon bulbs) because of the shape or it falls apart as you go so close to the source that a significant arc-angle is covered by the source itself meaning the law is no longer accurate. E.g. if you move ever closer to a light source with some diameter, at some point the result will look more and more like an area source and less like a point source. If you move so close that your entire horizon is covered by the light source it would be absurd to apply that formula since whatever is in front of the lamp gets the light from multiple directions now.
As such it is a great and useful simplification, but knowing the limits is equally important.
¹: this array tech is is used to get a more even level falloff from the combined wavefront so you don't have to blast off the ears of the front row while being barely audible in the back.
It's fun to puzzle through why optical reflections, active radar, and things like that are inverse R^4. (It's just two inverse-square laws composed together).
Dipoles, like magnetic fields and planetary tidal forces, decay as an inverse R^3. That's less intuitive.
Thanks for putting the effort in on this. The author pointed out that no LLMs were used in the production of the blog post. To be honest the first thing I did was put the blog post into an LLM and asked it to simplify. The LLM did a great job of providing a very clear explanation for me.
This is one of the super powers I find for LLMs. I am certain that this particular derivation spoke to the author and seemed like the obvious approach. In my experience math derivations a highly personal. What resonates with me will not resonate with someone else. An LLM allows an infinite number of variations. Thanks again to the author.
While we're on this... it is arguably even more befuddling to recognize that the information content potential of space apparently correlates with the surface area rather than volume https://en.wikipedia.org/wiki/Holographic_principle
It makes sense when you realize that all the information inside a black hole is present at its surface. An object itself approaches singularity, but to an outside observer it never crosses the event horizon, so everything "inside" (in its own view) is also "on the surface" to an outside observer.
Dimensions, and hence surfaces and volumes are part of our “tooling” used to understand the world (epistemic) rather than being intrinsic to nature. So the befuddling is related to our own interpretation.
I don't think that's true at all.
[flagged]
Maybe to signal the opposite. I've started using double dashes ever since LLMs stole my favorite symbol.