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When I went to listen to a dark matter talk at the observatory in Cape Town, my first response was: Why on earth do they keep insisting that the r^2 rule should

When I went to listen to a dark matter talk at the observatory in Cape Town, my first response was: Why on earth do they keep insisting that the r^2 rule should hold? Maybe a physicist can explain why, but as a mathematician my first quest would be to try to form a mathematical system that follows what we are observing, without introducing physical objects that we aren't actually seeing.

chopin 8y ago
You should look up MOND. However, it seems to have a hard time explaining all phenomena we see.
- mikorym 8y ago
  For those wondering: "Modified Newtonian dynamics (MOND) is a theory that proposes a modification of Newton's laws to account for observed properties of galaxies. It is an alternative to the theory of dark matter in terms of explaining why galaxies do not appear to obey the currently understood laws of physics." [1] Seems like this is (one approach to) exactly the question I asked! Even if proven not to lead to any satisfactory cosmological model, these kind of studies will at least tell you why the alternatives (to r^2) don't hold. [1] https://en.wikipedia.org/wiki/Modified_Newtonian_dynamics https://en.wikipedia.org/wiki/Modified_Newtonian_dynamics
phoenixstrike 8y ago
Because the r2 rule does hold, and it comes from something very fundamental about nature, the principle of least time. R2 comes directly out of the Lagrangian when you minimize the action. Its very fundamental to the universe and all observed phenomena. It has nothing to do with “introducing physical objects that we aren’t actually seeing.” Why do people on HN so like to make arrogant, spectator judgments on physics like this? The HN crowd is generally contrarian and likes to speak authoritatively on all topics regardless of actual knowledge, but it really seems to take a step up in physics threads. Everyone except the physicists seems like they have thought something original and obvious that the physicists have missed. The hubris, my god.
- weego 8y ago
  They just asked a question and put some context of the basis for the question. I think you read something that isn't there.
  - phoenixstrike 8y ago
    Oh please. You think saying something like “i dont understand why they keep saying this, but if it were up to me (as a mathematician not a physicist), I would be doing it this way, not that way...” is innocent? Its pretty much textbook passive aggressive condescension.
    - TeMPOraL 8y ago
      They're as much an armchair physicist by questioning the inverse-square rule as you're an armchair psychologist by invoking "passive aggressive condescension". Or do you actually have the necessary qualifications to make that diagnosis? Seriously though, it's a good question. Inverse-square rule is most likely well-founded, but reexamining such foundations is how you make theoretical progress - at least in mathematics, if I'm to believe Polya. I am not a mathematician itself, but it does seem like a pattern that generalizes well into solving theoretical problems in all domains ;).
    - pushpop 8y ago
      Sometimes the best way to understand a problem is to ask lots of questions. And the best way to get a meaningful response is to frame those questions with context. I don’t read any ego in the GPs tone - just an individual who was curious about a physics problem and wanted that explained in a language he understood (hence the mathematics context).
  - SiempreViernes 8y ago
    The were kinda rude about it though, ”why on earth” is not the humblest expression to prefix a question, especially when asking outside ones domiain knowledge. And the answer is simply that they are ill informed about r^2 being the best fit to all the data, popular talks just use rotation curves as a simple to explain justification and don’t bother with structure formation arguments and the like.
maxnoe 8y ago
It's not like physicists didn't try. Nobody really likes the idea of particles or other stuff that is impossible to measure. But until now, General Relativity has passed every experimental test and most alternative approaches to dark matter other than "Some heavy stuff not interacting electricly" is basically ruled out by one or another observation. There are solutions that fit the large scales pretty well (Cosmic Microwave Background, Galaxy Formation) but fail at the smaller scales (Orbital speeds) and vice versa.
- mikorym 8y ago
  Sorry for a perhaps simple question. Which part of GR requires that GR => r^2? I can try to think of why, but I'd be guessing. I understand that many things in GR follows from when we translate reference frames that there is a light speed constancy requirement. My exposure to physics is via category theory and specifically self-dual theories that apply to the quantum sized world.
  - mehrdadn 8y ago
    > Which part of GR requires that GR => r^2? It's over my head, but if you want to see the math, I think this link derives Newton's law from GR: https://math.uchicago.edu/~may/VIGRE/VIGRE2010/REUPapers/Tolish.pdf#page=14 https://math.uchicago.edu/~may/VIGRE/VIGRE2010/REUPapers/Tol...
  - jerf 8y ago
    "Which part of GR requires that GR => r^2?" Well, there's the answer that it's what seems to fit the facts. GR may be one of the primo-grade "unreasonable effectiveness of mathematics" examples as it captures rather complex (to human minds) behavior in a handful of carefully-defined symbols, but technically it's still an empirical theory first and mathematical theory second. Secondly, as others mention, there is the obvious answer that r^2 is how area changes as you increase the radius of a hypersphere in 3-D space; here I deliberately say "hypersphere" even though we're in 3-space here just to emphasize that it's particular to the the 3-D case. There is a theory that the reason gravity is so much weaker than the other forces is that it is "escaping" out in other dimensions we can't see, though that still leaves the stuff we can see expanding as inverse-square in what we can see. (One of my favorite crazy dark-matter theories is that it lies in those other dimensions, and the reason we can't capture any is that it is literally not "in our dimension". That said, it still can't be conventional baryonic matter, because dark matter doesn't clump anything like it does. If there was a "parallel universe" or 10 just like ours, except they gravitationally interact, we'd see things like stars in orbit around "nothing", because the stars in the parallel universes are interacting. That's not what the universe looks like.)
acqq 8y ago
It simply matches the measurements. Modern science has available modern technology to measure so much and with such a precision that any theory that attempts to be scientifically accepted has to match with the existing measurements. The only approach that is not "dark mater" which is not completely disqualifying itself on all scales (by not matching anything) is MOND, which is still more "let's try to find some formulas that can match at least some observations" than something that even now can be consistent with all observations and measurements. Still, where MOND doesn't match the observations, it fails not in the orders of the magnitudes, it even results in the wrong shape of the curve: https://arxiv.org/abs/1112.1320 https://arxiv.org/abs/1112.1320 "The biggest challenge facing MOND today is the shape of the matter power spectrum. The shape depicted in Fig. 1 is related to the acoustic oscillations observed in the CMB. If the Universe is dominated by dark matter, these matter oscillations, dubbed Baryon Acoustic Oscillations (BAO), are highly suppressed as the baryons fall into the potential wells created by dark matter, leaving only percent level traces of the primordial oscillations. In a no-dark matter model, on the other hand, the oscillations should be just as apparent in matter as they are in the radiation. Indeed, Fig. 1 illustrates that – even if a generalization such as TeVeS fixes the amplitude problem – the shape of the predicted spectrum is in violent disagreement with the observed shape." Or to summarize: https://medium.com/starts-with-a-bang/theres-a-debate-raging-over-whether-dark-matter-is-real-but-one-side-is-cheating-d25f28319d70 https://medium.com/starts-with-a-bang/theres-a-debate-raging... "On the scales of groups of galaxies, individual galaxy clusters, colliding galaxy clusters, the cosmic web, and the leftover radiation from the Big Bang, MOND’s predictions fail to match reality, whereas dark matter succeeds spectacularly. It’s possible, and perhaps even likely, that someday we will understand enough about dark matter to understand why and how the MOND phenomenon on the scales of individual galaxies arises. But when you look at the full suite of evidence, dark matter is practically a scientific certainty. It’s only if you ignore all of modern cosmology that the modified gravity alternative looks viable. Selectively ignoring the robust evidence that contradicts you may win you a debate in the eyes of the general public. But in the scientific realm, the evidence has already decided the matter, and 5/6ths of it is dark."
dnautics 8y ago
The r^2 rule should hold because there is a concept called "flux". Imagine a field of force emanating from the center of a sphere. Now twiddle the radius of the field. Presume the intensity of the field will be distributed evenly, but diluted by the net surface area of the sphere, that surface area will in proportion to the square of the radius. That said there are some fascinating alternative theories of gravity such as mond or the unruh effect which don't invoke an inverse swuare rule.
- mikorym 8y ago
  I have had rudimentary exposure to this and do find this perhaps one of the best intuitive approaches. Why is the rule not r^3 though? And what would the rule be in a 2-dimensional world?
  - maththrowawtays 8y ago
    Here is a diagram demonstrating what I think OP is trying to say for why it is r^2 in 3 dimensions: http://hyperphysics.phy-astr.gsu.edu/hbase/Forces/isq.html http://hyperphysics.phy-astr.gsu.edu/hbase/Forces/isq.html In 2 dimensions it would be just ~r.
    - mikorym 8y ago
      Right, so we are looking at area and not volume.
      - acqq 8y ago
        But still note, that even if some solution appears to be "pretty" or "obvious" mathematically for some specific scenario (like drawing the areas for inverse square rule, or looking at the galactic curve for a "fit" with MOND), it never means that it automatically follows that all of the measurements in all the experiments will match that. The "better" theory must cover more, not less of the measurements. The one that at the moment certainly covers the most is dark matter, and the alternatives simply cover much less (or as I've posted even somewhere result in the wrong shapes). Once some alternative manages to cover the observations and bring even more predictive power, that one will eventually be accepted (although sometimes "one funeral at a time" was needed), even if it's less "pretty" and less "obvious."
TomMckenny 8y ago
Not an analytic answer but I understand there are galaxies without dark matter. I imagine a theory that approaches r^2 on some galactic scales but not others would not very elegant.
appleflaxen 8y ago
what is the r^2 law in the context of dark matter? i read the post, and didn't see this term anywhere in it.
- mikorym 8y ago
  Dark matter is introduced to account for discrepancies with the r^2 rule when just accounting for "visible" matter.
  - acqq 8y ago
    It's more complex: dark matter theory beautifully predicted the results of the CMB measurements, and no alternative theory can match that even given a hindsight. https://en.wikipedia.org/wiki/Lambda-CDM_model https://en.wikipedia.org/wiki/Lambda-CDM_model
noselasd 8y ago
But that's what the physicists have tried (and not been successful at so far). The issue is that modifying how we think gravity and matter works is not just a change in a math equation. Changing those equations would also mean a whole chunk of modern physics, including general relativity is wrong - yet all those theories can be observed and confirmed in many very different ways to be consistent and valid. So if you change those equations to adjust for dark matter, they no longer give the result for many other phenomena that we observe.
Gravityloss 8y ago
There are theories. I don't claim to understand this but there are persistent physicists having driven alternate formulas for many many years. Example: http://riofriospacetime.blogspot.com/ http://riofriospacetime.blogspot.com/
nwallin 8y ago
They did try. Those alternative theories fall under an umbrella called MOND- MOdified Newtonian Dynamics. They were actually some of the first real proposals to solve the dark matter problem. Unfortunately, none of these theories panned out. The easiest to understand evidence against MOND is shown in the bullet cluster. The bullet cluster is a pair of galaxies in the process of colliding. They also happen to be gravitationally lensing galaxies behind them. While the xray and visible light observations show that gas clouds that make up the bulk of the normal matter in the galaxies stopped when they hit each other, (like two cars in a head on collision) the gravitational lensing shows that the overwhelming majority of the mass in the galaxies simply passed straight through, the way dark matter as particles would. So in order to accommodate the bullet cluster, MOND will need to show that empty space has some sort of momentum and memory, which is going to be much less elegant than a second hierarchy of particles which do not interact via the electromagnetic or strong nuclear forces. There are a very large number of other observations which also disprove MOND or variations on them, the bullet cluster is just the easiest to describe in a paragraph. It's actually very, very common for laypeople to come up with MOND as a solution for dark matter. I'm pretty sure I did when I first heard about dark matter. I don't think I've ever seen a dark matter post on HN or reddit that didn't include at least one post positing MOND, even if (especially if) they don't that the theory they came up with has a name, an acronym, and had already been thoroughly disproven by the scientific community. https://en.wikipedia.org/wiki/Modified_Newtonian_dynamics https://en.wikipedia.org/wiki/Modified_Newtonian_dynamics https://en.wikipedia.org/wiki/Bullet_Cluster https://en.wikipedia.org/wiki/Bullet_Cluster
- thaumasiotes 8y ago
  > even if (especially if) they don't [know] that the theory they came up with has a name, an acronym, and had already been thoroughly disproven by the scientific community. Without commenting on the merits, I think the theory "traditional kinematics need to be adjusted somehow at large scales" is a little too vague to have been disproven.
  - perl4ever 8y ago
    It seems pretty straightforward to me to say that has been disproven, for reasonable values of disproven, because you'd have to adjust it one way here and another way there.
zuminator 8y ago
Galaxies don't all have a proportionally equal amount of dark matter. So if you invented maths that would account for some dark matter, they would fail in other cases. You'd still have anomalous events that would require the type of ad hoc explanation that dark matter solves. Plus an extra superfluous theory on top of that.
- tylerjwilk00 8y ago
  The counter point is that with dark matter you can just keep adding more dark matter until it fits with observations. Seems a bit lazy and not very innovative.
GuB-42 8y ago
1- 1/r^2 is the most natural in 3D space. Think of it as the surface of en expanding sphere. A lot of things follow the inverse square law: electric fields, light intensity, sound intensity,... It is not a strong argument: there are other things that don't follow 1/r^2 and we are not even sure our universe really is 3D (though there is strong evidence for it). 2- Theories that don't involve 1/r^2 (MOND) exist, but they are not sufficient to match observations without involving dark matter. And following Occam's razor, since MOND introduces another parameter (the behavior of gravity on long scales), but doesn't simplify anything (dark matter is still there), it is not preferred. 3- While we don't know the details about dark matter, we "see" it in a way. We see it affect the movement of galaxies, and the gravitational lensing effect it has. It is just that we can't observe it with our preferred methods, which involve electromagnetism. I mean, we don't need to see a glass panel to know that it's there. Sure, we'd like to know more, and evidence of dark matter is weaker than our glass pane, but we simply don't have anything better right now.
- mikorym 8y ago
  A question on 1-: does introduction of intermediary objects stack well? What I mean, for instance: If the moon is between us and the sun, the light gets blocked out but gravity does not. Light intensity then does not stack well (for visible light frequencies) when there are intermediate objects. Or does the analogy here require a more formal transposition to GR and the distortion of spacetime?
  - Steuard 8y ago
    I'm not sure that I understand your question. Certainly all of our understanding of astronomical effects includes (and often depends on) the effects of intervening clouds of gas and dust: that hasn't been overlooked or omitted from any professional-level analysis. (My memory/guess is that in the presence of an absorptive medium, influences like light intensity will fall off as e^(-a*r)/r^2 rather than just 1/r^2. But the point is that we can model that absorptive term reasonably well, and the underlying 1/r^2 still results naturally from 3D space.) (Meanwhile, to a first approximation, intervening dense objects like planets or stars are rare enough to be negligible: even in a region like our solar system where such objects are most likely to be found, the available volume is overwhelmingly empty space.)
nabla9 8y ago
As a mathematician, you might want to start from Gauss's law for gravity (flux theorem for gravity). You need only the Gauss law and zero curl assumption to derive Newton's law of gravity.
- mikorym 8y ago
  It is interesting to see some of Gauss's side quests during his life and the scope of his interests. I remember reading that Gauss's (and his contemporaries') foray into statistics was greatly motivated by the central limit theorem. Every now and then one sees a question about least squares, asking why we don't usually use least cubes. [1] Abel said of Gauss: "He is like the fox who erases his tracks in the sand with his tail." [1] In such a discussion, if we are lucky to have a healthy discusion, we see references such as https://en.wikipedia.org/wiki/Gauss%E2%80%93Markov_theorem https://en.wikipedia.org/wiki/Gauss%E2%80%93Markov_theorem.
pchiusano 8y ago
You might be interested in quantized inertia http://physicsfromtheedge.blogspot.com/ http://physicsfromtheedge.blogspot.com/ which is a model that does away with the need for dark matter and that has been gaining some interest. For instance, author claims it replicates the success of MOND but also predicts where MOND and GR both fail (see wide binaries posts on the blog). I 100% agree that when the model doesn’t fit the data, making up some hitherto undetected kind of matter, distributed in just the right way to explain the discrepancies, is kinda odd... the QI author is constantly complaining about this.
- perl4ever 8y ago
  "making up some hitherto undetected kind of matter, distributed in just the right way to explain the discrepancies, is kinda odd" That seems totally backwards to me. Matter, by definition, is what is distributed spatially in just the right way to explain our observations. Physical laws, by definition, are not; they are universal. So why keep trying to explain what appears to be matter with adjustments to physical laws? It's like you kept seeing elephant footprints appear out of nowhere, but insisted that rather than believe in invisible elephants, the most parsimonious explanation is that they are a natural consequence of a corrected law of gravity. That is, to me, turning Occam's Razor inside out.
  - tylerjwilk00 8y ago
    Now, imagine no one has ever seen, felt, smelled, or heard the invisible elephant yet it makes up 85% of matter. I'd start to wonder if the tracks could be explained by another phenomenon.
    - perl4ever 8y ago
      I think, philosophically, dark matter has been "seen" as much as anything. Like, radio waves are invisible, but you accept that we've "seen" them, right? So-called visible light isn't what activates your visual cortex, it's a cascade of rube goldberg machinery triggered by photons; the inside of one's head is dark. Nothing is seen directly, but all the things which we think we've seen have various intermediate steps before we can "see" them. We've detected gravity waves - is that "seeing", and do people who doubt dark matter exists doubt the gravity wave detectors work?
kryptiskt 8y ago
It's very natural in a three-dimensional space that the force tapers off by r^2 as the area it has to cover increases by that. But Newton didn't take that for granted, he examined other laws and proved that there were no stable elliptical orbits under them.

chopin 8y ago

You should look up MOND. However, it seems to have a hard time explaining all phenomena we see.

mikorym 8y ago
For those wondering: "Modified Newtonian dynamics (MOND) is a theory that proposes a modification of Newton's laws to account for observed properties of galaxies. It is an alternative to the theory of dark matter in terms of explaining why galaxies do not appear to obey the currently understood laws of physics." [1] Seems like this is (one approach to) exactly the question I asked! Even if proven not to lead to any satisfactory cosmological model, these kind of studies will at least tell you why the alternatives (to r^2) don't hold. [1] https://en.wikipedia.org/wiki/Modified_Newtonian_dynamics https://en.wikipedia.org/wiki/Modified_Newtonian_dynamics

phoenixstrike 8y ago

Because the r2 rule does hold, and it comes from something very fundamental about nature, the principle of least time. R2 comes directly out of the Lagrangian when you minimize the action. Its very fundamental to the universe and all observed phenomena. It has nothing to do with “introducing physical objects that we aren’t actually seeing.” Why do people on HN so like to make arrogant, spectator judgments on physics like this? The HN crowd is generally contrarian and likes to speak authoritatively on all topics regardless of actual knowledge, but it really seems to take a step up in physics threads. Everyone except the physicists seems like they have thought something original and obvious that the physicists have missed. The hubris, my god.

weego 8y ago
They just asked a question and put some context of the basis for the question. I think you read something that isn't there.
- phoenixstrike 8y ago
  Oh please. You think saying something like “i dont understand why they keep saying this, but if it were up to me (as a mathematician not a physicist), I would be doing it this way, not that way...” is innocent? Its pretty much textbook passive aggressive condescension.
  - TeMPOraL 8y ago
    They're as much an armchair physicist by questioning the inverse-square rule as you're an armchair psychologist by invoking "passive aggressive condescension". Or do you actually have the necessary qualifications to make that diagnosis? Seriously though, it's a good question. Inverse-square rule is most likely well-founded, but reexamining such foundations is how you make theoretical progress - at least in mathematics, if I'm to believe Polya. I am not a mathematician itself, but it does seem like a pattern that generalizes well into solving theoretical problems in all domains ;).
  - pushpop 8y ago
    Sometimes the best way to understand a problem is to ask lots of questions. And the best way to get a meaningful response is to frame those questions with context. I don’t read any ego in the GPs tone - just an individual who was curious about a physics problem and wanted that explained in a language he understood (hence the mathematics context).
- SiempreViernes 8y ago
  The were kinda rude about it though, ”why on earth” is not the humblest expression to prefix a question, especially when asking outside ones domiain knowledge. And the answer is simply that they are ill informed about r^2 being the best fit to all the data, popular talks just use rotation curves as a simple to explain justification and don’t bother with structure formation arguments and the like.

maxnoe 8y ago

It's not like physicists didn't try. Nobody really likes the idea of particles or other stuff that is impossible to measure. But until now, General Relativity has passed every experimental test and most alternative approaches to dark matter other than "Some heavy stuff not interacting electricly" is basically ruled out by one or another observation. There are solutions that fit the large scales pretty well (Cosmic Microwave Background, Galaxy Formation) but fail at the smaller scales (Orbital speeds) and vice versa.

mikorym 8y ago
Sorry for a perhaps simple question. Which part of GR requires that GR => r^2? I can try to think of why, but I'd be guessing. I understand that many things in GR follows from when we translate reference frames that there is a light speed constancy requirement. My exposure to physics is via category theory and specifically self-dual theories that apply to the quantum sized world.
- mehrdadn 8y ago
  > Which part of GR requires that GR => r^2? It's over my head, but if you want to see the math, I think this link derives Newton's law from GR: https://math.uchicago.edu/~may/VIGRE/VIGRE2010/REUPapers/Tolish.pdf#page=14 https://math.uchicago.edu/~may/VIGRE/VIGRE2010/REUPapers/Tol...
- jerf 8y ago
  "Which part of GR requires that GR => r^2?" Well, there's the answer that it's what seems to fit the facts. GR may be one of the primo-grade "unreasonable effectiveness of mathematics" examples as it captures rather complex (to human minds) behavior in a handful of carefully-defined symbols, but technically it's still an empirical theory first and mathematical theory second. Secondly, as others mention, there is the obvious answer that r^2 is how area changes as you increase the radius of a hypersphere in 3-D space; here I deliberately say "hypersphere" even though we're in 3-space here just to emphasize that it's particular to the the 3-D case. There is a theory that the reason gravity is so much weaker than the other forces is that it is "escaping" out in other dimensions we can't see, though that still leaves the stuff we can see expanding as inverse-square in what we can see. (One of my favorite crazy dark-matter theories is that it lies in those other dimensions, and the reason we can't capture any is that it is literally not "in our dimension". That said, it still can't be conventional baryonic matter, because dark matter doesn't clump anything like it does. If there was a "parallel universe" or 10 just like ours, except they gravitationally interact, we'd see things like stars in orbit around "nothing", because the stars in the parallel universes are interacting. That's not what the universe looks like.)

acqq 8y ago

It simply matches the measurements. Modern science has available modern technology to measure so much and with such a precision that any theory that attempts to be scientifically accepted has to match with the existing measurements. The only approach that is not "dark mater" which is not completely disqualifying itself on all scales (by not matching anything) is MOND, which is still more "let's try to find some formulas that can match at least some observations" than something that even now can be consistent with all observations and measurements. Still, where MOND doesn't match the observations, it fails not in the orders of the magnitudes, it even results in the wrong shape of the curve: https://arxiv.org/abs/1112.1320 https://arxiv.org/abs/1112.1320 "The biggest challenge facing MOND today is the shape of the matter power spectrum. The shape depicted in Fig. 1 is related to the acoustic oscillations observed in the CMB. If the Universe is dominated by dark matter, these matter oscillations, dubbed Baryon Acoustic Oscillations (BAO), are highly suppressed as the baryons fall into the potential wells created by dark matter, leaving only percent level traces of the primordial oscillations. In a no-dark matter model, on the other hand, the oscillations should be just as apparent in matter as they are in the radiation. Indeed, Fig. 1 illustrates that – even if a generalization such as TeVeS fixes the amplitude problem – the shape of the predicted spectrum is in violent disagreement with the observed shape." Or to summarize: https://medium.com/starts-with-a-bang/theres-a-debate-raging-over-whether-dark-matter-is-real-but-one-side-is-cheating-d25f28319d70 https://medium.com/starts-with-a-bang/theres-a-debate-raging... "On the scales of groups of galaxies, individual galaxy clusters, colliding galaxy clusters, the cosmic web, and the leftover radiation from the Big Bang, MOND’s predictions fail to match reality, whereas dark matter succeeds spectacularly. It’s possible, and perhaps even likely, that someday we will understand enough about dark matter to understand why and how the MOND phenomenon on the scales of individual galaxies arises. But when you look at the full suite of evidence, dark matter is practically a scientific certainty. It’s only if you ignore all of modern cosmology that the modified gravity alternative looks viable. Selectively ignoring the robust evidence that contradicts you may win you a debate in the eyes of the general public. But in the scientific realm, the evidence has already decided the matter, and 5/6ths of it is dark."

dnautics 8y ago

The r^2 rule should hold because there is a concept called "flux". Imagine a field of force emanating from the center of a sphere. Now twiddle the radius of the field. Presume the intensity of the field will be distributed evenly, but diluted by the net surface area of the sphere, that surface area will in proportion to the square of the radius. That said there are some fascinating alternative theories of gravity such as mond or the unruh effect which don't invoke an inverse swuare rule.

mikorym 8y ago
I have had rudimentary exposure to this and do find this perhaps one of the best intuitive approaches. Why is the rule not r^3 though? And what would the rule be in a 2-dimensional world?
- maththrowawtays 8y ago
  Here is a diagram demonstrating what I think OP is trying to say for why it is r^2 in 3 dimensions: http://hyperphysics.phy-astr.gsu.edu/hbase/Forces/isq.html http://hyperphysics.phy-astr.gsu.edu/hbase/Forces/isq.html In 2 dimensions it would be just ~r.
  - mikorym 8y ago
    Right, so we are looking at area and not volume.
    - acqq 8y ago
      But still note, that even if some solution appears to be "pretty" or "obvious" mathematically for some specific scenario (like drawing the areas for inverse square rule, or looking at the galactic curve for a "fit" with MOND), it never means that it automatically follows that all of the measurements in all the experiments will match that. The "better" theory must cover more, not less of the measurements. The one that at the moment certainly covers the most is dark matter, and the alternatives simply cover much less (or as I've posted even somewhere result in the wrong shapes). Once some alternative manages to cover the observations and bring even more predictive power, that one will eventually be accepted (although sometimes "one funeral at a time" was needed), even if it's less "pretty" and less "obvious."

TomMckenny 8y ago

Not an analytic answer but I understand there are galaxies without dark matter. I imagine a theory that approaches r^2 on some galactic scales but not others would not very elegant.

appleflaxen 8y ago

what is the r^2 law in the context of dark matter? i read the post, and didn't see this term anywhere in it.

mikorym 8y ago
Dark matter is introduced to account for discrepancies with the r^2 rule when just accounting for "visible" matter.
- acqq 8y ago
  It's more complex: dark matter theory beautifully predicted the results of the CMB measurements, and no alternative theory can match that even given a hindsight. https://en.wikipedia.org/wiki/Lambda-CDM_model https://en.wikipedia.org/wiki/Lambda-CDM_model

noselasd 8y ago

But that's what the physicists have tried (and not been successful at so far). The issue is that modifying how we think gravity and matter works is not just a change in a math equation. Changing those equations would also mean a whole chunk of modern physics, including general relativity is wrong - yet all those theories can be observed and confirmed in many very different ways to be consistent and valid. So if you change those equations to adjust for dark matter, they no longer give the result for many other phenomena that we observe.

Gravityloss 8y ago

There are theories. I don't claim to understand this but there are persistent physicists having driven alternate formulas for many many years. Example: http://riofriospacetime.blogspot.com/ http://riofriospacetime.blogspot.com/

nwallin 8y ago

They did try. Those alternative theories fall under an umbrella called MOND- MOdified Newtonian Dynamics. They were actually some of the first real proposals to solve the dark matter problem. Unfortunately, none of these theories panned out. The easiest to understand evidence against MOND is shown in the bullet cluster. The bullet cluster is a pair of galaxies in the process of colliding. They also happen to be gravitationally lensing galaxies behind them. While the xray and visible light observations show that gas clouds that make up the bulk of the normal matter in the galaxies stopped when they hit each other, (like two cars in a head on collision) the gravitational lensing shows that the overwhelming majority of the mass in the galaxies simply passed straight through, the way dark matter as particles would. So in order to accommodate the bullet cluster, MOND will need to show that empty space has some sort of momentum and memory, which is going to be much less elegant than a second hierarchy of particles which do not interact via the electromagnetic or strong nuclear forces. There are a very large number of other observations which also disprove MOND or variations on them, the bullet cluster is just the easiest to describe in a paragraph. It's actually very, very common for laypeople to come up with MOND as a solution for dark matter. I'm pretty sure I did when I first heard about dark matter. I don't think I've ever seen a dark matter post on HN or reddit that didn't include at least one post positing MOND, even if (especially if) they don't that the theory they came up with has a name, an acronym, and had already been thoroughly disproven by the scientific community. https://en.wikipedia.org/wiki/Modified_Newtonian_dynamics https://en.wikipedia.org/wiki/Modified_Newtonian_dynamics https://en.wikipedia.org/wiki/Bullet_Cluster https://en.wikipedia.org/wiki/Bullet_Cluster

thaumasiotes 8y ago
> even if (especially if) they don't [know] that the theory they came up with has a name, an acronym, and had already been thoroughly disproven by the scientific community. Without commenting on the merits, I think the theory "traditional kinematics need to be adjusted somehow at large scales" is a little too vague to have been disproven.
- perl4ever 8y ago
  It seems pretty straightforward to me to say that has been disproven, for reasonable values of disproven, because you'd have to adjust it one way here and another way there.

zuminator 8y ago

Galaxies don't all have a proportionally equal amount of dark matter. So if you invented maths that would account for some dark matter, they would fail in other cases. You'd still have anomalous events that would require the type of ad hoc explanation that dark matter solves. Plus an extra superfluous theory on top of that.

tylerjwilk00 8y ago
The counter point is that with dark matter you can just keep adding more dark matter until it fits with observations. Seems a bit lazy and not very innovative.

GuB-42 8y ago

1- 1/r^2 is the most natural in 3D space. Think of it as the surface of en expanding sphere. A lot of things follow the inverse square law: electric fields, light intensity, sound intensity,... It is not a strong argument: there are other things that don't follow 1/r^2 and we are not even sure our universe really is 3D (though there is strong evidence for it). 2- Theories that don't involve 1/r^2 (MOND) exist, but they are not sufficient to match observations without involving dark matter. And following Occam's razor, since MOND introduces another parameter (the behavior of gravity on long scales), but doesn't simplify anything (dark matter is still there), it is not preferred. 3- While we don't know the details about dark matter, we "see" it in a way. We see it affect the movement of galaxies, and the gravitational lensing effect it has. It is just that we can't observe it with our preferred methods, which involve electromagnetism. I mean, we don't need to see a glass panel to know that it's there. Sure, we'd like to know more, and evidence of dark matter is weaker than our glass pane, but we simply don't have anything better right now.

mikorym 8y ago
A question on 1-: does introduction of intermediary objects stack well? What I mean, for instance: If the moon is between us and the sun, the light gets blocked out but gravity does not. Light intensity then does not stack well (for visible light frequencies) when there are intermediate objects. Or does the analogy here require a more formal transposition to GR and the distortion of spacetime?
- Steuard 8y ago
  I'm not sure that I understand your question. Certainly all of our understanding of astronomical effects includes (and often depends on) the effects of intervening clouds of gas and dust: that hasn't been overlooked or omitted from any professional-level analysis. (My memory/guess is that in the presence of an absorptive medium, influences like light intensity will fall off as e^(-a*r)/r^2 rather than just 1/r^2. But the point is that we can model that absorptive term reasonably well, and the underlying 1/r^2 still results naturally from 3D space.) (Meanwhile, to a first approximation, intervening dense objects like planets or stars are rare enough to be negligible: even in a region like our solar system where such objects are most likely to be found, the available volume is overwhelmingly empty space.)

nabla9 8y ago

As a mathematician, you might want to start from Gauss's law for gravity (flux theorem for gravity). You need only the Gauss law and zero curl assumption to derive Newton's law of gravity.

mikorym 8y ago
It is interesting to see some of Gauss's side quests during his life and the scope of his interests. I remember reading that Gauss's (and his contemporaries') foray into statistics was greatly motivated by the central limit theorem. Every now and then one sees a question about least squares, asking why we don't usually use least cubes. [1] Abel said of Gauss: "He is like the fox who erases his tracks in the sand with his tail." [1] In such a discussion, if we are lucky to have a healthy discusion, we see references such as https://en.wikipedia.org/wiki/Gauss%E2%80%93Markov_theorem https://en.wikipedia.org/wiki/Gauss%E2%80%93Markov_theorem.

pchiusano 8y ago

You might be interested in quantized inertia http://physicsfromtheedge.blogspot.com/ http://physicsfromtheedge.blogspot.com/ which is a model that does away with the need for dark matter and that has been gaining some interest. For instance, author claims it replicates the success of MOND but also predicts where MOND and GR both fail (see wide binaries posts on the blog). I 100% agree that when the model doesn’t fit the data, making up some hitherto undetected kind of matter, distributed in just the right way to explain the discrepancies, is kinda odd... the QI author is constantly complaining about this.

perl4ever 8y ago
"making up some hitherto undetected kind of matter, distributed in just the right way to explain the discrepancies, is kinda odd" That seems totally backwards to me. Matter, by definition, is what is distributed spatially in just the right way to explain our observations. Physical laws, by definition, are not; they are universal. So why keep trying to explain what appears to be matter with adjustments to physical laws? It's like you kept seeing elephant footprints appear out of nowhere, but insisted that rather than believe in invisible elephants, the most parsimonious explanation is that they are a natural consequence of a corrected law of gravity. That is, to me, turning Occam's Razor inside out.
- tylerjwilk00 8y ago
  Now, imagine no one has ever seen, felt, smelled, or heard the invisible elephant yet it makes up 85% of matter. I'd start to wonder if the tracks could be explained by another phenomenon.
  - perl4ever 8y ago
    I think, philosophically, dark matter has been "seen" as much as anything. Like, radio waves are invisible, but you accept that we've "seen" them, right? So-called visible light isn't what activates your visual cortex, it's a cascade of rube goldberg machinery triggered by photons; the inside of one's head is dark. Nothing is seen directly, but all the things which we think we've seen have various intermediate steps before we can "see" them. We've detected gravity waves - is that "seeing", and do people who doubt dark matter exists doubt the gravity wave detectors work?

kryptiskt 8y ago

It's very natural in a three-dimensional space that the force tapers off by r^2 as the area it has to cover increases by that. But Newton didn't take that for granted, he examined other laws and proved that there were no stable elliptical orbits under them.