@Joe_Bakhos, be careful though, some of these modifications can easily violate things like conservation of energy/momentum, Newton’s third law, etc. There is nothing wrong with theories without these properties, but one has to be prepared to give a coherent explanation! (For example, Einstein’s General Relativity does not conserve energy and momentum in general).
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@PdotdQ As you might expect, I’ve run into difficulties. First, I think that @dga471 is correct in that I will not be able to algebraicly demonstrate that the function becomes constant. I tried to graph a very simplified version of it; wanting to compare it to this:
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Might you answer what may seem to you to be a trivial question? What is the velocity function that gives the white, dashed, expected line in this picture? V = rad(gm/r) does not seem to be able to do this. V = rad(gm/r) starts very high and is always trending downwards. I understand that the actual function that they use must account for the dispersion of visible matter within the galaxy and that this must be the reason why the graph goes up initially. But I have not been able to find the equation that they use under kepler / newton assumptions to give this “expected” velocity curve?
As I mentioned,
This calculation is only valid far from the galaxy, where the gravitational acceleration due to the luminous part of the galaxy can be well approximated by a point source. Closeby, this calculation fails because of the “bulgy” parts of the galaxy. However, the idea is that far from the center, say at ~20kpc, the rotation curve will asymptote to the value given by
\frac{V^2}{R} = \vec{a}_{\rm{gravity}} \cdot \hat{r}
You can see this in the dashed line in the wikipedia picture you linked. Far from the center, the curve asymptotes to 1/\sqrt{r}.
Are you having problem integrating the forces from the womb? Did you try starting with a simple model such as a spherical or elliptical womb?
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@PdotdQ I am continuing to work on it, and will continue. My preliminary work suggests that reverse gravity under my equation would still be too weak to account for the flat curve – but I am trying different things.
While I continue to work, I thought of something relevant that is utterly beyond my capabilities to even guess if the equations might work, but you might have an intuition about it:
What if both MOND and my idea are on the right track? i.e. , what if I am on the right track about gravity ultimately reversing, but the true equation for it looks more like the MOND model?
In the mean time, I am continuing to work on my idea.
This is permissible if your theory does not possess any claims that is contrary to MOND. Note that when people say MOND, they actually mean the various relativistic generalization of MONDian theories, such as TeVeS. The original “MOND” is a phenomenological data fit that was abandoned because it does not conserve momentum/disobey Newton’s Third Law.
Of course, your theory cannot work with relativistic generalizations such as TeVeS, as these are relativistic theories with both time dilation and length contraction. Therefore, when you add MOND into your theory, you have to figure out what flavor of MOND you are actually adding - or whether this is even possible at all.
I don’t know where you are right now in your work, but if you have problems integrating the forces from a spherical womb, note that you can just copy the derivation of the Shell Theorem in Newtonian gravity. Just replace the Newtonian equation
\vec{F} = \frac{G Mm}{r^2} \hat{r} \; ,
with your equation in the derivation.
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@PdotdQ Thank you for the shell theorem hint. About MOND: I was not really saying in combination with my theory; I was assuming that my theory is wrong about everything EXCEPT the possibility of gravity reversing.
In that case I would be assuming that some version of MOND might be developed that incorporates gravity reversal without any reference to my ideas at all.
If you are just referring to MOND+gravity reversal, then this looks more promising, as then it might be incorporated to a relativistic MOND theory. Note that in this case, the length-scale where gravity reverses has to also be redefined to some covariant notion of length, as coordinate length is meaningless in relativistic theories such as TeVeS.
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@PdotdQ ah, I see; the joke is on me. I am assuming that the shell theorem would apply to repulsive gravity as well? I.e. the net force inside the shell is zero?
Yes for purely reversed gravity. For example, the shell theorem also works for a positively charged (electrically) particle trapped inside a cavity that is positively charged.
This is not the case for your force law, which is not purely reversed gravity, but includes switching the sign of gravitational acceleration depending on distance; for example, matter close to the womb will feel gravitational acceleration. However, if all of your womb is located at a significantly large distance that their gravity as felt by stars in the galaxy is purely repulsive, then yes, the shell theorem will state that a spherical womb does not affect the motion of stars in the galaxy.
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@PdotdQ 
Well, that gives me a lot to think about, to the say the least.
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@Joe_Bakhos, in case you did not see my edit:
This is important, because while a spherical womb that is located far from a galaxy cannot affect the motion of stars in the galaxy, a more general shaped womb could.
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@PdotdQ I am very grateful for your help. As I said, you have given me a lot to think about, i.e. shape of the womb and also the possibility that IF reverse gravity exists, then it might not be an inverse square law … which seems highly unlikely, to say the least.
As of right now, I’m betting my theory is not even close to the truth. But I will continue thinking about it. It’s fun!
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Indeed it is. One of my current projects include tests of modifications of gravity, so you are not alone in trying to modify gravity! Today, a huge number of physicists dedicate their lives in coming up with modifications of gravity and experiments that can test them, and literally billons of dollars had been spent in this endeavor. It is not true that contemporary physicists are happy with Einstein’s gravity and left it at that.
A book (although is VERY outdated now) that would introduce one to this field is Theory and Experiment in Gravitational Physics. I would say that if you want to continue in this field, you have to read that book - it gives a summary of what people have thought of with regards to modifying and testing gravity up to the early 80s (as I said it’s quite outdated now). Beware: this is not light reading for someone without a strong background in physics, so if you are interested in reading it you have to first pick up some prerequisites in more basic physics.
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I’m glad to see that you’re having fun thinking about the implications of your theory, even if it does not turn out to be something that revolutionizes physics. Good scientists rarely set out to do science in order to win prizes, become famous, or overturn all existing human knowledge - they are simply curious about nature and its hidden structure and delight in thinking about various possible ways to understand it, even if many of their theories and hypotheses turn out to be wrong later.
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@PdotdQ @dga471 So being stubborn, and still trying to salvage the idea gravity is dependent upon angle, I’ve tried this equation, which at first glance seems promising. I will see if it can flatten the rotation curve; since R is linear, maybe there’s hope, since this might solve both the flat curve and also Newton’s irritating hollow sphere. (I am sorry I still haven’t figured out how to do the equation add-in.)
The constant G for this equation is G = 3.687 X 10^-40
I am trying a new L, which is L = 2.8383 X 10^21 m
Now I realize I can fold the 2L in the bottom of the fraction up into the top constant, but I don’t want to do this because I want to keep playing with different Ls. So here is the equation:
F = Gm cosine[2arctan[r/(2L)]] / (r/(2L))
I am sorry; I was hasty; I may have messed up on the units. Anyways, this is the form of the equation I’m working on now – I will straighten out the units and constants tomorrow.
Unless there is a typo or I am not reading it correctly, this force is ~1/r even for small distances (e.g. solar system scales). Is this what you have in mind?
If so, this is problematic as solar system data is extremely well fitted by a 1/r^2 force. Indeed, Bertrand’s Theorem shows that only ~1/r^2 and ~r forces can have stable planetary orbits that are closed (elliptical or circular orbits). Small perturbation from these types of forces (e.g. general relativity) allows stable planetary orbits by having almost closed, almost elliptical orbits, but ~1/r is too much of a change.
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I may have made a mistake, but I’m too tired right now; I will check it further.
Well, I did not check what it comes to around the sun; all I checked was to see if acceleration due to gravity would come out to 9.81 m/s^2 on the surface of the earth. I will have to compare it to planetary orbits around the sun tomorrow. the units are meters, seconds, and kg
I read the initial post and skimmed much of the discussion, and while there are clearly more enlightened minds here than my own, I can point out one major issue – dark matter distribution.
The bullet cluster is a clear and unmistakable demonstration that dark matter has mass, inertia, and energy independent of baryonic matter. It does not appear that the modification to Newtonian dynamics you propose can adequately fit (let alone better explain) these observations.
Please correct me if I am wrong.
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@David_MacMillan I posted this a little ways above; easy to miss given all the pages of comments made:
"Also, while I’m working on my regression, I would like just to comment on some objections to reverse gravity that came up in other discussion threads.
One is the gravitational lensing in the bullet cluster (and some others). Generated images mapping the gravity show a center of gravity that is different from the visible mass. This has been considered direct proof of dark matter.
When I dove into the algorithms used to generate the gravity map however, I found an interesting thing: Up to 80% of the data had been deemed non-sensical and thrown out; meaning that whenever light was being curved in a way that the researchers deemed nonsense, they assumed it was due to some kind of interference and/or distortion on the way here.
When I questioned this, another person presented me with a picture of the supercluster Abell 2218, showing clearly visible gravitational distortion rings around the supercluster. So I read many articles about gravitational lensing in Abell 2218 – and I realized that the huge visible distortion ring can be explainable as the effect of reverse gravity. According to my theory reverse gravity IS summative over huge distances like this, while normal gravity is not.
Going further, I realized that the successful use of normal gravitational lensing in this super cluster just meant that there were smaller dense areas within the super cluster that were being used for normal lensing – but that the entire huge supercluster was not being used as a gravitational lense. "