In the previous post, we discussed how lensing data extend the Radial Acceleration Relation (RAR) seen in galaxy kinematics to very low accelerations. Let’s zoom out now, and look at things at higher accelerations and from a historical perspective.

This all started with Kepler’s Laws of Planetary Motion, which are explained by Newton’s Universal Gravitation – the inverse square law gbar = GM/r2 is exactly what is needed to explain the observed centripetal acceleration, gobs = V2/r. It also explains the surface gravity of the Earth. Indeed, it was the famous falling apple that is reputed to have given Newton the epiphany that it was the same force that made the apple fall to the ground that made the Moon circle the Earth that made the planets revolve around the sun.

The inverse square law holds over more than six decades of observed acceleration in the solar system, from the one gee we feel here on the surface of the Earth to the outskirts patrolled by Neptune.

Planetary motion in the radial acceleration plane. The dotted line is Newton’s inverse square law of universal gravity.*

The inverse square force law is what it takes to make the planetary data line up. A different force law would give a line with a different slope in this plot. No force law at all would give chaos, with planets all over the place in this plot, if, say, the solar system were run by a series of deferents and epicycles as envisioned for Ptolemaic cosmologies. In such a system, there is no reason to expect the organization seen above. It would require considerable contrivance to make it so.

Newtonian gravity and General Relativity are exquisitely well-tested in the solar system. There are also some very precise tests at higher accelerations that GR passes with flying colors. The story to lower accelerations is another matter. The most remote solar system probes we’ve launched are the Voyger and Pioneer missions. These probe down to ~10-6 m/s/s; below that is uncharted territory.

The RAR extended from high solar system accelerations to much low accelerations typical of galaxies – not the change in scale. Some early rotation curves (of NGC 55, NGC 801, NGC 2403, NGC 2841, & UGC 2885) are shown as lines. These probed an entirely new regime of acceleration. The departure of these lines from the dotted line are the flat rotation curves indicating the acceleration discrepancy/need for dark matter. This discrepancy was clear by the end of the 1970s, but the amplitude of the discrepancy then was modest.

Galaxies (and extragalactic data in general) probe an acceleration range that is unprecedented from the perspective of solar system tests. General Relativity has passed so many precise tests that the usual presumption is that is applies at all scales. But it is an assumption that it applies to scales where it hasn’t been tested. Galaxies and cosmology pose such a test. That we need to invoke dark matter to save the phenomenon would be interpreted as a failure if we had set out to test the theory rather than assume it applied.

It was clear from flat rotation curves that something extra was needed. However, when we invented the dark matter paradigm, it was not clear that the data were organized in terms of acceleration. As the data continued to improve, it became clear that the vast majority of galaxies adhered to a single, apparently universal+ radial acceleration relation. What had been a hint of systematic behavior in early data became clean and clear. The data did not exhibit the scatter that as was expected from a sum of a baryonic disk and a non-baryonic dark matter halo – there is no reason that these two distinct components should sum to the single effective force law that is observed.

The RAR with modern data for both early (red triangles) and late (cyan circles) morphological types. The blue line is the prediction of MOND: there is a transition at an acceleration scale to a force law that is universal but no longer inverse-square.

The observed force-law happened to already have a name: MOND. If it had been something else, then we could have claimed to discover something new. But instead we were obliged to admit that the unexpected thing we had found had in fact been predicted by Milgrom.

This predictive power now extends to much lower accelerations. Again, only MOND got this prediction right in advance.

The RAR as above, extended by weak gravitational lensing observations. These follow the prediction of MOND as far as they are credible.

The data could have done many different things here. It could have continued along the dotted line, in which case we’d have need for no dark matter or modified gravity. It could have scattered all over the place – this is the natural expectation of dark matter theories, as there is no reason to expect the gravitational potential of the dominant dark matter halo to be dictated by the distribution of baryons. One expects that not to happen. Yet the data evince the exceptional degree of organization seen above.

It requires considerable contrivance to explain the RAR with dark matter. No viable explanation yet exists, despite many unconvincing claims to this effect. I have worked more on trying to explain this in terms of dark matter than I have on MOND, and all I can tell you is what doesn’t work. Every explanation I’ve seen so far is a special case of a model I had previously considered and rejected as obviously unworkable. At this point, I don’t see how dark matter can ever plausibly do what the data require.

I worry that dark matter has become an epicycle theory. We’re sure it is right, so whatever we observe, no matter how awkward or unexpected, must be what it does. But what if it is wrong, and it does not exist? How do we ever disabuse ourselves of the notion that there is invisible mass once we’ve convinced ourselves that there has to be?

Of course, MOND has its own problems. Clusters of galaxies are systems$ for which it persistently fails to explain the amplitude of the observed acceleration discrepancy. So let’s add those to the plot as well:

As above, with clusters of galaxies added (x: Sanders 2003; +: Li et al. 2023).

So: do clusters violate the RAR, or follow it? I’d say yes and yes – the offset, thought modest in amplitude in this depiction, is statistically significant. But there is also a similar scaling with acceleration, only the amplitude is off. The former makes no sense in MOND; the latter makes no sense in terms of dark matter which did not predict a RAR at all.

Clusters are the strongest evidence against MOND. Just being evidence against MOND doesn’t automatically make it evidence in favor of dark matter. I often pose myself the question: which theory requires me to disbelieve the least amount of data? When I first came to the problem, I was shocked to find that the answer was clearly MOND. Since then, it has gone back and forth, but rather than a clear answer emerging, what has happened is more a divergence of different lines of evidence: that which favors the standard cosmology is incommensurate with that which favors MOND. This leads to considerable cognitive dissonance.

One way to cope with cognitive dissonance is to engage with a problem from different perspectives. If I put on a MOND hat, I worry about the offset seen above for clusters. If I put on a dark matter hat, I worry about the same kind of offset for every system that is not a rich cluster of galaxies. Most critics of MOND seem unconcerned about this problem for dark matter, so how much should a critic of dark matter worry about it in MOND?

*For the hyper-pedantic: the eccentricity of each orbit causes the exact location of each planet in the first plot to oscillate up and down along the dotted line. The extent of this oscillation is smaller than the size of each symbol with the exception of Mercury, which has a relatively high eccentricity (but nowhere near enough to reach Venus).

+There are a few exceptions, of course – there are always exceptions in astronomy. The issue is whether these are physically meaningful, or the result of systematic uncertainties or non-equilibrium processes. The claimed discrepancies range from dubious to unconvincing to obviously wrong.

$I’ve heard some people criticize MOND because the centroid of the lensing signal does not peak around the gas in the Bullet cluster. This assumes that the gas represents the majority of the baryons. We know the is not the case, and that there is some missing mass in clusters. Whatever it is, it is clearly more centrally concentrated than the gas, so we don’t expect the lensing signal to peak where the gas is. All the Bullet cluster teaches us is that whatever this stuff is, it is collisionless. So this particular complaint is a logical fallacy of the a red herring and/or straw man variety born of not understanding MOND well enough to criticize it accurately. Why bother to do that when you come to the problem already sure that MOND is wrong? I understand this line of thought extraordinarily well, because that’s the attitude I started with, and I’ve seen it repeated by many colleagues. The difference is that I bothered to educate myself.

A personal note – I will be on vacation next week, so won’t be quick to respond to comments.

Stacy McGaugh is an astrophysicist and cosmologist who studies galaxies, dark matter, and theories of modified gravity. He is an expert on low surface brightness galaxies, a class of objects in which the stars are spread thin compared to bright galaxies like our own Milky Way. He demonstrated that these dim galaxies appear to be dark matter dominated, providing unique tests of theories of galaxy formation and modified gravity. Professor McGaugh is currently the chair of the Department of Astronomy at Case Western Reserve University in Cleveland, Ohio, and director of the Warner and Swasey Observatory. Previously he was a member of the faculty at the University of Maryland, having also held research fellowships at Rutgers, the Department of Terrestrial Magnetism of the Carnegie Institution of Washington, and the Institute of Astronomy at the University of Cambridge after earning his Ph.D. from the University of Michigan.

29 thoughts on “The Radial Acceleration Relation starting from high accelerations”

  1. Stacy, these are wonderful graphs. They merit to be on the wall of every gravitational physicist.

    On the x-axis, is the measured value the radius r or is it g_bar? On the y-axis, is the measured value the speed v or is it some other quantity? If so, a graph with the axes y=g_obs=v^2/r (no change) and x=r (just relabeling) would look even more impressive for a physicist like me.

    If the value a_0 were due to the cosmological radius, as Milgrom asks, it should depend on the age of the universe, and thus increase with galactic distance. But it clearly does not. This is an awe-inspiring aspect by itself. It is mind-twisting.

    Does Verlinde’s entropic gravity reproduce the blue line or is there a difference? If I recall correctly, Verlinde avoids the time-dependency question as well. It is easy to find a model that reproduces a_0; but all such models suggest that a_0 depends on time (and thus on galaxy distance). But it does not. This issue is the biggest question of these wonderful graphs.

    In fact, these graphs possibly show one of the last open question of modern physics. A constant of nature (a_0) that nobody understands. A constant of nature (a_0) that should not be constant.

    Thank you for presenting this so clearly.

    Regards
    Klaus

    P.S. The other questions about the graphs are the usual ones. Are we completely, absolutely sure that there is no measurement effect plying a role? No effect of the speed of our own solar system? Are the galaxies in the graph spread out over the whole celestial sphere? Are we completely, absolutely sure that the v(r) measurements are reliable and typical for each galaxy? No selection bias of the stars yielding the v values? You probably have answered these questions a hundred times. Feel free not to answer.

    1. Yes – we appear to have a new constant of nature that is not understood. It is a “who ordered this?” moment that the physics community has met face-on with its head buried in the sand.

      There are lots of questions here that I’ll pass on answering here. Just to pick one: Verlinde’s entropic gravity asymptotes to MOND, so is close to MOND at low acceleration and Newton at high acceleration. However, the transition around a0 is perceptibly different, predicting an excess force that MOND does not, so it does not fit galaxy data as well as MOND does.

  2. Hi Stacy,

    Much has been said about MOND’s nonlinearity, but I would like to draw a new point to your attention and request your feedback.

    The well-known nonlinearity of the MOND RAR is that the RAR doesn’t work by summing over the acceleration contributions of individual kilograms. This is something I find irksome, but tolerable as the theory is developed. However, there is something that I find troublesome which is the low-acceleration cutoff.

    The RAR with the low-acceleration cutoff has a kind of arbitrariness in what local mass is evaluated before the low-acceleration cutoff is applied. For example, for the acceleration of a toy marble toward the center of the mass of the Earth, MOND automatically considers the whole mass of the Earth, but doing so ends up not treating the Earth as the sum of individual kilograms, each of whose acceleration contribution would typically invoke the MOND RAR.

    What is troublesome is that there is some hidden rule as to how the low-acceleration-cutoff can be applied.

    As another example, if I asked for the acceleration on a toy marble due to the contributions of the 8 planets, MOND would allow the vector sum the GM/r^2 contribution of each of the planets. But if I asked for the acceleration on a toy marble due to the contributions of the 8 planets and 1 kilogram of rock in the asteroid belt (<< 10^-10 m/s^2 and would invoke the MOND RAR), the vector sum of the accelerations would be incorrect.

    In other words, MOND has a hidden rule about vector summing acceleration contributions, and so does not give the universal acceleration on a test particle.

    Because of the arbitrariness of the low-acceleration condition, I would like to propose and request your feedback on something to succeed the MOND RAR, a program that can be run on every particle.

    const r_cutoff = 10^15 meters    // ~ 10,000 AU, this is a guess

        function calc_accel_for_particle() {
            a_near = 0
            a_far = 0

            for each mass within r_cutoff:
               a_near += Gm/r^2

            for each mass farther than r_cutoff:
               a_far += Gm/r^2
            a_far = sqrt(a_far * c^2/87Gly)

            a_total = a_near + a_far

            return a_total
        }

    This way, a particle’s universal acceleration, instant-to-instant, can be calculated.

    This also retains the plausible explanation for the extra gravity being the rest of the mass of the universe outside the galaxy, for a universe with the shape of a hypersphere and circumference 2pi * 13.8 Gly.

    The idea of a program with a discontinuity running for each particle is not elegant, but at least it is impartial to each particle mass.

    Sahil Gupta


    1. The a0 = c^2/87Gly ist an interesting coincidence

      The relationship a0 = c^2/87Gly is an interesting coincidence.

      But your program only works for the solar system (without fieldeffects). For the Milky Way, r_cutoff=10,000 AU is much too small…

      General:You need to reproduce MOND with your program, and MOND is about acceleration, not distance…

      1. But your program only works for the solar system 

        I disagree. Did you run through the program and get an output?

        MOND is about acceleration, not distance

        Yes, and a problem emerges from that. MOND has a hidden rule about vector summing acceleration contributions, and so does not give the universal acceleration on a test object.

        Consider an extragalactic planet. Some random planet around a star in a galaxy of the SPARC dataset. What is its total, instant acceleration? There’s a component in the direction of the host star and a component in the direction of the center of the galaxy. Right? Now, how is each component calculated, and how are they summed? If you truly applied MOND as written, based on the low-acceleration cutoff, to calculate the acceleration contribution by the host star, and if you were impartial about treating the star not as a “lump of star” but as individual units of mass, say, individual hydrogen atoms, then the acceleration contribution by the host star would be out of whack.

        In current MOND research, the host star is treated like a “lump of star” instead of individual units of mass, which is how the rest of the galaxy is treated. Do you understand that discrepancy of treatment?

        1. You are reproducing the struggles we encountered thirty years ago. Yes, the vector sum of all accelerations matters. Linear sums don’t work because the theory is nonlinear – you absolutely cannot treat a star as a simple sum of atoms. You need a Poisson solver, of which a number of codes exist. QUMOND as implemented in the Phantom of Ramses is probably the most accessible.

          1. I mentioned nonlinearity in my first post. I am aware of the 30-year struggle. But what I also mentioned in my first post was the trouble caused by the low-acceleration cutoff.

            And then I proposed the short program as a partial improvement. Do you have an opinion on the program? Does it have less error than the low-acceleration cutoff?

              1. I’m talking about the number 1 idea of MOND, that a special function is applied when accelerations are << a0. The rule behind “The MOND regime”.

                This low-acceleration cutoff or rule or limit, which then leads to terminology like “MOND regime”, “Newtonian regime”, and “Externally Imposed Newtonian Regime”, and various systems and sub-systems, and rules applying across systems overriding sub-systems and rules of sub-systems imposing on parent systems, are troublesome.

                Another way I can articulate the problem: For a planet’s total, instant acceleration, why is the acceleration component to the host star treated as a separate component to be summed to the galactic component? Why isn’t the host star treated as a portion of the galaxy, so there’d be no host star component but just one, whole galactic component? Because you know in advance that the acceleration to the star is > a0? That’s putting the answer into the prediction attempt. That’s not a real prediction.

                The priority has got to be a condensed formula to calculate the instant, total (universal) acceleration of a random planet as it moves around a star and galaxy.

                Can MOND produce that condensed formula (or program)?

            2. (I wanted to reply to Sahil’s reply below, but I think we reached the WordPress nesting limit, so I went up a couple of branches).

              I found MOND for Dummies quite helpful in my understanding of the application and implications of MOND. For the RAR, pages 14 and 15 are extremely useful.

              This is my current understanding (others can correct me if I go off base). There is no “cutoff”, just a Newtonian regime and a MOND regime and a transition between them, centred around a0. The interpolation function between the regimes is still an open question, but it is an interpolation and not a cutoff. The interpolation is a function that interpolates from Newtonian, where gobs = gbar, to MOND, where gobs = sqrt(gbar * a0). However, as far as I know there is no currently definitive interpolation function.

              I like to think of the interpolation function as an acceleration boost on Newton. Based on Stacy’s current Milky Way Mass Model, the Newtonian regime appears to extend down to a small multiple of a0, where the boost begins to become noticeable (i.e., gobs is noticeably greater than gbar). Below some fraction of a0, the MOND regime dominates with an asymptotic limit of gobs at sqrt(gbar * a0).

              I personally prefer the following interpolation function offered in M4D;

              v(x) = 1 / (1 – exp(-sqrt(a / a0)))

              obtained from curve fitting the RAR. This also has the correct limiting behaviour.

              Stacy’s new lensing data shows a quasi-Newtonian dip at quite low accelerations, but the error bars are still compatible with the curve fit and galaxies may not be completely isolated at that acceleration / distance (possibly EFE).

              There is currently no MOND theory explaining the MOND phenomenology, but the phenomenology is still phenomenal (pardon the pun). It is also predictive, galaxies didn’t have to behave that way. Galaxies are 10’s to 100’s of billions of stars executing an exquisite dance in suspected equilibrium over millions or billions of years (until another large dance troupe bumps into them). Modeled accurately by a simple formula based only on what we can see.

              I find the last paragraph on Page 15 of M4D particularly compelling.

              “… even though gbar is only ¼ that of gCDM, gbar directly
              predicts gCDM and the CDM is perfectly coupled to the baryonic mass.”

              Tail wagging the dog.

              Thanks again Stacy for maintaining this blog and putting out your latest research to the masses.

  3. Galaxy clusters seems to have the same problem with MOND as wide binaries. MOND seems to apply only to galaxy level complexity, that is its “natural” domain. On the other hand General Relativity seems to be very accurate on simple systems but fail at galaxy level complexity, obviously this failure invalidate GR use beyond galaxy level complexity, even more at “universe” level complexity.

    It will be extremely hard/painful for mainstream scientific thinking to acknowledge this reality, not surprising that the dark matter myth is receiving lots of funding while MOND related research is being defacto censored.

    1. I would not consider that the wide binary data is inconsistent with MOND, Banik’s supposed refutation has serious statistical problems, that have been discussed here and elsewhere.

      The main issue is that his model systematically underproduces dispersion in observed velocity, then to match the scatter in the data it needs a vast excess of unresolved multiples. Then by adding so many spurious multiples, this makes the model overpredict the observed velocities, which leads to underestimating the strength of gravity.

      In the MOND models the augmentation of gravity requires less unresolved multiples, but then the scatter is too small, and this reduces the likelihood of the MOND model appreciably.

      1. Reality has a hierarchical structure, when theories are developed they usually are targeting a very specific hierarchical level as they use very simple systems as starting points.

        Quantum mechanics in very simple quantum systems useless when applied to complex assemblies of quantum objects like living beings.
        General Relativity in very simple gravitational systems, fails at galaxy level complexity unless you introduce a fictitious dark matter, and beyond that is pure fiction.

        MOND at galaxy level complexity failing or inconclusive below or above that level.

        This is really not new but mainstream scientific thinking is hell-bent into using naive reductionism for everything, ignoring the ideas of Philip W. Anderson(More is different) and the implications of Chaitin mathematical results, like his heuristic principle: the results of a theory can’t be more complex than the theory itself implying among other things that complexity is a source of incompleteness/irreducibility.

        These results just reaffirm the objective fact that Reality has a hierarchical structure and this hierarchical structure is a manifestation of the emergence of new irreducible properties at each hierarchical level.

        Complexity is a boundary for the predictive/explanatory power of any theory. Any theory has a limited complexity range of applicability.

  5. The graphs show the overview beautifully, and summarise the puzzle. I know the results are less certain in clusters, and open to interpretation, but what about the evidence that clusters, and BCGs, use a different acceleration scale from a0, around 2e-9 ? We shouldn’t necessarily be fitting data to a predecided idea, that’s what’s often done with DM. If there is indeed a separate RAR at larger scales, the differences and similarities might be among the best clues we have. For instance, do small ellipticals use a0, while BCG ellipticals use 2e-9, or is that an oversimplification? Thanks for the post as usual.

    1. It seems that BCGs at the centers of clusters appear to share the discrepancy of the clusters of which they are part. Other ellipticals (large or small) do not show this behavior, being consistent with the usual value of a0.

  6. when you return

    arXiv:2407.18846 (astro-ph)

    [Submitted on 26 Jul 2024]Cosmological implications of the Gaia Milky Way declining rotation curve

    Even Coquery, Alain Blanchard

    …In a third step, we show that the usual MOND paradigm is not able to reproduce the declining part for a standard baryonic model. Finally, we examine whether the MOND theory can accommodate the declining part of the rotation curve when relaxing the characteristics of the baryonic components…Finally, a_0 is consistent with 0, with an upper limit of 0.53 10^{-10} m/s^{2} (95%), a value much lower than the above mentioned value usually advocated to explain standard flat rotation curves in MOND theory.

    problem with MOND theory.

  7. at the end og abstract: “Finally, a0 is consistent with 0, with an upper limit
    of 0.53 × 10−10 m/s2
    (95%), a value much lower than the above mentioned value usually advocated
    to explain standard flat rotation curves in MOND theory.”

    Do you believe them? Will you follow them?

    1. Stacy has already spoken at length about the supposed Keplerian decline in the Milky Way rotation speeds. For it to be correct, the Milky Way has to be unlike any other galaxy in the universe. That seems like something worth further investigation before taking as gospel.

      1. Also, Figure 1 is just copied from the Jiao 2023 paper. It only goes out to a radius of 26 kpc and has quite large error bars. Haven’t investigated the MOND model they are using, but in Stacy’s Milky Way baryonic mass model, the velocity descends from 222 km/s to 206 km/s in the same range, which is well within the error bars presented in the 2023 paper (if the error bars are even correct). Beyond 26 kpc, Stacy’s model does descend to 188 km/s at 152 kpc, but this matches MONDian (and not Keplerian) behaviour for the estimated baryonic mass.

        1. OK, “well within the error bars” was a bit of an overstatement, but to be fair, the chart is pretty small and has no grid lines. Nevertheless, Stacy’s model still comes close to the upper ends of the error bars in the 23.5 to 26.5 kpc range. It also does not resemble the MOND B2 model portrayed.

          For fun, I decided to have a look at the GAIA data for myself. I suspected that the GAIA data would be highly centred around our solar system (for obvious reasons). The Flatiron Institute has a really good GAIA DR3 data visualization and extraction site called FlatHUB. There are 1.8 billion sources in total, approximately 2% of the Milky Way galaxy. Around 470 million have a General Stellar Parameterizer for Photometry (GSP) solution, which among other things computes distance from parallax, magnitude, and spectra. Around 34 million sources have line of sight radial velocity, and about 26 million have both.

          Here is a visualization of these 26 million sources with respect to the galaxy (XY and Z). As suspected, the sources are clustered mostly within a few kpc of Sol.

          I had a look again at Wang et al (2022). They looked at sources with galactic longitude between 160 and 200 degrees (directly opposite to the galactic centre). They also used parallax only to calculate distance (i.e., not GSP). I found quite a bit of discrepancy between the GSP calculated distance and raw parallax calculated distance. Moreover, the GAIA data has an estimated systematic error on parallax of 0.05 mas, which is in the same order of magnitude as the expected parallax for sources in the 25 to 30 kpc radius range (17 to 22 kpc from Sol, i.e., 1/0.05 = 20 kpc + 8 kpc Sol = 28 kpc). On the other hand, the 26 million sources having both radial velocity and GSP distance do not extend more than 13 kpc from Sol in the anti-galactic centre direction, i.e., up to a radius of 21 kpc. This is likely why Wang et al and others used only parallax to extend beyond 20 kpc.

          Being just a interested amateur (my expertise is in terrestrial positioning), this can all be taken with a large chunk of salt. Nevertheless, this strikes me as a case of trying to use too little butter on too much bread. Notable in Wang et al (2022) is the sharp drop in Table 1 from the mid 190’s at 24.5 kpc to the mid 170’s from 25.5 kpc onwards. This looks to me like a systematic error. I also believe the error bars are underestimated.

          There may be a Keplerian decline in the Milky Way but I’m not convinced this can be definitively shown from the GAIA data. I think it gets pretty sketchy after a galactic radius about 20 kpc. My statistics professor always liked to say, “interpolation is dangerous, and extrapolation is insanity”. It is also a bit ironic that it is hardest for us to see our own galaxy, other likely similar galaxies we can see in their entirety appear to act strictly MONDian. Yes, I should have also looked at Jiao et al (2023), but I was already knee deep in Wang et al (2022) and it is also referenced by the MOND vs DM paper above.

    1. That sounds like a complicated system, so I’m not sure what to make of it. The collision appears to have been head-on and just happened, which is a rather special situation in both space and time. The initial velocity is crazy high for CDM; this is the sort of collision speed Angus & I predicted would happen in MOND. I don’t know what to make of “velocity space decoupling.”

  9. Hi all, only tangentially related to the above post; I’m trying to find a post on here by Indranil Banik where he lays out a theory of MOND plus neutrino dark matter. I’ve searched for Banik and neutrino but not found the post I’m remembering. Can someone help me?

  10. you might look at this and even blog

    arXiv:2408.00358 [pdf, html, other]

    Quasilocal Newtonian limit of general relativity and galactic dynamics

    Marco Galoppo, David L. Wiltshire, Federico ReComments: 6 pages, 2 figuresSubjects: General Relativity and Quantum Cosmology (gr-qc); Cosmology and Nongalactic Astrophysics (astro-ph.CO); Astrophysics of Galaxies (astro-ph.GA); High Energy Physics – Phenomenology (hep-ph)

    A new Newtonian limit of general relativity is established for stationary axisymmetric gravitationally bound differentially rotating matter distributions with internal pressure….. The self-consistent coupling of quasilocal gravitational energy and angular momentum leads to a modified Poisson equation.The solutions of the full system reproduce the phenonomenology of collisionless dark matter for disc galaxies, offering an explanation for their observed rotation curves. Halos of abundant cold dark matter particles are not required.

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