A LQG Legend Writes Paper Claiming GR Explains Dark Matter Phenomena

[ohwilleke]

In #62, #70, #76, I tried to identify Deur's methods of calculation. And a reminder, Ciotti #44 is the most thorough statement so far, of why one would not expect classical GR to produce such effects. So that's the gap one could try to bridge.

Also, even if that's not how GR works, one could try to design a modified gravity in which Deur's calculations *are* correct.

One of the things that I'm least clear about is whether Deur's method truly observes both the strong equivalence principle and the weak equivalence principle. I suspect that it observes only one but I'm not sure.

 

[PeterDonis]

QCD motivates the approach taken but isn't actually being used at all to make the calculations.

Even so, the QCD-like effects that Deur appears to be claiming should be much weaker for gravity than for QCD, as compared with the "Newtonian" component of the interaction, because the coupling constant for gravity is so much smaller, and the relative magnitudes of the effects should go like some power of the coupling constant.

 

[PeterDonis]

QCD motivates the approach taken

One major difference between QCD and gravity, though, is that the gauge group of QCD is compact, whereas the gauge group of gravity is not.

 

[kodama]

In #62, #70, #76, I tried to identify Deur's methods of calculation. And a reminder, Ciotti #44 is the most thorough statement so far, of why one would not expect classical GR to produce such effects. So that's the gap one could try to bridge.

Also, even if that's not how GR works, one could try to design a modified gravity in which Deur's calculations *are* correct.

one could try to design a modified gravity in which Deur's calculations *are* correct.--any suggestions for how to go do this ?

 

[kodama]

One major difference between QCD and gravity, though, is that the gauge group of QCD is compact, whereas the gauge group of gravity is not.

could you create a gravity that is like gr but with a compact gauge group

 

[mitchell porter]

could you create a gravity that is like gr but with a compact gauge group

Ashtekar variables describe general relativity in terms of a connection rather than a metric; and the connection is SU(2)-valued, and SU(2) is compact. But then LQG, etc, mostly use the complexification of SU(2), which is non-compact.

Perhaps someone would like to express Deur's formulas using Ashtekar's variables?

 

[PeterDonis]

the connection is SU(2)-valued

I'm not sure how that's true, but I'm not familiar with Ashtekar variables. If you have a good reference on those, that would be helpful.

That said, as I understand it, the gauge group of GR is the group of coordinate transformations, which is not compact. I don't think it matters whether the theory is expressed in terms of the metric or the connection.

 

[mitchell porter]

I said

Ashtekar variables describe general relativity in terms of a connection rather than a metric; and the connection is SU(2)-valued, and SU(2) is compact. But then LQG, etc, mostly use the complexification of SU(2), which is non-compact.

which is somewhat confused.

What I think is true, is that you can describe a Riemannian metric, and hence "Euclidean gravity", in terms of a connection valued in a compact group. But if you work in space-time, it's a Lorentzian signature, the metric is only "semi-Riemannian", and the connection will now take values in a non-compact group.

Ashtekar's original work indeed used a connection valued in a non-compact group. The interest was that the change of variables put the Hamiltonian constraint into a polynomial form resembling Yang-Mills theory. But having a quantum gauge theory based on a non-compact group is problematic.

Then Barbero argued that the quantum theory could be based on a real-valued (hence compact) SO(3) or SU(2) connection, at the price of the Hamiltonian constraint becoming non-polynomial again. Apparently this became the basis of most work in loop quantum gravity for a while. (Someone argued that the resulting theory is not actually a gauge theory, but I haven't read that paper.)

Much more recently, Peter Woit has been championing the idea that you could start with Euclidean quantum fields with an SO(4) local symmetry, factorize the SO(4) into two SU(2) factors, and use one SU(2) for a connection-based quantum gravity, and the other SU(2) for the weak gauge field of the standard model. Calculations in the empirical world of Lorentzian signature space-time would then be obtained as an analytic continuation, but the Euclidean theory would be fundamental. I think. It might be a distraction to mention this, but it's been discussed on some other threads recently.

@PeterDonis asked for references about the Ashtekar variables. I can't say that these are the best introduction, but you could try Wikipedia, Scholarpedia, and Ashtekar's original paper. Sections 3, 3.1 of Woit's paper may actually be a quick introduction to the ideas.

Returning to the issue of compactness, it now seems as if there are only two leading proposals for how to get general relativity from a compact gauge group. One is just to work in Euclidean signature. The other is Barbero's proposal, which is about selecting a compact subgroup within the non-compact group (apparently, the famous Immirzi parameter of loop quantum gravity specifies which copy of SU(2) inside SL(2,C) one is using?), and it's now unclear to me if it actually works.

 

[kodama]

I said

which is somewhat confused.

What I think is true, is that you can describe a Riemannian metric, and hence "Euclidean gravity", in terms of a connection valued in a compact group. But if you work in space-time, it's a Lorentzian signature, the metric is only "semi-Riemannian", and the connection will now take values in a non-compact group.

Ashtekar's original work indeed used a connection valued in a non-compact group. The interest was that the change of variables put the Hamiltonian constraint into a polynomial form resembling Yang-Mills theory. But having a quantum gauge theory based on a non-compact group is problematic.

Then Barbero argued that the quantum theory could be based on a real-valued (hence compact) SO(3) or SU(2) connection, at the price of the Hamiltonian constraint becoming non-polynomial again. Apparently this became the basis of most work in loop quantum gravity for a while. (Someone argued that the resulting theory is not actually a gauge theory, but I haven't read that paper.)

Much more recently, Peter Woit has been championing the idea that you could start with Euclidean quantum fields with an SO(4) local symmetry, factorize the SO(4) into two SU(2) factors, and use one SU(2) for a connection-based quantum gravity, and the other SU(2) for the weak gauge field of the standard model. Calculations in the empirical world of Lorentzian signature space-time would then be obtained as an analytic continuation, but the Euclidean theory would be fundamental. I think. It might be a distraction to mention this, but it's been discussed on some other threads recently.

@PeterDonis asked for references about the Ashtekar variables. I can't say that these are the best introduction, but you could try Wikipedia, Scholarpedia, and Ashtekar's original paper. Sections 3, 3.1 of Woit's paper may actually be a quick introduction to the ideas.

Returning to the issue of compactness, it now seems as if there are only two leading proposals for how to get general relativity from a compact gauge group. One is just to work in Euclidean signature. The other is Barbero's proposal, which is about selecting a compact subgroup within the non-compact group (apparently, the famous Immirzi parameter of loop quantum gravity specifies which copy of SU(2) inside SL(2,C) one is using?), and it's now unclear to me if it actually works.

does Euclidean quantum fields change the physics compared to LQG ?

 

[PAllen]

Don’t know if this has come up earlier in this thread, but there are n-body formulas at the PPN level that make no assumptions about spherical symmetry or a dominant central mass. I wonder, given modern computing power, if anyone has tried solving these for e.g. 50 bodies with initial conditions similar to a galaxy, with a central BH, and see what they lead to. Note, PPN formulas should include all nonlinear effects present except in strong fields and very high speed relative motion - and none of these should be relevant in a galaxy.

 

[kodama]

Don’t know if this has come up earlier in this thread, but there are n-body formulas at the PPN level that make no assumptions about spherical symmetry or a dominant central mass. I wonder, given modern computing power, if anyone has tried solving these for e.g. 50 bodies with initial conditions similar to a galaxy, with a central BH, and see what they lead to. Note, PPN formulas should include all nonlinear effects present except in strong fields and very high speed relative motion - and none of these should be relevant in a galaxy.

so what Does Deur predict

 

[PAllen]

so what Does Deur predict

I have no idea. I have never read Deur papers. But I do know about the use of PPN approximation to improve solar system approximation for all large bodies (including the large planetoids) way, way beyond what Newtonian gravity can achieve. And also, that PPN is able to predict GW wave forms for inspiralling BH with precision up until the final moments. It is used as a cross check on numerical relativity codes. But unlike numerical relativity it might just be possible to simulate e.g. 50 bodies with conceivable computer power.

Point is Deur makes claims about what classical GR would predict, but he does not demonstrate these claims. Using PPN approximation would be a possible way to verify or refute his claims.

 

[PeterDonis]

I wonder, given modern computing power, if anyone has tried solving these

I believe the paper referred to in post #44 of this thread did something like this (using the gravitomagnetic formalism, but AFAIK if the effects of many bodies are included this is mathematically equivalent to the formalism you refer to), and found, as mainstream opinion expected, that the GR corrections are too small to account for observed rotation curves with just the visible matter.

what Does Deur predict

As I understand it, Deur's claims fall, broadly speaking, into two categories:

(1) There are nonlinear effects in classical GR, amounting to large corrections to the Newtonian behavior for highly non-spherical cases, that are not properly accounted for in the usual models.

(2) There are non-perturbative effects in quantum gravity, analogous to things like gluon flux tubes in QCD, that can produce large corrections to the classical behavior but are not (obviously) taken into account in classical models.

Papers like the one linked to in post #44, IMO, cast serious doubt on Deur's claims in category 1 above. One could still argue that there are additional nonlinear effects that the formalisms used do not include, but such claims become increasingly unlikely as more and more detailed classical treatments are done.

The main issue as I understand it with Deur's claims in category 2 is that there is no well-defined theory behind them; they are just heuristics based on claimed similarities between quantum gravity (for which we have no well-defined theory at present) and QCD. These are interesting theoretical areas to look at, but in the absence of a well-defined theory from which definite predictions can be made, they remain speculative.

 

[ohwilleke]

Properly considering the five

parameters in the PPN formalism (ideally at the GR values of 1) should be what is necessary, although it is a bit hard (for me anyway) to tell precisely what the PPN formalism is disregarding.

But, this may not be right in light of this quotation from the introduction portion of A. Deur, "An explanation for dark matter and dark energy consistent with the standard model of particle physics and General Relativity." 79 Eur. Phys. J. C , 883 (October 29, 2019). https://doi.org/10.1140/epjc/s10052-019-7393-0

In GR, self-interaction becomes important for 𝐺𝑀/𝐿‾‾‾‾‾‾‾√GM/L large enough (L is the system characteristic scale), typically for 𝐺𝑀/𝐿‾‾‾‾‾‾‾√>𝑟𝑠𝑖𝑚10−3GM/L>rsim10−3 as discussed in Ref. [5] or exemplified by the Hulse-Taylor binary pulsar [8], the first system in which GR was experimentally tested in its strong regime, which has 𝐺𝑀/𝐿‾‾‾‾‾‾‾√=10−3GM/L=10−3. As in the case of QCD, self-interaction increases the binding compared to Newton’s theory. Since the latter is used to treat the internal dynamics of galaxies or galaxy clusters, its neglect of self-interaction may contribute to – or even create – the missing mass problem [4, 5, 9].

In Ref. [4] a non-perturbative numerical calculation based on Eq. (2) is applied in the static limit to spiral galaxies and clusters. A non-perturbative formalism (lattice technique) – rather than a perturbative one such as the post-Newtonian formalism – was chosen because in QCD, confinement is an entirely non-perturbative phenomenon, unexplainable within a perturbative approach.

I have taken a screen shot of Equation (2) and the related text to avoid having to format it with LaTex:

Reference [4] in the quoted material is A. Deur, “Implications of Graviton-Graviton Interaction to Dark Matter” (May 6, 2009) (published at 676 Phys. Lett. B 21 (2009)). This paper is only 11 pages including references, and is quite clear about the equations used, the way that those equations were derived, and exactly what assumptions are being made in doing so. The scalar approximation used to make it mathematically tractable is essentially making a static approximation that disregards gravitomagnetic effects, kinetic energy, electromagnetic flux, and pressure on the RHS of the Einstein field equations.

Reference [5] is to A. Deur, "Self-interacting scalar fields at high-temperature." 77 Eur. Phys. J. C, 412 (2017). https://doi.org/10.1140/epjc/s10052-017-4971-x

Reference [8] is to R.A. Hulse, J.H. Taylor, "Discovery of a pulsar in a binary system." 195 Astrophys. J. L51 (1975).

Reference [9] is to A. Deur, "A relation between the dark mass of elliptical galaxies and their shape", 438(2) Monthly Notices of the Royal Astronomical Society 1535–1551 (February 21, 2014). https://doi.org/10.1093/mnras/stt2293

 

[ohwilleke]

The main issue as I understand it with Deur's claims in category 2 is that there is no well-defined theory behind them; they are just heuristics based on claimed similarities between quantum gravity (for which we have no well-defined theory at present) and QCD. These are interesting theoretical areas to look at, but in the absence of a well-defined theory from which definite predictions can be made, they remain speculative.

The theory seems to be reasonably well defined and makes definite predictions. It may not be rigorously derived from first principles based upon the quantum field theory of a massless spin-2 graviton, and it may not have yet been thoroughly reviewed for theoretical consistency (something that it took almost three decades to do, for example, with renormalization in the SM after it started to be widely used), but the formulas are there and are possible to calculate with.

In particular, while the physical constant that modifies the self-interaction term ought to be possible to calculate from first principles using only Newton's constant and the speed of light, Deur actually uses the same data set that was used to determine the MOND acceleration constant to establish it without doing the first principles calculation, at least for purposes of the lattice calculation in Reference [4] cited in post #114 in this thread.

 

[PeterDonis]

the physical constant that modifies the self-interaction term

Isn't this just $16 \pi G$? (Or its square root?)

 

[ohwilleke]

Isn't this just $16 \pi G$? (Or its square root?)

In the spiral galaxy case, Deur's approach gives rise to the following formula, the first term of which is Newtonian gravity, and the second of which is the self-interaction term (ignoring higher order terms in an infinite series that are small by comparison):

F = GM/r2 + c^2(aπGM)^0.5/(2√2)r

where F is the effective gravitational force, G is Newton's constant, c is the speed of light, M is ordinary baryonic mass of the gravitational source, r is the distance between the source mass and the place that the gravitational force is measured, and a is a physical constant that is the counterpart of a(0) in MOND (that should in principle be possible to derive from Newton's constant) which is equal to 4*10^−44 m^−3s^2.

Thus, the self-interaction term that it modifies is proportionate to (GM)^0.5/r and is initially much smaller that the first order Newtonian gravity term in stronger fields, but it declines more slowly than the Newtonian term with distance until it is predominant.

 

[PeterDonis]

a is a physical constant that is the counterpart of a(0) in MOND

Ah, ok, got it.

 

[mitchell porter]

One comment on Deur's "Self-interacting scalar fields at high-temperature" (mentioned in @ohwilleke's #114, this thread). It cites arXiv:0709.2042 (Deur's reference 15) as evidence that QCD can be approximated by a scalar field, in a way which motivates Deur's own scalar approximation of GR. But this cited paper has been criticized for a reason I gave in #76: the validity of the scalar approximation requires that some physical influence ("constraint forces") prevents all the other degrees of freedom from coming to life.

I guess that Deur's reasoning may be found in his "Implications of Graviton-Graviton Interaction to Dark Matter": the $T^{00}$ component of the stress-energy tensor dominates "in the stationary weak field approximation", and therefore the $\phi^{00}$ component of the gravitational $\phi$ field (its relation to the metric is that $g_{\mu \nu} = (e^{k \phi})_{\mu \nu}$) should similarly dominate. This is an assumption that one might want to scrutinize.

 

[strangerep]

(Sigh.) I'm reaching a point where I'm starting to suspect think Deur's expansion method might be BS. He never seems to explain it properly afaict. (If you think you know a paper where he does explain it extensively, please tell me.)

E.g., in his 2009 paper (arXiv:0901.4005), he says he expands in powers of $h_{\mu\nu} = g_{\mu\nu} - \eta_{\mu\nu}$, but rescales $h_{\mu\nu} \to h_{\mu\nu}/\sqrt{M}$, or $h_{\mu\nu} \to h_{\mu\nu}/\sqrt{G}$. However, the 1st order solution for the original (non-rescaled) $h_{\mu\nu}(G)$ must be the Newtonian solution, and that is always $O(G)$.

Newtonian solutions for disk galaxies with physically realistic exponential radial mass distributions have been around for ages (see the treatment in galaxiesbook.org for a readable account). The solutions always have a $G$ at the front.

So in the 2nd-order Einstein equations we have 2nd-order derivatives of $h(G^2)$, but the $\Gamma\,\Gamma$ terms in $R_{\mu\nu}$ cannot contain $h(G^2)$ because a $\Gamma$ must contain at least one $\partial h(G)$. The $\Gamma\,\Gamma$ could of course contain quadratic expressions in $\partial h(G)$. So the 2nd-order Einstein equations
are of the form $$\partial^2 h(G^2) ~+~ h(G) \partial^2 h(G) ~+~ \partial h(G) \partial h(G) ~=~ 0 ~.$$ But this only gives an expansion of the physical (dimensionless) metric in powers of $G$. I don't see where an $h$ (of any order) can enter that only involves $\sqrt{G}$ (without a sleight-of-hand rescaling).

 

[PAllen]

I am not sure what the current status of this is, but it seems to me that a classical (non quantum) modified gravity theory to reduce reliance on dark matter must be more like Bekenstein’s approach. A theory that conflicts with reproducible tests or with SR is simply a no go:

https://arxiv.org/abs/astro-ph/0403694

 

[ohwilleke]

One comment on Deur's "Self-interacting scalar fields at high-temperature" (mentioned in @ohwilleke's #114, this thread). It cites arXiv:0709.2042 (Deur's reference 15) as evidence that QCD can be approximated by a scalar field, in a way which motivates Deur's own scalar approximation of GR. But this cited paper has been criticized for a reason I gave in #76: the validity of the scalar approximation requires that some physical influence ("constraint forces") prevents all the other degrees of freedom from coming to life.

I guess that Deur's reasoning may be found in his "Implications of Graviton-Graviton Interaction to Dark Matter": the $T^{00}$ component of the stress-energy tensor dominates "in the stationary weak field approximation", and therefore the $\phi^{00}$ component of the gravitational $\phi$ field (its relation to the metric is that $g_{\mu \nu} = (e^{k \phi})_{\mu \nu}$) should similarly dominate. This is an assumption that one might want to scrutinize.

A paper by independent authors confirms that scalar approximations can reproduce experimental tests:

We construct a general stratified scalar theory of gravitation from a field equation that accounts for the self-interaction of the field and a particle Lagrangian, and calculate its post-Newtonian parameters. Using this general framework, we analyze several specific scalar theories of gravitation and check their predictions for the solar system post-Newtonian effects.

Diogo P. L. Bragança, José P. S. Lemos “Stratified scalar field theories of gravitation with self-energy term and effective particle Lagrangian” (June 29, 2018) (open access) (pre-print here).

 

[ohwilleke]

I am not sure what the current status of this is, but it seems to me that a classical (non quantum) modified gravity theory to reduce reliance on dark matter must be more like Bekenstein’s approach. A theory that conflicts with reproducible tests or with SR is simply a no go:

https://arxiv.org/abs/astro-ph/0403694

I believe that Bekenstein's approach failed an experimental test or two a few years ago.

 

[PAllen]

A paper by independent authors confirms that scalar approximations can reproduce experimental tests:

Diogo P. L. Bragança, José P. S. Lemos “Stratified scalar field theories of gravitation with self-energy term and effective particle Lagrangian” (June 29, 2018) (open access) (pre-print here).

Actually, the paper says at least one weak field experimental test has not yet been replicated in this class of theories, and that none of the strong field tests have yet been reproduced.

 

[andresB]

Arguing that GEM doesn't work.

[Submitted on 20 Jul 2022]

On the rotation curve of disk galaxies in General Relativity​

Luca Ciotti (Dept. of Physics and Astronomy, University of Bologna (Italy))

Comments:	16 pages, 4 figures, ApJ, accepted
Subjects:	Astrophysics of Galaxies (astro-ph.GA); General Relativity and Quantum Cosmology (gr-qc)
Cite as:	arXiv:2207.09736 [astro-ph.GA]

According to Towards a full general relativistic approach to galaxies, the approximation is not valid at galactic scales

"Since the speeds of stars in galaxies are much smaller than the speed of light and gravity is assumed to be “weak” far from the central region, the general consensus is that the Newtonian limit of the Einstein equations is applicable in this setting. Therefore, full GR is not usually considered to be a viable solution. However, the matter is far more delicate than what it might seem at first glance.

Indeed, though in the presence of low velocities and weak gravitational fields the Newtonian approximation is certainly valid everywhere locally, it turns out not to be valid anymore globally in spatially extended rotating systems, such as galaxies. The reason for this lies in the dynamical nature of the gravitational field, which in such systems manifests itself primarily through the dragging effect due to the off-diagonal elements of the metric, which, in general, are of the same order of magnitude of the diagonal ones."

 

[PAllen]

According to Towards a full general relativistic approach to galaxies, the approximation is not valid at galactic scales

"Since the speeds of stars in galaxies are much smaller than the speed of light and gravity is assumed to be “weak” far from the central region, the general consensus is that the Newtonian limit of the Einstein equations is applicable in this setting. Therefore, full GR is not usually considered to be a viable solution. However, the matter is far more delicate than what it might seem at first glance.

Indeed, though in the presence of low velocities and weak gravitational fields the Newtonian approximation is certainly valid everywhere locally, it turns out not to be valid anymore globally in spatially extended rotating systems, such as galaxies. The reason for this lies in the dynamical nature of the gravitational field, which in such systems manifests itself primarily through the dragging effect due to the off-diagonal elements of the metric, which, in general, are of the same order of magnitude of the diagonal ones."

Actually, you have something backwards. The paper Luca Ciotti is published after and references the paper you linked (actually, it references a more thorough successor paper by the primary author), and claims to refute these papers. Also, its overall argument isn't just that GEM doesn't work, it is that the whole program of GR doesn't need dark matter is not plausible because GEM analytically can be proven (see the reference Mashoon papers) to encompass all first order corrections to Newtonian gravity, and all higher order corrections are provably smaller, in this regime. Note that unlike QCD, we have an exact classical field theory for GR. GEM is derived by Mashoon analytically with provable error bounds. (It is equivalent to first order post-Newtonian, with linear simplification @PeterDonis and I discussed; but all the error bounds from this are computable).

 

[PeterDonis]

the off-diagonal elements of the metric, which, in general, are of the same order of magnitude of the diagonal ones

A rough back of the envelope calculation does not seem to bear this out for the Milky Way galaxy.

First, metric coefficients are dimensionless numbers, with $1$ being a "large" value and the flat spacetime value for diagonal elements, and $0$ being the flat spacetime value for off diagonal elements. So what we are actually interested in are the relative magnitudes of the corrections to the elements.

Roughly speaking, the corrections to the diagonal elements are of order $2M / R$, and the corrections to the off diagonal elements for a rotating system are of order $4 J / R^2$. Here I am using geometric units, $M$ is the total mass of the system, $J$ is its total angular momentum, and $R$ is its characteristic distance scale or "size".

We can easily do a rough estimate of these numbers for the Milky Way. The geometric mass $M$ is given by $G M_\text{conv} / c^2$, and the geometric angular momentum $J$ is given by $G J_\text{conv} / c^3$. The "conv" values are the values in SI units. In SI units we have (assuming 300 billion solar masses for the Milky Way) $M_\text{conv} = 6 \times 10^{41}$ and $J_\text{conv} = 10^{67}$. So we obtain for the geometric values (in meters), assuming a rough "size" for the Milky Way of 30,000 light years (roughly the distance of the solar system from the center):

$$
M = 4.5 \times 10^{14}
$$

$$
J = 2.5 \times 10^{31}
$$

$$
R = 2.8 \times 10^{20}
$$

This then gives

$$
\frac{2M}{R} = 3.2 \times 10^{-6}
$$

$$
\frac{4 J}{R^2} = 1.3 \times 10^{-9}
$$

So the off diagonal correction is more than 3 orders of magnitude smaller than the diagonal correction.

 

[strangerep]

The paper Luca Ciotti is published after and [...] and claims to refute these papers.

For the benefit of others, here is the abstract from the Ciotti paper.

Luca Ciotti,
"ON THE ROTATION CURVE OF DISK GALAXIES IN GENERAL RELATIVITY"
arXiv: https://arxiv.org/abs/2207.09736

Abstract:

Recently, it has been suggested that the phenomenology of flat rotation curves observed at large radii in the equatorial plane of disk galaxies can be explained as a manifestation of General Relativity instead of the effect of Dark Matter halos. In this paper, by using the well known weak field, low velocity gravitomagnetic formulation of GR, the expected rotation curves in GR are rigorously obtained for purely baryonic disk models with realistic density profiles, and compared with the predictions of Newtonian gravity for the same disks in absence of Dark Matter. As expected, the resulting rotation curves are indistinguishable, with GR corrections at all radii of the order of $v^2 /c^2 \approx 10^{−6}$. Next, the gravitomagnetic Jeans equations for two-integral stellar systems are derived, and then solved for the Miyamoto-Nagai disk model, showing that finite-thickness effects do not change the previous conclusions. Therefore, the observed phenomenology of galactic rotation curves at large radii requires Dark Matter in GR exactly as in Newtonian gravity, unless the cases here explored are reconsidered in the full GR framework with substantially different results (with the surprising consequence that the weak field approximation of GR cannot be applied to the study of rotating systems in the weak field regime). In the paper, the mathematical framework is described in detail, so that the present study can be extended to other disk models, or to elliptical galaxies (where Dark Matter is also required in Newtonian gravity, but their rotational support can be much less than in disk galaxies).

 

[kodama]

Actually, you have something backwards. The paper Luca Ciotti is published after and references the paper you linked (actually, it references a more thorough successor paper by the primary author), and claims to refute these papers. Also, its overall argument isn't just that GEM doesn't work, it is that the whole program of GR doesn't need dark matter is not plausible because GEM analytically can be proven (see the reference Mashoon papers) to encompass all first order corrections to Newtonian gravity, and all higher order corrections are provably smaller, in this regime. Note that unlike QCD, we have an exact classical field theory for GR. GEM is derived by Mashoon analytically with provable error bounds. (It is equivalent to first order post-Newtonian, with linear simplification @PeterDonis and I discussed; but all the error bounds from this are computable).

does this refute Deur ?

 

[PAllen]

does this refute Deur ?

No, because IMO Deur is simply a modified gravity theory, and he is wrong that it is equivalent to GR. It is better than most modified gravity theories in that it has a physical motivation independent of fitting prior data. It also has promise ( due its construction) to match GR strong field tests. Of prior MOND family theories that I know of, only Bekenstein’s was promising in this area. On the other hand, @strangerep , above, has provided some other reasons to doubt the plausibility of Deur.

 

[PeterDonis]

IMO Deur is simply a modified gravity theory

Some of Deur's papers appear to propose a modified gravity theory, but not all of them; some of them, at least to me, appear to claim that there are effects in standard GR that are not taken into account in the standard analysis of galaxy rotation curves. It is not always easy to tell which position Deur is taking, though, and some of his claims that appear on the surface to be of the latter type look speculative to me, like the analogies he draws between standard GR and QCD.

 

[PAllen]

Some of Deur's papers appear to propose a modified gravity theory, but not all of them; some of them, at least to me, appear to claim that there are effects in standard GR that are not taken into account in the standard analysis of galaxy rotation curves. It is not always easy to tell which position Deur is taking, though, and some of his claims that appear on the surface to be of the latter type look speculative to me, like the analogies he draws between standard GR and QCD.

Right, and my belief is that his models that he claims are based on standard GR are simply not. To my knowledge, he never derives anything starting from the EFE or, manifold plus Minkowskian metric, or ADM formalism for evolution from initial conditions. Of course, I might have missed where he does any of these things, but if he doesn’t, his claims to being based on standard GR are unfounded.

 

[PeterDonis]

To my knowledge, he never derives anything starting from the EFE or, manifold plus Minkowskian metric, or ADM formalism for evolution from initial conditions.

This is my understanding as well: all of his claims about, for example, analogies between GR and QCD, as far as I can tell, are heuristic only and do not rest on any actual derivation.

 

[ohwilleke]

This is my understanding as well: all of his claims about, for example, analogies between GR and QCD, as far as I can tell, are heuristic only and do not rest on any actual derivation.

At least most of the time, Deur's analysis starts with a General Relativistic Lagrangian, rather than the EFE.

 

[PeterDonis]

At least most of the time, Deur's analysis starts with a General Relativistic Lagrangian, rather than the EFE.

Do you mean he starts with the GR Lagrangian? Or just a Lagrangian that includes the standard GR terms, but also has others?

 

[PAllen]

Looking at https://arxiv.org/abs/1709.02481, I do see reference to a GR Lagrangian, but I don’t see anything actually derived from it. Further concerning is that the Hulse-Taylor binary pulsar is cited as an example of when self interaction is significant. But the Hulse -Taylor is quantitatively modeled by second order post Newtonian approximation. Inspiralling BH wave forms are successfully modeled quantitatively by anything greater than 3d order Post-Newtonian approximation. All of this is consistent with the paper discussed earlier establishing that GR cannot account for the effects claimed by Deur. Thus I remain convinced that to the extent that his model is successful, it is actually a modified gravity model; and for whatever reason he refuses to accept this.

Anyway, the whole argument is being repeated - unlike QCD, there is an exact classical field theory, for which the post Newtonian approximation has been fully validated by full numeric relativity, including error bounds.

 

[ohwilleke]

Do you mean he starts with the GR Lagrangian? Or just a Lagrangian that includes the standard GR terms, but also has others?

The GR Lagrangian. Just sloppy writing.

 

[wumbo]

Re the Ciotti paper: Can general relativity play a role in galactic dynamics?

We use the gravitoelectromagnetic approach to the solutions of Einstein's equations in the weak-field and slow-motion approximation to investigate the impact of General Relativity on galactic dynamics. In particular, we focus on a particular class of the solutions for the gravitomagnetic field, and show that, contrary to what is expected, they may introduce non negligible corrections to the Newtonian velocity profile. The origin and the interpretation of these corrections are discussed and explicit applications to some galactic models are provided. These are the homogeneous solutions (HS) for the gravitomagnetic field, i.e. solutions with vanishing matter currents.

Provides a pretty thorough counter-argument to it IMO.

I'm surprised this wasn't worked out before. Linearized gravity is well known. You can't on one hand tell me that gravity propagates at a finite speed and on the other tell me it's irrelevant at cosmological distances. Trivially, there's frame dragging inside a spherical shell of mass in GR that has absolutely no connection to anything Newtonian. The cavalier approach to turning a weakly hyperbolic set of equations into an elliptic set has always to struck me as odd. Cooperstock has an example using the van Stockum cylinder of dust: https://doi.org/10.1142/S021827181644017X

It doesn't have to explain every use of dark matter to be valid. It should be a signal to take approximations to GR with far deeper care. Numerical relativity is sorely needed.

 

[PAllen]

Re the Ciotti paper: Can general relativity play a role in galactic dynamics?
Provides a pretty thorough counter-argument to it IMO.

I'm surprised this wasn't worked out before. Linearized gravity is well known. You can't on one hand tell me that gravity propagates at a finite speed and on the other tell me it's irrelevant at cosmological distances. Trivially, there's frame dragging inside a spherical shell of mass in GR that has absolutely no connection to anything Newtonian. The cavalier approach to turning a weakly hyperbolic set of equations into an elliptic set has always to struck me as odd. Cooperstock has an example using the van Stockum cylinder of dust: https://doi.org/10.1142/S021827181644017X

It doesn't have to explain every use of dark matter to be valid. It should be a signal to take approximations to GR with far deeper care. Numerical relativity is sorely needed.

Apparently, it is not so simple. No one has used numerical relativity for these cases. This debate goes through https://arxiv.org/abs/2205.03091, from May of this year, followed by https://arxiv.org/abs/2207.09736, cited earlier claiming to refute this, followed just this past November by https://arxiv.org/abs/2211.11815, which you cite above. Clearly, this debate is ongoing among the field's experts.

 

[wumbo]

Apparently, it is not so simple. No one has used numerical relativity for these cases. This debate goes through https://arxiv.org/abs/2205.03091, from May of this year, followed by https://arxiv.org/abs/2207.09736, cited earlier claiming to refute this, followed just this past November by https://arxiv.org/abs/2211.11815, which you cite above. Clearly, this debate is ongoing among the field's experts.

Numerical relativity is just a fast way to avoid the paper back and forth -- faithfully simulate it and see what happens. Would clear up the mystery pretty quickly.

You don't need to be a GR expert to linearize the EFE, it's a standard GR intro exercise. The papers are quite readable to anyone familiar with PDEs and perturbation theory, no expertise needed. It's really more about lack of rigor in doing the perturbation analysis and its consequences being negligible (or not).

 

[PeterDonis]

You don't need to be a GR expert to linearize the EFE

Linearizing the EFE wouldn't be sufficient to resolve the issue, since the claims in question are that nonlinear effects are much more significant (as in, orders of magnitude more significant) in galaxies than standard cosmology assumes.

 

[PAllen]

Numerical relativity is just a fast way to avoid the paper back and forth -- faithfully simulate it and see what happens. Would clear up the mystery pretty quickly.

If numerical relativity for galactic rotation were simple, someone would have done it.

You don't need to be a GR expert to linearize the EFE, it's a standard GR intro exercise. The papers are quite readable to anyone familiar with PDEs and perturbation theory, no expertise needed. It's really more about lack of rigor in doing the perturbation analysis and its consequences being negligible (or not).

Obviously this characterization is not reasonable. All authors are top notch GR experts, this debate has been ongoing for at least 15 years, and is still not resolved to the level of clear consensus. However, among GR experts I know personally, the view of Ciotti is the one most believe.

 

[wumbo]

Linearizing the EFE wouldn't be sufficient to resolve the issue, since the claims in question are that nonlinear effects are much more significant (as in, orders of magnitude more significant) in galaxies than standard cosmology assumes.

Not entirely. Linearized EFE includes the gravitomagnetism bit that also has a pretty decent influence. See this explanation. In short, it's suppressed by a factor of 1/c^2 and would otherwise be negligible, but there's a wrench in the works since it changes the characteristics of the PDEs critically (hyperbolic to elliptic). Hence the ask for numerical relativity simulations of the whole EFE as a tiebreaker.

If numerical relativity for galactic rotation were simple, someone would have done it.

Obviously this characterization is not reasonable. All authors are top notch GR experts, this debate has been ongoing for at least 15 years, and is still not resolved to the level of clear consensus. However, among GR experts I know personally, the view of Ciotti is the one most believe.

I mean, yeah, that's the point of scientific debate right? The difference here is that the math is easy and the argument are comprehensible to anyone with PDE and perturbation theory experience. You don't need to be an expert on general relativity to work through the argument, which is unusual.

FWIW, the standard derivation of linearized gravity is very cavalier about declaring terms negligible and needs more rigor.

Ciotti's paper is correct nevertheless, see the November 2022 response that agrees but makes the point that he didn't consider relevant homogeneous solutions which matter.

 

[PeterDonis]

Provides a pretty thorough counter-argument to it IMO.

I'm not so sure. Fig. 1 in the paper you cite does show numerically different velocity profiles vs. the Newtonian ones as a function of the parameter $\lambda$, which measures the "strength" of the GEM effects, and the corrections, as the authors state, are around 10% to 15%, so not negligible. But all of those profiles have the same general shape as the Newtonian one. None of the profiles are flatter than the Newtonian one once the "peak" is reached, which is what would be required to help reduce the disconnect between the visible matter and observed rotation curves without adding dark matter to the model. Indeed, if anything they are less flat, meaning that these corrections make the problem worse, not better.

 

[PeterDonis]

Ciotti's paper is correct nevertheless, see the November 2022 response that agrees but makes the point that he didn't consider relevant homogeneous solutions which matter.

Per my post #144 just now, if these homogeneous solutions do indeed matter, it is by showing that with the GEM corrections in these solutions, the disconnect between visible matter and observed rotation curves is worse than in the Newtonian approximation, not better.

 

[PeterDonis]

https://arxiv.org/abs/2205.03091

I don't understand equation 76 in this paper, which appears to be a crucial one. This equation says that in what is called the "strong gravitomagnetic limit", the quantity $\chi$ is of order $c^0$, the same as $\gamma^2 ( - H )$. But in the general equation for the low energy expansion of $\chi$, equation 55, the leading term is of order $c^{-1}$. So I don't understand where an order $c^0$ term would come from.

 

[PeterDonis]

See this explanation. In short, it's suppressed by a factor of 1/c^2 and would otherwise be negligible, but there's a wrench in the works since it changes the characteristics of the PDEs critically (hyperbolic to elliptic).

Where is this discussed in the paper you reference here?

 

[wumbo]

Where is this discussed in the paper you reference here?

That paper is just a link to background on GEM and linearized gravity, which has qualitatively different behavior to Newtonian gravity despite being linear. It's a response to this:

since the claims in question are that nonlinear effects are much more significant (as in, orders of magnitude more significant

which is wrong. The effects are from linear equations (take a look yourself!) and need not be more significant, just big enough to violate the underlying assumptions of the perturbation expansion you do in the Newtonian limit.

The following sentence is my interpretation of the critical difference between what is normally done and what linearized GR keeps around. Poisson's equation is elliptic. The linearized GR equations are hyperbolic, which is required to preserve causality. That's something an intro PDE course covers, and an intro perturbation theory course covers the issues with doing the usual c -> infinity rule.

If you demand a PDF that goes into detail read this

 

[PeterDonis]

The effects are from linear equations

The GEM effects are. But some of the effects that Deur is claiming (and a paper by Deur was what started this thread) are not.

(take a look yourself!)

Take a look at the entire thread before snarking.

The following sentence is my interpretation

Ok. I'm not sure I agree with it, but discussion of personal interpretations is off topic. At least I'm clear now that I don't need to look in the paper itself for those claims.

If you demand a PDF that goes into detail read this

I'm not sure how this paper is relevant to what we're discussing.

 

[timmdeeg]

It is noticeable that several authors take reference to Gravitomagnetism in order to explain the observed flat galactic rotation curves, e.g.

On the gravitomagnetic origins of the anomalous flat rotation curves of spiral galaxies​

https://www.sciencedirect.com/science/article/abs/pii/S1384107618301970?via=ihub

Galactic rotation curve and dark matter according to gravitomagnetism​

https://link.springer.com/article/10.1140/epjc/s10052-021-08967-3

Galactic Dynamics in General Relativity: the Role of Gravitomagnetism
https://arxiv.org/pdf/2112.08290.pdf

On the rotation curve of disk galaxies in General Relativity​

https://arxiv.org/abs/2207.09736

while Alexandre Deur seems to be quite alone arguing the gravitational field has an energy and hence gravitates too leading to field self-interaction:

Relativistic corrections to the rotation curves of disk galaxies
https://arxiv.org/pdf/2004.05905.pdf

The reason could be that GR experts do trust Gravitomagnetism but don't trust Deur's field self-interaction which based on the heuristic that "GR and QCD have similar Lagrangians".

Now let me come to my questions: e.g. G.O.Ludwig says:

Near the origin, where the gravitational field did not build up yet, the rotation curve shows a linear rise. Farther away from the origin the rotation speed shows a transition to a nearly constant value. At large distances the gravitomagnetic field is sufficiently intense to balance the decaying gravitational and centrifugal forces. Although the relativistic effects are weak (with a beta ratio of the order of 1/2000), the nonlinear coupling provides the mechanism that drives the transition in the rotation profile.

Is the mentioned "nonlinear coupling" thing without controversy or rather an interpretation of the author?

According to Deur the field self-interaction decreases with increasing sphericity of elliptic galaxies:

An important point for the present article is that the morphology of the massive structures in which gravity may be trapped determines how effective the trapping is: the less isotropic and homogeneous a system is, the larger the trapping is. For example, this implies a correlation between the missing mass of elliptical galaxies and their ellipticities. The correlation was predicted in [4] and subsequently verified in [9].

Do the Gravitomagnetism paper predict something similar or something else which could find support by observation?

​