# Tully-Fisher & DM via Modified Symmetries -- Arraut

• mitchell porter
mitchell porter
Gold Member
haushofer said:
For my PhD-thesis I constructed different theories of gravity by gauging the underlying Lie algebras describing the spacetime symmetries.
Do you have any comment on this proposal to change the conserved quantity in Kepler's law at large scales?

"The Tully-Fisher's law and Dark Matter effects derived via modified symmetries" by Ivan Arraut

See equation 9, contrasted with equation 6.

haushofer
mitchell porter said:
"The Tully-Fisher's law and Dark Matter effects derived via modified symmetries" by Ivan Arraut

See equation 9, contrasted with equation 6.
Arraut's eq(6) is ##L = r^2 \dot\phi## (where I'm using an overdot abbreviation for ##d/d\lambda##).

Arraut's eq(9) is ##L^2/r = \gamma^2 ##, which he says is "the new conserved quantity replacing eq(6)". He seems to think that ##\gamma## is a velocity, but it's not. (Check the dimensions.)

He then says:
Arraut said:
The result (9) is equivalent to suggest that the conserved quantity at galactic scales is not the angular momentum but rather the velocity, which is what has been perceived at galactic scales in agreement with the observations [8, 18–21].
(Refs 18-21 are Milgrom's original 1983 papers).

Arraut seems to think that (tangential) velocity being independent of radius is the same thing as "conserved". This shows that he doesn't understand what "conserved" means.

In the next paragraph he says
Arraut said:
Since the velocity is the new conserved quantity, then the expression (6) is replaced by the new conserved quantity ##\gamma = r \dot\phi##, which is incompatible with his eq(6).
which is incompatible with his eq(6).

A bit later, between eqs (11) and (12),
Arraut said:
[...] the result (11), can be only obtained if the metric at galactic scales suffers a modification on the angular part such that ##r^2 d\Omega^2 → r d\Omega^2## . This is the necessary
or the velocity to become a conserved quantity (instead of angular momentum) [...]
This is nonsense. You can't arbitrarily replace the term ##r^2 d\Omega^2## in a spherical line element ##ds^2## by ##r d\Omega^2## -- the dimensions are wrong.

More broadly, he seems to think that MOND is mainly concerned with Kepler's 2nd law (conservation of angular momentum) whereas in fact MOND is more closely related to a modification of Kepler's 3rd law.

These mistakes are sufficiently elementary that, with regret, I'm inclined to call "crackpot" on Arraut's paper.

arXiv:2310.19894

MONDified Gravity

Authors: Martin Bojowald, Erick I. Duque

strangerep said:
This is nonsense. You can't arbitrarily replace the term ##r^2 d\Omega^2## in a spherical line element ##ds^2## by ##r d\Omega^2## -- the dimensions are wrong.

More broadly, he seems to think that MOND is mainly concerned with Kepler's 2nd law (conservation of angular momentum) whereas in fact MOND is more closely related to a modification of Kepler's 3rd law.

These mistakes are sufficiently elementary that, with regret, I'm inclined to call "crackpot" on Arraut's paper.
I'll check the details later, but this is indeed very strange. You would expect some constant with a length/inverse mass dimension appear in the metric then.

I have read Arraut's paper. It is true that \gamma is not the velocity... Yet still the calculation is correct and the proposal is interesting. The replacement r^2d\Omega^2\to rd\Omega^2 is possible if a parameter multiplies r or if r now does not scale anymore as the ordinary distance. The idea might need some arrangements but it is valid...

kodama and weirdoguy
IDAG said:
I have read Arraut's paper. It is true that ##\gamma## is not the velocity...
Arraut's eq(10), i.e., ##\gamma = r d\phi/d\lambda##, sure looks like a velocity to me. although this is totally inconsistent with his eq(9).

IDAG said:
Yet still the calculation is correct
Which calculation, specifically? A mathematically inconsistent calculation cannot be "correct".

IDAG said:
The replacement ##r^2d\Omega^2\to rd\Omega^2## is possible if a parameter multiplies ##r##
That thought occurred to me too, but it's not what Arraut actually does.
[Edit: I'll analyze that a bit more in a subsequent post.]

IDAG said:
or if r now does not scale anymore as the ordinary distance.
If ##r## is not the ordinary radial coordinate, then what is it?

IDAG said:
The idea might need some arrangements but it is valid...
So,... the idea needs fixing but it's valid?? That's not a sensible statement.

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haushofer said:
You would expect some constant with a length/inverse mass dimension appear in the metric then.
Let's explore this possibility...

Arraut's proposal is that, at galactic scales, the usual angular term ##r^2 d\Omega^2## in the spherical line element becomes ##r d\Omega^2##. This is dimensionally wrong, so let us postulate a universal length scale (constant) ##{\mathcal R}## with dimensions of length, such that the angular term is ##{\mathcal R} r d\Omega^2## at large ##r##.

Now let's do a sanity check. For small ##r##, (i.e., planetary scale and below), we must have ##{\mathcal R} r \ll r^2## in order to recover familiar ##r^2 d\Omega^2## at "small" ##r##. That implies $${\mathcal R} ~\ll~ r$$in that regime, all the way down to microscopic ##r##. Hence the value of ##{\mathcal R}## must be tiny indeed, else we would see non-Newtonian dynamics at planetary scale, lab scale, and quantum scale.

But, on the scale of the outer parts of galaxies, we'd need ##{\mathcal R} r \gg r^2## in order for ##{\mathcal R} r d\Omega^2## to dominate. That implies ##{\mathcal R} \gg r## when ##r## is of the order of galactic radii.

These 2 requirements are obviously contradictory.
[Edit: Perhaps the situation is not so simple -- see my later post in this thread.]

(And besides, such focus on a length scale ignores the observational fact that the transition to deep-MOND behaviour is determined by an acceleration scale, not a length scale.)

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Well, I cannot judge because in GR there are many things which look trivial but they are not. I think that probably the author in the paper just set the parameter to one for the sake of simplicity. We do the same when we set the speed of light to one and then mix up space and time, which for some people might look inconsistent but it is not. For me that's a minor issue which does not deserve discussion. I think that Arraut refers to \gamma as the one-form of the angular velocity, in the sense of the proposed new metric. Thinking in this way, I think that the result is consistent but it is intriguing for me that he did not mention these details. It would be nice to see the nature of the parameter multiplying r (replacing r^2) and I would like to see the explanation with some presentation of the author someday...

Suppose we invent a (dimensionless) interpolation function ##f = f(r/{\mathcal R})## and postulate that the angular term in the line element looks like this: $${\mathcal R} \; f(r/{\mathcal R}) \; r \, d\Omega^2 ~,$$where the function ##f## must satisfy the following conditions: $$f ~\to~ r/{\mathcal R} ~~~~ \mbox{for}~~ r \ll {\mathcal R} ~,~~~~~~ \mbox{and}~~ f ~\to~ 1 ~~~~ \mbox{for}~~ r \gg {\mathcal R} ~.$$ Then we'd have $${\mathcal R} \; f(r/{\mathcal R}) ~\to~ r ~~~~ \mbox{for}~~ r \ll {\mathcal R} ~,~~~~~~ \mbox{and}~~{\mathcal R} \; f(r/{\mathcal R}) ~\to~ {\mathcal R} ~~~~ \mbox{for}~~ r \gg {\mathcal R} ~.$$Do functions like ##f## exist? Sure, e.g., $$f(x) ~:=~ \frac{x}{1+x} ~,~~~~ \mbox{since then:}~~~ {\mathcal R} \, f(r/{\mathcal R}) ~=~ \frac{r {\mathcal R}}{r +{\mathcal R} } ~.$$Another example:$$f(r/{\mathcal R}) ~:=~ 1 - \exp(-r/{\mathcal R}) ~,$$since for large ##r## the exponential approaches ##0##, hence ##f \to 1##. For small ##r/{\mathcal R}## the leading term in the expansion of the exponential dominates, leaving ## f \to r/{\mathcal R}##.

There are many such interpolation functions, just as there are in MOND, but one must find such a function with steep enough transition so that the effects of ##{\mathcal R}## are too small for (current) detection at solar system scale, but show up at outer galactic scale. (This still blithely ignores the fact that observations favour an acceleration scale, not a length scale. But let's continue a little longer and see what happens.)

The next task is to examine what happens in the geodesic equation with such a line element...

No, my initial instinct seems to have been correct...

Starting from a Schwarzschild-like energy Lagrangian
$$\def\Rc{{\cal R}} \def\Lc{{\cal L}} \Lc ~:=~ g_{\mu\nu} u^\mu u^\nu ~=~ -\left(1 - \frac{2m}{r} \right) c^2 \dot t^2 ~+~ \left(1 - \frac{2m}{r} \right)^{-1} \dot r^2 ~+~ \rho r (\dot\theta^2 + \sin^2\!\theta\,\dot\phi^2)~, ~~~~~~~~~ \left[\, m := \frac{GM}{c^2} \,\right] ~,$$$$\mbox{where}~~ \rho ~=~ \rho(r,\Rc) ~,~~~ ~~~\mbox{such that}~~ \rho \to r ~~\mbox{for}~ r \ll \Rc \;, ~~~\mbox{and}~~ \rho \to \Rc ~~\mbox{for}~ r \gg \Rc \;.$$The standard method for computing (timelike) geodesic orbits (circular with ##\ddot r = 0##) gives
$$0 ~=~ - c^2 \frac{m}{r^2} ~-~ \left( \frac{2m \rho}{r} + m \rho' \right) \,\dot\phi^2 ~+~ \frac12 (\rho + r \rho') \dot\phi^2 ~,$$and there's no Tully-Fisher in there (which would need a relationship between ##\dot\phi^4## and ##M## for large ##r##).

Separately, I also found another issue with Arraut's paper. On p3 where he goes from ##V_1(r)## in eq(13) to ##\nabla V_1(r)## in eq(14) he seems to avoid differentiating ##\gamma## wrt ##r##, even though his ##\gamma## is r-dependent.

OK, enough of this.

To close off this thread, a paper has just appeared on the arxiv pointing out more formally the errors in Arraut's paper.

ohwilleke and haushofer
I saw it too. It seems pretty bad.

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