When I discuss with various physicists, I am surprised by the very low

awareness about Nottale’s scale relativity work, and I would like to

know if you have an explanation. First, I let me give a quick

overview, since I have to assume that readers here are not familiar

with the ideas (this was apparently not discussed much on `sci.physics.research` since

about 2002).

Nottale starts with the following observation: there is a non-

dimensionless constant **hbar** that appears in Schrödinger’s equation.

Just like **c** in Maxwell’s equation suggested that there was a

“speed” (the speed of light) that did not obey the “normal” additive

combination of speed at the time (*v = v1+v2*), the existence of hbar

suggests that Planck’s length is similarly invariant under the

“normal” multiplicative combination of scales (*s = s1 * s2*). For

special relativity, the solution was to state that c was indeed

invariant, and to use the Lorentz transform as the correct combination

law. Scale relativity, taking the log of scales, applies the same

reasoning. It postulates that Planck’s length is indeed invariant by

scaling, and that the correct law for a change of scale is (after

taking the log) similar to Lorentz (Nottale called it non-Galilean).

Nottale developed the idea quite a bit. I will refer your to his articles or his book (Fractal Space time and Microphysics).

The consequence of the postulate is that we need to give up

differentiability of space-time, which gives space-time a “fractal”

structure.

What I find particularly thought-provoking is the number of

cosmological predictions and retrodictions that this simple hypothesis

leads to:

- Large scale stellar structures (look at the pictures in section

4.4.3 of astro-ph/0310036, see also classification in astro-ph/0310031) without any need for dark matter - More than 3/4 of the anomalous Pioneer acceleration (gr-qc/0307042),
- An interesting toy model of the electron.
- Quantization of solar and extrasolar planetary systems.
- Prediction of the distribution of planets in extrasolar systems.

That last result alone, which is a true prediction (made in 1996, still verified today as far as I know), should in my opinion have received a lot of attention. But my impression is that this work is largely ignored. There is a striking contrast between string theory, which has yet to make a single firm prediction that is not flatly contradicted by experiments, but receives a lot of media coverage, and scale relativity, which made a number of pretty detailed predictions that were later confirmed, but which nobody seems to know about.

I don’t know why this is. Maybe the fractal mathematics he uses is just too bizarre and nobody wants to learn it. On the other hand, for me at least, it does not require the same gigantic leap of faith (“it works therefore it’s mathematically valid”) as, say, the renormalization group (to take something that deals with similar issues).

So, any idea why physicists do not seem interested by this approach? Is the work published in the wrong journals? Does Nottale have insufficient connections within the research community? Is the work clearly wrong but I can’t see it?

Perhaps there are vested identities and interests (conscious or otherwise) in suppressing a communicable lens that has the potential to unveil curruption and political misinformation campaigns that are involved in synthetic (having natural basis but tangential to it now), bureaucratic, hierachical power structures… well in my opinion… the “man” is going to be seen without his clothes on, like it or not. Nature will revel in its own bitter-sweet nature… go play… conspire… create… or if you feel oppressed, work to make the space for it (in a recursively respectful manner of course)! =)

“So, any idea why physicists do not seem interested by this approach? Is the work published in the wrong journals? Does Nottale have insufficient connections within the research community? Is the work clearly wrong but I can’t see it?”Maybe it’s because most physicists only like the Gillette razor and think that Ockham’s is obsolete ? Just kidding…Whatever the reasons, let’s give Laurent Nottale’s theory a little bit more visibility on the web. I’m currently working on a blog dedicated to the theory of scale relativity : The Scale Relativity CornerIt’s still under construction and there are only a few articles posted by now, but I’ll add more on a regular basis if my work, wife and sons allow me to do so… :o)

Thanks for the notification. I added a link in the sidebar.Another reason not mentioned in the article is that it looks like Nottale’s web site is hard to access from outside Europe or France. I heard several people complain that they could not download the papers. And many of his older papers are not on arXiv.

In the list of 5 scientific predictions that you attibute to the theory of Scale Relativity, there is, I think, one error in what concerns the paper which analyses the Pionner space probe anomaly. In this work Laurent Nottale does not use the Scale Relativity but the version of the General Relativity theory which incorporates the cosmological constant and satisfies so, according Nottale, the Mach Principle. In this aricle, Nottale refers to the work of J.L. Rosales and J.L. Sanchez-Gomez, eprint gr-qc/9810085 (RS) (1999) to justify that the anomaly which affect the Pionner probe does not affect the planets of the solar system. I think that the 4 others predictions listed in your article, are well related to the theory of scale relativity.

Hi Alexandre,Your reading of the articles is perfectly legitimate, since Nottale does not explicitly refer to the scale relativity theory in the article on the Pioneer anomaly. In a sense, it’s good, it means that he does not use a bigger hammer than necessary.However, I still consider that the article makes an implicit reference to scale relativity because the cosmological constant is interpreted as a manifestation of an invariant scale, an assumption that is one of the foundations of scale relativity.In other words, I see this article as using scale relativity, just as any article where c is assumed to be constant for any observer implicitly uses Einstein’s special relativity.The exact boundary remains debatable, and again, your opinion is just as reasonable as mine.

Hi Christophe,I analyse your answer and you will see that I stick to the idea to separate this article from the 4 others. It is not only a matter of opinion but mainly a question of rational analysis of the work done by Nottale. I agree that Scale Relativity is fascinating by the vision it gives of the unity of quantum and relativity theories and of the geometrical origin of the quantum concepts. But if we don’t need it to explain at least a part of the Pionner anomaly and if General Relativity is the right frame to provide this explanation, this shoud be said. In any case it is the merit of Nottale to give all these results. I answer now point by point to your message :(Chris) Your reading of the articles is perfectly legitimate, since Nottale does not explicitly refer to the scale relativity theory in the article on the Pioneer anomaly. In a sense, it’s good, it means that he does not use a bigger hammer than necessary.(Alex) Scale relativity, in its present state, does not aim to replace General Relativity which is still the only theory able to adress questions like universe shape, space expansion, Mach hypothesis . So today Scale relativity is by no means a bigger hammer than General Relativity for this kind of prediction. (Chris) However, I still consider that the article makes an implicit reference to scale relativity because the cosmological constant is interpreted as a manifestation of an invariant scale, an assumption that is one of the foundations of scale relativity.(Alex) The most general solution of the Einstein equation of the General Relativity includes the so called cosmological constant. It is the physical dimension of this constant which implies that the square of its inverse is a constant length. So there is no need of the hypothesis of the existense of an horizon of scale (invariant scale) as it is done in the scale relativity, to introduce the cosmological constant, General Relativity and its mathematics are sufficient for that. (Chris) In other words, I see this article as using scale relativity, just as any article where c is assumed to be constant for any observer implicitly uses Einstein’s special relativity.(Alex) No, the assumption c constant is not sufficient to imply implicitly the Einstein’s special relativity, for example the invariance of the speed of light was initialy explained by physicists by a contraction of the matter. We need also at least the relativity principle. So the existence of the cosmological constant and so of a constant length, in General Relativity, is not sufficient to imply the Scale Relativity. By the way Christophe, I like very much your blog which shows that you have a very large spectrum of interests and I am sure that this discussion will not end like the one with Sylvain Poirier !

Alexandre,You wrote:(Alex) No, the assumption c constant is not sufficient to imply implicitly the Einstein’s special relativity, for example the invariance of the speed of light was initialy explained by physicists by a contraction of the matter. We need also at least the relativity principle. So the existence of the cosmological constant and so of a constant length, in General Relativity, is not sufficient to imply the Scale Relativity.Of course I know that, but you misinterpreted what I wrote. My point was simply that in any modern article written by someone aware of modern physics, unless stated otherwise, the assumption that c is constant is to be interpreted as indicating that the article is written on the background of special or general relativity. Similarly, an article written by Nottale where a constant immutable scale appears is, in my opinion, written with scale relativity in the background.Unlike you, I consider this a matter of opinion more than one of “rational analysis”, because how you or I prefer to classify articles and relate them to one another is, I think, something really personal.Finally, I’d like to point out that the topic of the cosmological constant is all but settled in general relativity. Einstein himself called it his greatest blunder, and even if more precise measurements later lead to a revival of the idea, the precise value or the precise meaning of this constant remain a matter of debate to this day.In particullar, while the cosmological constant dimension is related to distance, I have not seen many articles that actually compute the equivalent of Nottale’s “cosmic length scale Lv”. Maybe you can point me to other such articles, that might explain why you have a different standpoint on this topic than I have.

Hi Christophe,(Chris) Of course I know that, but you misinterpreted what I wrote. My point was simply that in any modern article written by someone aware of modern physics, unless stated otherwise, the assumption that c is constant is to be interpreted as indicating that the article is written on the background of special or general relativity. Similarly, an article written by Nottale where a constant immutable scale appears is, in my opinion, written with scale relativity in the background.Unlike you, I consider this a matter of opinion more than one of “rational analysis”, because how you or I prefer to classify articles and relate them to one another is, I think, something really personal.(Alex) Ok, If in the meaning of “background” you incorporate the psychological state of the writer at the time he writes his article, you may be right. The mainstream of the currently accepted ideas for you and me or the personnal vision which belongs to Nottale, play certainly a big role in the way our thought is elaborated. I know that in the domain of research, the best way people communicate is often by informal discussion which does not overemphasize rigor. Moreover it seems that the great ideas have as much their origins in the worlds of emotions and dream that in the one of rationality. But the world of science is traditionally more accustomed to see the final ideas only exposed on a rational mode with an embargo on all what is certainly important but too personnal and certainly prone to misunderstandings. It come to me the example of a discussion which is related on you blog and where one of the interlocutor seems unable to stay for more of one sentence without beeing submerged by his emotions. We cannot say that such an attitude really helps an effective communication. But this is certainly an extreme case. To come back to your classification of articles, you make finally the psychological hypothesis that even if all technical tools used in the analysis of the Pionner anomaly belongs to the theory of General Relativity, it is not innocent that the main idea has arised in the context of a mind prepared by the creation and practice of Scale Relativity. This is a strong conjecture, but why not and we should ask to M. Nottale who has certainly a wise opinion on what he did (it is not sure with the Unconscious of Freud … and M. Nottale has certainly something else to do).(Chris) Finally, I’d like to point out that the topic of the cosmological constant is all but settled in general relativity. Einstein himself called it his greatest blunder, and even if more precise measurements later lead to a revival of the idea, the precise value or the precise meaning of this constant remain a matter of debate to this day.(Alex) You right, the cosmological constant has a long history and as Nottale explains, it seems that the measurement of the cosmic microwave radiation background performed by the space mission WMAP confirms its non nul value. Next space mission in preparation : Planck, should improve its knowledge. (Chris) In particullar, while the cosmological constant dimension is related to distance, I have not seen many articles that actually compute the equivalent of Nottale’s “cosmic length scale Lv”. Maybe you can point me to other such articles, that might explain why you have a different standpoint on this topic than I have.(Alex) You can find page 17 of the article ” Scale Relativity: A Fractal Matrix for Organization in Nature”, the following :“We have suggested that log-Lorentz dilation transformations were also relevant at very large scales. In this case the invariant scale becomes a maximal length-scale, invariant under dilations, that we have identied with the length-scale L = Lambda^(-1/2) that can be constructed from the cosmological constant (which is a curvature, i.e. the inverse of the square of a length). Such an identication brings new light about the nature and the value of the cosmological constant …”This article is dated June 2007 and can be found on the Web page of M. Nottale.It is clearly written that the identification of the length derived from the cosmological constant and of a large scale horizon is a suggested conjecture. I think that this conjecture is certainly justified if, even if a formal theory of a space time manifold wich combines the general scale relativity and the classical general relativity is not available, it can lead to the prediction of new phenomenas which can actualy be checked by measurement in astrophysics. Christophe, I wish you a happy new year 2008 which will bring a lot of good news from the Scale Relativity Theory.

Hi Christophe,(Chris) Of course I know that, but you misinterpreted what I wrote. My point was simply that in any modern article written by someone aware of modern physics, unless stated otherwise, the assumption that c is constant is to be interpreted as indicating that the article is written on the background of special or general relativity. Similarly, an article written by Nottale where a constant immutable scale appears is, in my opinion, written with scale relativity in the background.Unlike you, I consider this a matter of opinion more than one of “rational analysis”, because how you or I prefer to classify articles and relate them to one another is, I think, something really personal.(Alex) Ok, If in the meaning of “background” you incorporate the psychological state of the writer at the time he writes his article, you may be right. The mainstream of the currently accepted ideas for you and me or the personnal vision which belongs to Nottale, play certainly a big role in the way our thought is elaborated. I know that in the domain of research, the best way people communicate is often by informal discussion which does not overemphasize rigor. Moreover it seems that the great ideas have as much their origins in the worlds of emotions and dream that in the one of rationality. But the world of science is traditionally more accustomed to see the final ideas only exposed on a rational mode with an embargo on all what is certainly important but too personnal and certainly prone to misunderstandings. It come to me the example of a discussion which is related on you blog and where one of the interlocutor seems unable to stay for more of one sentence without beeing submerged by his emotions. We cannot say that such an attitude really helps an effective communication. But this is certainly an extreme case. To come back to your classification of articles, you make finally the psychological hypothesis that even if all technical tools used in the analysis of the Pionner anomaly belongs to the theory of General Relativity, it is not innocent that the main idea has arised in the context of a mind prepared by the creation and practice of Scale Relativity. This is a strong conjecture, but why not and we should ask to M. Nottale who has certainly a wise opinion on what he did (it is not sure with the Unconscious of Freud … and M. Nottale has certainly something else to do).(Chris) Finally, I’d like to point out that the topic of the cosmological constant is all but settled in general relativity. Einstein himself called it his greatest blunder, and even if more precise measurements later lead to a revival of the idea, the precise value or the precise meaning of this constant remain a matter of debate to this day.(Alex) You right, the cosmological constant has a long history and as Nottale explains, it seems that the measurement of the cosmic microwave radiation background performed by the space mission WMAP confirms its non nul value. Next space mission in preparation : Planck, should improve its knowledge. (Chris) In particullar, while the cosmological constant dimension is related to distance, I have not seen many articles that actually compute the equivalent of Nottale’s “cosmic length scale Lv”. Maybe you can point me to other such articles, that might explain why you have a different standpoint on this topic than I have.(Alex) You can find page 17 of the article ” Scale Relativity: A Fractal Matrix for Organization in Nature”, the following :“We have suggested that log-Lorentz dilation transformations were also relevant at very large scales. In this case the invariant scale becomes a maximal length-scale, invariant under dilations, that we have identied with the length-scale L = Lambda^(-1/2) that can be constructed from the cosmological constant (which is a curvature, i.e. the inverse of the square of a length). Such an identication brings new light about the nature and the value of the cosmological constant …”This article is dated June 2007 and can be found on the Web page of M. Nottale.It is clearly written that the identification of the length derived from the cosmological constant and of a large scale horizon is a suggested conjecture. I think that this conjecture is certainly justified if, even if a formal theory of a space time manifold wich combines the general scale relativity and the classical general relativity is not available, it can lead to the prediction of new phenomenas which can actualy be checked by measurement in astrophysics. Christophe, I wish you a happy new year 2008 which will bring a lot of good news from the Scale Relativity Theory.

Christophe,May be the best article on the subject of scale relativity and conjectures for cosmology is the revised 2006 of “Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity.” Chapter 7 Prospect, Section 7.1: “Scale Relativity and Cosmology”, pp. 283-307 (PDF). The Pdf of this article is named : LIWOS7-1cor.pdf . It is available on the WEB page of L. Nottale. The subject of the cosmological constant is deeply adressed. This very interesting article ends with the following honest warning : “We conclude this section by once more stressing the fact that the above results should be considered as tentative. A coherent scheme appears to emerge, but it remains to be demonstrated that the application of our approach to cosmology can be made consistent with the firmly established constraints of general relativity”.

Alexandre,Thanks for the pointers. But the articles you pointed me to are written by Nottale. My point is that the connection between a constant scale and the cosmological constant is nothing obvious, and is certainly unexplainable without the technical background of scale relativity. There are no observer-invariant lengths in general relativity, which is why I am surprised that you keep insisting that there is nothing more than general relativity in the article about Pioneer.Also, I was not talking about the psychological background, but the technical background. When a modern scientific article uses “c is constant” as a part of a reasoning, I don’t need to question the author’s frame of mind to know that the article uses special relativity, unless the author specifically states otherwise.So, to use a rational analysis:1) I am not aware of articles connecting a constant length to curvature that were not written by Nottale.2) This connection requires scale relativity, because a constant length makes no sense in “standard” general relativity.3) The Pioneer article refers to the Lv constant, a constant scale related to curvature.4) Therefore, the Pioneer article goes above and beyond standard general relativity, and uses a small piece of scale relativity.Again, you are entitled to a different opinion, but I think that mine is also reasonable, and that based on the data I have, it is a rational analysis. I do not need any kind of psychological analysis to reach my conclusion, but my knowledge of who actually wrote the article does reinforce the analysis above.

Alexandre,As an aside, you seem to know these topics quite well. Any papers or blog or web of yours you could point me to?ThanksChristophe

Hi Christophe,I answer your last reaction. I wish you will understand better my position.I one of my messages I wrote :”The most general solution of the Einstein equation of the General Relativity includes the so called cosmological constant”. This formulation is misleading and I should rather have written : “The most general equation for General Relativity includes the so called cosmological constant”. To give just an idea of what that means just consider that the first equation set by Einstein was : Gab = 8 *Pi *Tab where Gab is computed from the lorentz metric gab of the space-time which defines its geometry and Tab is the stress-energy tensor which caracterises the matter-energy distribution in space time and is also, as Gab, dependant on the lorentz meric gab. Now as you may be know, when this equation is solved for gab and matter-energy distribution, it describes a dynamic space-time which did not satisfied at all Einstein for who the universe had to be static. Then Einstein proposed to slighly change his equation by taking instead of the left term Gab, a linear combination of Gab and gab which gave the second equation: Gab + Lambda * gab = 8 *Pi *TabThis change, based on the introduction of the so called cosmological constant Lambda, was justified by mathematical reasons to be the most sound generalisation of the first equation. Moreover this change answered the unsatisfaction of Einstein, as with an adequate choice of the constant Lambda, the new equation accepted the gab solution of a static universe.Now it is important, Christophe, to understand that a distance between two events of the space-time, when it is defined, is imposed by the lorentz metric gab. The inverse square of the constant Lambda by itself, called invariant cosmic length-scale in the article on Pionner anomaly, does not give at all, in the theory of General Relativity, the value of any particular observed distance between two points of the universe even if it has the physical dimension of a length. Now I will answer your 4 points.(Chris)Thanks for the pointers. But the articles you pointed me to are written by Nottale. My point is that the connection between a constant scale and the cosmological constant is nothing obvious, and is certainly unexplainable without the technical background of scale relativity.(Alex) For me things are simpler :* Cosmological constant is the name of the coefficient in front of gab in the equation of Einstein.* Constant scale is the name given to the inverse of its square (it has not the meaning of an observed/real length).* The relation is : inverse of square. General Relativity is sufficient to cope with these points. There is no other special semantic to these denominations of “Cosmological Constant” or “Constant Scale” in General Relativity which would require Scale Relativity for example. To be complete, it can also be seen as a constant density of energy and negative pressure when identified to the vaccuum energy of the quantum theory. (Chris) There are no observer-invariant lengths in general relativity, which is why I am surprised that you keep insisting that there is nothing more than general relativity in the article about Pioneer.(Alex) I agree with you for the first point : the only case I know of an observer-invariant length in general relativity is in the frame of a static universe (for which the cosmological constant has been specifically introduced but which is likely not existing as being unstable). For the second point, the article on Pionner never introduces a real observed invariant length in its text. To summarize, the General Relativity accepts the wording “invariant length defined at the scale of the universe” but it would not give sense to the wording “real/observed invariant length …”. You can refer to page 3 fifth paragraph of the article: “In standard general relativity, Lambda…” to have an illustration of this point.(Chris)Also, I was not talking about the psychological background, but the technical background. When a modern scientific article uses “c is constant” as a part of a reasoning, I don’t need to question the author’s frame of mind to know that the article uses special relativity, unless the author specifically states otherwise.(Alex) Ok to let the psychological background on the side (Chris) So, to use a rational analysis:1) I am not aware of articles connecting a constant length to curvature that were not written by Nottale.(Alex) I agree if you are talking of a real/observed constant length. I disagree if you are talking of an element of the theory which has the physical dimension of a length but is not not a prediction of a length in the real/physical world (physical length prediction is made through the metric gab as written in my introduction).(Chris) 2) This connection requires scale relativity, because a constant length makes no sense in “standard” general relativity.(Alex) A constant length as the “Constant Scale” make sense within the standard General Relativity as long as it is not a length prediction made through the space metric gab. There is so no need of Scale Relativity if we are not talking of a measured/observed length.(Chris) 3) The Pioneer article refers to the Lv constant, a constant scale related to curvature.4) Therefore, the Pioneer article goes above and beyond standard general relativity, and uses a small piece of scale relativity.(Alex) Lv constant is not an observed/real length linked to the metric gab in the Pioner article. So this article does not use more than the General Relativity for its technical development and conclusion which is to explain the 3/4 of the observed speed deviation of the space probe. Sorry,I have no web page or article to refer to.

Alex wrote: I answer your last reaction. I wish you will understand better my position.I think that I understand your position. It is simply that I do not entirely subscribe to it. I know how the cosmological constant appears in General Relativity and why it was introduced. My only observation is that interpreting this as a manifestation of a constant length of nature is scale-relativistic, much like interpreting c as the actual manifestation of a constant speed (the speed of light) instead of some constant related to epsilon0 and mu0 is relativistic. It does not depend on whether in that particular document, c is an observed speed or not. Similarly, stating that Lv is an invariant length is enough for me, even if the article does not say what length this could be or how we could measure it.