Can Non-Differentiable Space-Time Be Described Mathematically?

[nomadreid]

(First: to the moderator: I am not sure if this should go here or in general relativity, because it spans both topics.)
Google "non-differentiable space-time" and one gets lots of interesting papers. But I am missing one essential idea that I am sure should be obvious: these papers continue to use equations and formulas which are, or are based on, differential equations. If one is investigating a non-differentiable space, how can you apply formulas that require that your space be differentiable?

 

[arkajad]

If one is investigating a non-differentiable space, how can you apply formulas that require that your space be differentiable?

If you give an example of a particular place in a particular paper that is annoying you - then, perhaps, you will likely get some help. The devil hides in the details.

 

[nomadreid]

Thanks for the reply. I will take the example of the paper "Scale Relativity and Fractal Space-Time:Applications to Quantum Physics,Cosmology and Chaotic Systems" by
L. Nottale, at luth2.obspm.fr/~luthier/nottale/arRevFST.pdf. In the first section on "Mathematical tools" he explains, roughly, that one deals with differentiable approximations to non-differentiable continuous functions. He sort of sticks to this approach in the first three sections. Then, in Section 4, he starts with the Schrödinger Equation and a number of others, justifying briefly this use by saying that quantum mechanics is statistical in nature anyway. This justification seems suspect to me. He continues with electroweak interactions in which the justification gets less clear; in the section on gravitation he uses the field equations with apparently the justification that either the effects are on the quantum level, see above, or are on too large a scale to worry about them. In section 7.2.4 onward, the justification is basically: see above, although the equations are different. We have what the Germans call die Qual der Wahl, so many choices that I am not sure which one to pick; take an example of your choice from section 4 onwards. Thanks.

 

[arkajad]

Well, it happened in the past that people were doing right things for wrong reasons. So, skip the parts with "justifications" and look for the real content, what is there that is new (if anything) that is being offered. First check if the math is right and consistent. If the math is right, then perhaps there is some new physics lurking, or perhaps not.

 

[humanino]

I personally can tell you that precisely this question (as well others) have been asked to Nottale, and his failure and/or refusal to provide an answer (in this and other cases) is the main reason people feel there is a mismatch between his professional career and his personal account in his books. Take for instance the Einstein Hoffman Infeld method. Nottale's point of view is that he must basically take it as a postulate. Many do not feel it is satisfactory, especially for a "theory" with such wide ambitions as his.

Anyway, scale relativity does not hold. It is badly broken.

 

[arkajad]

Many do not feel it is satisfactory, especially for a "theory" with such wide ambitions as his.

The truth does not depend on the number of votes. Till now you did not provide any particular error in the math. Skip the philosophy, look at the math. Either it is right or it is wrong. Moreover, quite often a wrong math can be fixed - think of Dirac's delta and the theory of distributions that came only later.

Justifications can be wrong, sometimes the proof of a theorem is wrong until someone finds the way of fixing it.

 

[humanino]

Yes, I agree that science is not a democratic process. In fact I did point to a mathematical dead-end if you read carefully. But also, I do not (and the majority does not) care about debunking Nottale's scale relativity. The regularity of the functions we use is not a physical, measurable, meaningful constraint. Whether a function is continuous or not, you can not tell, nobody can. Debunking has been done already many times (here in french). We tried to communicate with him. We asked him why he does not care to use a form of breaking of conformal symmetry, as anybody else (already) does. Frankly, Nottale is a very nice human being, but he is mostly a dreamer.

Anyway, since you like his theory so much, let us talk seriously : can you please provide for us which publication your opinion is based on ? Please keep in mind that we are talking about serious theoretical physics journals. Then we will talk about the math in those papers.

 

[arkajad]

Please keep in mind that we are talking about serious theoretical physics journals.

The quality of a particular research have nothing to do with the "seriousness" of a journal. Wrong and crappy papers have been published in serious journals, serious scientist were writing good referee reports about papers they never really read or understood, and innovative and important papers have been published in "non-serious" journals or just on internet. You need to give up your prejudices - they are good for being promoted and respected at work, but not so good in discussing the issues of science itself.

As for "debunking", either the debunking was right, and then there is no point of discussing it, or it can be debunked, then there is no point of quoting it. The devil is always in the details. Bring the details in, and they can be discussed in a scientific way.

 

[nomadreid]

Interesting discussion. First: humanino, thank you for your useful answers, especially the link: I read French, and have started going through the many links inside that link. (They could have been a bit more succinct in their expression, but that is beside the point.)

Arkajad: also thank you for your answers. I agree with you that the article is best judged on its own merits, not by the quality of the journal it is published in, but since checking the maths in an article is time-consuming, it is true that one tends to have more confidence in the editors of certain journals than others -- although there comes a blooper sometimes there too. But that is not my main point. You are right that sometimes the maths is better than its interpretation, but my question as to the justification has not to do with its interpretation, but rather the mathematics insofar as a function which is applied outside of its domain is just as false as a calculation error, even though more insidious. (The link provided by humanino provides a couple of examples of calculation errors, some serious, some not.) This is the point of mathematics that I was asking for in my question, whether his system is indeed self-consistent in postulating a domain for certain functions which they cannot have. But my question was not a rhetorical question: perhaps the domains can be extended or restricted, as the case may be, so that there is no problem, but in the article cited he seems to only hand-wave to say that they can. Of course there can be solid mathematics to back up hand-waving, as is clear to any reader of mathematics books ("it is easy to show...we leave the proof to the reader...etc.), so I was asking if someone could present me with a link where this solid mathematics exists, so that I can check it, as you suggest.

However, I am glad that I opened up this Pandora's Box with my question, because better people than me (I am not a physicist, alas) seem to have found weak points in the theory that I did not notice. I look forward to reading further in humanino's link.

 

[arkajad]

Think of poor James Clerk Maxwell. He had the idea that there is some geometro-algebraic structure in his equations. There was no math available to express this structure at that time. So he got involved with quaternions - which was not the right structure. He was messing up with quaternions for a while, with no real gain and with some truoubles. Maxwell died in 1892. A year before William Kingdon Clifford published his famous (today) „Applications of Grassmann's Extensive Algebra”. That was only the beginning, because even Clifford didn't go far enough. Only today, using Clifford algebra rather than just quaternions, using Clifford's "vectors" and "rotors", we can write all Maxwell's equations as just one equation. But it took 100 years! So, quite often people have have glimpses of some future developments and they try to apply whatever math is available for them at any given time. Quite often they err. Someone will fix the errors later. Or, the idea was wrong from the very start. You can't tell. The only thing you can is to check the math and either you find errors - then it is your duty to point them out, or you are able to fix some holes - then you sometimes you will be rewarded and sometimes you will be scolded!

 

[humanino]

You need to give up your prejudices - they are good for being promoted and respected at work, but not so good in discussing the issues of science itself.

Please keep in mind PF's guidelines

It is against our Posting Guidelines to discuss, in most of the PF forums or in blogs, new or non-mainstream theories or ideas that have not been published in professional peer-reviewed journals or are not part of current professional mainstream scientific discussion.

This is not the independent research forum. You may choose to ignore the fact that Nottale managed to publish most of his "theoretical physics" papers only in astrophysics journals. It remains an "experimental" fact which others can interpret as well.

Again, if you care to provide reference to a specific published paper, then we discuss the maths. He has plenty of interpretation, but very little math unfortunately. The debunking I provided in french for instance comes from a mathematician.

 

[arkajad]

This "debunking" has not been published in a respectable journal, therefore, according to the rules of this forum, is irrelevant.

 

[humanino]

This "debunking" has not been published in a respectable journal, therefore, according to the rules of this forum, is irrelevant.

This is true !

I did not mean to sabotage the discussion by requesting a published reference however.

 

[nomadreid]

Arkajad: As long as we are referring to history, one may refer to the fact that at first the calculations provided by the Copernican system were inferior in correlation to data to those provided by the Ptolemaic system, but the Copernican system won out because it was more elegant, and only in time proved its empirical power. The elegance lay primarily in its simplification of the mathematical framework. On the other hand, there is another kind of elegance, a theory that unifies previously distinct phenomenon, which is why Newton's or Maxwell's theories were so successful. The second kind of elegance can win out over the first one, which is why string theory, which on one side is mathematically ugly and on the other side would have great unifying power, is so appealing. It is this latter type of elegance which attracted me to the concept of fractal space-time. As you say, the next step would be to correct any of its minor mathematical faults. However, if the very base of the theory is mathematically incorrect, then one cannot just tweak the theory, but must throw it out. Whereas humanino's source was not from a scientific journal, I believe it was you that pointed out that an idea should be judged on its merits; hence that source deserves examining. My original question was whether the bases of the mathematics were valid; you do not accept the source of a negative answer; hence perhaps you have a source which provides an answer, whether positive or negative? I would still be grateful for further links on this subject (links are preferable because I do not presently have access to a good scientific library).

 

[LukeD]

In non-commutative geometry (which is a generalization of differentiable space to certain cases that would usually be considered non-differentiable), smooth geometries can be used as approximations to the non-smooth geometry.

If someone seems to be "doing their math wrong", then there is usually more math to be done. Non-differentiable geometry is current mathematical research.
In the future, people will likely have standard ways of doing these things, but they don't yet.

 

[nomadreid]

To LukeD's comment: indeed, one of the reasons I was attracted to Nottale's paper was that I know that there are people who attack this sort of thing with a lot more care, such as in Alain Connes' book "Noncommutative Geometry" (which the author has kindly put on his website www.alainconnes.org for a free download). But since Connes' careful methods are different to the ones that Nottale seems to be hand-waving about, I was and am interested in whether there is some place in which Nottale's paper's mathematical methods were properly justified and/or smoothed out. So far I have not received any positive replies, and I am wary of physical theories based on shaky or dubious mathematical foundations.
Anyway, thanks LukeD for making that point.