Scale invariance, conformal symmetry and general relativity
rlolder....@amherst.edu
fractal cosmological models, because there seems to be good empirical
justification for such a paradigm: www.amherst.edu/~rloldershaw .
I suspect that Hermann Weyl started out on the right track when he proposed
that the next major step in the special relativity-> general relativity-> ?
evolution would be fundamentally involved with the concept of scale.
He reasoned that since neither space nor time is absolute, why should scale
be absolute? He was led in the direction of conformal geometry wherein length
ratios and angles (shapes) are preserved in transformations, but there are no
absolute length scales, only relative length scales.
His original plan foundered on two problems: 1. lengths appeared to depend on
the "history" of the object, and 2. massive particles appeared to have
variable masses. His ideas evolved into gauge invariance and the rest is
history.
I think there is sufficient reason to reopen the original issue, and
especially to consider the concept of *discrete* conformal symmetries. I am
not sure if anyone else is interested in discussing the discrete self-
similarity observed in nature and its relationship to the discrete dilation
invariance of a discrete conformal symmetry. But if there are intrepid
natural philosophers out there who want to think about "something completely
different", I would welcome some discussion.
Reviewing some background material would probably make sense and I have a
very short and simple paper at www.arxiv.org/ftp/physics/papers/0701132.pdf .
My website at www.amherst.edu/~rloldershaw offers a large and varied source
of information on discrete fractal cosmology and offers considerable
empirical motivation/justification for discrete cosmological self-similarity.
I would be delighted to discuss anything even tangentially related to the
issues discussed in the paper and at the website.
Rob Oldershaw
rlolder....@amherst.edu
> I am interested in discrete self-similar models of nature, i.e., discrete
> fractal cosmological models, because there seems to be good empirical
> justification for such a paradigm: www.amherst.edu/~rloldershaw .
>
> Reviewing some background material would probably make sense and I have a
> very short and simple paper at www.arxiv.org/ftp/physics/papers/0701132.pdf
Sorry the address for the paper had a glitch.
The correct address is, I think:
http//arxiv.org/ftp/physics/papers/0701132.pdf
Rob
FrediFizzx
news:guest.20070218201911$0f...@news.killfile.org...
That link does not work either. It is better to post a link to the
abstract page when referencing to arXiv.org.
http://arxiv.org/abs/physics/0701132
Fred Diether
Christophe de Dinechin
> I am interested in discrete self-similar models of nature, i.e., discrete
> fractal cosmological models, because there seems to be good empirical
> justification for such a paradigm: www.amherst.edu/~rloldershaw .
You may be interested in the work of Laurent Nottale in that space. He started publishing about
"Scale relativity" around 1992, if I recall correctly. You can find a number of papers on arXiv, he also
published a book ("Fractal Space-time and microphysics", ISBN 978-9810208783).
> I suspect that Hermann Weyl started out on the right track when he proposed
> that the next major step in the special relativity-> general relativity-> ?
> evolution would be fundamentally involved with the concept of scale.
While I was enthused with Nottale's formulation of "scale relativity" years ago, I now believe that
describing it as "scale" is too restrictive. Instead, I now suggest focusing on the more general idea of
"relativity of measurements" (http://cc3d.free.fr/tim.pdf), scale-dependency being one aspect
among many of "measurement process dependency".
Something that this approach does not capture as well as yours, however, are "emergent properties",
what you call the fact that nature is organized in a hierarchical manner. That in itself is a "revolution"
of physics in progress, I believe. Probably too many authors to list, though ;-)
> I think there is sufficient reason to reopen the original issue, and
> especially to consider the concept of *discrete* conformal symmetries. I am
> not sure if anyone else is interested in discussing the discrete self-
> similarity observed in nature and its relationship to the discrete dilation
Count me in ;-)
--
Thanks
Christophe
rlolder....@amherst.edu
>
> That link does not work either. It is better to post a link to the
> abstract page when referencing to arXiv.org.
>
> http://arxiv.org/abs/physics/0701132
Thanks for the corrected link and the advice to stick to the Abstracts Page
when citing papers on the arXiv preprint site.
Rob
rlolder....@amherst.edu
>
> You may be interested in the work of Laurent Nottale in that space.
> focusing on the more general idea of
> "relativity of measurements" (http://cc3d.free.fr/tim.pdf),
> scale-dependency being one aspect
> among many of "measurement process dependency".
Thank you very much for your response.
I have attempted several times to understand Nottale's theory. The
introductions to papers and his book sound exciting, but once he starts
explaining the ideas I start to have trouble. Fundamental to his theory is
absolute cutoffs to nature's hierarchy on both large and small scales. I
seriously doubt the wisdom of these assumptions. Then he draws an analogy
between velocities and changing scales. I find this bizarre and
incomprehensible. Perhaps he has discovered something valuable, but it does
not fit at all with the way I see nature.
Speaking of the latter, I thought I might slowly review the fundamental
observational features I see in nature, as a prelude to more detailed
technical questions about nature's geometry. Also, I think it would be a good
idea to proceed in steps of 2-3 ideas per post so that potential readers do
not get overwhelmed by information.
I. Nature is a global hierarchy. One can classify and order objects according
to their masses. From the smallest particles at about 10^-28 g to the largest
superclusters of galaxies at about 10^52 g, we have a mass range of about 80
orders of magnitude.
II. The global hierarchy is highly discrete. Three narrow bands, comprised of
only 15 orders magnitude out of the full 80 orders of magnitude, contain
99.9999% of all observable matter in the observable universe. We can call
them the Atomic, Stellar and Galactic Scales.
Note very carefully: each one of these 5-order-of-mag Scales accounts for at
least 99.99% of all observed mass! That is correct. Each one separately
accounts for 99.99%. What does this tell you? One thing is that one cannot
have a serious cosmological paradigm that does not have a fundamentally
global and discrete hierarchical structure. This needs to be given very
considerable mulling over. It is one of nature's most important properties,
and yet most scientists do not seem to be well aware of it.
III. The global hierarchy is probably infinite. Lastly for today, I would
argue that if the global hierarchy is capped off at the "top" and "bottom",
then that puts us sort of in the middle (roll over Copernicus). Even though
it cannot be proven, I would bet my very life that the hierachy extends way
beyond our current observational limits in both the "down"
and "up" "directions". Anything less is way too anthropocentric!
Out of space/time for today; tomorrow: nature's fundamental self-similarity.
Rob
rlolder....@amherst.edu
> Out of space/time for today; tomorrow: nature's fundamental
> self-similarity.
So if:
I. nature is a global hierarchy,
II. the global hierarchy is divided into discrete Scales, and
III. the global hierarchy is probably infinite,
the obvious next question is what is the relationship between the Atomic,
Stellar and Galactic Scales, which completely dominate the observable portion
of nature's hierarchy.
IV. Discrete Cosmological Self-Similarity. Mandelbrot has shown that *within*
the different Scales there is a wealth of evidence for fractal self-
similarity. It is a hallmark of the design of almost everything in nature, as
discussed in Paper #12 of the "Selected Papers" section of
www.amherst.edu/~rloldershaw .
a. The Discrete Scales Are Self-Similar. In addition to all this self-
similarity within the Scales, it appears that the Scales themselves are
highly self-similar to one another. This would mean that each Scale is
organized in an analogous way and that for each class of system on Scale N
there is a self-similar analogue on Scale N +/- 1.
b. The Self-Similar Scale Transformation Equations. After about 8 years of
struggling to figure out how the scaling works, I found that there is one
unique scaling to nature's hierarchy. It is embodied in the following
discrete scaling equations.
R = Kr
T = Kt
M = K^Dm
where R,T and M are lengths, times and masses of anything on Scale N and
r,t,m are their self-similar counterparts on Scale N-1. The dimensionless
scaling constants are: K = 5.2 x 10^17, D = 3.174 and K^D = 1.70 x 10^56. All
other scaling properties can be derived from this length, time, mass scaling.
Details on the derivation of these scaling equations can be found in Papers
#1 and #2 of the "Selected Papers" section of www.amherst.edu/~rlolderahaw .
c. Testing the Paradigm. At www.amherst.edu/~rloldershaw there is a huge
amount of information that develops the idea of Discrete Cosmological Self-
similarity. One link called "Successful Predictions/Retrodictions" lists 33
basic observational properties (galactic radii, pulsar spin periods, ...)
that agree with the proposed scaling of the paradigm. The crucial test of the
whole paradigm is that it predicts that the galactic dark matter is in the
form of Kerr-Newman black holes with masses of 8 x 10^-5 solar masses, 0.145
solar masses and 0.58 solar masses (Stellar Scale analogues of e^-, p^+ and
He^+2) which dominate the Atomic Scale.
This highly abbreviated overview of the Self-Similar Cosmological Paradigm
will conclude tomorrow with my guess as to what it might mean in terms of
fundamental physics if we were to find that nature is an infinite global
hierarchy divided into discrete and highly self-similar Scales.
Rob
rlolder....@amherst.edu
> This highly abbreviated overview of the Self-Similar Cosmological Paradigm
> will conclude with my guess as to what it might mean in terms of
> fundamental physics if we were to find that nature is an infinite global
> hierarchy divided into discrete and highly self-similar Scales.
Say, for the sake of argument, that the key dark matter mass spectrum
prediction of the SSCP is verified by the GLAST gamma-ray experiments and/or
microlensing experiments. One would certainly want to consider the following
possibility.
V. Perhaps nature's discrete Scales have exact self-similarity and therefore
are completely equivalent (except for purely relative space-time-mass scale).
We are far into the realm of hypothesis and speculation (fantasy?), but it is
observational evidence which guided us here.
Let us consider the implications of V.
1. In nature there would ne no absolute size, time or mass scales (except
*within* individual Scales).
2. Each Scale has its own, equally valid, cgs units (*relative* magnitudes
related by the discrete self-similar transformations).
3. All Scales have the same organization and equivalent classes of
constituents.
4. The laws of physics are identical on all Scales.
5. The Principle of General Covariance must be expanded: laws of physics are
independent of arbitrary choices of coordinates or reference frames such that
the laws of physics do not depend on space, time, orientation, state of
motion, or *discrete cosmological Scale*.
6. Reductionism is a "fool's errand" (except within individual Scales).
One could call this set of ideas Discrete Scale Relativity and it would be
the ? in SR -> GR -> ?.
That ends the brief run through the conceptual paradigm. Now the difficult
question is: How does one proceed towards a mathematical formalism that
expresses the conceptual and empirical content of Discrete Scale Relativity?
This goes back to the first post in this thread, wherein the possibility of
reviving Weyl's program for introducing relativity of scale and also unifying
GR and EM. Clearly Conformal Geometry has a major role in developing a
formalism for DSR. Tomorrow I hope to explore the issue of whether one needs
discrete conformal geometry to model DSR, or whether the conformal space-time-
mass geometry of nature is continuous , but *stable* solutions only occur at
discrete multiples of L,T,M,Q, etc. values.
Feel free to jump in with advice, criticism, comments on related issues, etc.
My solitary wanderings through darkness and light have been fun, but a little
company would be most welcome at this point. Especially since I am a complete
dunce when it comes to mathematics.
Ken S. Tucker
...
> Feel free to jump in with advice, criticism, comments on related issues, etc.
> My solitary wanderings through darkness and light have been fun, but a little
> company would be most welcome at this point. Especially since I am a complete
> dunce when it comes to mathematics.
Well somewhere along the line symbolic logic
is necessary. I guess you're familiar with
Weyl's paper "Gravitation and Electricity"
(See Principle of Relativity pg.200).
Having studied your stuff for a awhile, I think
you're introducing the "quantization" of scale,
or perhaps quantization of gauge. By clicking
through terminology you might key into a path.
You may need to develope a math quite apart
from physics, somewhat like Newton developed
calculus, then applied it to physics. Maybe
query sci.math to see if something like that
has been done.
I've studied "running constants" which appear
to be scale dependant. If I find something I
will certainly post about it.
Regards
Ken
Christophe de Dinechin
> Thank you very much for your response.
>
> I have attempted several times to understand Nottale's theory. The
> introductions to papers and his book sound exciting, but once he starts
> explaining the ideas I start to have trouble. Fundamental to his theory is
> absolute cutoffs to nature's hierarchy on both large and small scales. I
> seriously doubt the wisdom of these assumptions.
By absolute cutoff, I assume you mean the existence of a "smallest scale"? The
idea is really the same as for special relativity. I hope I'm not rehashing
something that you already understood...
In STR, based on the idea that speed measurements are always relative to some
observer, a new composition rule for speed replaces the old v=v1+v2, where an
absolute cutoff speed c appears. That's actually a good thing, because such an
"absolute speed of propagation" c also appeared out of constants (epsilon0 and
mu0) in Maxwell's equations. Such an observer-independent speed was
incompatible with the pre-relativistic combination of speeds.
In scale relativity, Nottale points out that scale measurements are also
relative to one another (you measure a rod with another rod), and that by the
same reasoning, the traditional combination of scales s=s1*s2 (or taking the
log, log(s)=log(s1)+log(s2)) is not the only possible one. It is equally
possible to postulate a "non-Galilean" combination law for scales with a
cutoff (same formulation as Lorentz but in log form). The benefit is the same
as in STR, it explains why there would be an absolute length (Planck's length,
or equivalently, hbar) showing up in Schroedinger's equation.
Of course, one is tempted to point out that we can change speed easily, but we
can't grow or shrink. To clarify this point, Nottale often uses the term
"resolution" instead of "scale". So I think he's saying that, everything else
being the same, a photon with twice the energy gives us twice the resolution,
so it's like zooming in by a factor of 2. His starting point can be rephrased
doesn't that means that the Planck scale does not change when we double the
resolution?
> Then he draws an analogy
> between velocities and changing scales.
Did my short explanation clarify the rationale for the analogy? The two key
points are: scale measurements are always relative to one another, and we have
some constant that we need to explain, which relates to an absolute scale.
> I find this bizarre and
> incomprehensible. Perhaps he has discovered something valuable, but it does
> not fit at all with the way I see nature.
It is certainly surprising initially, but not more so than the existence of an
absolute speed limit must have been for physicists in 1905.
> Note very carefully: each one of these 5-order-of-mag Scales accounts for
> at least 99.99% of all observed mass! That is correct. Each one separately
> accounts for 99.99%. What does this tell you? One thing is that one cannot
> have a serious cosmological paradigm that does not have a fundamentally
> global and discrete hierarchical structure. This needs to be given very
> considerable mulling over. It is one of nature's most important properties,
> and yet most scientists do not seem to be well aware of it.
I certainly wasn't aware of it, thanks for bringing that up.
A comment about the way you phrased it: naively, if each of the orders of
magnitude accounts for 99.99% of the mass, then we have close to 500% of the
mass in the end ;-) I think you mean that if we look at the size of objects,
it "clusters" into small ranges among all the possible sizes, is that right?
> III. The global hierarchy is probably infinite.
I don't know if I convinced you with the short explanation above, but Nottale
would definitely make the opposite hypothesis, that Planck's scale or
something close to it is the theoretical limit. All I'm saying is: don't bet
your life on it just yet ;-)
--
Regards,
Christophe
rlolder....@amherst.edu
> Having studied your stuff for a awhile, I think
> you're introducing the "quantization" of scale,
> or perhaps quantization of gauge.
Yes, I think you have identified a key issue. If nature's hierarchy is
divided into discrete Scales (... Atomic, Stellar, Galactic, ...), then the
global space-time-mass geometry of nature seems to have this recursive worlds-
within-worlds property and it can perhaps be thought of in terms of
fundamental "particles" that are allowed to have quantized global scaling.
> You may need to develope a math quite apart
> from physics, somewhat like Newton developed
> calculus, then applied it to physics.
It seems to me that we are saturated with mathematical formalisms, a
veritable cornucopia. But perhaps we may need to apply existing math in ways
that have not been fully explored yet (as was the case for Riemannian
geometry and General Relativity).
Conformal geometry, with its capacity for Dilation Invariance, seems like a
promising start for exploring math applications to Discrete Scale Relativity.
Maybe the new approach will be to consider discrete dilation invariance.
Maybe the fact that conformal geometry allows fundamental "particles" to
have "variable" masses (say discrete values that are given by the DSR scaling
laws) is not a problem, but the way nature really works.
I hope to put together some preliminary comments on application of Conformal
Geometry to the Self-Similar Cosmological Paradigm in the near future.
Thanks for your comments,
Rob
rlolder....@amherst.edu
> I certainly wasn't aware of it, thanks for bringing that up.
>
> A comment about the way you phrased it: naively, if each of the orders of
> magnitude accounts for 99.99% of the mass, then we have close to 500% of
> the
> mass in the end ;-) I think you mean that if we look at the size of
> objects,
> it "clusters" into small ranges among all the possible sizes, is that
> right?
Actually each Scale accounts for more than 99% of the mass of the observable
universe and the total mass does not add up to more than 100%. That is the
nature of the global hierachical organization.
Here is how it works. Virtually all observable mass in in galaxies. Virtually
all of the observable mass of galaxies is in the form of Stellar Scale
objects. Virtually all of the observable mass of Stellar Scale objects is in
the form of Atomic Scale particles. It takes a bit of thinking to grasp the
nature of global hierarchical organization. I think it is one of the most
important, and most under-appreciated, observational facts about nature.
> > III. The global hierarchy is probably infinite.
>
> I don't know if I convinced you with the short explanation above, but
> Nottale
> would definitely make the opposite hypothesis, that Planck's scale or
> something close to it is the theoretical limit. All I'm saying is: don't
> bet
> your life on it just yet ;-)
For a completely different calculation of what I think is the correct Planck
Scale, go to www.amherst.edu/~rloldershaw and click on "Technical Notes",
then click on "A Revised Planck Scale". In a manner of speaking I have
already bet my life on this stuff; no going back now!
I will study your explanation of Nottale's reasoning and may have questions
in the future.
rlolder....@amherst.edu
> I will study your explanation of Nottale's reasoning and may have questions
> in the future.
I have now had time to read and think about your explanation of what Nottale
is proposing.
1. I understand what we are doing (and why) when we combine velocities in
both the Galilean and Lorentzian ways.
I do not understand the physical meaning of combining scales? What are we
after here and why? Can you give me a definite conceptual physical example of
what we are trying to do by "combining scales"?
v1 + v2 makes sense, as does its relativistic counterpart.
scale = L1 or L2 or T1 or T2
scale ratio = L1/L2 or T1/T2
What does s = s1*s2 mean? Why would we multiply scales?
Bottom line here: I see quite clearly the analogy he is attempting between
Lorentz transformations for velocities and analogous relativistic (and
logarithmic) transformations in "scale". The question is: does doing this
make any physical sense? How does going through the abstract mathematical
gymnastics lead us to understand something better. A simple, clear,
conceptual application of the ideas has always seemed to me to be lacking. I
need a carefully worked out example that shows how his approach provides a
better explanation or a better fit to observations. Because without that, it
looks like mostly posturing and arm-waving to me.
2. Regarding the Planck Scale "cutoff"(an abomination) to nature, I think
Nottale postulates such a cutoff as an axiom, then goes through a lot of
prestidigitation and ends up with the Planck length, and says: see nature has
a cutoff. I strongly suspect that Nottale's evidence for a Planck Scale
cutoff involves circular reasoning. I am willing to be persuaded otherwise
but only by a step-by-step logical argument. I bet Nottale cannot do this and
I am fairly sure that high-energy physicists have not done it either.
If you want to read something more meaningful, I repeat my recommendation to
go to www.amherst.edu/~rloldershaw , click on "Technical Notes" and click
on "A Revised Planck Scale". If anyone wants to discuss (no barking, thank
you) this radically revised understanding of the Planck Scale, I am at your
service.
Rob
Ken S. Tucker
> "Ken S. Tucker" <dyna...@uniserve.com> writes:
>>Having studied your stuff for a awhile, I think
>>you're introducing the "quantization" of scale,
>>or perhaps quantization of gauge.
>
>
> Yes, I think you have identified a key issue. If nature's hierarchy is
> divided into discrete Scales (... Atomic, Stellar, Galactic, ...), then the
> global space-time-mass geometry of nature seems to have this recursive worlds-
> within-worlds property and it can perhaps be thought of in terms of
> fundamental "particles" that are allowed to have quantized global scaling.
Hi Rob,
1st a comment, I quote you,
"self-evident, but often under-appreciated, fact that nature is
organized in a hierarchical manner,"
The words like "self-evident" is rather subjective,
"fact that nature", is elevating your theory/conjecture,
pre-maturely, even if it right;-).
>>You may need to develope a math quite apart
>>from physics, somewhat like Newton developed
>>calculus, then applied it to physics.
>
>
> It seems to me that we are saturated with mathematical formalisms, a
> veritable cornucopia. But perhaps we may need to apply existing math in ways
> that have not been fully explored yet (as was the case for Riemannian
> geometry and General Relativity).
Well I think you need clear principles underlying
"hierarchical manner", or clearly worded postulate(s).
You see, I'm unable to say, "you're on to something"
because you may have a "numeralogical coincidence".
For example, how did you arrive at your first few
"hierarchial" equations?
> Conformal geometry, with its capacity for Dilation Invariance, seems like a
> promising start for exploring math applications to Discrete Scale Relativity.
> Maybe the new approach will be to consider discrete dilation invariance.
> Maybe the fact that conformal geometry allows fundamental "particles" to
> have "variable" masses (say discrete values that are given by the DSR scaling
> laws) is not a problem, but the way nature really works.
>
> I hope to put together some preliminary comments on application of Conformal
> Geometry to the Self-Similar Cosmological Paradigm in the near future.
Looking forward.
> Thanks for your comments,
> Rob
Your welcome.
Ken
rlolder....@amherst.edu
Hermann Weyl, writing in the book Philosophy of Mathematics and Natural
Science, explores his thoughts on relativity, scale, symmetry groups, etc. He
is somewhat torn between his intuition, which suggests to him that scale,
like space and time, is relative and relational, and his knowledge of Atomic
Scale systems that appear to have absolute scale.
In the end, being a good natural philosopher he allows observations to carry
the day, commenting:
1. "But the facts of atomism teach us that *length is not relative but
absolute*. The atomic constants of charge and mass of the electron and
Planck's quantum of action h fix an absolute standard of length, ...".
2. 'The group of *physical* automorphisms (10 parameter Poincare group?) is
therefore smaller than the full group of possible geometric automorphisms (15
parameter conformal group), and that fact "proves conclusively that *physics
can never be reduced to geometry* as Descartes had hoped"'.
However, if the ideas [I - V] introduced in this thread are correct, then
conclusions #1 and #2 are no longer valid and the questions of relative scale
and the geometrization of physics are wide open again.
a. For the Atomic Scale there is indeed "absolute scale", but given the
infinite number of Scales implied by Discrete Scale Relativity,
the "absoluteness" is lost as soon as you start thinking of nature as a
whole, or as a large number of Scales. When we consider all of nature,
neither the charge and mass of the electron , nor h, are absolute anymore.
There are an infinite number of sets of q, m or h values. One set for each
cosmological Scale. And no set is more fundamental, real or elementary than
any of the others. The cosmological Scales are identical except for purely
relative scale.
b. It is clear that if Discrete Scale Relativity is correct, then the group
of physical automorphisms is larger than Weyl concluded. Therefore it is
possible that additional physical laws and/or phenomena might be understood
within a purely geometrical model, as was the case with gravitation. Given
the Self-Similar Cosmological Paradigm, it is far too early to rule out the
geometrization of EM, spin/torsion or anything else.
The question I will be pursuing in the near future is whether DSR says that
nature's geometry involves the full 15-parameter conformal group of
symmetries, or whether nature is better modelled by just the Poincare Group,
extended to include (discrete) Dilation Invariance.
Whether DSR makes sense or not will be decided definitively when the true
nature of the dark matter is revealed empirically.
Robert L. Oldershaw
www.amherst.edu/~rloldershaw
Ken S. Tucker
...
> There are an infinite number of sets of q, m or h values.
I've studied quite alot about the "fundamental charge q"
and "Planck's Constant h" being constant and invariant,
based on experiment and observation.
Things like atomic spectra being the same in our local
labs as they are from distant galaxy's.
What evidence do you have to suggest those values differ
on the basis of scale? (you provided some, so that's ok).
Maybe I should ask, what physically measureable invariants
and constants are true in the universe you propose?
Regards
Ken
rlolder....@amherst.edu
> Maybe I should ask, what physically measureable invariants
> and constants are true in the universe you propose?
It is important to distinguish whether we talking intra-Scale (i.e., within
one particular Scale) or inter-Scale (i.e., considering all, or at least
several, Scales).
Within a Scale like the Atomic Scale you get all the constants you are
familiar with, except that G' for the AS is not the Newtonian constant G, but
is about 10^38 times larger.
Taking the inter-Scale perspective, the things that remain invariant are
velocities (especially c) and dimensionless constants. These remain invariant
for all Scales. Also conformal transformations preserve angles and shapes, so
morphology, in general, is Scale invariant (note S, not s).
Taking the full Discrete Scale Relativity perspective, the constants on each
Scale (e.g., e, h, c, G, ...) are identical on all Scales, so long as we
remember that the *units* that they are expressed in are related by the Self-
Similar Scale transformation Equations. Or if you prefer you can keep our
conventional cgs and perform the Scale tranformation on the numerical value
of each "constant". Same scaling rules; the only difference is whether you
apply them to the "6.67 x 10^-8" or the cm^3/gsec^2.
We are used to thinking that G has one and only one value. The concept that
there are an infinite number of G values, differing on each Scale by a factor
of about 10^38, or all the same values if we scale the units instead of the
number, takes some getting used to. At www.amherst.edu/~rloldershaw I have
just added a paper on Discrete Scale Relativity as Paper #12 of the "Selected
Papers" section. It is short and goes through the argument above in a little
more detail. Discrete conformal symmetry is some pretty cool stuff and has
the potential for an unprecedented unification of physics - worth the effort,
I think.
Rob
Robert
wrote:
>
> You may be interested in the work of Laurent Nottale in that space. He started publishing about
> "Scale relativity" around 1992, if I recall correctly. You can find a number of papers on
Hi Christophe,
I am sincerely interested in exploring Nottale's ideas on scaling in a
little more depth.
In my post of Feb 23 in this thread there is a section (1) that
describes my problems with understanding what Nottale is trying to do
with the concept of scale/resolution.
I would appreciate any help you, or anyone else, might offer regarding
the questions and issues raised in that 2/23 post.
Thanks,
Rob
Christophe de Dinechin
> rlolder....@amherst.edu writes:
> 1. I understand what we are doing (and why) when we combine velocities in
> both the Galilean and Lorentzian ways.
>
> I do not understand the physical meaning of combining scales? What are we
> after here and why? Can you give me a definite conceptual physical example of
> what we are trying to do by "combining scales"?
> v1 + v2 makes sense, as does its relativistic counterpart.
>
> scale = L1 or L2 or T1 or T2
>
> scale ratio = L1/L2 or T1/T2
>
> What does s = s1*s2 mean? Why would we multiply scales?
This is, I believe, the most common problem with scale relativity, and
I tried to address it by giving the example of a photon with twice the
energy. Apparently, it did not work, so let me try another example.
Imagine that I write my physics based on a reference meter. I find for
example some laws for gravitation and electrogmagnetism that look like
F=K/r^2. I can then find a physical process that splits my reference
meter in two, and replace my meter with a reference "half". Now, where
I had "one meter", I now have "two halves". For simplicity, let's not
change the unit of forces. Then, any measurement that used to have
numeric value r now has numeric value 2r. So the law F=K/r^2 only
works if I replace K with K/4.
This process where I replace a given measurement at resolution "one
meter" with another measurement at resolution "one half" and, in doing
so, adjust laws of physics accordingly, is, I believe, what Nottale
calls scaling. Trivially, it can be generalized to replacing
measurement at resolution \lambda with measurement at resolution \mu.
The physical process corresponding to a change of resolution can
combine. I think this may be where you have trouble, so I will try to
e. For instance, in our example, the process we used was "cut the
reference solid in half". If we were using photons, the process would
be "cut the wavelength in half" which, assuming speed of propagation
does not depend on frequency, is equivalent to "double the frequency"
and, in turn, if the law E=h.nu holds, to "double the energy". Well,
physics limits permitting, nothing prevents me from cutting the
resulting reference solid in half again, or from cutting the
wavelength in half again.
Now, if you cut the reference in half, the numerical distance
measurement transforms as r'=2r. If you cut again, you transform again
r''=2r'. So, very naturally, you are led to r''=4r, where 4=2x2. This
is the meaning of s=s1*s2.
I gave above a "resolution" interpretation. Now, back to the "scale"
interpretation. If A is 2 times smaller than B, and if B is 2 times
smaller than C, then you also would say that A is 4 times smaller than
C. This is the "scale" intepretation of the s=s1*s2 law, and I believe
this is what Nottale started with.
> Bottom line here: I see quite clearly the analogy he is attempting between
> Lorentz transformations for velocities and analogous relativistic (and
> logarithmic) transformations in "scale". The question is: does doing this
> make any physical sense?
It will not make any physical sense as long as your mental model of
measurements in physics is based on them only approximating a single,
flat, continuous "reality". That's because in real numbers, if I
multiply by 2 and then again, I always multiply by 4 in the end.
Now, let me give an example to try to justify that you must give up
this ideal mental model. Start with 1 kg of lead, and 1kg of hydrogen.
They have the exact same mass, we can measure that very accurately.
Now, put together the exact same number N of kilos of hydrogen and
kilos of lead, where N is large enough to make a star. If the hydrogen
star ignites a fusion, do you still think that your two bodies have
exactly the same mass? And if not, doesn't that mean that the original
equality of mass at the 1kg range no longer holds when you multiply
each side by a few trilliions?
> How does going through the abstract mathematical
> gymnastics lead us to understand something better.
What I really liked about Nottale was not the mathematics (I actually
find a few flaws in the detailed logic, like assuming that you can
change scale without changing the definition of mass). What I really
liked was the physical insight.
>A simple, clear,
> conceptual application of the ideas has always seemed to me to be lacking. I
> need a carefully worked out example that shows how his approach provides a
> better explanation or a better fit to observations.
Look at http://luth2.obspm.fr/~luthier/nottale/ukpredic.htm for a list
of predictions and results. I think my favorite one is the prediction
of a quantization of orbitals in solar systems. As far as I know, this
was a true case of a prediction confirmed by later observations.
> Because without that, it
> looks like mostly posturing and arm-waving to me.
Without that, it *would* look like mostly posturing, yes :-)
> 2. Regarding the Planck Scale "cutoff"(an abomination)
I'm afraid the words "an abomination" indicate you aleady know the
answer. It is not an abomination, it is an hypothesis.
> to nature, I think
> Nottale postulates such a cutoff as an axiom, then goes through a lot of
> prestidigitation and ends up with the Planck length, and says: see nature has
> a cutoff.
That is not the way I read his work. Instead, Nottale postulates a law
for scale combination that he knows results in a cutoff (we know that
because Einstein, Lorentz, Poincarre used the same technique for light
speed more than a century ago). And then he proceeds to do a lot of
prestidigitation to demonstrate that this not only introduces a cutoff
(expected and hoped for), but also a new set of "fractal" properties
for space time which happen to explain a number of things as well or
better than other theories.
> I strongly suspect that Nottale's evidence for a Planck Scale
> cutoff involves circular reasoning.
Circular reasoning is always possible and necessary between equivalent
postulates (like "The speed of light c is constant" and "there is a
constant c appearing in the Maxwell equations describing light").
Nottale goes back and forth between different equivalent formulation,
that does not invalidate the reasoning as long as there is a well
defined "root postulate" anchoring the whole.
Regards
Christophe
Robert
wrote:
> >A simple, clear,
> > conceptual application of the ideas has always seemed to me to be lacking. I
> > need a carefully worked out example that shows how his approach provides a
> > better explanation or a better fit to observations.
>
> of predictions and results. I think my favorite one is the prediction
> of a quantization of orbitals in solar systems. As far as I know, this
> was a true case of a prediction confirmed by later observations.
Thanks, I appreciate your efforts to make Nottale's ideas more clear
to me. I will read this post slowly and carefully. Maybe I'll have
some further questions.
One thing that should be mentioned immediately is that the
"application" you describe above may have several big problems.
1. The Bode/Titus "Law" of planetary spacings has been known for
centuries. There have been very many numerical and theoretical
attempts to "explain" the heuristic observations of apparent
regularities in the spacings.
2. I suspect that Nottale has played with his heavy mathematical
machinery until he gets an approximate fit. To date, no one has
achieved an exact fit. If someone could do that it would be very big
news in physics/astronomy and everybody would know about it.
Maybe Nottale is on to something important, and no one is more
interested in fractals, self-similarity and extending relativity than
me so I am motivated in this direction, but his successes seem
contrived and the theoretical foundation remains opaque to me. I will
keep trying to increase my understanding of his work.
Thanks again,
Rob
Christophe de Dinechin
> 1. The Bode/Titus "Law" of planetary spacings has been known for
> centuries. There have been very many numerical and theoretical
> attempts to "explain" the heuristic observations of apparent
> regularities in the spacings.
The Bode-Titius law only applies to the solar system. What was *not*
known was that a generalized law would apply to all stellar systems.
Nottale predicted such a law, which was since then confirmed from the
first few extrasolar planetary observations.
> 2. I suspect that Nottale has played with his heavy mathematical
> machinery until he gets an approximate fit.
Why this negative bias? Skepticism is about remaining neutral, not
starting biased against.
> To date, no one has
> achieved an exact fit. If someone could do that it would be very big
> news in physics/astronomy and everybody would know about it.
Well, I think it is pretty big news. As ot why only a few know about
it... Here is my theory, sorry if I digress:
First, a lot of people have a negative reaction to anything big in
physics, something like: "you are no Einstein". Well, logically, you
don't need to be Einstein to make progress in physics, and besides,
Einstein was not that good at math initially ;-) So why do we keep
hearing that kind of argument?
If you don't mind, I will use your previous posts as an example: you
keep writing that a) you do not understand Nottale's theory (which is
fine), and b) that you suspect it's incorrect. Logically, it would
seem that you can suspect part or all of the theory is incorrect (as I
do) *because* you understand it. Conversely, it would seem logical to
reserve judgement until you understand it. But your reaction
exemplifies how the human brain actually works: for eons, it needed to
take quick decisions (should I flee) with incomplete data (I have not
positively proven that this tiger is awake), and this behavior sticks
today :-) This is why being a true skepting is so difficult. Most
people, scientists and myself included, have a strong bias against
anything that changes their previous postulates. If you want to see
*me* biased, try convincing me that QM and GTR are on solid ground :-)
Second, big progress in physics is likely to come from a serious
paradigm shift, for instance a big change in our mathematical
apparatus. Why? Because that's how it happened before: Newton,
Hamilton, GTR, QM... But that's not what most physicists will tell
you. I read recently (could not find it again, sorry) that physics was
due for a major paradigm shift, and that this would probably come from
a new form of the Hamiltonian. Well, new forms of Hamiltonian is how
QM progressed in the past 50 years, how is that a major paradigm
shift?
As you point out, Nottale's mathematics is "different" (in his case,
the heavy use of fractals). Most of us have spent a lot of time
learning some pretty darn complicated math, and are a bit reluctant
learning a completely new system just for one theory. That would be a
big waste of time if the theory turns out to be bogus. Worse yet,
being back at the student level makes it more difficult for us to
judge the validity of this or that proof, it makes reading much
slower, and so on. Overall, it's painful.
Third, the current status of physics publications does not favor
breakthrough innovation. The peer review process itself favors the
status quo. The reviewers are ordinary people, with bias towards what
they already accepted, and a new theory is not something they already
accepted. Another problem is that the review is not subjected to the
same scrutiny as the paper itself, unlike in some public fora like
Slashdot (http://www.slashdot.org). So the standard of review is not
guaranteed. By the way, did you know that Einstein was probably
subjected to peer review only once (http://www.physicstoday.org/vol-58/
iss-9/p43.html)? A lot of good science did happen without anonymous
peer review. There was a lot of non-anonymous peer review, though:
Einstein or Dirac knew who did not agree with them, and had fruitful
back and forth discussions.
This compounds with the metric by which most scientists are measured,
which is number of publications and/or number of citations. You get
much better metrics by publishing on a well known topic (favors
pubication) and if the majority is likely to agree with your
conclusions (favors citations). Guess what, people adjust their
behavior according to how they are being measured.
The only big counterweight is a social effect: either you are well
known, or you convinced someone who is well known, like a famous
teacher, or your theory reaches some critical mass to become socially
acceptable. But otherwise, the net effect of the above situation is a
very large number of relatively low key publications that nobody has
the time to read because they are so busy writing their own...
Regards
Christophe
======================================= MODERATOR'S COMMENT:
One goal of this group is to counterweight a little bit the 'lateral damages' of science industry :-)
Oh No
>If you want to see *me* biased, try convincing me that QM and GTR are
>on solid ground :-)
General relativity is on solid ground except for the affine connection,
which Einstein himself challenged in his attempts at unified field
theory. For some reason physicists in the main have ignored that and
suggested Einstein had lost it. It is, however, very difficult to see
that a sensible physical theory could be put together which does not
incorporate the general principle.
Von Neumann made valiant attempts to put QM on solid ground, but
physicists in the main seem to ignore that too. It is a pity that,
although Einstein and Von Neumann were at Princeton together there does
not seem to have been any meeting of minds, the one ignoring quantum
theory and the other ignoring relativity. In fact the common ground
between these two theories, as theories of measurement, is very solid
indeed. That which cannot be reconciled between them, the affine
connection and the Schrodinger equation as fundamental, is that which
needs to be modified if they are to be unified. The remaining parts of
the theories are both solid, and sufficient, for a resolution imv.
Regards
--
Charles Francis
moderator sci.physics.foundations.
substitute charles for NotI to email
Robert
wrote:
>
> Why this negative bias? Skepticism is about remaining neutral, not
> starting biased against.
"Skepticism is about remaining neutral"??? Surely you jest!
The way I see it, the optimum attitude for the natural philosopher, if
not the physicist, is a combination of open-mindedness and skepticism.
That is, precisely, being willing to listen to anyone's ideas with an
open mind, but preserving a reasonable level of skepticism (in the
standard definition) until the ideas survive the gamuts of (1) logical
consistency, (2) retrodictive testing and (3) genuine predictive
testing.
>
>
> Well, I think it is pretty big news. As ot why only a few know about
> it... Here is my theory, sorry if I digress:
Your comments make me think that you do not sufficiently distinguish
retrodictions and the much more rare definitive prediction/testing of
truly excellent science.
Do I want to discuss Nottale's work further? Not unless we discuss
specific predictions and specific results, in reasonable detail.
Robert L. Oldershaw
www.amherst.edu/~rloldershaw
Christophe de Dinechin
> On Feb 28, 5:19 am, "Christophe de Dinechin" <christo...@dinechin.org>
> wrote:
>
>
>
> > Why this negative bias? Skepticism is about remaining neutral, not
> > starting biased against.
>
> "Skepticism is about remaining neutral"??? Surely you jest!
Skeptic: a person inclined to question or doubt *all* accepted
opinions (emphasis on "all" is mine). As soon as you start with a
bias, i.e. you do not doubt each proposition equally, you are no
longer truly a skeptic.This is particularly true if you tend to doubt
"new" opinions more than "accepted" ones. I did not mean to imply more
nor less.
You are right that the word has taken a much more negative meaning
now, maybe "a person inclined to question or doubt other's opinions".
But this is an unfortunate use of the word.
> > Well, I think it is pretty big news. As ot why only a few know about
> > it... Here is my theory, sorry if I digress:
>
> Your comments make me think that you do not sufficiently distinguish
> retrodictions and the much more rare definitive prediction/testing of
> truly excellent science.
I gave the link http://luth2.obspm.fr/~luthier/nottale/ukpredic.htm,
and indicated that my favorite result was the prediction of the
structure of stellar systems. See http://luth2.obspm.fr/~luthier/nottale/ukresult.htm
for details. The prediction was made first in 1996, the data
confirming it became available only years later (Nottale uses data
collected as of 2005 on the page). This is not a retrodiction.
Naturally, good science begins with retrodictions as a sanity test. So
it's good that Nottale lists also a good body of retrodictions. But
that's not what makes the work excellent. I believe my standards for
"excellent science" are close to yours.
Regards
Christophe
Robert
wrote:
>
> Naturally, good science begins with retrodictions as a sanity test. So
> it's good that Nottale lists also a good body of retrodictions. But
> that's not what makes the work excellent. I believe my standards for
> "excellent science" are close to yours.
Yes you are probably right, and we are highly motivated to understand
nature.
Therefore, if you spend as much time studying my paradigm at
www.amherst.edu/~rloldershaw as you have devoted to Nottale's website,
then two things will happen.
1. You will have a much better feeling for whether Nottale's Scale
Relativity or Discrete Scale Relativity is a better paradigm for how
nature is actually organized and works.
2. I can promise you that if you understand Discrete Scale Relativity,
you will experience a vision of nature that is, so-to-speak, the
Mandelbrot Set times a googol.
When you, or anybody else, wants to talk about specifics, I am ready
and waiting.
Robert L. Oldershaw
www.amherst.edu/~rloldershaw
Christophe de Dinechin
> Therefore, if you spend as much time studying my paradigm atwww.amherst.edu/~rloldershawas you have devoted to Nottale's website, then two things will happen.
Rest assured that I already did spend some time trying to understand
your approach.
> 1. You will have a much better feeling for whether Nottale's Scale
> Relativity or Discrete Scale Relativity is a better paradigm for how
> nature is actually organized and works.
The two approaches are very different. Very broadly, your approach is
empirical (we observe some self-similarity, is it a general pattern?),
whereas Nottale's is much more aximoatic (what if scale covariance was
written in a non-galilean way?).
Let me take an example to illustrate. Both Nottale and yourself find a
similarity between a stellar system and an atomic system. But you do
it by postulated similarity (e.g.if red dwarfs behave like Helium
atoms, then this law we observe between isotopes of Helium should
apply), Nottale computes it as a result of equations describing large-
scale chaotic systems (see http://luth2.obspm.fr/~luthier/nottale/arA&A322.pdf).
This might, in the end, be two facets of a same intuition. Nottale's
equations might end up predicting exactly the similarity you
postulate. What is the best paradigm: the observation that apples and
oranges fall at the same speed, or Newton's law of gravitation? Both
are two sides of the same coin.
> 2. I can promise you that if you understand Discrete Scale Relativity,
> you will experience a vision of nature that is, so-to-speak, the
> Mandelbrot Set times a googol.
That is also a pretty good description of Nottale's fractal space-
time, where the dimension varies with the resolution you use to look
at it :-)
> When you, or anybody else, wants to talk about specifics, I am ready
> and waiting.
At this point, my first question remains unanswered. Why postulate
that "The SSCP proposes that nature's hierarchy extends far beyond our
current observational limits on both large and small scales, and is
probably unbounded in terms of scale" (from your "Main concepts"
page), when 1) we cannot prove it, and 2) the fact that it's actually
unbounded seems to play very little role in your reasoning.
Another way to phrase the question: how many of your retrodictions are
invalidated if the "probably unbounded in terms of scale" phrase is
removed?
Christophe
Robert
wrote:
>
> That is also a pretty good description of Nottale's fractal space-
> time, where the dimension varies with the resolution you use to look
> at it :-)
>
> that "The SSCP proposes that nature's hierarchy extends far beyond our
> current observational limits on both large and small scales, and is
> probably unbounded in terms of scale" (from your "Main concepts"
> page), when 1) we cannot prove it, and 2) the fact that it's actually
> unbounded seems to play very little role in your reasoning.
>
> Another way to phrase the question: how many of your retrodictions are
> invalidated if the "probably unbounded in terms of scale" phrase is
> removed?
I can answer these related questions together. The Discrete Fractal
Paradigm (aka Self-similar Cosmological Paradigm, found at
www.amherst.edu/~rloldershaw ) does not require that nature's
hierarchy be infinite, nor are any of its successful predictions or
retrodictions nullified by going to a finite hierarchy paradigm.
However, when you go to Discrete Scale Relativity, which is an
interpretation of the empirical evidence for the SSCP, and which
proposes that the Scales are not only self-similar, but further that
they are fully equivalent (except for purely relative scale), then
nature's hierarchy must be infinite. The reasoning is that only for an
infinite hierarchy can you have exact self-similarity among analogues
on different Scales. The proof involves a one-to-one matching of the
internal Scales of any given system and its analogue on the next lower
Scale, in something vaguely resembling Cantor's diagonal proof of the
infinity of real numbers.
Nottale's paradigm, because of its anthropocentric cutoffs, involves
at best merely an infinitessimal segment of the the full infinite
hierarchy of nature.
My arguments for an infinite hierarchy are mostly natural philosophy:
cutoffs violate the spirit of Copernicus/Kepler/Galileo/Bruno/ N. de
Cusa/etc.; an infinite hierarchy offers far more symmetry; a truncated
hierarchy is exceedingly ugly, and an infinite hierarchy makes
Discrete Scale Relativity possible.
When the concept of an infinite, discrete self-similar hierarchy first
occurred to me on 12/21/76 my first reaction was: of course, this
means a formal relativity of the discrete cosmological Scales. For
many years I did not dare but hint at that possibility since it is
really fairly shocking when you first consider the natural
consequences and full implications. It also guaranteed rejection
letters. My initial reticence has vanished. I feel that anyone who
truly understands Discrete Scale Relativity will realize that it is a
unique and worthy successor to General Relativity and that it is the
right paradigm for our next big step forward in natural philosophy.
However, if you would like nature to give you an unambiguous empirical
verdict on the SSCP, then feel free to wait for the dark matter test
to be completed. Those who understand the SSCP have known since about
1985 what the dark matter is. I expect convincing results within 0.7 -
5 years, via GLAST and microlensing experiments. Much remains to be
done, and surely many subtleties remain to be discovered (like exactly
how EM fits into the paradigm). But I feel very confident that if we
could suddenly fast-forward 20 years, DSR would be our cosmological
paradigm.
Robert L. Oldershaw
www.amherst.edu/~rloldershaw
Robert
wrote:
> acceptable. But otherwise, the net effect of the above situation is a
> very large number of relatively low key publications that nobody has
> the time to read because they are so busy writing their own...
I think you have identified some real problems that make physics seem
like the proverbial Tower of Babel, wherein progress is difficult
because everybody is speaking in 'different languages' and no one is
listening to anyone else anyway.
This probably has been a problem throughout the history of science (it
took the authority of Max Planck and one or two other phyicists to get
the physics community to seriously consider special relativity;
initially 99% of the community preferred that it would quietly go
away).
The question that really interests me is whether the situation has
markedly deteriorated in recent decades to the point where physics is
getting hard to distinguish from New Age babble (e.g., string theory
and "landscapes"), or whether science always just *seems* to be
bumbling toward chaos, but is always saved by self-correcting
mechanisms.
My guess is that both scenarios mentioned above are partially true,
that T.S. Kuhn had it down nearly right, and that we are on the brink
of a major paradigm-shift. That is why the babble has reached a
cacophony. If this assessment is true, temporary sanity and coherence
will return in the near future with some type of new and unifying
paradigm that represents a major break from the past.
Ever the optimist,
Robert L. Oldershaw
www.amherst.edu/~rloldershaw
Robert
>
I think that I have come up with an interesting preliminary and
conceptual answer to my geometry problem: how to geometrically model a
discrete fractal cosmos as discussued at www.amherst.edu/~rloldershaw
.
1. John Baez on a sci.physics.research thread (tutorial on symmetry)
once commented that Diffeomorphism Invariance (aka General Covariance
as in GR) is the ultimate symmetry (the largest symmetry group
relevant to physics).
2. If you extend Diffeomorphism Invariance to include relativity of
scale, as seems to be required in an infinite discrete self-similar
cosmos, then Diffeomorphism Invariance now includes Dilation
Invariance (no more absolute units).
3. The "vacuum" equations (source-free) of EM and GR have the *full*
Diffeomorphism Invariance, now including relativity of scale.
4. Now you ask, where does the discrete Scales part come in? And here
is the jist of the possible answer. When you solve the coupled EM+GR
equations (allowing for relativity of scale), then the *stable*
solutions will form a discrete hierarchy. This is the general way in
which nature's hierarchy is divided into discrete cosmological Scales,
and the individual Scales are further divided into discrete
hierarchical families of *stable* particles and systems.
5. The classical fractal phenomena are mostly in evidence in the
InterScale domains.
6. Conformal geometry is neat stuff, and fun to think about, and the
conformal invariance of EM and GR pointed the way to relativity of
scale, but it is Diffeomorphism Invariance (extended to include
complete relativity of scale [i.e., units]) that rules in this
paradigm.
The above run-through of the proposed answer is abbreviated and still
somewhat tentative, but my intuition tells me we are getting very
close to the goal of a unified cosmological paradigm that would
explain all domains of physics with one unified theoretical framework.
Thanks for your help in this quest,