Why did Galileo build his analysis of the strength of animal bone for an animal of increased weight from a scaling standpoint and not a dimensional standpoint? The scaling he used is induced by dimension? Did dimensional capacity raise issues that seemed best avoided? Did the principle of dimensional capacity raise issues irrelevant for the new science of strength of materials? The dimensional standpoint seems involved in metabolic scaling, Brownian motion, and dark energy (expanding space), the cosmic scale factor, among other modern questions. Hence the question, implied in the article:
Preprint Size, scaling, and invariant ratios
My guess is that it makes relative assessments that much easier. Using dimensional characteristics implies a renewed calculation at each particular size being assessed. Using scaling allows relative characteristics to be quickly assessed.
We use many relative characteristics in engineering EG relative density or specific gravity.
When selecting materials in engineering we commonly compare relative strength, relative density, refractive index and a host of similar characteristics to select materials quickly and efficiently.
|so if I want to judge a flea compared to an elephant, I can compare size against strength against weight to judge which animal is best for a particular use.
There's no difference, since the scaling argument is nothing more and nothing less than dimensional analysis in the case at hand.
There's no such notion as ``dimensional capacity'' however, nor does dimensional analysis imply anything about the scale of dark energy.
The existence of the cosmological constant isn't the result of a scaling argument, but of a symmetry (general coordinate invariance). Scaling implies that the solutions of Einstein's equations can only depend on a particular-dimensionless-combination of the cosmological constant and Newton's constant, whose only property that is independent of its value is its sign.
Dear Robert,
The answer to your first question
Why did Galileo build his analysis of the strength of animal bone for an animal of increased weight from a scaling standpoint and not a dimensional standpoint?
is immediate: weight is proportional to the volume ( three dimensions) while strenght on the bond is proportional to the surface ( two dimensions). Therefore if you change the scale by 10, for instance, the weight will do it 1000 while the force only 100 which means that there is not invariance on the conformal invariance.
Brownian motion is a pure effect due to the temperature effect ( (mean kinetic energy) on a small particle within a bath of particles of the same size in thermodynamic equilibrium and the dark matter nobody knows what is, less how it evolves under a change of scale of the Universe.
Ian's answer above seems correct. One primary postulate about the universe is that is self-similar in scale (fractal). Therefore, the math that applies to our scale also applies to cosmology and the small of quantum mechanics. What are very small effects at our scale may be very important at smaller scales. What is major at our scale may be only minor variation at the cosmological scale. The STOE is based on this.
Self-similarity≠dimensional analysis. What Galileo did was dimensional analysis-identifying the constants that do have dimensions and can, therefore, be used as units of measure.
Self-similarity means that there doesn't exist any dimensionful quantity and, therefore, that its choice is arbitrary and nothing physical can depend on it; whereas in Galileo's case, the identification of dimensionful quantities meant that the physics does depend on the relation of the other dimensionful quantities with the values of the units.
New dimensionful constants can't be ``guessed''-they're discovered, by studying the dynamics. The cosmological constant and Newton's constant are dimensionful quantities-their presence can't be obtained by any siimilarity argument. Same for Planck's constant. That mathematics is the framework that applies to any scale doesn't have any content, because the mathematical description does depend on the scale, through corresponding quantities.
There's no such notion as ``our scale'', unless a dimensionful constant isn't defined beforehand that defines it.
While Newton's constant, Planck's constant and the speed of light in vacuum are sufficient to define corresponding length, time and mass scales doesn't mean that something new, necessarily, occurs at those scales (indeed, the fact that these scales are independent of the cosmological constant might imply that they're missing out on something, since there's no reason to assume that Newton's constant is non-zero, but the cosmological constant vanishes); anymore than the existence of a finite value for the wavelength of a wave implies that something new happens at that scale-interference effects occur at multiples of the wavelength; but which multiple isn't singled out.
@Stam Nicolis
I agree with you in precise terms. and basic formal logic would always imply that we use dimensional analysis to recalculate where we have sufficient measurements.
However, in practice in cosmology, we can only assess the properties of stars and galaxies through the use of scale and similarity.
This has proven remarkably successful in explaining the broad sweep of stellar properties, but it has to be acknowledged that there are always problems and dissimilarities that need further explanation.
The practical application of science into engineering is fraught with exactly the same issues and, to link back to the original query, Galileo only has rough approximations to work with in the assessment of animals.
Galileo never thought that gravitation were going to permit a change of scale in the Universe and even he thought the inverse employing the effect between weight and bones resistance.
At present what we have is a map of what "could be the distribution of the dark matter in our Universe" using weak gravitational lensing. The scale changes don't play any role in this representations at all. The most important part corresponds to have a cross-correlated mass map based with maps of galaxy and cluster samples in the same dataset looking for the statistical significance. This is interesting but without any relationship with an scale physical consequence that were not known: on what scales are matter and geometry coupled by Einstein’s equations?. In standard cosmology, the FLRW metric is assumed to exactly describe the average growth of the Universe on arbitrarily large scales. However, in a generally inhomogeneous universe as described in timescape cosmology, this is no longer the case.
Dark matter follows to be nothing more than an hypothesis and what is a problem is if our gravitational models are enough for explaining the whole Universe making a comparition of such scale with ours.
Daniel Baldomir
Daniel
I would be interested to see a reference to the gravity map you mention. I was not aware that the LIGO facility (or any other for that matter) was advanced enough to create a skymap.
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