## Wednesday, June 10, 2009 ... /////

### Changes of dimensionful quantities are unphysical

Juliane Dalcanton asked the question whether anyone cares about the publications in Nature these days.

Some people submit their papers not only into the fast arXiv, in order to be the first ones and to share the results quickly, but they still want to use the power of journal Nature to influence a broader public which requires the papers to be embargoed before they're printed. So in reality, all experts know about the new paper (from the arXiv) but they must pretend that they don't.

I agree with Julianne that it is an ethically questionable decision to restrict the freedom of speech of the readers of the arXiv if the only possible gain is a personal gain of the authors. But I guess that the "embargoed" label cannot possibly have any legal (or moral) strength unless you have signed a contract with Nature - so everyone is free to ignore it.

Dmitry Podolsky wanted to write about certain new papers describing the observations of the most distant known object in the Universe but he had to remove the links. I am not sure whether he was legally bound to do so, but he did so, anyway.

Does journal Nature still matter? Well, I think it surely influences a broader educated public, in a similar way as major newspapers. But I don't think it is important as a resource for experts in the fields of science that I am familiar with. In those contexts, it is just a popular magazine.

The main topic: units

At Cosmic Variance, Count Iblis mentioned a very interesting story about a simple paper by Mike Duff who in 2002 wanted to correct some common myths that affect a huge part of the popular literature about physics and most science journalists.

Duff's paper was rejected by Nature but not because it was about an obvious topic that every good (in physics) high school student understands (and I surely understood them when I was 12 or so) but because two referees have independently "disagreed" with Duff, to use a polite verb for their acting as complete idiots.

Disappointingly enough, Paul Davies shared the stunning misunderstanding with the referees.

Duff's point was simple and obvious: only the values (and time variation) of dimensionless quantities have a universal physical meaning. Whether or not dimensionful quantities are changing with time depends on our choice of units (and how these units are changing with time relatively to other units). And because units depend on arbitrary social conventions, their hypothetical "change" or "constancy" has no invariant meaning.

The referees and Paul Davies couldn't possibly understand this trivial fact. They were claiming how "fundamental" it is to find out whether the speed of light was changing with time, and all this nonsense. The difference of their opinions clearly exists in the case of all dimensionful quantities, not just the speed of light.

They agreed with this point and interpreted it as a reason to think that their opinion about the "fundamental importance of constancy of the speed of light" had to be right because all of their understanding of physics would otherwise collapse. What the "universality" of the questions about the dimensionful quantities actually meant was that their knowledge of all physics had actually been collapsed already.

With different units, various quantities may be constant or variable. Let's look at kilograms, meters, and seconds. Until 1960, one second was defined as a fraction of the solar day (that was required to be 24 hours long) while centuries ago, one meter was defined as a fraction of a meridian (whose circumference was required to be 40,000 kilometers). Finally, one kilogram was defined to be approximately the weight of 1 liter (one cubed decimeter) of water under certain conditions.

As our measurements were improving, people found more accurate and reliable ways to define the units. With these new prototypes, they were able to show that their previous ideas about the length of solar days or meridians or the density of water were not quite accurate. The new, more accurate units allowed them to express the length of the solar day or the meridian or the density of water - and the figures were accurate numbers that were demonstrably different from 24 hours, 40,000 km, and 1000 kg/m^3 - constants that were previously considered "holy" or at least true by definition. Spectroscopy turned out to be the most accurate methodology to measure times and distances.

So until 1983, meters and seconds were defined as multiples of wavelengths and periods of electromagnetic waves emitted by two independent atomic transitions. That changed in 1983 when the meter was redefined to be 1/299,792,458 of a light second (measured in the vacuum). With this new definition, the speed of light is guaranteed to be constant in these units, namely 299,792,458 m/s. Period.

It should be obvious that with different units, the speed of light in the very same world could vary. For example, if one second were defined as a fraction (1/86,400) of the solar day while one meter would be defined by spectroscopy, the speed of light would fluctuate because of the irregularities (and deceleration) of the rotation of our blue planet. The fluctuations of such a value of the speed of light would tell us nothing about the deep questions about the light or the spacetime; they would only reflect mundane, low-energy facts about astronomy.

The definition that keeps "c" constant by definition can be used in any world, regardless of the time-dependence of some parameters of physics. One can impose as many choices of this kind as the number of independent units ("N") we use - for example "1=c=hbar=G=epsilon_{0}=k_{Boltzmann}=N_{Avogadro}". Let me omit candelas which are too non-fundamental, because of their desire to match the sensitivity of the human eye, and the U.S. dollar that may collapse soon under the weight of the current irresponsible U.S. socialist government and that has never been a part of the SI units, anyway.

In fact, a physicist should make these (or similar) choices, in order to remove the "N" universal quantities (or functions of time) that are unknown but that are also completely unphysical, unnecessary, and unmeasurable. At some moments in the past, "N" was higher. For example, before Joule found that heat was energy, people had to use different units for heat and for work. Here, you should imagine an essay explaining all revolutions in physics as explanations why certain constants are naturally equal to one.

The only problem that could affect - but almost certainly doesn't - the modern definition of the speed of light is that it could be ambiguous because light at different frequencies could propagate by different speeds while the definition doesn't specify the frequency. They almost certainly propagate by the same speed, because of both direct experiments and indirect theoretical arguments based on Lorentz symmetry and consistency conditions.

But even if the speed of light were frequency-dependent, we could choose a preferred frequency - e.g. the light emitted by those popular atomic transitions - and define (one meter so that we could see) the speed of this light to be a fixed value, for example c = 299,792,458 m/s. A priori, meters and seconds are independent, so it is always possible to impose one condition (in fact, two conditions) restricting their values.

Only the dimensionless parameters such as the fine-structure constant or mass ratios have an invariant meaning that doesn't depend on "social conventions". It's very obvious that extraterrestrial aliens wouldn't be using our second or our meter - for example because their planet has a different size and different speed of rotation. They would also fail to use the relation between cubed meters and kilograms because water wouldn't be their most important compound. They could also use a non-decimal, "bosonic" numeral system if they had 26 rather than 10 fingers. ;-)

If the dimensionless constants of Nature are really constant, the ratio of our units and the units used by an advanced extraterrestrial civilization would almost certainly be constant. After all, all of us could naturally use the same Planck units.

But if some dimensionless quantities were changing with time, we would have a couple of equally natural conventions telling us what should be kept fixed (while other things have to change). String theorists know such things very well. For example, distances in perturbative string theory can be measured in the string units or the Planck units (among related options); these units multiplicatively differ by a power of the string coupling constant (the exponential of the dilaton) which may be variable (however, not in the realistic, stabilized vacua).

The difference between the Planck length and string length even influences general relativity: the whole metric tensor may be redefined and multiplied by a power of the coupling constant (which is a function of spacetime coordinates). Such an operation switches us from the string frame (with distances measured in string lengths) to the Einstein frame (with distances measured in the Planck length) or vice versa. These two frames differ by the coefficient of the Einstein-Hilbert term, "R", in the action. The Einstein frame has a constant pre-factor while the string frame has exp(-2.phi) prefactor, arising from the "sphere" diagram.

These comments are no difficult subtleties of string theory; they're much simpler than that. The possibility of having several unit systems - in which some things can look constant in one system but variable in another system - can appear in much simpler models. Duff himself mentioned Planck units, Stoney units, and Schrödinger units.

Amusingly enough, Magueijo who has written lots of papers with silly titles like the "variable speed of light" was found by Duff to realize that only the change of dimensionless quantities was physical: Duff thought that Magueijo had chosen the titles to be more attractive for the dumb journalists who liked to talk about a changing speed of light - but he meant some physical changes of dimensionless quantities. That's what Duff thought until he had seen a more recent paper that proved that Magueijo misunderstood what units meant in physics, too.

So I think that Magueijo is among the people who completely misunderstand this point and a huge portion of his papers are just boasting about this rudimentary ignorance.

Unfortunately, there are more famous people who are confused. Lev Okuň wrote some texts that influenced me when I was kid. Nevertheless, in a 2001 trialogue with Duff and Veneziano, he argued that one can't say that 1=c=hbar=G because the right-hand sides are dimensionful: instead, he proposed an arrow, and similar nonsense. Well, if we use units where "1=c=hbar=G", they're manifestly dimensionless: and be damn sure that we can use such units.

Even Veneziano was extremely confused. He thought that one couldn't use "c,hbar,G" as conversion factors because they already had a different meaning (when things are O(1) in natural units, new phenomena occur). Well, this is actually the same meaning. Exactly because "hbar" measures the typical angular momentum or action where quantum mechanics fully shows its muscles, it is a natural conversion factor and the conversion may be done (it can be done even with unnatural factors). It's just not true that these two things are incompatible.

Veneziano also says that aliens (in his case, female aliens) will agree with our values of "c" and other constants. How is that possible? Well, because they had to adopt (and be told about the definition of) our units, our methodology, and our everything else. But in that case, they're no independent extraterrestrial aliens: they're just our secretaries. This is totally missing the point which is that their values of "c" will be different before they're forced to adopt our units, methods, and Christianity. ;-) After all, we don't need aliens to define new units whose ratio with the old ones may be time-dependent; as mentioned above, people on this planet have done it many times in the past.

These "controversies" are examples of situations in which my pedagogical skills are chased out by the fear of stupidity. How can sensible people possibly disagree about such simple and obvious matters? I just think that every physicist must be able to sort out these things by herself: there's no room for a "leadership" by teachers. If someone isn't able to independently understand that the "size" or "time-dependence" of a unit is a human convention, and so is inevitably any statement that depends on such units, she or he can't possibly be able to reliably figure out anything else about the real world, can she?

Indeed, I must admit, Veneziano's discovery of string theory doesn't quite fit my picture. ;-) Well, the message is that even if you are confused by simple things such as the meaning of units in physics, you may discover something as big as string theory!

#### snail feedback (1) :

I think I have to rephrase my initial question: Suppose a spinning, charged black hole (a \neq 0, q \neq 0) is taken to be a voluminous fuzz.

Using Boyer-Lindquist coordinates, would some fuzz exist at -r < 0?

It seems that the outer horizon is spherical in Boyer-Lindquist coordinates, though the angular components of the Kerr-Newman metric produce a total surface area greater than that given by the Schwarzschild angular components. e.g., 4 pi (r+^2 + a^2) instead of 4 pi r+^2.

Above and beyond this, the conversion from Boyer-Lindquist coordinates to Cartesian coordinates is inherently different than the conversion from spherical Schwarzschild to Cartesian coordinates.

The spherical Boyer-Lindquist outer horizon seems to convert into an oblate spheroid in Cartesian coordinates.

The singular point r = 0 in Boyer-Lindquist coordinates seems to convert into a ring of radius a in Cartesian coordinates. To fill in the region bound by the radius of the ring, a negative value of r = -a seems to be required in order to reach the Cartesian origin.

Is any of this correct?

I am posting this in slow comments so that it doesn't spam up the fast comments if it's unsuitable.