*This post is for the special "fools, failures and frauds" edition of The Giant's Shoulders.*

The early 20th century was clearly an exciting time to be a physicist. In 1905, Einstein published his special theory of relativity, radically revising human concepts of space and time. In the same year, and the same issue of the Annalen der Physik, Einstein really sparked the "quantum revolution" with his explanation of the photoelectric effect, an explanation that would scramble existing preconceptions of the nature of light and matter and would eventually shake the deterministic foundations of physical theory.

By the teen years of the 1900s, it must have seemed to many physicists that no idea was too crazy to possibly be true! Furthermore, the simplicity and elegance of Einstein's relativity must have suggested to scientists that the secrets of the universe remaining to be discovered would be of the same sort of "beautifully obvious" form.

One researcher who was seduced by this sort of thinking was Richard C. Tolman. In 1914, he published a paper on a new physical principle that he referred to as "the principle of similitude". In Tolman's own words, his principle represented a new form of "relativity of size", which "provides a very simple and general method for obtaining conclusions as to the form of functional relations connecting physical magnitudes."

Tolman's theory was bold, it was powerful... and it didn't really work out. It is a great example of a failed theory, and even more fascinating because its proponent was no crackpot, and its insights turned out to have some practical use in the end. There's even a whiff of a conspiracy surrounding similitude, which I will describe at the end of the post!

This post is closely related to the idea of dimensions in physics; if you're not familiar with this concept, check my earlier post here.

Richard C. Tolman (1881-1948) was still a relatively¹ young researcher when he published the principle of similitude, having earned his Ph.D. in chemical engineering from MIT in 1910. His most significant research accomplishments were in the fields of thermodynamics and statistical mechanics, though he later extended this research into relativity and cosmology. He wrote influential textbooks on both relativity and statistical mechanics, which are still available today, and he was apparently the first to describe the concept of "relativistic mass" in 1912. By 1914, he was apparently well-versed in relativity theory, and only 3 years later he would publish his first book on the subject.

So what is the principle of similitude? In his first paper², Tolman states it as follows:

The fundamental entities out of which the physical universe is constructed are of such a nature that from them a miniature universe could be constructed exactly similar in every respect to the present universe.

The hypothesis is, in essence, that there is a way to "shrink" the universe such that the change would be completely indistinguishable to a person within it! This is a quite strong statement: it implies not only that the *form* of the equations of physics would remain the same under such a transformation, but that the fundamental physical constants would also keep the same values!

Tolman argues that his principle of similitude allows one to derive the functional form of various laws of physics, based only on the idea that the equations must remain invariant under a similitude transformation.

So, how does it work? Tolman begins by imagining two universes, our current universe and a shrunken one. An observer in our universe is labeled \(O\), and a comparable observer in the shrunken universe is labeled \(O'\). Let us suppose that each observer has a meter stick from his respective universe, and the two measure the length of the same object. If \(O\) measures a length \(l\), and \(O'\) measures a length \(l'\), the relationship between the two measurements may be expressed in the form,

\(l'=xl\),

where \(x>1\) is the scale difference between our universe and the miniature one. The quantity \(l'\) is larger than \(l\) because observer \(O'\) has a shorter meterstick than \(O\).

Now things get interesting and/or weird! Tolman's principle of similitude argues that the laws of physics in two universes must be indistinguishable. That means that both observers must have exactly the *same* value for the speed of light: \(c = 3times 10^8\) meters/second. If our two observers, standing side by side, measure the speed of the same photon (light particle), they must both get the same result. This means, according to Tolman, that their units of time must also be scaled. Since the speed \(v\) of an object is defined as distance/time, this means that the two observers must have their measured intervals of time related by

\(t'=xt\).

This idea of scaled times for different observers is very reminiscent of special relativity, and this is clearly what Tolman had in mind. With this scaled time, we can say that the two observers measure the same result for velocity,,

\(v'=v\).

Though the two observers measure the same velocity, the difference in their units of time means that they will measure different values for the acceleration of an object. Acceleration has units of meters/second*second, which means that when both observers measure the acceleration of the object, their results differ by

\(a' = frac{1}{x}a\).

And at this point, Tolman was just getting warmed up! Turning to electricity, he considered the measurement of electric charge by the observers. The discreteness of electric charge had been well-established thanks to the discovery of the electron in 1896 and the Millikan oil drop experiment of 1909; because of this discreteness, Tolman hypothesized that the two observers must measure the same value of electric charge, i.e.

\(e' =e\).

With the transformation of electric charge specified, Tolman could now use Coulomb's law to specify the transformation property of mass. Coulomb's law tells us that the force between two charges \(e_1\) and \(e_2\) is proportional to the product of the charge strengths and inversely proportional to the distance \(l\) between them; using Newton's law, \(F=ma\), we may write

\(displaystyle ma = frac{e_1e_2}{l^2}\).

According to Tolman's principle of similitude, the observer \(O'\) must also find Coulomb's law obeyed, so that

\(displaystyle m'a' = frac{e_1'e_2'}{l'^2}\).

We have expressions describing the transformation of all quantities except \(m'\); substituting these expressions into the above, we find that mass must transform as

\(displaystyle m' = frac{1}{x}m\).

Some final relevant transformations derived by Tolman are for energy,

\(displaystyle E' = frac{1}{x}E\),

and temperature,

\(displaystyle T'=frac{1}{x}T\).

The energy transformation follows from the dimensions of energy -- \(Esim [m][l]^2/[t]^2\) -- and the temperature transformation follows from the equipartition theorem, that states that the average energy of a particle in thermodynamic equilibrium is proportional to temperature³.

It's hard to say what good *any* of this is supposed to do for us at this point. An astute reader will notice that this technique seems incredibly ad hoc -- there's no experimental evidence to justify the "principle of similitude", and the choice of invariant versus non-invariant quantities seems a bit of a stretch, though theoretically justified. The real question is whether "similitude" can do anything for us. Here Tolman surprises us, and shows that a few simple but nontrivial equations of physics can seemingly be derived using similitude reasoning.

The simplest of these derivations involves the ideal gas law, which states that the product of the pressure and volume of an ideal gas is proportional to the temperature of the gas. This is usually written as

\(PV = nRT\),

where \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles of gas particles, and \(R\) is the ideal gas constant. Tolman asks if we can derive the temperature dependence of \(PV\) using similitude. To demonstrate this, he assumes that the product \(PV\) has the form

\(PV = f(T)\), (1)

where \(f(T)\) is a function of \(T\) to be determined. We expect the formula should be completely unchanged in the shrunken universe, so that

\(P'V'=f(T')\).

Pressure has dimensions of force per area, and therefore

\(displaystyle P' = frac{1}{x^4}P\).

Volume has dimensions of length cubed, which means that it transforms under similitude as

\(V' = x^3V\).

If we use the preceding two expressions to rewrite the product of \(P'V'\), we get

\(PV = xf(T')\).

If \(f(T')=T'\), then we may use \(T'=T/x\) and reproduce function form of Eq. (1)! It seems that similitude results in the correct expression for the ideal gas law.

A less trivial example is the calculation of the energy density of an ideal blackbody. A blackbody is, ideally, an object that absorbs all energy that is incident upon it -- a perfect absorber. The body heats up and reradiates its energy in the form of thermal radiation. Though "perfect" blackbodies don't exist, stars and planets can be treated crudely as blackbodies, and the universe itself acts very nearly like a perfect blackbody!

Tolman considers how the density of energy (energy/volume) of a "hohlraum" (laboratory blackbody) depends on temperature. The energy density \(u=E/V\) is assumed to satisfy the relation

\(u=F(T)\),

and must satisfy the same equation in the shrunken universe,

\(u'=F(T')\).

But since \(E'=E/x\) and \(V'=x^3V\), we may readily find that

\(u=x^4F(T')\).

This implies that the energy density must be proportional to temperature to the fourth power:

\(u propto T^4\),

which is the proper result for Stefan's law!

There is one notable failure that appears in Tolman's first paper, but he attempts to make lemonade out of the lemon that is the law of gravitation. This law may be written as

\(displaystyle F = Gfrac{m_1m_2}{l^2}\), (2)

where \(m_1\) and \(m_2\) are the masses being attracted, and \(G\) is the universal gravitational constant. If we look at the equation in the shrunken universe, we have

\(displaystyle F'=Gfrac{m_1'm_2'}{l'^2}\).

This equation, however, is not invariant; converting it back into units in our universe gives,

\(displaystyle F=Gfrac{m_1m_2}{x^2l^2}\).

The equation is different from Eq. (2)! Similitude arguments don't apply to gravity.

Tolman apparently agonized over this flaw in his principle. He wrote,

Of course this absurd conclusion might merely mean that the principle of similitude is not universally true. If, however, we have accepted the principle, there are two possible solutions of the problem. In the first place gravitational action may really be proportional not to mass but to some quantity which is itself more or less accidentally proportional to mass, and which like electrical charge appears of the same magnitude both to observer \(O\) and to observer \(O'\). A second possible solution is that the attraction of gravitation does not depend merely on the masses of the attracting bodies and the distance between them, but also on the properties of some mechanism by which gravitational action is produced. The search for the true nature of gravitational action will now become an important problem of physics, and the principle of similitude will be a criterion for judging the correctness of proposed solutions.

Tolman does provide other examples where similitude seems to work, both in his original paper and in several follow-ups reaching as far as 1921^{4}; furthermore, at least one other author managed to claims to have derived known results from the principle^{5}.

However, the operative word in that sentence is "known"; Tolman's similitude seems to work reasonably well to derive results that are already known by other means, but doesn't seem to work very well in providing genuinely new insights. Furthermore, any insights it does provide would be suspect; as we have noted, there is at least one glaring example where the technique doesn't work (gravity), and any new similitude results derived would be similarly suspect.

As to why it works at all, an observant reader will note that the method is suspiciously similar to the method of dimensional analysis, which I discussed at length in a previous post. Equations in physics have to be dimensionally consistent, and must have the same fundamental quantities on both sides of the equation; that is, an equation that may in short be expressed as "velocity = velocity" is good, but "velocity = mass" is bad. Using this observation, it is possible to derive the functional behavior of certain physical quantities. For instance, one can determine a formula for the "lift" of an aircraft wing (dimensions \([M][L]/[T]^2\)) in terms of its surface area (\([L]^2\)), velocity (\([L]/[T]\)), and air density (\([M]/[L]^3\)).

Tolman's technique, which essentially "tags" different dimensional quantities with their own transformation properties, seems to be a crude and modified form of dimensional analysis. This was not lost upon other authors of Tolman's time; both Tatiana Ehrenfest-Afanassjewa^{6} (mathematician and wife of physicist Paul Ehrenfest) and Edgar Buckingham^{7} ("father" of dimensional analysis) wrote papers arguing that Tolman's effect is nothing more than this. Buckingham's paper is worth quoting; after developing his technique for solving equations by dimensional analysis, he writes:

In his article... Mr. Richard C. Tolman announces the discovery of a new principle and illustrates its value in reasoning about the forms of physical equations, by treating several examples. The statement of the principle is couched in such general terms that I have difficulty in understanding just what the postulate is, but it seems to me to be merely a particular case of the general theorem given in the foregoing section. Mr. Tolman selects length, speed, quantity of electricity, and electrostatic force as the four independent kinds of quantity which suffice for his purposes, and after subjecting them to four arbitrary conditions, he proceeds to find the conditions to which several other kinds of quantity are subject in passing from the actual universe to a miniature universe that is physically similar to it. Now I do not know whether the developments set forth above have even been published in just this form, but it is certain that they are merely consequences of the principle of dimensional homogeneity, which is far from being either new or unfamiliar. The unnecessary introduction of new postulates into physics is of doubtful advantage, and it seems to me decidedly better, from the physicist's standpoint, not to drag in either electrons or relativity when we can get on just as well without them.

This passage sums up the problem with similitude: it works in very much the same way as dimensional analysis, and can be used to derive many of the same results. The introduction of the idea as a new physical principle seems unnecessary and overly confusing. It is confusing enough to me that I am unable to decide whether it is exactly the same as dimensional analysis, or slightly modified and more limited version of it. Tolman himself wrote a response to Buckingham in which he argued that it is a fundamentally distinct concept^{8}. However, even if it is new, it is fundamentally limited in the fact that, like dimensional analysis, it can only be used to derive relatively simple physical results, all of which, as far as I can tell, can be done much more convincingly in other ways.

Though the concept didn't really take off, and seems to have been bordering on the verge of the nonsensical, Tolman's "similitude" doesn't seem to have hurt his career. As I noted, he went on to write well-respected books and was a significant researcher in relativity and cosmology. I'm actually glad to see that; the scientific world is often too unforgiving of those who sincerely make a mistake or propose a failed hypothesis, and that harsh judgment can have a stifling effect on people with new ideas. Tolman seems to have come out none the worse for wear.

Curiously, however, the term "similitude" is used to this day in engineering, in a sense closely related to that of Tolman's! When designing large, complicated structures such as ships, it helps to first construct small-scale versions to test for design flaws. This can be non-trivial, however, as the physical effects do not scale properly just by shrinking the size of the object. It is not possible, for instance, to shrink the density of water that the model will float in. By making transformations similar to those of Tolman's, however, one can design a model that possesses similar geometric (shape), kinematic (motion) and dynamic (force) behavior to the actual size object. It is unclear if the term "similitude" came from Tolman's usage; it seems more likely to have come from Buckingham and Lord Rayleigh, both of whom co-opted the term "similitude" to refer to "dimensional analysis".

There is one curious footnote to this discussion of similitude: I first came across the term while browsing the archives of the American Physical Society, and found Ehrenfest-Afanassjewa's paper. When I went to seek out Tolman's original paper on the subject, I found instead the document given at this link. (I don't think the APS will sue me for publishing this.) Even more bizarre, Tolman's 1915 paper isn't listed *at all* on the APS website.

Is this just an oversight on the APS's part? It seems like an incredible coincidence to have two of Tolman's papers on similitude unavailable, but it also seems hard to imagine that the APS would omit his work deliberately, unless someone was trying to "whitewash" Tolman's history. I don't really have an opinion on the matter, but I found the omission really odd.

Well, if you've stuck with me this long, I hope you've enjoyed this post! Describing Tolman's principle of similitude required a bit more math than I like to include in my posts, but that math hopefully illustrated how even a misguided idea can have a veneer of plausibility to it, especially when some of the math works!

***************************

¹ "Relatively"; tee hee. I crack myself up.

² R.C. Tolman, "The principle of similitude," Phys. Rev. 3 (1914), 244-255.

³ Actually, in Tolman's original paper he mistakenly treats temperature of having dimensions of energy. He gives the revised argument in a follow-up paper.

^{4} R.C. Tolman, "The specific heat of solids and the principle of similitude," Phys. Rev. 4 (1914), 145-153, R.C. Tolman, "The principle of similitude and the entropy of polyatomic gases," J. Am. Chem. Soc. 43 (1921), 866-875.

^{5} S. Karrer, "Tolman's transformation equations, the photoelectric effect and radiation pressure," Phys. Rev. 9 (1917), 290-291.

^{6} T. Ehrenfest-Afanassjewa, "On Mr. R.C. Tolman's 'Principle of similitude'," Phys. Rev. 8 (1916), 1-11.

^{7} E. Buckingham, "On physically similar systems; illustrations of the use of dimensional equations," Phys. Rev. 4 (1914), 345-376. This is in fact the classic, original paper on dimensional analysis.

^{8} R.C. Tolman, "The principle of similitude and the principle of dimensional homogeneity," Phys. Rev. 6 (1915), 219-233.

A heads-up: some of your equations and sentences are garbled (at least in my browser). From a bit after eqn 1 to a bit before your big Stefan's Law reveal the post is not followable.

"When designing large, complicated structures such as ships, it helps to first construct small-scale versions to test for design flaws. This can be non-trivial, however, as the physical effects do not scale properly just by shrinking the size of the object. It is not possible, for instance, to shrink the density of water that the model will float in. By making transformations similar to those of Tolman’s, however, one can design a model that possesses similar geometric (shape), kinematic (motion) and dynamic (force) behavior to the actual size object."

Tell that to the Mythbusters! Every time they make some sort of model to test an effect I yell at my TV, "No, those forces/masses/whatever don't scale that way!"

Thanks! I think I've fixed the LaTeX now. I'm not used to the double-$ way of typesetting LaTeX on the new blog...

Well, you

cantreat temperature as having units of energy, if you choose the size of your degree such that Boltzmann's constant equals 1. Indeed, this is just one part of the fundamental law of sufficiently theoretical physics:$k_B = c = hbar = pi = 1.$

Indeed! Though it seems that that choice of units was inconsistent with Tolman's other unit choices, as he himself seemed to think in his later arguments. In the end, it doesn't make or break his theory any more than it was already...

It's difficult to tell, is Tolman's theory equivalent to a statement that the laws of physics are scale-invariant wrt spacetime?

I would say "no"; Tolman really had something else in mind. On the other hand, you're right -- it's actually quite hard to tell exactly what Tolman was getting at, even when reading his papers on the subject! (This post gave me a bigger headache than any in recent memory.)

This does look just like dimensional analysis in the units we work with in particle physics: set c = hbar = 1, so measure length and time in inverse energy units, energy and momentum in the same units, etc. Of course, Newton's constant has dimensions 1/energy^2, which is why he ran into a problem when he arbitrarily decided not to rescale it!

Yeah, the more I look at it (and I'm actually trying not to -- I spend so much time thinking about this that the word "similitude" gives me a headache) it seems that it is really just dimensional analysis with a particular choice of units. This is more or less what Ehrenfest said, though Tolman strongly argued against it.

The engineering principle, to my understanding, is that of you do your analysis with dimensionless quantities. You choose your fundamental quantities (length, mass, charge, etc), THEN develop your relationships with by non-dimensionalizing (divide lengths by characteristic lenth, divide charge by abs(electron charge), etc). If done properly, things scale, and you get these lovely dimensionless quantities like Reynolds numbers to play with.

Per your description, Tolman's idea is not basic fundamental analysis. It is scaling in terms of a characteristic length scale, similar to the way that Lorentz is scaling in terms of velocity. Dimensional analysis asks "What combination of units match this behavior?", not "What does this look like if I change some quantity?"

Sorry about the delay in replying -- I wanted to wait until server issues were straightened out.

That seems to be the crux of it. T. Ehrenfest-Afanassjewa states,

It's a challenge to understand that from Tolman's own papers simply because he himself didn't really know what he was doing!