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The second group contained in the classical construction of the elasticity surface belongs to a SL(2, R) type structure, and is actually represented in the original construction of the surface only by a subgroup: the subgroup of inversions. However, this structure is by no means a stranger when it comes to the problems of light. If the mathematical apparatus we create is to fit our needs, then one might say that this algebraic structure is one of the main consequences of the fact that we see the universe as we do, and test our knowledge as we test it.

Indeed, one of the most reliable criterion for the selection the so-called ‘laws of radiation’ (actually the only one we are aware of!) is Wien’s displacement law. It shows that the spectral density of energy of the blackbody radiation (u_ν), as a function of frequency (ν) and absolute temperature (T), must be of the functional form

(1)

where Φ is an arbitrary but universal function. The importance of this law in the theoretical realm was mainly manifested by the fact that the theoretical research was now focused exclusively on finding the universal function Φ, or in selection of some constants along the way of construction of this function.

There are many ways of obtaining this law but, true to the property that gives its name, – i.e. the property of displacement of the maximum of spectral density of radiation with the absolute temperature – it can be obtained by arguments of dimensional analysis (Buckingham, 1912). These arguments sweep away one of the first impressions when surveying the different methods of demonstration, namely that it might be only an approximate law. Better yet, it shows that if it is an approximate law then it is certainly due to our definition of the quantities related to blackbody radiation. This obviously raises the question of the scale of validity of the laws of blackbody radiation, for they are validated with experimental data referring to confined spaces, and are used – and not only occasionally, in latter times! – to systematize astrophysical data, which obviously refer to spaces out of our capability of confinement.

Assume therefore that the light is confined in the space of a variable container that evolves adiabatically, remaining similar to itself, in a precise sense: along this evolution all of the characteristic lengths defined by us and involved in the physics of radiation from our container, remain similar by a scale factor (Ţiţeica, 1982). This way we limit the arbitrariness of our definition of quantities involved in the process to the lengths only, which is quite a substantial gain. Assume further that the factor of similitude is the same no matter of the direction in space (we therefore have isotropy). Now, obviously everything in our container undergoes transformations induced by the deformation of the container itself. If we accept that the behavior of a quantity is dictated by its very nature, then the behavior of lengths reproduce the behavior of the very linear dimensions of the container. So if ‘κ’ is the scale factor of the lengths, we must have:

(2)

no matter what ‘l’ means physically: a geometrical length proper, a wavelength, a velocity×time, and so on. According to this philosophy, the volume of the container will go as the third power of length, so that

(3)

Now as we assume that the process of changing of the enclosure is adiabatic, one can prove that the product of the volume and the cube of absolute temperature should remain unchanged, so that:

(4)

On the other hand, we can decide the character of the frequency when undergoing such an adiabatic change of the enclosure. For this we have at our disposal the definition

(5)

where λ is the wavelength and ‘c’ is the light speed in vacuum. As in equation (5) we have explicitly a velocity, it seems that our philosophy breaks down at this point: we have to introduce time characteristics that are not at our disposal at any scale of space. Here, however, our experience helps, by the fact that it shows that the speed of light is an absolute constant. One of the meanings of this experience is by a scale invariance independent of time, like in Maxwell’s theory of light where the speed of light is calculable from fundamental properties of empty space. Therefore, the equation (5) can be read as saying something about the behavior of the frequency:

(6)

The equations (4) and (6) show then that (ν/T) is invariant with respect to the adiabatic evolution of the enclosure containing the radiation. Therefore it is always the same no matter of the space scale of the enclosure, be it the laboratory or the whole universe. On the other hand, it is easy to see that, inasmuch as the enclosure does not change energy with the external world, the density of this energy behaves like the inverse of a cubic length:

(7)

This means that (u_ν(T)/ν³) is also an invariant with respect to this type of transformation. So a universal law of radiation should be a function relating the two invariants, which is obviously the case with the Wien’s displacement law (1).

The transformation underlying the thermodynamics of light is therefore only a particular case of a general scale transformation of lengths – the homographic, or Möbius, transformation. It can be written in the form

(8)

where α, β, γ, δ are four real parameters, not all of them essential in the characterization of the homographic action of a 2´2 real matrix. Indeed, one can see that, for instance, the Wien displacement law used the transformation (8) with β = γ = 0 and κ = α/δ, while the Fresnel elasticity surface is obtained with the help of transformation for α = δ = 0 and R² = β/γ. Even in the general case such a transformation is defined by only three parameters – the so-called essential parameters – in view of the fact that it is actually a ratio between two first-degree binomials. These parameters are usually so chosen that the infinitesimal variant of the transformation appears as a second order polynomial in the variable. The necessity of such a representation will become more obvious if we bring in a classical example, by the way of illustration of a certain understanding of the fundamental issues.

This example comes naturally with the classical discovery of Galileo Galilei that the fall of bodies toward the Earth’s surface is a uniformly accelerated vertical motion. This motion is described by the well-known quadratic equation in time. However, there are facts of which we are nowadays aware, but Galileo’s epoch was not, namely that the gravitational acceleration varies with the height above Earth’s surface, and that the vertical motion is actually a radial motion toward the Earth’s center. These facts allow us to infer that, when considered to a more extended space scale, the free fall toward Earth should have a more general equation of motion. It is in view of this philosophy that we may choose to write the radial equation of motion, for instance in the ‘general’ form of the ratio of two uniform motions

(9)

Here r is the radial coordinate measured from the center of Earth. This equation may be taken as meaning that both the velocity and the initial position of the motion from the numerator vary, but they are measured by the uniform motion from the denominator. Of course, we don’t know if this equation is true or not: the two parameters of the uniform motion of the numerator can vary in some other ways. Then what, might one ask, is the reason for choosing this particular functional form as a general equation of motion? There is indeed a very good reason: what we know for sure is that we need to choose an equation of motion describing the free fall of bodies, for which the Galileo’s discovery should be only a special case dictated by the idea of ‘smallness’ of parameters related to the space scale. Indeed, the Galileo’s experiments cover a only limited expanse of the universe: that part accessible to man at that epoch, i.e. close to the Earth’s surface. Therefore the classical equation of vertical motion discovered by Galilei should appear as only an approximate variant of the true, universal we should say, equation of motion. And indeed, the equation of the accelerated vertical motion:

(10)

appears as a version of the equation (9) whereby the parameters a, b, c are ‘small’ parameters. Only this way can one say that the Galileo’s observations illustrate the situation ‘near’ the Earth’s surface, a fact brought explicitly into physics first by Newton, and later on by Einstein.

Here is the reason that compels us to consider the equation of motion (9) as a transformation of lengths. In construction of his classical system Newton used the time as defined by a uniform rectilinear motion. When referred to a particle, such a motion can have an arbitrary velocity, but in considering the time “true and mathematical time of itself” Newton felt the necessity of taking it as “duration”, disregarding the accidental momentary definition whereby time is characterized by a uniform motion. If one holds the meaning of time given by measurement though, one has to conceive of the equation (9), first and foremost, as of a transformation between two lengths: one of them represents the distance with respect to the center of Earth, the other, the accidental one, is the length defining the time. Now, our experience demands, as it always did, ‘standardization’ so to speak, which brings one to the idea of defining the time by a unique uniform motion. It is here the point where the general theory of relativity steps into scenario. It can be considered as having the direct lineage with the classical dynamical theory on many counts. Among these, the essential one is the fact that it came to realize what ‘small parameters’ mean quantitatively. Indeed, in general relativity the ‘smallness’ is judged with respect to the magnitude of the speed of light, a standard left undetermined by the classical dynamics.

The standard parameterization that makes the infinitesimal variant of (8) or (9) a second order polynomial, and which we adopt throughout this page, is the following:

(11)

This choice is justified by the fact that the equation has very important ‘standard’ properties. First of all, for the origin of the space of parameters (a¹, a², a³), we have the identity transformation x′ = x, as one can easily see. Secondly, for very small values of those parameters, a Taylor expansion to the first order gives

(12)

making the difference (x′–x) a ‘Galilei equation’. But the main point we want to make out of this approach is revealed by a Taylor expansion of a function of the variable x. In the first order it is

(13)

Here X₁, X₂, X₃ are the differential operators

(14)

satisfying the commutation relations

(15)

These reveal a SL(2, R) structure, which we will take as standard throughout present work, as we said. One can thus declare that the Galilei physics is characterized by a fuzzy sphere in the sense of John Madore (Madore, 1991) with respect to an Einsteinian theory of the three-dimensional space we are inhabiting.

Speaking of time transformation though, in a purely geometric case, the invariance properties related to the free fall are to be better illustrated with reference to a group acting in two variables: the time as well as the radial coordinate. The statistical aspect is thus somehow implicit, inasmuch as all the known results of the theory are taken nowadays as referring specifically to quantum mechanical problems (de Alfaro, Fubini, Furlan, 1976). However, it is not at all less true that this approach allows us to build a physics strictly based on classical phenomenology, for this physics is related to some known classical models. Indeed, in this case the group (1) would have, as we said, to be extended to a group acting on the time as well as on the radial coordinate. This action, abundantly used by Elie Cartan in order to illustrate his method of repère mobile, along with the previous homographic action (Cartan, 1951), is given by two equations

(16)

In the parameterization given by equation (11) the infinitesimal counterparts of these equations are

(17)

Thus, instead of (13) the Taylor expansion in the first order will be here

(18)

with the operators X_k given this time by equations

(19)

where the numerical coefficients are appropriately adjusted, in order to cope with the structure from equation (15). We have therefore to deal here with a realization of this structure in two variables: the time and the radial coordinate. This time the equation of motion should be searched for in some other direction: it is an expression of a certain invariance. Indeed, the functions which are invariant by this group are given by the solution of the equation

(20)

Therefore the most general invariant function of the group is a function of the general solution of the characteristic equation of (20):

(21)

The interpretation of these transformations does not (and actually could not) rely upon accelerated radial motion, but on the uniform radial motion. Indeed, the invariant function defined on this group carries some special meaning. It can be obtained here as a consequence of the free motion in space, which is described by the vector equation:

This means that the equation of radial component of the free motion should be described by equation

(22)

which obviously corresponds to the invariant function from equation (21). Remarkably enough, as an equation of a classical motion, besides the trivial free motion of course, this equation is also satisfied by a charged particle moving around a Poincaré magnetic pole (Poincaré, 1896). It can thus be, for instance, the direct motion of an electron plunging into a nucleus, or better yet, it can describe the accretion of matter in general, at least at a certain stage.

But for now we are merely interested in issues related to a classical phenomenology. From this point of view, we can interpret the transformation group given in equation (16). Recall that the classical time is always dictated by a uniform rectilinear motion, or by a ratio of such motions, which can explain the first transformation (16). As to the second transformation, it expresses the idea that the radial coordinate itself is also referred to such a uniform motion, the very same occurring in the transformation of time. Therefore the equation (22) offers a new interpretation of the equation of motion as an invariant function on the group.

References

Alfaro, V. de, Fubini, S., Furlan, G. (1976): Conformal Invariance in Quantum Mechanics, Il Nuovo Cimento, Vol. 34A, pp. 569–611

Buckingham, E. (1912): On the Deduction of Wien’s Displacement Law, Bulletin of the Bureau of Standards, Washington, Vol. 8, pp. 545 – 557

Cartan, E. (1951): La Théorie des Groupes Finis et Continus et la Géométrie Différentielle Traitées par la Méthode du Repère Mobile, Gauthier-Villars, Paris

Madore, J. (1991): The Fuzzy Sphere, Preprint LPHTE, Orsay 91/09; Classical and Quantum Gravity, Vol. 9, pp. 69–80 (1992)

Poincaré, H. (1896): Remarques sur une Expérience de M. Birkeland, Comptes Rendus de l’Académie des Sciences de Paris, Vol. 123, pp. 530–533

Ţiţeica, Ş., (1982): Termodynamics, Editura Academiei, Bucureşti (in Romanian)