On the Principle of Similarity
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FROM CURRENT LITERATURE.
On the principle of similarity.
(Richard C. Tolman. The Principle of Similitude. Phys. Review, 3, p. 244. 1914).
The author establishes a new physical principle of the following kind: for every real physical world there is conceivable a similar miniature world in which the same general physical laws exist as in the real one. The real observer is provided with a meter, a clock, and other measuring instruments. The imaginary observer of the micro-world is provided with a shorter unit of length than the meter, a clock with an altered period, etc. If the first, measuring a physical length, finds the value \(l\), then in the second it is \(l'\), with \(l' = xl\). Similarly, for the interval of time between two events: \(t' = xt\). Having established these relations, the author shows that from them there follow a number of others, for example: for pressure \(p' = p/x^4\), for volume \(V' = x^3 V\), for absolute temperature \(T' = T/x\). The principle of similarity formulated above makes it possible to find relations between various physical quantities. For example, an ideal gas is taken. It is asked how the product of pressure by volume depends on the absolute temperature. Suppose that the general law is
\[ pV = F(T), \]
where \(F\) is an unknown function. By the principle of similarity, for the micro-world it must be
\[ p'V' = F(T'), \]
whence:
\[ pV = xF(T/x). \]
Since \(x\) is any number, the only solution of this functional equation compatible with \(pV = F(T)\) is
\[ pV = kT, \]
i.e. the Mariotte—Gay-Lussac law is obtained.
By the same method, Tolman establishes the independence of the heat capacity of an ideal gas from temperature, the Stefan–Boltzmann law, the dependence of the density of electrostatic energy on the field strength, the dependence of the electromagnetic mass of a spherical electron on its radius, and the proportionality between the magnitude of the radiation of an electron and the square of its acceleration. Wien’s law, although it cannot be derived from the principle of similarity, does not contradict it. Maxwell’s electromagnetic equations likewise agree with the principle. Newton’s law of attraction in the formulation \(f=km_1m_2/l^2\) contradicts the principle of similarity, whence the author concludes that the formulation of the law is not sufficiently general.
K. Shaposhnikov.
(Richard C. Tolman. The Specific Heat of Solids and the Principle of Similitude. Phys. Review, 4, p. 145, 1914).
The author, from the principle of similarity as he has established it, derives Debye’s well-known formula for the heat capacity of a solid or liquid elastic body, which Debye obtains from quantum theory. He defines the energy of a body as the energy of standing longitudinal and transverse waves that arise in the body at equilibrium; he indicates the functional dependence of the energy on two arguments: the frequency of oscillations and the temperature; then he expresses the various wave frequencies, according to the known formulae of the theory of elasticity, in terms of the constants: the coefficient of compression \(\chi\), Poisson’s coefficient \(\sigma\), the volume \(V\), and the density \(\rho\). One functional dependence of the energy on the temperature and these constants is obtained. The principle of similarity gives another. From comparison of the arguments of the two functions, a conclusion is drawn as to the form of the function itself. The proof of Debye’s formula proposed by the author is entirely independent of quantum theory.
K. Shaposhnikov.
(Richard C. Tolman. The Principle of Similitude and the Principle of Dimensional Homogeneity. Phys. Review 4, p. 219, 1915).
In this article the author shows that his principle of similarity is essentially distinct from the principle of dimensions. Thus the equation of state of an ideal gas and the Stefan–Boltzmann law can be obtained from the principle of similarity, but cannot be derived from the principle of dimensions. The former presupposes full relativity in the measurement of physical quantities (the Relativity of Size), whereas the latter is not connected with this. The author further points out that the application of the principle of similarity has certain limits.
K. Shaposhnikov.
(T. Ehrenfest-Afanassieva. On Mr. R. C. Tolman’s “Principle of Similitude.” Phys. Review, 8, p. 1, 1916; Richard C. Tolman. Note on the Homogeneity of Physical Equations. Phys. Review 8, p. 8, 1916).
Both authors analyze in detail the conditions which Tolman’s system of units of the micro-world must satisfy in order that, in the latter, the physical equations of the real world should be homogeneous in the dimensional sense.
K. Shaposhnikov.