The Relativity of Size
Richard C. Tolman's Principle of Similitude
2013 November 11
Readers of this blog may recall that in June it featured The Relativity of Space by Henri Poincaré. One of the many arguments advanced in that 1906 paper -- and that for which it is best remembered -- is the claim that space exhibits what is now called "dilational invariance". In Poincaré's words:
"Suppose that in the night all the dimensions of the universe become a thousand times greater; the world will have remained similar to itself, (giving to the word similitude the same meaning as in Euclid, Book VI). Only what was a meter long will measure thenceforth a kilometer, what was a millimeter long will become a meter. The bed whereon I lie and my body itself will be enlarged in the same proportion. When I awake to-morrow morning, what sensation shall I feel in the presence of such an astounding transformation? I shall perceive nothing at all. The most precise measurements will be incapable of revealing to me anything of this immense convulsion, since the measures I use will have varied precisely in the same proportion as the objects I seek to measure."
Many early specialists in relativity agreed, few with more enthusiasm than Richard Chace Tolman. In 1914 Tolman was just 33 years old, his greatest contributions to cosmology and statistical mechanics (and his tutelage of Linus Pauling, the beginning of American dominance in quantum chemistry) still in the future. His attention was focused largely on special relativity -- in 1916 he would become one of the several physicists who realised that the existence of tachyons might permit construction of a time machine. The question of the day was how the special theory could be made less special -- by generalisation to include gravity, but perhaps in other ways. Perhaps along the lines suggested by Poincaré ...
THE PRINCIPLE OF SIMILITUDE
by Richard C. Tolman
[Physical Review 3, 244 (1914)]
Tolman's words are in bold.
The purpose of the following article is to present some considerations which appear to have validity throughout the field of physical science. Our conclusions will all be drawn from a single postulate which we shall call the principle of similitude. This new principle may be stated 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.
For particular kinds of dynamical systems a somewhat similar hypothesis was advanced by Newton [ Principia, lib. II., prop. 32] but we shall see that any complete development of the consequences of our postulate is dependent both on a knowledge of the electron theory and the theory of relativity.
We shall find that our principle provides a very simple and general method for obtaining conclusions as to the form of functional relations connecting physical magnitudes. As examples of the method we shall first deduce a number of relations in various branches of physics which will be found in agreement with those that can be obtained by more specific methods of attack. We shall then use the principle for obtaining a conclusion as to the nature of gravitational action. In a later article we shall use the principle of similitude in deriving a formula for the specific heat of homogeneous isotropic elastic solids. In the future we may regard the principle of similitude at least as a temporary criterion for the correctness of physical theories which may be advanced.
In order to derive the desired conclusions from our postulate let us consider two observers, O and O′, provided with instruments for making physical measurements. O is provided with ordinary meter sticks, clocks and other measuring apparatus of the kind and size which we now possess, and makes measurements in our present physical universe. O′, however, is provided with a shorter meter stick, and correspondingly altered clocks and other apparatus so that he could make measurements in the miniature universe of which we have spoken, and in accordance with our postulate obtain exactly the same numerical results in all his experiments as O does in the analogous measurements made in the real universe.
UNITS OF LENGTH.
Let the meter stick used by O′ be shorter than that used by O in the ratio 1 : x. Hence if O and O′ should both measure the same given distance they would find for it respectively the lengths l and l′ connected by the relation [Eq. 1]
l′ = xl
UNITS OF TIME.
We may next inquire as to the units of time employed by the two observers. Since by our postulate we could construct for O′ a universe which would appear to him exactly the same as the actual universe does to O, it is obvious that the velocity of light in free space must measure the same both for O and O′.
The possibility of making the velocity of light appear the same to O′ as to O by filling O′'s space with an ether having different properties from that of O being excluded, (since in accordance with the theory of relativity free space contains no ether), the only way in which this can be possible is for O′ to use a shorter unit of time than does O, shorter in the ratio of 1 : x. This shorter unit of time will then exactly compensate for the shorter meter stick, and O′ will thus obtain the same numerical value for the velocity of light as does O. If now O and O′ should both measure the same interval of time they would find for it respectively the number of seconds t and t′ connected by the relation [Eq. 2]
t′ = xt
UNITS OF VELOCITY AND ACCELERATION.
Since we have seen that O′ uses units of length and time both of which are shorter than those of O in the ratio 1 : x, it is evident that any given velocity will appear the same both to O and to O′ giving us the relation, [Eq. 3],
v′ = v
Furthermore, since acceleration has the dimensions [l][t]-2 it is evident that their measurements of a given acceleration will be connected by the relation [Eq. 4]
a′ = a / x
UNITS OF ELECTRICITY.
We must next inquire as to the relation between measurements of a given electrical charge as made by the two observers. In accordance with the electron theory we may accept the general principle that electricity is not a continuum but exists in definite amounts, each elementary charge being that of the electron. From this fundamental point of view the proper way to measure an electrical charge is to count the number of electrons which it contains, the fundamental unit of electricity will be the charge of the electron, and the magnitude of a charge will be expressed as an integral number.
Now it is evident that if O and O′ should examine the same electrical charge and count the number of electrons which it contains they would necessarily arrive at the same result, and hence if e and e′ are their values for the magnitude of a given charge, we shall have the relation [Eq. 5]
e′ = e
UNITS OF MASS.
In order to obtain a relation between the units of mass employed by O and O′ we may consider how a simple electrostatic experiment would appear to the two observers. Consider two charges of electricity e1 and e2 placed on suitable bodies and separated by a considerable distance l. From Coulomb's law the force between the charges is e1e2/l2, and hence if we release one of the bodies which has the mass m it will obtain the acceleration a as given by the following equation, [Eq. 6]:
ma = e1e2/l2
We may suppose the quantities in this equation to have been measured by observer O. If O′, however, should also observe the same experiment, it is evident from the principle of similitude that he too would have to find Coulomb's law obeyed and would obtain the relation [Eq. 7]
m′a′ = e′1e′2/l′2
We have, however, transformation equations for all the quantities in this equation except m′. These equations, Nos. (1), (4) and (5), give us on substitution the relation [Eq. 8]
m′(a/x) = e1e2/(xl′)2
Combining with equation (6) we obtain the desired relation between the measurements of a given mass as made by the two observers, namely [Eq. 9]:
m′ = m/x
FURTHER TRANSFORMATION EQUATIONS.
We have now obtained transformation equations for the fundamental magnitudes, length, time and mass, and can hence obtain a whole series of further equations for force, temperature, etc., by merely considering the dimensions of the quantity in question.
Since force has the dimensions [m][l][t]-2 we shall obtain the transformation equation [Eq. 10]
f ′ = f/x
Energy and absolute temperature both have the dimensions of [m][l2][t]-2, leading to the transformation equations [Eq. 11]
E′ = E/x
and [Eq. 12]
T′ = T/x
It should be pointed out that the transformation equation for energy has the same form as for mass, which agrees with the requirements of the theory of relativity, which has made mass and energy identical.
For area we shall evidently obtain the transformation equation, [Eq. 13],
S′ = x2S
For volume, [Eq. 14],
V′ = x3V
For pressure, [Eq. 15],
P′ = P/x4
For density of mass or energy, [Eq. 16],
u′ = u/x4
For frequency, Eq. 17],
ν′ = ν/x
DETERMINATION OF THE FORM OF FUNCTIONAL RELATIONS FROM THE THEORY OF SIMILITUDE.
Having obtained the above transformation equations for physical measurements made by the two observers O and O′, we may make use of them for finding the necessary form of a number of relations between physical magnitudes. Our general method of procedure will be to consider some construct which could exist either in the actual universe or in the miniature universe which to observer O′ appears the same as the actual universe. It is evident from the principal of similitude that the properties of this construct will have to obey the same general laws, whether measured by observer O or by observer O′ while a further condition will be imposed upon the magnitude of these measurable properties by the transformation equations which we have just developed. These two sets of conditions will permit the attainment of definite information as to the necessary form of the functional relation connecting the measurements of different properties of the construct.
THE PROPERTIES OF AN IDEAL GAS.
Let us first consider an ideal gas made up of rigid, elastic, material particles. It is obvious that such a construct would appear to be an ideal gas both to observer O and to observer O′, although in accordance with equation (9) the mass of each particle would appear to O′ to be m′ = m/x, where m is the mass as it appears to O.
The Law of Charles. --- Suppose now we are interested in the way in which the pressure-volume product of such a gas would vary with the temperature, we have, [Eq. 18],
PV = F(T)
where F(T) is the unknown function whose form we wish to determine. If there is a general law connecting the pressure-volume product and the temperature of an ideal gas, it is evident from the principle of similitude that this law must also apply to measurements made by observer O′, and hence we shall also have
P′V′ = F(T′)
where F is the same function as in the previous equation. Substituting for the accented letters their values as given by transformation equations, Nos. (15), (14) and (12), we obtain
PV/x = F(T/x)
and combining with equation (18) we have
F(T) = xF(T/x)
Since x may be any number the only solution of this functional equation is F(T) = kT where k is some constant which leads to the relation
PV = kT
In other words, we have derived from the principle of similitude the law of Charles for an ideal gas.
The Specific Heat of an Ideal Gas. --- Let us suppose that the energy of such an ideal gas is dependent merely on the temperature. We have, [Eq. 19],
E = F(T)
and from the principle of similitude
E′ = F(T′)
Substituting transformation equations (11) and (12) we have,
E/x = F(T/x)
and combining with (19) we have,
E = F(T) = xF(T/x)
a functional equation for which the only solution is,
E = kT
where k is some constant.
This proves that the energy content of such an ideal gas is proportional to its temperature, or that the specific heat is independent of the temperature, a relation which is known to hold for gases which can be considered as composed of elastic, rigid, material particles.
THE PROPERTIES OF THE HOHLRAUM.
A hohlraum [i.e. blackbody] is another construct which would obviously appear as such both to observer O and observer O′, although in accordance with our transformation equations its temperature and the frequency of the radiation which it contains would appear different to the two observers.
The Energy Density in Thermodynamic Equilibrium. --- Consider for example a hohlraum which is in thermodynamic equilibrium; we may determine the law connecting the energy density and the temperature. We have, [Eq. 20],
u = F(T)
where F is the unknown function whose form we wish to determine. And from the principle of similitude we also have,
u′ = F(T′)
But from the transformation equations (16) and (12) we obtain.
u/x4 = F(T/x)
Combining with (21) we have,
u = x4F(T/x) = F(T)
and the only solution for this functional equation is
u = aT4
where a is some constant. Hence we see that the principle of similitude has led to Stefan's law for the energy density of a hohlraum.
Distribution of Radiation. --- It is also possible to obtain from the principle of similitude some information concerning the distribution of energy among the different wave-lengths. Let uν = du/dν be the rate of change of the energy density with the frequency. We have, [Eq. 21],
uν = du/dν = F(ν, T)
and from the principle of similitude,
u′ν′ = du′/dν′ = F(ν′, T′)
Substituting transformation equations (16), (17) and (12) we obtain,
uν/x3 = F(ν/x, T/x)
uν = F(ν, T) = x3F(ν/x, T/x)
Unfortunately, this functional equation has no unique solution; it is important to point out, however, that a particular solution of our equation is the functional relation
uν = F(ν, T) = ν3F(T/ν)
which Wien has shown to be a necessary condition for any radiation equation.
THE PROPERTIES OF THE ELECTROMAGNETIC FIELD.
The principle of similitude leads to simple proofs of a number of important relations in the theory of electromagnetism.
Energy Density of an Electrostatic Field. --- Suppose, for example, we wish to determine how the density of the energy u in an electrostatic field depends upon the field strength E. We have, [Eq. 22],
u = F(E)
and from the principle of similitude,
u′ = F(E′)
Now the field strength E has the dimensions of force per unit charge so that by applying transformation equations (16), (10) and (5) we obtain
u / x4 = F(e / x2)
and by combining with equation (22) we have,
u = F(E) = x4 F(e / x2)
for which the only solution is
u = k E2
where k is a constant. In other words, the energy density of an electrostatic field is proportional to the square of the field strength. By similar methods we could show that the energy density of a magnetic field is proportional to the square of the magnetic field strength.
There are many other electromagnetic relations upon which light is thrown by the theory of similitude. We shall content ourselves, however, by pointing out that the five fundamental equations of electromagnetic theory
Curl H = 4πk + dE/ct
Curl E = - dH/ct
div E = 4πσ
div H = 0
F = E + (v/c) × H
are in complete accord with the principle of similitude as will be seen by the application of the transformation equations which we have presented.
For example, consider the first of these equations. If the principle of similitude is correct we must also have as a true equation,
Curl H′ = 4πk′ + dE′/c′t′
Now the Curl operation is essentially a differentiation with respect to length, and the transformation equation for magnetic field strength is the same as for force, so that we could put
Curl H′ = (1/x) Curl (H/x2) = (1/x3) Curl H,
For current density our transformation equations would evidently give us
k′ = (1/x3)k
and for dE′/c′t′ we can evidently write
Substituting above, we obtain
(1/x3)Curl H = (1/x3)[4πk + dE/ct]
but this equation evidently reduces to the one we started with, thus showing no conflict with the requirements of the theory of similitude.
PROPERTIES OF THE ELECTRON.
The principle of similitude permits us to draw two interesting conclusions as to the properties of the electron. We may consider an electron as a sphere of radius r containing the unit quantum of electricity.
Relation between Mass and Radius of an Electron. --- We may now determine how the mass of an electron would depend on its radius. We have,
m = F(r),
and from the principle of similitude, we have [Eq. 23]:
m′ = F(r′).
Applying transformation equations (9) and (1) we have,
m / x = F(xr)
Combining with (23) we obtain
m = xF(xr) = F(r),
for which the only solution is
m = k / r
where k is a constant. Hence, according to the principle of similitude, the mass of the electron would be inversely proportional to the radius, a relation which can also be obtained by the more elaborate calculations of electromagnetic theory.
Radiation from an Electron. --- We may also determine with considerable ease the relation between the energy radiation from an electron and its acceleration. We have as [Eq. 24]
dE / dt = F(a)
and from the principle of similitude,
dE′ / dt = F(a′)
Substituting transformation equations (11), (2) and (4) we have
dE / dt = x2F(a)
and combining with (24) we obtain
F(a) = x2F(a)
for which the only solution is
dE / dt = ka2
where k is a constant. We thus see that the rate of emission of energy from an accelerated electron is proportional to the square of the acceleration.
THE THEORY OF GRAVITATION.
In what has preceded we have shown that the principle of similitude provides a simple method for obtaining relations in the most diverse fields of physical science. These relations can all be obtained, however, by the more specific methods of attack used in the particular branches of science under consideration. We shall now point out that in the field of gravitation theory an acceptance of the principle of similitude will lead to quite new conclusions.
Science has long been troubled by questions as to the mechanism by which gravitational forces are produced. On the one hand, almost countless hypotheses have been advanced to explain gravitation by the action of moving corpuscles, ether waves, or some electromagnetic influence, while, on the other hand, it has been warmly urged that gravitational attraction is an inherent property of the mass of a body, and that, having found in Newton's law an exact description of the way in which this attraction acts, any search for a mechanism by which the force is produced is meaningless. An acceptance of the principle of similitude, however, will force us to believe that the gravitational attraction between two bodies is not merely a function of the masses of the bodies and the distances between them, but must depend on something else as well, perhaps, for example, on the properties of some intervening medium.
To prove our point let us assume that the gravitational attraction between two bodies does depend merely on their masses m1 and m2 and the distance l between them. We have from Newton's law [Eq. 25]
f = k m1 m2 / l2
But if our assumption that gravitation depends merely on the physical entities m1, m2 and l is correct, it is evident from the principle of similitude that we must have for the same system
f ′ = k m1′ m2′ / l′2
But substituting transformation equations (10), (9) and (1) we obtain
f = k m1 m2 / (xl)2
[Typographical error in original corrected.]
an equation which does not agree with the one with which we originally started, No. (25).
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 of the problem 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.
Let us suppose, for example, that the force of gravitation depends not only on the masses of the gravitating bodies and the distance between them, but on the magnitudes, A, B, C, etc., of some properties of a gravitational mechanism. We shall then have,
f = F(A, B, C ...) m1 m2 / l2
and from the principle of similitude,
f ′ = F(A′, B′, C′ ...) m1′ m2′ / l′2
Let us assume that the transformation equations for A, B, C, etc., are of the form A′ = xaA, B′ = xbB, etc., we may then obtain from (10), (9) and (1)
f = F( xaA, xbB, xcC ...) m1 m2 / (xl)2
F(A, B, C ...) = F( xaA, xbB, xcC ...) / x2
as an equation which must be fulfilled by a successful hypothesis for the explanation of gravitational attraction.
(The writer first conceived the idea of the principle of the relativity of size eight or nine years ago, and for more than a year has been engaged in a definite attempt to draw useful conclusions from its corollary the principle of similitude. His progress has always been stopped, however, by the apparent failure of the phenomena of gravitation to meet the criterion of similitude. It is now hoped that the above treatment of the gravitational problem successfully removes this difficulty.)
In the preceding article we have seen that the principle of similitude can be used for the derivation of a large number of physical relations. The methods to be applied have the advantage of great simplicity and generality, but the disadvantage of not providing any information as to the magnitude of the numerical constants which enter the equations.
We have also seen that the principle of similitude could be of use for testing new physical theories which may be advanced.
In conclusion we may point out that our fundamental postulate is, as a matter of fact, a relativity principle. Indeed it might be called the principle of the relativity of size.
Our postulate states that the fundamental physical entities are of such a nature that from them a miniature universe could be constructed exactly similar in every respect to the present universe, and in the transformation equations which we have developed we have shown just what changes would have to be made in lengths, masses, time intervals, energy quantities, etc., in order to construct such a miniature world. If, now, throughout the universe a simultaneous change in all physical magnitudes of just the nature required by these transformation equations should suddenly occur, it is evident that to any observer the universe would appear entirely unchanged. The length of any physical object would still appear to him the same as before, since his meter sticks would all be changed in the same ratio as the dimensions of the object, and similar considerations would apply to intervals of time, etc. From this point of view we see that it is meaningless to speak of the absolute length of an object, all we can talk about are the relative lengths of objects, the relative duration of intervals of time, etc., etc. The principle of similitude is thus identical with the principle of the relativity of size.
January 18, 1914.