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The Relativity of Size

Richard C. Tolman's *Principle of Similitude*

2013 November 11

[Photo by "Freddie10538", 2006]

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 e_{1} and
e_{2} placed on suitable bodies and separated by a considerable distance l.
From Coulomb's law the force between the charges is
e_{1}e_{2}/l^{2},
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 = e_{1}e_{2}/l^{2}
**

**
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′_{1}e′_{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) =
e_{1}e_{2}/(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][l^{2}][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′ = x^{2}S
**

**
For volume, [Eq. 14],
**

**
V′ = x^{3}V
**

**
For pressure, [Eq. 15],
**

**
P′ = P/x^{4}
**

**
For density of mass or energy, [Eq. 16],
**

**
u′ = u/x^{4}
**

**
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/x^{4} = F(T/x)
**

**
Combining with (21) we have,
**

**
u = x^{4}F(T/x) = F(T)
**

**
and the only solution for this functional equation is
**

**
u = aT^{4}
**

**
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 u_{ν} = du/dν
be the rate of change of the energy density with the frequency.
We have, [Eq. 21],
**

**
u_{ν} = 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,
**

**
u_{ν}/x^{3}
= F(ν/x, T/x)
**

**
u_{ν} = F(ν, T)
= x^{3}F(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
**

**
u_{ν} = F(ν, T)
= ν^{3}F(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 / x^{4} = F(e / x^{2})
**

**
and by combining with equation (22) we have,
**

**
u = F(E) = x^{4} F(e / x^{2})
**

**
for which the only solution is
**

**
u = k E^{2}
**

**
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/x^{2}) = (1/x^{3}) Curl H,
**

**
For current density our transformation equations would evidently give us
**

**
k′ = (1/ x^{3})k
**

**
and for dE′/c′t′
we can evidently write
**

**
(1/ x^{3}) dE/ct
**

**
Substituting above, we obtain
**

**
(1/ x^{3})Curl H = (1/x^{3})[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 = x^{2}F(a)
**

**
and combining with (24) we obtain
**

**
F(a) = x^{2}F(a)
**

**
for which the only solution is
**

**
dE / dt = ka^{2}
**

**
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
m_{1} and m_{2}
and the distance l between them. We have from Newton's law
[Eq. 25]
**

**
f = k m_{1} m_{2} / l^{2}
**

**
But if our assumption that gravitation depends merely on the physical
entities m_{1}, m_{2} and l
is correct, it is evident from the principle of similitude
that we must have for the same system
**

**
f ′ = k m_{1}′ m_{2}′ /
l′^{2}
**

**
But substituting transformation equations (10), (9) and (1) we obtain
**

**
f = k m_{1} m_{2} / (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 ...)
m_{1} m_{2} / l^{2}
**

**
and from the principle of similitude,
**

**
f ′ = F(A′, B′, C′ ...)
m_{1}′ m_{2}′ /
l′^{2}
**

**
Let us assume that the transformation equations for A, B, C, etc., are
of the form A′ = x^{a}A,
B′ = x^{b}B, etc., we may then obtain
**from (10), (9) and (1)

**
f = F( x^{a}A, x^{b}B, x^{c}C ...)
m_{1} m_{2} / (xl)^{2}
**

**
**that is**
**

**
F(A, B, C ...) =
F( x^{a}A, x^{b}B, x^{c}C ...)
/ x^{2}
**

**
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.)
**

**
**

** CONCLUSION.**

**
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.
**

**
BERKELEY, CAL.,
January 18, 1914.
**

**
**