Dimensional angles and universal constants

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Am. J. Phys., 66, 814 (1998)
Dimensional angles and universal constants
Jean-Marc Lévy-Leblond
Physique théorique, Université de Nice
F-06108 Nice Cedex, France
K. R. Brownstein very sensibly advocates treating angles “squarely” by endowing
them with a specific physical dimension1. Introducing, as he does, a constant of
proportionality between angles and arclength-to-radius ratios, in fact is but a case of
the procedure commonly followed when writing the universal laws of physics (that
is, laws which are supposed to hold for all physical phenomena). Such a law always
expresses the identification of previously independent notions, each one having its
own dimension — and unit. A (universal) constant is then required to make explicit
this identification, and to allow for the conversion between units2. Standard examples
are the Planck constant
h
in quantum theory, as unifying (for instance) energy and
frequency according to the formula
E=h
ν
, or the limit velocity
c
(better to be called
“Einstein constant”) which underlies the very structure of space-time by allowing a
(universal) interconversion of space and time intervals according to
Δ
x=c
Δ
t
(or the
identification of mass and energy, through
E=mc2
). Once one gets accustomed to
the new theory, the core of which is expressed by such a law, it becomes natural to
use a simpler system of dimensions and units, in which the universal constant is
taken as unity, thereby taking for granted the conceptual unification it symbolizes.
This is the reason why constants such as
h
and
c
have practically disappeared from
the explicit formulas of quantum theory or relativity in the scientific literature, which
amounts, in the case of
c
, to measuring time in years and distances in light-years, or
masses in electron-volts.
The point, now, is that, besides these ‘modern’ and conspicuous constants, there are
a host of other ones, hidden in the very foundations of physics. These ‘archaic’
constants express laws of physics so ancient and general that we no longer think of
them as expressing an equivalence between different concepts3. As an example, the
isotropy of space is so deeply imbedded in the formalism of physics, that we do not
consider using different units (and dimensions) for horizontal and vertical distances
— unless the concrete situation compels us to do so; plane pilots use miles
(horizontally) and feet (vertically), as they have better not to mix up the two
directions4. Or, still, we usually forget that areas and length-squared, or volumes and
length-cubed, are not the same thing — unless one is forced to deal with archaic
systems of units, where, for instance, volumes and length cubes, measured by
independent units, are related by universal constants such as 6.229… gallon/(foot)3.
In order to make the point more explicit, consider the intermediate situation,
between the ‘archaic’ and ‘modern’ universal constants, furnished by the ‘classical’
ones, expressing well-established conceptual unifications, but still surviving as
nontrivial conversion factors ; think for example of the Joule constant
J
in the heat-
work equivalence
W=JQ
or the Boltzmann constant
k
in the temperature-thermal
energy equivalence
ε
=kT
.
One may now see that there indeed is a universal law of physics (that is, of geometry
as physics of space) which relates an angle
θ
to the ratio of the arclength
s
it

2
subtends on a circle with the radius
R
of the circle:
θ
=
∠
(s/R)
. The constant ∠ is a
universal constant (of the archaic type), with a specific physical dimension, that of an
angle, and numerical value ∠
=1rad
or 57.3°, etc. Of course, this constant ∠ (why not
call it “ang”?) is related to Brownstein’s one
!
, by ∠
=!−1
, and formally equivalent
to it — although it seems to me that using the constant ∠ is rather more natural. As
in the cases previously mentioned (distances, volumes, etc.), the maintaining of this
explicit constant, or its disappearing into oblivion through taking it as unity, is a
matter of convenience. In any case, it is seen that endowing angles with a physical
dimension of their own is a quite natural procedure which fits in the usual scheme of
physics.
It is a pleasure to thank the Istituto di Fisica of the University of Rome (“La
Sapienza”), and especially Dr Gianni Battimelli, for their hospitality.
Electronic mail: jmlevleb@math.unice.fr
1. Kenneth R. Brownstein, “Angles — Let’s treat them squarely”, Am. J. Phys. 65,
605-614 (1997).
2. Jean-Marc Lévy-Leblond, “On the Conceptual Nature of Physical Constants”, Riv.
Nuovo Cimento 7, 187 sqq. (1977).
3. See ref. 2.
4. It is perhaps worth mentioning here a simple case in elementary mechanics where
it is useful to distinguish horizontal and vertical distances, as their equivalence plays
no role in the physical situation. Consider the range
d
of a ballistic projectile fired
with initial velocity
v
at an angle
α
with the horizontal in the gravity field
g
of the
Earth. Dimensional analysis, with a single magnitude for distances (vertical and
horizontal altogether) suffices to establish that
d=f(
α
)v2/g
, where the function
f
is unknown. Suppose now that we do distinguish between horizontal distances
(such as
d
) and vertical ones (which enter the dimensional expression for
g
);
characterizing then the velocity, not by its magnitude
v
and direction
α
, but by its
horizontal and vertical components
vx
and
vz
, it is easy to see that dimensional
analysis (with one more dimension!) leads to
d=Kvxvz/g
, with only an unknown
constant
K
. In other words, by recognizing the irrelevance of the isotropy of space in
this specific situation, we are enabled to obtain the exact functional dependence of
f
as
f(
α
)=Ksin
α
cos
α
.