Re: Radians, dimensions, & explanations

[David Bowman <dbowman@TIGER.GEORGETOWNCOLLEGE.EDU>]

Regarding Joel's comment:
> This is not unique to the *radian*, but would be true of any
> dimensionless unit; would it not?

It is certainly true that he radian is not unique in regard to its
dimensionless status.  Other dimensionless units include units of solid
angle, i.e. the steradian, units of information such as the nat, decimal
digit, bit, byte, gbyte, etc. and units for logarithmic measures of
intensity ratios such as bels, decibels, stellar magnitude units, and
Richter scale points.  We usually consider the source charges (flavor &
color) for the weak and strong interactions as dimensionless units as
dimensionless multiples of the elementary quantum of such quantities.  For
example we measure strangeness, topness, and blue-colorness in natural
units of dimensionless multiples of their elementary quantum units.  This
would presumably be the case for electrical charge sources for the
electromagnetic interaction as well if not for the prior conventions of
history (esu, coulomb, etc.) made possible by the long range nature of
that particular gauged interaction.

Physical (engineering) dimensions of quantities are not objective actual
things of the real world; they are merely a bookkeeping device set up by
the conventions of whatever measurement system of units used.  We
typically associate a single physical dimension for each physical
quantity used as a base unit for the measurement system.  For instance,
in the SI system the physical dimensions are various products of powers
of length, time, mass, thermodynamic temperature, electric current,
etc..  In this system the quantity of force has the dimension of
mass*length/time^2.  The unit of force, the newton, is a named label for
kg*m/s^2.  But in the U.S. customary engineering system which uses the
foot, second, and the pound-force as base units the physical dimensions
involve products of powers of length, time, force, etc..  In this system
mass is measured with a derived unit and has the dimension of
force*time^2/length, and the unit, the slug, represents a named label for
lbf*s^2/ft.

We thus see that the (engineering) dimension of a given physical quantity
depends on what base quantities make up the irreducible dimensional
factors, and these base quantities are measured using the base units of
the measurement system.  In the SI system angular momentum and action
have the dimension of mass*length^2/time, but in the natural unit system
of elementary particle physics (where h-bar==1) these quantities are
dimensionless.  Also in the SI system speed and velocity have the
dimension of length/time, but in the natural unit system of relativity
(where c=1) these quantities are dimensionless slopes in spacetime
because time and length have the same dimension in that system.

It seems that the main reason why the founders of the MKSA system (that
eventually became the SI system) decided to make another base quantity,
electric current, and the ampere a base unit for representing electrical
quantities is that with the introduction of such an extra base quantity
all electrical quantities then come out with integer power dimensions.  If
such a base unit/dimension were not adopted then most electrical
quantities would have half-integer powers (of the 3 mechanical base
dimensions & units) in their dimensions & units.

If John Mallinckrodt ever gets his way and his idea of making the radian
a base unit and giving planar angle its own dimension is ever adopted by
a large number of people, then quantities like action and angular
momentum would have different dimensions and work and torque likewise
would have different dimensions as well.  It's not clear to me why this
would be a good idea (although it may have a limited pedagogical value in
distinguishing different quantitieswith different dimensions).  It seems
to me that any benefit would be overshadowed by all the extra
complications that occur in the extra factors of powers of angles that
would appear in the dimensions of various quantities.  My preference is
to tend to go with as few base dimensions and corresponding base units as
possible.  It seems most natural to me to consider angles, whether
measured in cycles, gradients, radians, degrees, arcmin, arcsec, or
microradians, as a dimensionless sort of quantity.  I also conceptually
prefer to think of entropy as dimensionless and of temperature and energy
to have the same dimension as each other. A dimensionless entropy would
do a better job, I believe, of illustrating that entropy is fundamentally
an amount of information, and information tends to be thought of as
a dimensionless quantity.  I like the Planck unit system that has no
base units or dimensions at all and, correspondingly, requires that all
physical quantities be then dimensionless.  This system forces an
emphasis on the fundamental connections in physics as understood by
modern physical theory.  Giving action a fundamental meaning as a
phase angle of the complex quantum amplitude for some process, and
giving time and space the same dimension as each other--thus emphasizing
the unity of the spacetime manifold--are just two conceptual connections
suggested to us by quantum mechanics and relativity respectively.  These
connections are similar in spirit to the unification (and simplification)
that took place in classical thermodynamics when (because of the results
of Joule's experimental calorimetric work) the equivalence of heat to
dissipated work allowed us to consider both heat and work as both having
the same dimension (of energy), and measure both of them in energy
units, rather than cluttering up the equations of thermodynamics by
putting the unit conversion factor (in the first law of thermodynamics)
between heat units (i.e. calories), and work units (i.e. joules).
Setting h-bar==1, c==1, and k==1 in quantum mechanics, relativity, and
statistical mechanics, respectively, corresponds to similar theoretical
unifications and simplified (near) identifications among related
quantities, and is like the closer identification of heat with work
and energy suggested by the experimental work of Joule.

Ludwik wrote:
> Every physical quantity can be expressed quantitatively in the
> dimentionless form, if we want. Take temperature, for example.
> Instead of saying 25 degrees C I can say 0.25 (one quarter of the
> temperature difference between boiling and freezing). Or take the
> difference of potentials. Instead of saying 48 volts I can say "four
> times as much as in my fully loaded car battery". Or a charge of
> seven electrons. Or an energy of 10, meaning ten times as much
> as that of one galon of water rased to an elevation of one furlong.

Hear, hear.

> Why not? Why yes?

Yes.  For why, see above.

David Bowman
dbowman@georgetowncollege.edu