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Re: factor-label analysis (was: relativity question)



On Tuesday, Jun 10, 2003, at 17:10 US/Eastern, cliff parker wrote:
>
>> "In SR unites, the light-second is considered to be equivalent to
>> the second of time, and both units are simply referred to as
>> seconds. This means that these units can be canceled if one appears
>> in the numerator and the other in the denominator. For example, in
>> SI units, velocity has units of meters per second; but in SR units
>> it has units of seconds per second = unitless(!)"

On 06/10/2003 05:45 PM, Joe Heafner wrote:
>
> Ratios are DIMENSIONLESS, but not necessarily UNITLESS.

Quite right.

To see how this ties in to what I was saying about
convenient units: A convenient unit for measuring
speeds in the classroom might be the nano-c. It's
about one foot per second.

In units where c=1, a nano-c is, as Joe says, dimensionless
but not unitless. It is entirely analogous to measuring
angles in degrees or milliradians. They are dimensionless,
but you don't want to confuse them with each other, or
with an entire radian, which is the conventional "unit"
angle.

As I have remarked previously, c is to boosts as a
radian is to rotations.

===========================

There is an unstated question here that might be
bothering people: what about dimensional analysis?

People are accustomed to being able to do three
dimensional analysis checks on every formula:
the LHS and the RHS have to agree as to length,
time, and mass.

But if we measure length and time in the same units,
then we lose one check -- or do we?

It may help to distinguish three concepts:
-- dimensional analysis
-- factor-label method
-- scaling arguments

Dimensional analysis is commonly used to check
formulas for dumb mistakes. It is useful but
not nearly as powerful as the other items on the
list. It loses some of its power when we choose
units such that c=1. But this is a trivial loss.

More important is the factor-label method. It tells
us we should not write "12" to denote a length;
we should write "12 inches" or "12 furlongs" or
whatever. If you follow this method, you will
never commit the sort of errors that can be caught
by dimensional analysis. This is vastly more
powerful that plain dimensional analysis, and it
keeps all of its power even when c=1.

Scaling arguments are even more powerful. The
factor-label method is basically just mathematics
and logic; scaling arguments belong to physics.
They are very powerful in the hands of wizards
(and occasionally problematic in the hands of
sorcerer's apprentices). We will not discuss it
further here.

As far as I can tell, the usual "proofs" of the
Buckingham Pi theorem are not very useful, because
they assume the existence of certain physically
"independent" units. Often they claim a long list
of independent units, such as kilogram, metre, second,
ampere, and many others. But what if metre and second
are not independent? Do we throw out the conclusions
of the "theorem"? Perhaps we do. Perhaps the
theorem only describes dimensional analysis, not the
more-powerful factor-label method. Actually it's
worse than that, because I'm not sure _any_ of those
units are physically independent. You can't arbitrarily
redefine the ampere independently of the kg, m, and s,
for instance. So perhaps the theorem doesn't apply to
real physics at all.

I think about the factor-label method in the same
way I think about objects in an object-oriented
programming language such as C++. I can define as
many objects as I want. I can define an object that
has a magnitude and a "tag". The "tag" could be kg,
second, metre, foot, inch, furlong, or whatever. We
can defer the question as to objects with different
tags can be interconverted. If we have a conversion
rule, we can use it. If not, we do without.

If I have an expression in feet and I want something
in inches, I can apply the conversion, multiplying
by unity (12in/ft).

If we learn that a boost mixes length and time in
a particular ratio, we can write equations like

1 nanosecond = 12 inches (1)

which is correct; it is just like writing

1 foot = 12 inches (2)

Even though each equation has different tags (different
units) on each side, there is no error and no ambiguity.
Because of the tags, there is only one way to interpret
the quantities involved.

If you have a length measured in nanoseconds and you
want it measured some other way, you can always
multiply by the appropriate conversion factor.