Re: dimensionless units

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

This note concerns John Denker's comments about my (nonserious) proposal
to append an inverse angle dimension to the Levi-Civita [epsilon] symbol.

After further consideration about my proposal, I would like to make an
amendment to it.  Rather than append the inverse angle dimension directly
to the [epsilon]_i_j_k symbol, suppose we include it as a separate
dimensioned scalar factor that multiplies it.  This factor would be a
universal constant of nature (let's call it 'R') and have the value of
R = 1 rad^-1 = [pi]/180 deg^-1, etc.  This constant would be similar in
concept to the dimensioned constants [mu]_0 and N_A which are needed in
the SI system because of the introduction of the base units of the ampere
and the mole in that system.

When writing a physics equation that traditionally involves a cross
product we change the multiplication rule by inserting a factor of the
constant R into the product expession.  Thus, when we write the vector
product expression L = r x p , what we *mean* by it is:

L_i = (R*[epsilon]_i_j_k)*(r_j)*(p_k)  .

This change has the effect of allowing any previous formulae relating
products of Levi-Civita [epsilon]s to Kronecker [delta]s and their
products remain unaffected by the change.  We only include the R factor
when we do a cross product.

> 1) In 3D, there are _two_ ways in which angles are related to vectors:
>   a) As David mentioned, there is the cross product, |A| |B| sin theta,
> which can be written in terms of the Levi-Civita symbol epsilon_i_j_k.
>   b) There is also the dot product, |A| |B| cos theta, which can be
> written in terms of the Kronecker symbol delta_i_j.
>
> It seems to me that attaching dimensions to the sines will not be
> sufficient;  one also needs to go after the cosines.

With my amended version the cross product magnitude becomes
|A|*|B|*R*sin([theta]) so the sine function itself (coming from the
interplay of the components due to the [epsilon]) doesn't get any
dimension.  The dimension is separate in its own factor of a constant of
nature.

> 2) The Levi-Civita symbol can be used for things other than cross
> products.  For instance, we have the following mathematical identity:
>         epsilon_i_m_n  epsilon_j_m_n  =  2  delta_i_j
> where we are invoking the Einstein summation convention for repeated indices.
>
> Now it's pretty clear that we need to have a Kronecker delta that is
> dimensionless.  Therefore David's dimensionful epsilon symbol cannot
> possibly be the whole story; ....

As mentioned above the amended version fixes this.

> 3) I think it should be possible to fix up the bug mentioned above.  But it
> won't be automatic.

The most pressing problems mentioned by John seem to be now fixed.

> It will involve checking all the formulas of physics
> to see which are covariant, which are contravariant, and which are
> invariant with respect to a change in what units (radians .. cycles ..
> degrees ...) are used to measure angles.

Not an inviting prospect--especially for a nonserious proposal.

> On the third hand, that wouldn't be a bad thing to do!

No comment.

> Here's a start:
>
> a) The formula for centrifugal field
>         A = R omega^2
> should be replaced by
>         A = R omega^2 radian^-2

The formula for the centrifugal field is:
(vector) A(r) = - [omega] x ([omega] x r).  As such, the radian^-2 factor
is built-in in (both versions of) my proposal to the definition of the
cross product, and it cancels the 2 radian factors coming from the
[omega] factors (in rad/s), so A(r) is a normal acceleration field with
usual units & dimensions.

> b) The formula for the number of times (S) a bell is struck by a crank
> rotating at given frequency (f) during a given period (t)
>         S = f t
> should be replaced by
>         S = f t cycle^-1

This case doesn't seem to be very related to the previous one's related to
angular measure.  In this case the linkage results in a 1 - 1 relationship
between cycles of the crank and bongs of the bell, so S (bongs of bell) =
S (cycles of crank) = f*t.  The answer in this case *ought* to have the
cycles remain uncancelled so they can be reinterpreted as bell bongs.

David Bowman
David_Bowman@georgetowncollege.edu