Arg. Y'=9192631770, with Hz system [X':9192631770] relative to [X:1] on the Caesium standard. Sorry for the multi-multi-post... On Fri, Jan 31, 2020 at 9:53 AM Brad Klee <bradklee@gmail.com> wrote:
In my experience, the most common response is a reminder that: "If you want to accomplish practical work in science and engineering, you shouldn't waste time on corner cases, such as fractal curves etc." This could be regional bias.
The problem you mention about torque and energy, I don't see it as a dimensional problem, but rather as arising from macroscopic / microscopic divide. Torque should actually be lumped in with force. Similarly it is possible to show, using Lagrange multipliers, that angular and linear momentum can conserve together, see for example [1]. I would think that these sort of extra [L] factors usually arise from working with extensive bodies.
As for your assertion about co-vectors, I'm skeptical and have not had time to read your book. One counter argument is that dimensions should be swept into the metric itself. In Dirac notation, if (v_1|v_2) = (v_1|g_0|v_2), then change of basis can be accomplished by g_0 -> g_1 = k_1*k_2*g_0, then (v_1|g_1|v_2) = k_1*k_2*(v_1|v_2), with k1, k2 dimensional factors.
However, a stronger counterargument is to entirely forget about linear algebra, and to propose another model. Here is a brief description of the projective system I mentioned earlier:
The real projective line has coordinates [X:Y] with absolute magnitude M=X/Y. Let us associate Y=1 with a fixed scale of measurement, for example Cesium's hyperfine frequency split, 9192631770 Hz, and let us associate X with, say, the frequency of an electric dipole oscillating in a constant electric field. One acceptable definition of measurement is that for any possible measurement of X, the absolute value of M must remain constant. The projective coordinate system allows covariant change of Y, i.e. change of units.
In practice, the unit system [X:1] is very rarely used, with Hz a more common variant, though less precise. To change units we have to solve the projective equation [1:1] = [9192631770:Y'], thus Y' = 1/9192631770. Now a measurement of X' in Hz has a representation [X':1/9192631770], whose absolute value is equal to another measurement in the [x:1] system.
It is clear that any number Y, Y', Y'' etc. is a dimensional unit, for example: 1 = 1 Caesium cycle / 9192631770 Hz, 1 = 1 KHz / 1000 Hz, etc. In teaching I would almost never mention the analogy to projective geometry, but always emphasize the point that changing units is equivalent to multiplication by unity, a dimensionless factor.
I plan to write all this up more clearly in the future. That could be a chance for me to look through your book or to think about corner cases, but as I said, I'm initially skeptical.
Cheers,
--Brad
[1] http://physics.princeton.edu/~mcdonald/examples/2cylinders.pdf Historical note: This problem was not "originally suggested" by me, it was "originally solved" by me. Kirk's first solution, which accompanied a journal rejection, was incorrect. I corrected him, and he published his subsequent work online. I have the emails. * * * They owe me a cracker at Princeton! * * *
On Thu, Jan 30, 2020 at 9:57 PM George Hart <george@georgehart.com> wrote:
I find that even people who work with dimensions all the time can have surprisingly different feelings about such questions and what the structure of the algebra of dimensions might be.
As with angle and dimensionlessness, torque and energy appear to have the have the same exponents in their dimensional terms (length^2 * mass^1 * time^-2 * charge^0) yet different physical roles in which they are never added, suggesting standard algebras of dimensions do not quite capture everything of interest even for defining something as basic as when addition of two quantities is defined.
These sorts of issues pop up in my book, but are not the central focus. As far as I know, no one has a fully satisfactory framework even for scalars. There is another level of richness when considering the dimensions of vectors, matrices, etc. But to the original point: including dimensions within the linear algebra brings out very clearly the difference between a vector and a co-vector.
George http://georgehart.com