From Osher Doctorow
In dimensional analysis, Buckingham's PI theorem says that if there
are n variables and r fundamental dimensions ("reference dimensions"),
then a dimensionally homogeneous equation w n variables and r
fundamental dimensions can be reduced to an equation between (n - r)
independent dimensionless products. The most useful case is when n -
r = 1, since then we have one dimensionless product which products of
powers of the n variables equals.
When there are 11 dimensions, as happens in Superstring theory as well
as in the "Final Equation" of this thread with 4 force dimensions, 3
length dimensions, and 4 time dimensions, then it only requires one
more dimension to complete the symmetry of 4 length dimensions, but it
also requires only one non-fundamental variable, call it V, to be
added to 11 variables each having a different single fundamental
dimension, to make 12 variables, so that:
1) 12 - 11 = 1
and one dimensionless product will be found.
For example, it makes sense that there are 4 variables V1, V2, V3, V4
respectively having dimensions Fg, Fem, Fs, Fw (gravitation,
electromagnetic force, strong/color force, and weak force), and 4
variables V5, V6, V7, V8, each having respective dimensions of T1
(forward time), T2 (backward time), T3 (fast time), T4 (slow time).
There are also 3 variables having respective length dimensions Lx (=
V9), Ly ( = V10), and Lz ( = V11), namely for example 3 perpendicular
distances relevant to a problem, or alternatively the 3 projections of
the radius of a sphere on the x axis, y axis, or z axis
respectively. Now let's add as our twelfth variable V12 the radius
of the sphere in question, which for example could be centered at some
particle or the center of some finite string, etc. Then by
Buckingham's PI theorem we can be sure that:
2) V1^a V2^b V3^c V4^d V5^e V6^f V7^g V8^h V9^i V10^j V11^k V12^l = k
(dimensionless product)
Osher Doctorow
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