From Osher Doctorow
Dimensional analysis usually works, but so do separate phases even
though they may occasionally change (for example, from liquid to gas or
liquid to solid). Could Sir Isaac Newton have formulated both
gravitation and antigravitation "explicitly" and our ideas about
dimensional analyis obscure that fact?
Let's take another look at the two force equations that we know from
Sir Isaac:
1) F = d(mv)/dt
2) F = Gm1m2/r^2
We know that (2) is the classical equation of gravitation, so shouldn't
antigravitation if it exists be embodied in (1) as a special case?
And how do we get from (2) to (1)?
The obscuring fact or principle has to do with G in (2). What if G is
actually "universally 1", that is to say isn't in the equation? We
then get the pair of equations:
3) F = d(mv)/dt
4) F = m1m2/r^2
In this and earlier threads of mine, the notion of variable exponents
has been used, so let's see what happens if the general equation of
force F really is:
5) F = m^u L^v T^w, u, v, w variable (possibly "piecewise constant on
different intervals")
Now look at the dimensions [F] of F in (3) and (4):
6) [F} = MLT^(-2) in (3)
7) [F] = M^2 T^(0) L^(-2) in (4)
Let's use the arrow --> only here to indicate how the dimensions change
from (6) to (7):
8) T^(-2) --> T^0
9) L --> L^(-2)
10) M --> M^2
The quantity /2/ seems involved in all three of (8)-(10), but there's
another surprise:
11) 2 - 1 = 1 (difference of exponents of M in (10))
12) 0 - (-2) = 2 (difference of exponents of T in (8))
13) -2 - 1 = -3 (difference of exponents of L in (9))
The absolute values here from (11) to (13) are 1, 2, 3, which leads us
to ask why. Well, we know that length in "ordinary" 3 space and 3+1
quantum gravity has dimension 3. We know that mass in "ordinary"
scenarios has dimension 1. That leaves time. According to (12), T
seems somehow associated with 2. Since time is about the least
"intuitively grasped" dimension in ordinary scenarios, could time in
ordinary scenarios have actually the dimension 2, that is to say is
"ordinary" time 2-dimensional? Ultimately, in the Universal Code (see
my thread on that), time should be 3-dimensional like length, but in
ordinary scenarios 2 seems plausible.
Most important perhaps, to get from M to M^2 in gravitation, you join
or interact or interface masses, and in the reverse direction you
separate them. "Containment" versus "separation"? We'll see,
hopefully.
Osher Doctorow
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