In the SI system, the dimension of all physical quantities is a combination of the 7 fundamental quantities such as

Name Unit Symbol

Distance Meter L

Mass Kilogram M

Time Second T

Current Ampere I

Light Candela J

Heat Kelvin K

Concentration Mole N

Any physical quantity Q can then be dimensionally expressed as

[Q] = L

^{a}M

^{b}T

^{c}I

^{d}J

^{e}K

^{f}N

^{g} With exponents, a
g, representing the influence of each constituent on the final quantity and the sign of the exponent indicating direct or inverse (1/x) proportionality. The notation [Q] means the dimension of Q.

It has long been suspected that it should be possible to derive the dimension of physical quantities from space and time only. Maxwell even stated that :

"If, as in the astronomical system, the unit of mass is defined with respect to its attractive power, the dimensions of [M] are L3T−2"

Unfortunately, it is easy to demonstrate that this conclusion is invalid. Not only because it is easy to prove that [GM]=L3T-2 and that absolutely nothing in current knowledge supports the assumption that G, the gravitational constant, is dimensionless. But also because this version fails the basic test of the cartesian product :

1) If we agree that the dimension of physical quantities can be derived from space-time only

2) Then, by definition, all physical quantities are part of the Cartesian product of the space and time sets

3) If we build a matrix that presents the Cartesian product of Planck space and time sets (Lpx * Tpy, with x and y = 0 to infinity)

4) Then all Planck values must appear on this matrix

The point 1 is our initial hypothesis.

Maxwell says [M] = L3T-2 (the dimension of all other quantities can be derived from [M]).

The point 2 has to be true by definition as the Cartesian product represents all combinations of the two initial sets (Lpx and Tpy).

The point 3 is what I did to represent visually this Cartesian product.

The point 4 is the key to all the reasoning.

If point 4 is false, then point 1 must be false also ... and mass is really a dimension in itself.

But if point 1 is true, then point 4 must also be true.

Then we ask ourselves: Do we see the Planck quantities appearing as expected at specific locations predicted by the Maxwell version. The answer is no. So inevitably, the Maxwell version is clearly invalid.

Mathematically, we can also say that if [M]=L3T-2 (and so [G]=1, i.e dimensionless), then automatically [F]=L4T-4 because F=Ma (the use of F is only an example, we could use any other value instead). Now, in a Planck's world, we should have then Fp=Lp4Tp-4, the same way we have GMp=Lp3Tp-2. Using a calculator, we accurately find GMp=Lp3Tp-2, but we fail to find Fp=Lp4Tp-4. Finding GMp is mandatory because we know for sure that it corresponds to Lp3Tp-2. On the other hand, finding Fp is related to wether [M]=L3T-2 is true or not. Because we do not find Fp=Lp4Tp-4, we can categorically say that [M]=L3T-2 is false.

While playing with these ideas in my research, I found something very strange ...

http://www.losangeinformatique.com/physics/dimension_of_physical_quantities.htmQuestions or comments are welcome

Regards,

Lho