# The Naked Scientists Forum

### Author Topic: The dimensions of physical quantities  (Read 10048 times)

#### Lho

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##### The dimensions of physical quantities
« on: 29/09/2009 03:38:25 »
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] = LaMbTcIdJeKfNg

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 ...

Questions or comments are welcome :)

Regards,
Lho

#### LeeE

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• Posts: 3382
##### The dimensions of physical quantities
« Reply #1 on: 30/09/2009 20:42:00 »
I think there is much of interest in what you've done there, but it's still based upon treating distance (L) and time (T) as entirely different entities, and to get to where I think you're trying to get to, you'll need to unify them.

I'm sorry that I can't be more helpful, but I think you're following a path worthy of further exploration.

#### Mr. Scientist

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##### The dimensions of physical quantities
« Reply #2 on: 01/10/2009 20:59:40 »

#### Lho

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• Posts: 3
##### The dimensions of physical quantities
« Reply #3 on: 02/10/2009 02:30:37 »
Hi,

Thanks for taking the time to reply.

@LeeE
Thanks for your interest in this work. I'm not sure I want to go there (unifying length and time):) My initial idea was to build a visual matrix that can be used for very fast dimensional analysis. I describe this in: newbielink:http://www.wbabin.net/science/hollo2.pdf [nonactive]

@Mr. Scientist
Thanks for commenting. I went to see the page ... When talking about G, I presume that you refer to the section calles "Dimensionless physical constants" at the bottom. It seems to be the only place where they explicitely mention it. In this case, I must say that I'm a bit disapointed. The section seems to mix the notions of numerical value and dimension. While the title of the section implies that G is a dimensionless quantity, nothing is provided here to support it. it is an assertion. Then the section explains the concept of natural units where some quantities see their numerical value "reduced" to 1 ... but their dimension is still the same, whatever it is.

More radically (mathematically speaking), I think the demonstration I made using the cartesian product completely and absolutely invalidates the possibility that G could be dimensionless, don't you think?

Regards,
Lho

#### Mr. Scientist

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• Thanked: 2 times
##### The dimensions of physical quantities
« Reply #4 on: 02/10/2009 08:33:19 »
Hi,

Thanks for taking the time to reply.

@LeeE
Thanks for your interest in this work. I'm not sure I want to go there (unifying length and time):) My initial idea was to build a visual matrix that can be used for very fast dimensional analysis. I describe this in: www.wbabin.net/science/hollo2.pdf

@Mr. Scientist
Thanks for commenting. I went to see the page ... When talking about G, I presume that you refer to the section calles "Dimensionless physical constants" at the bottom. It seems to be the only place where they explicitely mention it. In this case, I must say that I'm a bit disapointed. The section seems to mix the notions of numerical value and dimension. While the title of the section implies that G is a dimensionless quantity, nothing is provided here to support it. it is an assertion. Then the section explains the concept of natural units where some quantities see their numerical value "reduced" to 1 ... but their dimension is still the same, whatever it is.

More radically (mathematically speaking), I think the demonstration I made using the cartesian product completely and absolutely invalidates the possibility that G could be dimensionless, don't you think?

Regards,
Lho

I'm still trying to understand the greater part of this - can i see this matrix in full form?

#### Lho

• First timers
• Posts: 3
##### The dimensions of physical quantities
« Reply #5 on: 02/10/2009 19:46:44 »
Hi Mr. Scientist,

I'm still trying to understand the greater part of this - can i see this matrix in full form?

The initial intent was only to build a dimensional analysis matrix ... I had to justify the position of the initial quantity (the mass for ex. the other are derived from it) ... My initial matrix was compliant with what I call the Maxwell version ([M]=L3T-2). But not only I discovered that there is nothing in current knowledge that supports this, I also demonstrated that it is impossible ... I continued my research and found what I'm now describing :)

The matrix as a tool for dimensional analysis
Now, I think that the concept of a dimensional analysis matrix(see the link I mentionned in my previous post) is very interresting in itself. In fact, I built the first one when I was at school (electronics) and used it for a long time. It worked perfectly by allowing me to be absolutely sure of the validity of any equation of any domain of physics ... quite good for a student :)
I also found something about "remembering/knowing" the position of all quantities ... but this is the subject of my next paper ...
The fact that my initial matrix was based on the Maxwell version (and so is invalid) didn't really matter because whatever the position you choose for the initial quantity, the position of all other is dimensionnaly coherent, so the matrix can still be used for its dimensional analysis benefits.

The space time dimensions of physical quantities
Knowing the dimension of physical quantities not only helps to perform dimensional analysis, but it also helps to understand their relations within the same domain or across physical domains .This is something that analysis of units doesn't provide ... somehow, units are meaningless ... a "Tesla" doesn't say anything about the relation of the magnetic field and other quantities. The same is true for the "Weber" (Flux). On the other hand, dimensional analysis shows the relation between the field and the flux ([flux]=[field]L2). The further reduction to only space and time makes it even easier to see these relations ... although what it tells is sometimes weird.

The determination of fundamental quantities
The SI uses 7 fundamental quantities to derive the dimensions of all others, so the feasability/accuracy of the determination of the quantity is important. As you probably know, the mass is currently "mesured" from the artefact located in Paris (BIPM) ... this causes problems like distorsions with the material loosing electrons. So, one of the goal of the BIPM is to determine the mass differently (watt scale or Klitzing+josephson for example). It seems that the BIPM should be very happy to find a way to determine mass from space and time. It would be very simple and only the accuracy of these two quantities should be enhanced.

So, I think these three points are the main interrest of it all.

You can find the Excel matrix at:

Questions and comments are welcome :)

Regards,
Lho

#### The Naked Scientists Forum

##### The dimensions of physical quantities
« Reply #5 on: 02/10/2009 19:46:44 »