# LOG#073. The G2 system.

The second paper I am going to discuss today is this one:

http://inspirehep.net/record/844954?ln=en

In Note on the natural system of units, Sudarshan, Boya and Rivera introduce a new kind of “fundamental system of units”, that we could call G2 system  or the Boya-Rivera-Sudarshan system (BRS system for short). After a summary of the Gamov-Ivanenko-Landau-Okun cube (GILO cube) and the Planck natural units, they make the following question:

Can we change the gravitational constant $G_N$ for something else?

They ask this question due to the fact the $G_N$ seems to be a little different from $h, c$. Indeed, many researchers in quantum gravity use to change $G_N$ with the Planck length as fundamental unit! The G2 system proposal is based in some kind of twodimensional world. Sudarshan, Boya and Rivera search for a “new constant” $G_2$ such as $G_2/r$ substitutes $G_N/r^2$ in the Newton’s gravitational law. $\left[G_2\right]=L$ in this new “partial” fundamental system. Therefore, we have

$F_N=G_2Mm/r$

and the physical dimensions of time, length and mass are expressed in terms of $G_2$ as follows (we could use $\hbar$ instead of h, that is not essential here as we do know from previous discussions) :

$T=c^{-4}hG_2$

$L=c^{-3}hG_2$

$M=c^2/G_2$

In fact, they remark that since $G_2$ derives from a 2+1 dimensional world and Einstein Field equations are generally “trivial” in 2+1 spacetime, $G_2$, surprisingly, is not related to gravitation at all! We are almost “free” to fix $G_2$ with some alternative procedure. As we wish to base the G2 system in well known physics, the election they do for $G_2$ is the trivial one ( however I am yet thinking about what we could obtain with some non-trivial alternative definition of $lates G_2$):

$\boxed{G_2=\dfrac{c^2}{M_P}=G_N/L_P \approx 4.1\cdot 10^{24}MKS=4.1\cdot 10^{25}CGS}$

and any other equivalent expression to it. Please, note that if we fix the Planck length to unit, we get $G_N=G_2$, so it is equivalent to speak about $G_2$ or $G_N$ in a system of units where Planck length is set to the unit. However, the proposal is independent of this fact, since, as we said above, we could choose some other non-trivial definition for $G_2$, although I don’t know what kind of guide we could follow in those alternative and non-trivial definition.

The final remark I would like to make here is that, whatever we choose instead of $G_N$, it is ESSENTIAL to a quantum theory of gravity, provided it exists, it works and it is “clear” from its foundational principles.

See you in my next blog post!

# LOG#072. The hG system.

Brazil is experimenting an increase of scientific production. Today, I am going to explain this brazilian paper http://arxiv.org/abs/0711.4276v2 concerning the number of fundamental constants.

The Okun cube of fundamental constant, firstly introduced by Gamov, Ivanenko and Landau, has raised some questions about what we should consider as fundamental “unit” we already had but now with more intensity. I mentioned the trialogue of fundamental constants between Veneziano, Duff and Okun himself more than a decade ago. Veneziano argued that 2 fundamental constants were well enought to fix everything. However, it is not the “accepted” and “more popular” approach in these days, but the brazilian paper about defends such a claim!

What do they claim? They basically argue that what we need is a convention for space and time measurements and nothing else. Specifically, they say that every physical observable $\mathcal{O}_i$ with $i=1,2,\ldots$ can be expressed as follows:

(1) $\boxed{\mathcal{O}_i=\Omega_i \sigma^{\alpha_i}\tau^{\beta_i}}$

and where $\alpha_i,\beta_i,\Omega_i$ are pure dimensionless numbers, while $\sigma, \tau$ denote “basic units” of space and time. We could argue that these two last “fundamental units” of “space and time” were “quanta” of “space” and “time”, the mythical “choraons” and “chronons” some speculative theories of Quantum Gravity seem to suggest, but it would be another different story not related to this post!

After introducing the above statement, they discuss 2 procedures to measure with clocks and rulers, what they call $G$-protocol and $h$-protocol. They begin assuming some quantity in the CGS system (note that the idea is completely general and they could use the MKSA or any other traditional system of units):

(2) $\boxed{\mathcal{D}_i= \Delta_i T^{\alpha_i}L^{\beta_i}M^{\gamma_i}}$

where $\Delta_i,\alpha_i,\beta_i,\gamma_i$ are dimensionless constants. And then, the 2 protocols are defined:

1st. G-protocol. Multiply the above equation (2) by $G^{\gamma_i}$ and identify $\mathcal{O}_i$ with $\mathcal{D}_i G^{\gamma_i}$ or $\mathcal{O}_i^{(G)}$. Rewriting all the physical quantities and laws in terms of this protocol in terms of $\mathcal{O}_i^{(G)}$ instead $\mathcal{D}_i$  we gain some bonuses:

i) The unit M from CGS “vanishes” or is “erased” from physical observables.

ii) G disappear from every physical law.

iii) Masses being measured in $cm^3/s^2$ imply that from Newton’s gravitational law $g=-G_Nm/d^2$ we deduce that

$\boxed{g=-m^{(G)}/d^2}$

where $m^{(G)}=mG$ are units with physical dimension $L^3T^{-2}$. $G_N$, the gravitational constant, is some kind of conversion factor between mass and “volume acceleration” $L^3/T^2$. This G-protocol applied to the Planck constant provides

$\boxed{h^G=hG}$

and it has dimensions of $L^5/T^3$.

2nd. h-protocol.  From equation (2), if we divide by $h^{\gamma_i}$ and we identigy $\mathcal{D}_i/h^{\gamma_i}$ with $\mathcal{O}_i^{(h)}$ we get the so-called h-protocol. The consequences are:

i) M units disappear from physical laws and quantities, as before.

ii) h is erased and vanishes from every equation, law and quantity.

iii) Masses are measured in units of $s/cm^2$, e.g., from the Compton equation we get in the h-protocol

$\Delta \lambda= \dfrac{1}{m^{(h)}c}\left(1-\cos\theta\right)$

and where $m^h=m/h$ are units of mass in the h-protocol with dimensions $T/L^2$. Therefore, h is the conversion factor between inverse areolar velocity $s/cm^2$ and mass $g$. In this protocol the inverse of the Compton length measures “inertia”, and indeed this fact fits with some recent proposals to determine a definition of kg independent from the old MKSA pattern (the famours iridium “thing”, which is know now not to have a 1 kg mass). Moreover, we also get that

$G^{(h)}=Gh$

and

$G^{(h)}=h^{(G)}$

The two protocols can be summarized in a nice table

They also derive the mysterious relations between charge and mass that we saw in the previous post about Pavšič units, i.e., they also derive

$e^{(G)}=2\cdot 10^{21}m_e^{(G)}$

and it is equivalent to $e=\kappa_0 m_e$. Somehow, and electron is more electrical/capacitive than gravitational/elastic!

Finally, in their conclusions, they remark that two constants, $(c, h^{(G)})$ instead three $(c,h,G_N)$ seems to be well enough for physical theories, and it squashes or squeezes the Gamov-Ivanenko-Landau-Okun (GILO) cube to a nice plane. I include the two final figure for completion, but I urge you to read their whole paper to get a global view before you look at them.

Are 2 fundamental constants enough? Are Veneziano (from a completely different viewpoint) and these brazilian physicists right? Time will tell, but I find interesting these thoughts!

See you soon in another wonderful post about Physmatics and system of units!