# 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!