LOG#073. The G2 system.

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


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 two dimensional (2d) 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!

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