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https://www.ias.ac.in/article/fulltext/pram/073/06/0961-0968

Fundamental constants; gravitation.

We propose to substitute Newton’s constant $G_{N}$ for another constant $G_{2}$, as if the gravitational force would fall off with the $1/r$ law, instead of the $1/r^{2}$; so we describe a system of natural units with $G_{2} , c$ and $\hbar$. We adjust the value of $G_{2}$ so that the fundamental length $L = L_{\text{Pl}}$ is still the Planck’s length and so $G_{N} = L \times G_{2}$. We argue for this system as (1) it would express longitude, time and mass without square roots; (2) $G_{2}$ is in principle disentangled from gravitation, as in (2 + 1) dimensions there is no field outside the sources. So $G_{2}$ would be truly universal; (3) modern physics is not necessarily tied up to $(3 + 1)$-dim. scenarios and (4) extended objects with $p = 2$ (membranes) play an important role both in M-theory and in F-theory, which distinguishes three $(2, 1)$ dimensions.

As an alternative we consider also the clash between gravitation and quantum theory; the suggestion is that non-commutative geometry $[x_{i} , x_{j}] = \Lambda^{2} \theta_{ij}$ would cure some infinities and improve black hole evaporation. Then the new length 𝛬 shall determine, among other things, the gravitational constant $G_{N}$.

Luis J Boya¹ Cristian Rivera¹ E C G Sudarshan²

1. Departamento de Física Teórica, Universidad de Zaragoza, E-50009 Zaragoza, Spain
2. Physics Department, University of Texas, Austin 78712, USA

Published

December 2009

Manuscript received

25 February 2009

Manuscript revised

9 June 2009

Accepted

22 June 2009