Conformally Compactified Minkowski Spacetime and Planck Constant

If the geometrical system of units


Introduction
Minkowski spacetime, the spacetime of special relativity (S.R.), is the simplest solution of the vacuum Einstein equations for general relativity (G.R.).As such, it is subject to the mathematical process of conformal compactification [1] [2], leading to a new-and therefore distinct-spacetime known as the Penrose diagram of the original space.This process is applied to many other solutions of the Einstein equations, in particular to black hole solutions, in the vacuum case or with the presence of matter [3].In all cases, the obtained spaces are distinct from the starting ones and could, in principle, be considered as non-physical.However they globally exhibit with total clarity the light cones or causal structure of the original spaces.We shall restrict our discussion to the 4-dimensional Minkowski spacetime, here denoted by M.
The conformal compactification process necessarily requires the introduction, at an intermediate step, of a length scale L which can not be arbitrary, since unavoidably it appears in the definition of the dimensionless coordinates of the final Penrose space (diagram) P. Though not unique [4], the mostly accepted as Journal of High Energy Physics, Gravitation and Cosmology natural unit of length is the Planck lenght given by (G is Newton constant,  is the reduced Planck constant, and c the velocity of light in vacuum, respectively associated with gravitation, quantum mechanics, and S.R.).In G.R., which is a classical theory-and therefore also in S.R.-the most commonly used system of units is the geometrical system of units (g.s.u.) defined by which leads to Notice that the g.s.u. is not the natural system of units (n.s.u.) in which 1 is the Boltzmann constant associated with thermody- namics and statistical mechanics).
The systematic and well known step by step derivation of the Penrose space P corresponding to M (Section 2) exhibits, beyond the causal structure of M, a hidden and/or a novel fact: the unavoidable presence of  , the essence of quantum physics, at least if the g.s.u. is used and Pl L is considered the funda- mental length scale.
Conclusions (remarks and questions) are presented in Section 3.

Penrose Space (P) of Minkowski Space (M)
, , , , , , , , , with (except for the usual coordinate singularities): where The coordinate transformation: We pass from two null coordinates , v u ′ ′ and two spacelike coordinates , θ ϕ to one timelike coordinate τ and three spacelike coordinates , , ρ θ ϕ through Through the conformal transformation (v) : cos cos ds τ ρ We Which is represented in Figure 1.

S
). P is represented in Figure 2.
Remarks: 1) In the whole procedure we passed successively through four distinct spacetimes: cc M , M  , conf M , and P. 2) P has the dimensionless 4-volume

Discussion and Conclusion
Using the coordinate transformations (ii), (iii) and (iv), the constant conformal transformation (ii)', and the trigonometric identities   .The presence of L is unavoidable which, unless one chooses a universal scale, leaves behind a degree of arbitrariness in the metric for P, contrary with the statement in Ref. [6] that the scale L is irrelevant.With u and v are null coordinates: constant u (v) lines represent outgoing (ingoing) light rays.Rescaling the metric with Pl L L = =  we have the constant conformal transformation
a question remains open: Does the above discussed facts indicate the existence of some hidden and/or unknown relation between Minkowski spacetime and quantum mechanics?