Conformally Compactified Minkowski Spacetime and Planck Constant ()
1. 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 natural unit of length is the Planck lenght given by
(1)
(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
(2)
which leads to
(3)
Notice that the g.s.u. is not the natural system of units (n.s.u.) in which
(
is the Boltzmann constant associated with thermodynamics 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
is considered the fundamental length scale.
Conclusions (remarks and questions) are presented in Section 3.
2. Penrose Space (P) of Minkowski Space (M)
Let
,
, and
be M in cartesian, spherical, and lightcone coordinates; the corresponding coordinate transformations are:
(4)
with (except for the usual coordinate singularities):
(5)
(6)
,
,
,
,
,
, with metric
(7)
where
. Since the metric coefficients of
and
are null i.e.
, u and v are null coordinates: constant u (v) lines represent outgoing (ingoing) light rays.
Rescaling the metric with
we have the constant conformal transformation
(8)
leading to
with metric
(9)
The coordinate transformation:
(10)
leads to
with metric
(11)
We pass from two null coordinates
and two spacelike coordinates
to one timelike coordinate
and three spacelike coordinates
through
(12)
leading to
with metric
(13)
Through the conformal transformation
(14)
We obtain
with metric
(15)
Which is represented in Figure 1.
in Figure 1 is a 2-sphere with radius
:
.
(vi) The final step is to add the boundary of
,
(16)
with
: future (past) timelike infinity,
: spatial infinity, and
,
,
, respectively the future (past) null infinity, leading to the conformally compactified Minkowski spacetime
, with
(17)
including the boundary, and topology
(18)
where
and
the 3-sphere with unit radius (
). P is represented in Figure 2.
Remarks: 1) In the whole procedure we passed successively through four distinct spacetimes:
,
,
, and P. 2) P has the dimensionless 4-volume
.
3. Discussion and Conclusion
Using the coordinate transformations (ii), (iii) and (iv), the constant conformal transformation (ii)', and the trigonometric identities
and
[5] , we can express
and
in terms of t, r and
:
(19)
Any choice of length scale L distinct from
would leave unchanged the global picture of P. However, the requirement of a physical origin for L (and
is the most natural and universal) and the use of the g.s.u. in special and general relativity, force the appearance of
in the time (
) and radial (
) coordinates of the Penrose diagram.
If one insists with a dimensionful metric for P, one should multiply both sides of Equation (15) by
obtaining
Figure 2. Penrose space of Minkowski space.
(20)
with
,
, and
. 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
in the g.s.u., (20) becomes
(21)
We conclude that 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?
Acknowledgements
The author thanks Daniel Núñez at PCF-UNAM, México, for useful discussions; and Oscar Brauer at the University of Leeds, United Kingdom, for drawing Figure 1 and Figure 2.