September 30, 2008

Singularities as Cosmological Limits

Singularities as Cosmological Limits

An interesting set of representations of the concept of limit in the natural world are the space-time singularities appearing in general relativity, especially the notorious "initial" singularity associated to the "big-bang". One cannot blame the modern scientists for being uncomfortable with the "initial singularity" - note that Lemaitre's idea of an expanding universe appeared initially to Einstein "abominable" from a physical point of view.

Thus, repeated attempts have been made in order to "resolve" the initial singularity and advance "no-boundary proposals". One of them is the Hartle-Hawking model making use of the "imaginary time". This model suggested that a four-dimensional euclidean sphere (with none of the four dimensions being temporal) may be used to "smoothen" the big-bang singularity, the implication being that the universe does not have a boundary and just unfolded out of a "timeless state". Theological controversies were generated by the Hawking's model. Hawking himself questioned "the need for a Creator".

On the other hand, the idea of an universe without boundary is not in itself inconsistent with the idea of a Creator, which transcends any space-time-matter manifold, real or imaginable, with, or without singularities. This being said, we have to point out that what the Hartle-Hawking's model may suggest is that there may be more to creation than "just" big-bang.

Surprisingly, Alexei Nesteruk develops a method of "apophatic monodualism" which allows the contemplation of the Creator even in the case of cosmological models without a boundary such as Hartle-Hawking's. Nesteruk's approach applies to such "ultimate theories" precisely because somehow they cannot avoid incorporating entities of different ontological status in the corresponding models - for example, in the Hartle-Hawking model, one cannot assign ontological reality to a timeless mathematical construct like an "imaginary time 4-sphere".

Nesteruk suggests that the apophatic opposition between a "thesis" stating that Hawking's euclidean 4 sphere exists and is absolutely necessary for the existence of the sensible (visible) universe, and an "antithesis" stating that Hawking’s 4-sphere does not exist due to its different ontological status, points out to a "diaphora" in creation, indicative of the ex-nihilo creation of both the sensible (visible, detectable) and the intelligible (pertaining to the mathematical ideas) universe through the agency of the transcendent God, as mentioned in the Symbol of Faith. Thus, detecting the apophatic opposition in a model such as Hartle-Hawking's may contribute, in fact, to a deeper, mystical understanding of God.

Another model incorporating an atemporal "beginning" of the universe, this time a non-commutative, C*-algebra-based model of unification of general relativity and quantum mechanics was suggested by a group involving, among others, M. Heller (the 2008 Templeton Prize winner, considers the tendency to equate the creation of the universe with with the "big-bang" singularity to be simplistic), W. Sasin, L. Pysiak and Z. Odrzygozdz. Their unification model suggests that, deeper than the Planck scale, the "classical" geometric concepts of space-time (that is, operating in the "classical general relativity") no longer operate and, in that "non-commutative limit", the distinction between "singular" and "non-singular" states disappears.

The ordinary space, time and singularities from general relativity, as well as the "ordinary quantum mechanics" appear through a "phase transition" taking place at the Planck length level, from a non-commutative, non-local, atemporal geometry which is the geometry of a non-commutative algebra defined on a transformation groupoid, a world that is under the incidence of the "free (non-commutative) probability" theory to the classical, "commutative world", with singularities, and the ordinary quantum mechanics and its associated type of randomness.

We suggest that Nesteruk's apophatic monodualist method applies in this case as well, and the detectable diaphora in creation points, again, towards the Creator of the "visible" (that is, "classical" relativity and quantum mechanics) and the "invisible" (non commutative algebra regime of atemporal existence).

(excerpts from Florin Caragiu's presentation "Passing to the Limit – Landmarks for a Christian Hermeneutical Dialogic Approach", delivered at the 2008 Congress on the dialogue between Science and Religion in the Orthodox World, held in Bucharest)

September 24, 2008

Congress 2008