Dear Professor Mashkevich,

I am very happy to read your fascinating article [Ref. 1], in which you introduced the universal cosmological time. Thank you very much!

My efforts stem from brain science and psychology. It seems to me that the outstanding challenge of mind-brain problem boils down to finding a solution to a well-known problem in your field of expertise. If we look at the Einstein equation, we see that two ontologically different entities, matter and geometry, are correlated via their bi-directional "talk",

http://members.aon.at/chakalov/Einstein.html#talk

I've been trying, since January 1972, to find a physical solution to the mind-brain problem, such that the mind would play the role of geometry in Einstein's GR, and would satisfy the following requirements: the mind has to be both detached from matter, to preserve its ontologically different nature, and linked to it, in order to communicate with its brain via a bi-directional "talk".

I believe this task can indeed be traced back to the nature of the
infinitesimal,

http://members.aon.at/chakalov/Bell.html

http://members.aon.at/chakalov/Pestov.html#continuum

It seems to me that we need to use Synthetic Differential Geometry,
since I cannot envisage an exact solution to the problem of 'too much freedom' in 3+1-D spacetime,

http://members.aon.at/chakalov/Carlip.html#3D

I wonder what would be your opinion on this issue. The opinion of your colleagues will be highly appreciated, too.

Kindest regards,

Dimi Chakalov
--
http://God-does-not-play-dice.net

Reference

[Ref. 1] Vladimir Mashkevich, General Relativity and Quantum Jumps: The Existence of Nondiffeomorphic Solutions to the Cauchy Problem in Nonempty Spacetime and Quantum Jumps as a Provider of a Canonical Spacetime Structure, gr-qc/0403056

p. 2: "The breakdown of the diffeomorphic connectedness implies the necessity for canonical complementary conditions. It is quantum jumps that provide the latter. Nonlocal quantum jumps click out a universal cosmological time t [e] T, so that spacetime manifold has a canonical global structure: M^4 = T × S where S is a cosmological space. The canonical complementary conditions are of the form dt = g(d/dt, ·), which corresponds to a synchronous frame.
...

p. 9: "3. Quantum jumps and a canonical spacetime manifold structure

"In this section, we show that quantum jumps provide a canonical
structure of spacetime manifold and, by the same token, canonical
complementary conditions.
...

p. 10: 3.4. A canonical spacetime manifold structure

"A quantum jump of the state vector gives rise to a set of events -
jumps of 'gij . Those events are, by definition, simultaneous, which
allows for synchronizing clocks and thereby furnishing a universal time. The latter, in its turn, implies the product spacetime manifold:

[xxx]

The one-dimensional manifold T is the universal cosmological time, the three-dimensional manifold S is a cosmological space.
...

p. 11: "In view of the global character of time t, these conditions are global and, in fact, coordinate-free.

"We may rewrite (3.5.5) in the explicitly coordinate-free form:

[xxx]

In connection with this, we quote Weyl [18]: "The introduction of
numbers as coordinates ... is an act of violence whose only practical
vindication is the special calculatory manageability of the ordinary
number continuum with its four basic operations."
...

"The problem of the underdetermination of the Einstein equation is
resolved by the canonical complementary conditions, which are provided by nonlocal quantum jumps. Thus, quantum jump nonlocality not only does not contradict relativity, but it is essential for general relativity to be a complete theory. Quantum jumps occur in nonempty spacetime -- just where the underdetermination problem arises."
 

Note: It seems to me that Prof. Vladimir Mashkevich does not like spacetime singularities, both naked and hidden by some "event horizon", not to mention the metaphysical belief in Penrose's Cosmic Censorship Conjecture (V. Mashkevich, gr-qc/0011090).

Please pay attention to his equation at p. 11, which is written in explicitly coordinate-free form. Could this be a solution to the whole bundle of pathologies of the classical spacetime manifold (I. Raptis, gr-qc/0110064)?

If we are to abandon the presentation of spacetime continuum in the standard differential geometry, can we work out a solution to the paradox of the infinitesimal in the framework of Synthetic Differential Geometry (SDG)? How does one observe 'no bananas' or 'empty space' (J. Fearns, quant-ph/0202079)?

My 'no bananas' proposal is about a new reference object, called 'global mode of spacetime', but perhaps Prof. Vladimir Mashkevich has already cut the Gordian Knot of quantum gravity. That would be really fantastic!
 

D. Chakalov
March 16, 2004