Subject: The completeness of the Einstein equation and
the Cauchy problem
Date: Tue, 16 Mar 2004 12:56:19 +0200 From: Dimi Chakalov <dimi@chakalov.net> To: Vladimir Mashkevich <Vladimir_Mashkevich@qc.edu> CC: Alexander Lisyansky <alexander_lisyansky@qc.edu>, mash@mashke.org, pestov@thsun1.jinr.ru, carlip@physics.ucdavis.edu, pullin@baton.phys.lsu.edu, jbell@uwo.ca 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 mindbrain problem boils down to finding a solution to a wellknown 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 bidirectional "talk", http://members.aon.at/chakalov/Einstein.html#talk I've been trying, since January 1972, to find a physical solution to the mindbrain 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 bidirectional "talk". I believe this task can indeed be traced back to the nature
of the
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,
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
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, grqc/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
p. 10: 3.4. A canonical spacetime manifold structure "A quantum jump of the state vector gives rise to a set
of events 
[xxx] The onedimensional manifold T is the universal cosmological
time, the threedimensional manifold S is a cosmological space.
p. 11: "In view of the global character of time t, these conditions are global and, in fact, coordinatefree. "We may rewrite (3.5.5) in the explicitly coordinatefree form: [xxx] In connection with this, we quote Weyl [18]: "The introduction
of
"The problem of the underdetermination of the Einstein
equation is
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, grqc/0011090). Please pay attention to his equation at p. 11, which is written in explicitly coordinatefree form. Could this be a solution to the whole bundle of pathologies of the classical spacetime manifold (I. Raptis, grqc/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, quantph/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
