Dear Dr. Kijowski,

I read your "Canonical gravity and gravitational energy", and thought that you might be the right person to help me understand the ambiguities in the definition of gravitational energy,

http://God-does-not-play-dice.net/Will.html#3

I'm wondering, what contradiction would have been be reached if the gravitational energy were a *local observable*. In other words, why is this ban from Mother Nature?

I will appreciate the opinion of Dr. Chrusciel as well. Please be assured that I'll keep your feedback strictly private and confidential.

Regards,

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

Note: Let me quote from the introductory section of the paper mentioned above:

Katarzyna Grabowska and Jerzy Kijowski, Canonical gravity and gravitational energy, in: Differential Geometry and Its Applications, Proc. Conf., Opava (Czech Republic), August 27-31, 2001, Silesian University, Opava, 2001, pp. 261-274,
http://8icdga.math.slu.cz/PDF/261-274.pdf

"There is a lot of ambiguities in the definition of gravitational energy. A textbook version of the Legendre transformation, which is often used to derive Hamiltonian formalism from the Lagrangian field theory, leads to a somewhat paradoxical result: gravitational energy vanishes modulo boundary terms. The same textbook version of the Canonical Field Theory (used, e.g., as a starting point for second quantization of Electrodynamics) is only "volume sensitive" but not "boundary sensitive". This means that boundary phenomena are simply neglected. But here, in Gravity Theory, neglecting boundary terms means neglecting everything. Some authors improve this version of Canonical Gravity by imposing extra requirements on the energy functional in the asymptotically flat case (see, e.g., [1]).
--
[1] A. Ashtekar, Lectures on Non-Perturbative Canonical Gravity (World Scientific, Singapore, 1991).

"In this way gravitational Hamiltonian is defined as "zero + boundary corrections". These corrections are, however, often obtained not by a universal procedure, well defined for any field theory (e.g., electrodynamics), but via ad hoc improvements, which make no sense outside of Gravity Theory."

I hope to hear from Jerzy Kijowski and Piotr Chrusciel. See the opinion of Albert Einstein and my last email to Tomohiro Harada. As to the Lectures by A. Ashtekar, see my last email to Karel Kuchar.

More at GR18 in 2005.
 

D. Chakalov
July 11, 2004

P.S. I'm not sure if Jerzy Kijowski would be interested in answering my question, but let me put my cards on the table before he or Piotr Chrusciel decide to reply.

I believe the reason why the gravitational energy cannot be a local observable is that it is being determined by the Holon, which belongs to the postulated global mode of spacetime. The latter is absent in Einstein's GR, and hence physicists need to make some truly ridiculous statements, such as "gravitational energy vanishes modulo boundary terms." Jerzy Kijowski is aware of this problem: "But here, in Gravity Theory, neglecting boundary terms means neglecting everything." Sure. You neglect the Holon. It is like 'the context of a sentence'. It is not located in the local mode of spacetime. Hence the gravitational energy cannot be a local observable. It's not "a simple localisation of energy".

To understand the difference between the global and local modes of spacetime, recall that in the non-relativistic QM we cannot, even in principle, locate the famous 'quantum state' in the local mode of spacetime. Hence the measurement problem, and the two postulates introduced 'by hand', von Neumann's projection postulate and Born rule. But these postulates say nothing about the 'quantum state' that must exist before we measure it. We boldly presume that immediately prior the instant of measurement, the system has been in some 'quantum state'. Trouble is, we cannot, even in principle, locate this immediately prior quantum state in the local mode of spacetime, and hence cannot, even in principle, develop a relativistic theory of the so-called collapse. See a detailed description of this immediately prior quantum state in:

Daniel T. Gillespie, Quantum Mechanics Primer: An Elementary Introduction to the Formal Theory of Non-Relativistic Quantum Mechanics. New York: International Textbook Co., 1973, pp. 49-58.

On the other hand, this immediately prior quantum state must exist. Where? In the Holon, that is, in the postulated global mode of spacetime.

Going back to Einstein's GR, the global mode of spacetime is needed to explain the so-called dark energy, since it too lives in the Holon. Can't locate it in the local mode of spacetime. No way. The explanation is here.

Here's my question again: "I'm wondering, what contradiction would have been be reached if the gravitational energy were a *local observable*. In other words, why is this ban from Mother Nature?"
 

D. Chakalov
July 11, 2004