On Sep 11, 2010, at 9:52 PM, Paul Zielinski wrote:
Yes it is a very basic example, but it serves to show that there need not be anything mysterious about the physics inside a volume being "encoded" in the physics occurring at an enclosing boundary. In the case of the divergence theorem, the key is a simple conservation principle. My general point here is that once it is recognized that the gravitational vacuum is a physical system with objective physical properties, it is no suprise that such properties include thermodynamic ones. Once that is admitted, then there is no reason why a thermodynamic holographic principle should not be understood in terms of some more basic intuitively transparent principle, as in the simple example of the divergence theorem in fluid mechanics.
Yes, I agree, however on the cosmic scale the important boundary is in our future not our past! This is what is missing from all the papers in the field.
On Sep 11, 2010, at 1:57 PM, Paul Zielinski wrote:
Exactly.
On Sat, Sep 11, 2010 at 1:55 PM, Brian Josephson wrote:
--On 11 September 2010 13:48:14 -0700 Paul Zielinski wrote:
Naive question: Is Gauss' Theorem (Divergence Theorem) a holographic
principle?
Naive answer: there's something called Green's theorem which tells you the potential in the interior of a region if you know it on the boundary, assuming Laplace's eqn. applies. So you might surmise that the holographic principle might apply in any sufficiently constrained situation. Is there more to it than that?
Brian
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Clearly, Gauss's theorem etc is related to the hologram principle, it is the poor boy's version.
The more general theorem is in terms of Cartan's generalized Stoke's theorem
http://en.wikipedia.org/wiki/Stokes'_theorem
(dw|W) = (w|&W)
where little omega w is a p-form, d^2 = 0, d(little omega) is a p+1 form, BIG Omega W is a p+1 co-form (domain of integration or "chain)
&(BIG Omega) is the p-coform boundary of BIG Omega. d & are dual.
The quasi-Dirac (BRA|KET) is the de Rham integral.
Quite, obviously if little omega w is a geometrodynamical field like a gravitational tetrad 1-form then d(tetrad) is something like a torsion field 2-form.
However, we need to use the "covariant" BIG D (in spite of what the Captain of the Pinafore sings ;-) ).
HMS Pinafore by W. S. Gilbert and Arthur Sullivan
Jun 4, 2005 ... I never, never use, Whatever the emergency; Though "bother it" I may. Occasionally say, I never use a big, big D —. Chorus. What, never? ...
math.boisestate.edu
D = d + (spin connection)/
D^2 =/= 0 even though d^2 = 0.
So we have something like a HOLOGRAM SURFACE 2-form D(gravity tetrad) with an interior BULK 3-form D^2(gravity tetrad) =/= 0 because of the SPIN CONNECTION. This looks something like a hologram principle for geometrodynamics?
However, not any Damn boundary surface will do the trick. It must be a causal horizon in curved spacetime with pixelated thermodynamics area/entropy & conjugate Hawking temperature as in Tamara Davis's Fig 1.1 (2004 PhD online).
All the pseudo-physics nonsense of using holography in 5D and more seems silly to me unless they actually explain some real data in particle physics that cannot be explained otherwise - if so I will eat my words and change my opinion.