Witten's News
Here is a bookmark on the new work of Witten on exact black hole solutions and drinking moonshine. Matti Pitkanen has an informative post on the connection with TGD, and of course our colleague Motl also has a post.
An interesting fact about the Leech lattice is discussed by Tony Smith: the neighbours of the origin in 24 dimensional surreal number space form a Leech lattice. Recall that the surreal numbers give a tree structure to the reals, and as we know, trees are more fundamental than real numbers.
Speaking of canonical constructions for the reals, recall that in using associahedra, we came across integral arguments for the Riemann zeta function. But the full Riemann hypothesis requires us to understand the function on the complex plane. The surreal tree is a binary logic construction of the reals. A Space Time complexification could give us a notion of complex number.
An interesting fact about the Leech lattice is discussed by Tony Smith: the neighbours of the origin in 24 dimensional surreal number space form a Leech lattice. Recall that the surreal numbers give a tree structure to the reals, and as we know, trees are more fundamental than real numbers.
Speaking of canonical constructions for the reals, recall that in using associahedra, we came across integral arguments for the Riemann zeta function. But the full Riemann hypothesis requires us to understand the function on the complex plane. The surreal tree is a binary logic construction of the reals. A Space Time complexification could give us a notion of complex number.
posted by Kea | 10:31 AM
4 Comments:
Mach's relationism isn't proved. He claimed that the earth's rotation is just relative to the stars. However, the Foucault pendulum proves that absolute rotation can be determined without reference to the stars. Mach could only counter that objection by his faith (without any evidence at all) that if the earth was not rotating, but the stars were instead rotating around it, then the Coriolis force would still exist and deflect a Foucault pendulum.
To check Mach's principle, instead of the complex situation of the earth and the stars (where the dynamical relationship is by consensus unknown until there is a quantum gravity to explain inertia by the equivalence principle), consider the better understood situation of a proton with an electron orbiting it.
From classical mechanics, neglecting therefore the normal force causing exchange (equilibrium) radiation which constitutes fields, there should be a net (non equilibrium) emission of radiation by accelerating charge.
If you consider a system with an electron and a proton nearby but not in orbit, they will be attracted by Coulomb's law. (The result of the normal force-causing exchange radiation.)
Now, consider the electron in orbit. Because it is accelerating with acceleration a = (v^2)/r, it is continuously emitting radiation in a direction perpendicular to its orbit; i.e., generally the direction of the radiation is the radial line connecting the electron with the proton.
Because the electron describes a curved orbit, the angular distribution of its radiation is asymmetric with more being emitted generally towards the nucleus than in the opposite direction.
The recoil of the electron from firing off radiation is therefore in a direction opposite to the centripetal Coulomb attraction force. This is why how the "centrifugal" effect works.
What is so neat is that no loss kinetic energy occurs to the electron. T.H. Boyer in 1975 (Physical Review D, v11, p790) suggested that the ground state orbit is a balance between radiation emitted due to acceleration and radiation absorbed from the vacuum's zero point radiation field caused by all the other accelerating charges which are also radiating in the universe surrounding any particular atom.
H.E. Puthoff in 1987 (Physical Review D v35, p3266, "Ground state of hydrogen as a zero-point-fluctuation-determined state") assumed that the Casimir force causing zero-point electromagnetic radiation had an energy spectrum
Rho(Omega)d{Omega} = {h bar}[{Omega}^3]/[2(Pi^2)*(c^3)] d{Omega}
which causes an electron in a circular orbit to absorb radiation from the zero-point field with the power
P = (e^2)*{h bar}{Omega^3}/(6*Pi*Epsilon*mc^3)
Where e is charge, Omega is angular frequency, and Epsilon is permittivity. Since the power radiated by an electron with acceleration a = r*{Omega^2} is:
P = (e^2)*(a^2)/(6*Pi*Epsilon*c^3),
equating the power the electron receives from the zero-point field to the power it radiates due to its orbit gives
m*{Omega}*(r^2) = h bar,
which is the ground state of hydrogen. Puthoff writes:
"... the ground state of the hydrogen atom can be precisely defined as resulting from a dynamic equilibrium between radiation emitted due to acceleration of the electron in its ground-state orbit and radiation absorbed from zero-point fluctuations of the background vacuum electromagnetic field, thereby resolving the issue of radiative collapse of the Bohr atom."
This model dispenses with Mach's principle. An electron orbiting a proton is not equivalent to the proton rotating while the electron remains stationary; one case results in acceleration of the electron and radiation emission, while the other doesn't.
The same arguments will apply to the case of the earth rotating, or the stars orbiting a stationary earth, although some kind of quantum gravity/inertia theory is there required for the details.
One thing I disagree with Puthoff over is the nature of the zero-point field. Nobody seems to be aware that the IR cutoff and the Schwinger requirement for a minimum electric field strength of 10^18 v/m, prevents the entire vacuum from being subject to creation/annihilation loop operators. Quantum field theory only applies to the fields above the IR cutoff, or closer than 10^{-15} metre to a charge.
Beyond that distance, there's no pair production in the vacuum whatsoever, so all you have is radiation. In general, the "zero-point field" is the gauge boson exchange radiation field which causes forces. The Casimir force works because long wavelengths of the zero-point field radiation are excluded from the space between two metal plates, which therefore shield one another and get pushed together like two suction cups being pushed together by air pressure when normal air pressure is reduced in the small gap between them.
Puthoff has an interesting paper, "Source of the vacuum electromagnetic zero-point energy" in Physical Review D, v40, p4857 (1989) [note that an error in this paper is corrected by Puthoff in an update published in Physical Review D v44, p3385 (1991)]:
"... the zero-poing energy spectrum (field distribution) drives particle motion ... the particle motion in turn generates the zero-point energy spectrum, in the form of a self-regenerating cosmological feedback cycle. The result is the appropriate frequency-cubed spectral distribution of the correct order of magnitude, this indicating a dynamic-generation process for the zero-poing energy fields."
What interests me is that Puthoff's calculations in that paper tackle the same problems which I had to deal with over the last decade in regards to a gravity mechanism. Puthoff writes that since the radiation intensity from any charge will fall as 1/r^2, and since charges in shells of thickness dr will have an area of 4*Pi*r^2, the increasing number of charges at bigger distances offsets the inverse square law of radiation, so you get a version of Obler's paradox appearing.
In addition, Puthoff notes that:
"... in an expanding universe radiation arriving from a particular shell located now at a distance was emitted at an earlier time, from a more compacted shell."
This effect tends to make Obler's paradox even more severe, because the earlier universe we see at great distances should be more and more compressed with distance, and ever brighter.
Redshift of radiation emitted from receding matter at such great distances solves these problems.
However, Puthoff assumes that some already known metric of general relativity is correct, which clearly is false because of the redshift of gauge bosons in an expanding universe will weaken the gravitational coupling constant between receding (distant) masses, a fact that all widely accepted general relativity metrics totally ignore!
Sorry for the length of this comment, and feel free to delete this comment (I'll put a copy on my blog).
Duh, is this what you are hinting at with those honeycombs.
So what? I don't get it. :-]
Hi nosy snoopy. Well, actually, we've already pretty well figured out that diagram, and it's not really that interesting. Ask kneemo.
Nigel, PLEASE stop doing this. I might start deleting long posts.
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