Black Hole Entropy in terms of Mass. Note Quote.

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If M-theory is compactified on a d-torus it becomes a D = 11 – d dimensional theory with Newton constant

GD = G11/Ld = l911/Ld —– (1)

A Schwartzschild black hole of mass M has a radius

Rs ~ M(1/(D-3)) GD(1/(D-3)) —– (2)

According to Bekenstein and Hawking the entropy of such a black hole is

S = Area/4GD —– (3)

where Area refers to the D – 2 dimensional hypervolume of the horizon:

Area ~ RsD-2 —– (4)

Thus

S ~ 1/GD (MGD)(D-2)/(D-3) ~ M(D-2)/(D-3) GD1/(D-3) —– (5)

From the traditional relativists’ point of view, black holes are extremely mysterious objects. They are described by unique classical solutions of Einstein’s equations. All perturbations quickly die away leaving a featureless “bald” black hole with ”no hair”. On the other hand Bekenstein and Hawking have given persuasive arguments that black holes possess thermodynamic entropy and temperature which point to the existence of a hidden microstructure. In particular, entropy generally represents the counting of hidden microstates which are invisible in a coarse grained description. An ultimate exact treatment of objects in matrix theory requires a passage to the infinite N limit. Unfortunately this limit is extremely difficult. For the study of Schwarzchild black holes, the optimal value of N (the value which is large enough to obtain an adequate description without involving many redundant variables) is of order the entropy, S, of the black hole.

Considering the minimum such value for N, we have

Nmin(S) = MRs = M(MGD)1/D-3 = S —– (6)

We see that the value of Nmin in every dimension is proportional to the entropy of the black hole. The thermodynamic properties of super Yang Mills theory can be estimated by standard arguments only if S ≤ N. Thus we are caught between conflicting requirements. For N >> S we don’t have tools to compute. For N ~ S the black hole will not fit into the compact geometry. Therefore we are forced to study the black hole using N = Nmin = S.

Matrix theory compactified on a d-torus is described by d + 1 super Yang Mills theory with 16 real supercharges. For d = 3 we are dealing with a very well known and special quantum field theory. In the standard 3+1 dimensional terminology it is U(N) Yang Mills theory with 4 supersymmetries and with all fields in the adjoint repersentation. This theory is very special in that, in addition to having electric/magnetic duality, it enjoys another property which makes it especially easy to analyze, namely it is exactly scale invariant.

Let us begin by considering it in the thermodynamic limit. The theory is characterized by a “moduli” space defined by the expectation values of the scalar fields φ. Since the φ also represents the positions of the original DO-branes in the non compact directions, we choose them at the origin. This represents the fact that we are considering a single compact object – the black hole- and not several disconnected pieces.

The equation of state of the system, defined by giving the entropy S as a function of temperature. Since entropy is extensive, it is proportional to the volume ∑3 of the dual torus. Furthermore, the scale invariance insures that S has the form

S = constant T33 —– (7)

The constant in this equation counts the number of degrees of freedom. For vanishing coupling constant, the theory is described by free quanta in the adjoint of U(N). This means that the number of degrees of freedom is ~ N2.

From the standard thermodynamic relation,

dE = TdS —– (8)

and the energy of the system is

E ~ N2T43 —– (9)

In order to relate entropy and mass of the black hole, let us eliminate temperature from (7) and (9).

S = N23((E/N23))3/4 —– (10)

Now the energy of the quantum field theory is identified with the light cone energy of the system of DO-branes forming the black hole. That is

E ≈ M2/N R —– (11)

Plugging (11) into (10)

S = N23(M2R/N23)3/4 —– (12)

This makes sense only when N << S, as when N >> S computing the equation of state is slightly trickier. At N ~ S, this is precisely the correct form for the black hole entropy in terms of the mass.

Spirit is Matter on the Seventh Plane; Matter is Spirit – on the Lowest Point of its Cyclic Activity; and Both — are MAYA. Note Quote.

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In the 1930s the scientist Sir James Jeans wrote:

the tendency of modem physics is to resolve the whole material universe into waves, and nothing but waves. These waves are of two kinds: bottled-up waves, which we call matter, and unbottled waves, which we call radiation or light. If annihilation of matter occurs, the process is merely that of unbottling imprisoned wave-energy and setting it free to travel through space. These concepts reduce the whole universe to a world of light, potential or existent . . . . — The Mysterious Universe

The idea of matter being crystallized light echoes what H. P. Blavatsky wrote half a century earlier in The Secret Doctrine, where she speaks of “that infinite Ocean of Light, whose one pole is pure Spirit lost in the absoluteness of Non-Being, and the other, the matter in which it condenses, crystallizing into a more and more gross type as it descends into manifestation” (The Secret Doctrine). Material particles, she said, were infinitely divisible centers of force, and matter could therefore exist in infinitely varying degrees of density. Our physical senses have been evolved to perceive only one particular plane of matter, which is interpenetrated by countless other worlds or planes invisible to us because composed of ranges of energy-substance both finer and grosser than our own.

Modern science has analyzed matter down to the point where it vanishes into wisps of energy. Energy is said to be a measure of motion or activity. But motion of what? It is a truism that there can be no motion without something that moves. Scientists in the last century believed that wave-motion took place in a universal medium called the ether. This hypothesis was abandoned because the ether proved to be chemically and physically undetectable, and science was left with the unlikely idea that waves are transmitted through “empty space.”

Modern physicists believe that underlying the material world there is a quantum field, also called the quantum void or vacuum. The quantum field is said to be “a continuous medium which is present everywhere in space” (The Tao of Physics) and matter is said to be constituted by regions of space in which the field is extremely intense. Scientists assert that the quantum field is non-material, but deny that it is mere nothingness. Paul Davies states that the quantum void is not inert and featureless but throbbing with energy and vitality, a seething ferment of “Virtual” particles and “ghost” particles. (Superforce) It therefore seems to be actually a form of ether, which is non-material only in the sense that it is not composed of physical matter. Rather than material particles being “knots of nothingness,” as Davies calls them, they may therefore be seen as vibrations in an etheric medium composed of a subtler, superphysical grade of substance. The same reasoning applies to all the other “non-material” fields and forces postulated by science.

Everything is relative. Physical matter is condensed energy, but what for us is energy would be matter for beings on a higher plane than ours, as is suggested by the fact that energy does not exist in a continuous flow but is composed of discrete units or quanta. Likewise, the energy on the next plane would be matter to an even higher plane. The loftiest form of energy in any particular hierarchy of worlds is what we call spirit or consciousness. As H. P. Blavatsky put it: “Spirit is matter on the seventh plane; matter is Spirit – on the lowest point of its cyclic activity; and both — are MAYA.” (The Secret Doctrine). To say that spirit and matter are “maya” or illusion does not mean that they do not exist, but that we do not understand them as they really are. Any particular plane of energy-substance can be understood only with reference to superior, causal planes. Everything — from atom to human, from star to universe — is the expression of something higher.

Throughout the ages, sages and seers have suggested that hidden within the phenomenal world in which we live there are inner worlds of reality — astral, mental, and spiritual — and that the physical world is but a pale shadow of the spiritual world. These inner worlds cannot be investigated with physical instruments, but only by delving into the depths of our own minds and consciousness, and this requires many lives of self-purification and self-conquest. Scientists using only materialistic methods are in no position to deny point-blank the possibility of such higher planes.

Most scientists, in fact, now believe that some 90% of the matter in the universe exists in a state unknown to them; it is called “dark matter” because it is physically unobservable, and its existence is known of only by its gravitational effects. Such matter is suggestive of the higher subplanes and planes postulated by theosophy, which are composed of matter of increasingly slower rates of vibration and are therefore beyond our range of perception. Given scientists’ confessed ignorance of most of the matter in the universe and their inability to explain satisfactorily the evolution of life and consciousness and the “laws of nature” along materialistic lines, any suggestion that they are on the verge of discovering the innermost secrets of nature or of reducing the mystery of existence to a single equation is premature to say the least!

In theosophical philosophy, the physical universe is regarded as no more than a cross section through infinitude. Universal nature is composed of worlds within worlds within worlds, filled full of conscious, living beings at infinitely varying stages of their evolutionary awakenment. Our finite minds cannot embrace the infinite. As G. de Purucker says in his Fundamentals of the Esoteric Philosophy, we can do no more than to try and form a simple conception of the Boundless All: never-ending life and consciousness in unceasing motion everywhere. The ancients, he says, were never so foolish as to try to fathom infinitude. They recognized the reality of being and let it go at that, knowing that an ever-expanding consciousness and an ever-growing understanding of existence is all that we can ever attain to during our eternal evolutionary journey through the fields of infinitude.

Osteo Myological Quantization. Note Quote.

The site of the parameters in a higher order space can also be quantized into segments, the limits of which can be no more decomposed. Such a limit may be nearly a rigid piece. In the animal body such quanta cannot but be bone pieces forming parts of the skeleton, whether lying internally as [endo]-skeleton or as almost rigid shell covering the body as external skeleton.

Note the partition of the body into three main segments: Head (cephalique), pectral (breast), caudal (tail), materializing the KH order limit M>= 3 or the KHK dimensional limit N>= 3. Notice also the quantization into more macroscopic segments such as of the abdominal part into several smaller segments beyond the KHK lower bound N=3. Lateral symmetry with a symmetry axis is remarkable. This is of course an indispensable consequence of the modified Zermelo conditions, which entails also locomotive appendages differentiating into legs for walking and wings for flying in the case of insects.

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Two paragraphs of Kondo addressing the simple issues of what bones are, mammalian bi-lateral symmetry, the numbers of major body parts and their segmentation, the notion of the mathematical origins of wings, legs and arms. The dimensionality of eggs being zero, hence their need of warmth for progression to locomotion and the dimensionality of snakes being one, hence their mode of locomotion. A feature of the biological is their attention to detail, their use of line art to depict the various forms of living being – from birds to starfish to dinosaurs, the use of the full latin terminology and at all times the relationship of the various form of living being to the underlying higher order geometry and the mathematical notion of principle ideals. The human skeleton is treated as a hierarchical Kawaguchi tree with its characteristic three pronged form. The Riemannian arc length of the curve k(t) is given by the integral of the square root of a quadratic form in x’ with coefficients dependent in x’. This integrand is homogenous of the first order in x’. If we drop the quadratic property and retain the homogeneity, then we obtain the Finsler geometry. Kawaguchi geometry supposes that the integrand depends upon the higher derivatives x’’ up to the k-th derivative xk. The notation that Kondo uses is:

K(M)L,N

For:

L Parameters N Dimensions M Derivatives

The lower part of the skeleton can be divided into three prongs, each starting from the centre as a single parametric Kawaguchi tree.

…the skeletal, muscular, gastrointestinal, circulation systems etc combine into a holo-parametric whole that can be more generally quantized, each quantum involving some osteological, neural, circulatory functions etc.

…thus globally the human body from head through trunk to limbs are quantized into a finite number of quanta.