# Black Hole Entropy in terms of Mass. Note Quote.

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.

All perceptible matter comes from a primary substance, or tenuity beyond conception, filling all space, the akasha or luminiferous ether, which is acted upon by the life giving Prana or creative force, calling into existence, in never-ending cycles all things and phenomena – Nikola Tesla

Teilhard de Chardin:

In the eyes of the physicist, nothing exists legitimately, at least up to now, except the without of things. The same intellectual attitude is still permissible in the bacteriologist, whose cultures (apart from substantial difficulties) are treated as laboratory reagents. But it is still more difficult in the realm of plants. It tends to become a gamble in the case of a biologist studying the behavior of insects or coelenterates. It seems merely futile with regard to the vertebrates. Finally, it breaks down completely with man, in whom the existence of a within can no longer be evaded, because it is a subject of a direct intuition and the substance of all knowledge. It is impossible to deny that, deep within ourselves, “an interior” appears at the heart of beings, as it were seen through a rent. This is enough to ensure that, in one degree or another, this “interior” should obtrude itself as existing everywhere in nature from all time. Since the stuff of the universe has an inner aspect at one point of itself, there is necessarily a double to its structure, that is to say in every region of space and time-in the same way for instance, as it is granular: co-extensive with their Without, there is a Within to things.

Both Indian thought and modern scientific thought accept a fundamental unity behind the world of variety. That basic unitary reality evolves into all that we see around us in the world. This view is a few thousand years old in India; We find it in the Samkhyan and Vedantic schools of Indian thought; and they expound it very much on the lines followed by modern thought. In his address to the Chicago Parliament of Religions in 1893, Vivekananda said:

All science is bound to come to this conclusion in the long run, Manifestation and not creation, is the word of science today, and the Hindu is only glad that what he has been cherishing in his bosom for ages is going to be taught in more forcible language, and with further light from the latest conclusions of science.

The Samkhyan school uses two terms to represent Nature or Pradhana: Prakrti denoting Nature in its unmodified state, and Vikrti denoting nature in its modified state. The Vedanta similarly speaks of Brahman as the inactive state, and Maya or Shakti as the active state of one and the same primordial non-dual reality. But the Brahman of the Vedanta is the unity of both the spiritual and the non-spiritual, the non-physical and the physical aspects of the universe.

So as the first answer to the question, ‘What is the world?’ we get this child’s answer in his growing knowledge of the discrete entities and events of the outer world and their inter-connections. The second answer is the product of scientific thought, which gives us the knowledge of the one behind the many. All the entities and events of the world are but the modifications or evolutions of one primordial basis reality, be it nature, space- time or cosmic dust.

Although modern scientific thought does not yet have a place for any spiritual reality or principle, scientists like Chardin and Julian Huxley are trying to find a proper place for the experience of the spiritual in the scientific picture of the universe. When this is achieved, the scientific picture, which is close to Vedanta already, will become closer still, and the synthesis of the knowledge of the ‘without’ and the ‘within’ of things will give us the total view of the universe. This is wisdom according to Vedanta, whereas all partial views are just pieces of knowledge or information only.

The Upanishads deal with this ‘within’ of things. Theirs in fact, is the most outstanding contribution on this subject in the human cultural legacy. They term this aspect of reality of things pratyak chaitanya or pratyak atman or pratyak tattva; and they contain the fascinating account of the stages by which the human mind rose from crude beginnings to clear, wholly spiritual heights in the realization of this reality.

How does the world look when we view it from the outside? We seek an answer from the physical sciences. How does it look when we view from the inside? We seek an answer from the non-physical sciences, including the science of religion. And philosophy, as understood in the Upanishadic tradition, is the synthesis of these two answers: Brahmavidyā is Sarvavidyāpratishthā, as the Mundaka Upanishad puts it.

क्षेत्रक्षेत्रज्ञयोर्ज्ञानं यत्तज्ज्ञानम् मतं मम

kṣetrakṣetrajñayorjñānaṃ yattajjñānam mataṃ mama

“The unified knowledge of the ‘without’ and the ‘within’ of things is true knowledge according to Me, as Krishna says in the Gita” (Bhagavad-Gita chapter 13, 2nd Shloka).

From this total viewpoint there is neither inside nor outside; they are relative concepts depending upon some sort of a reference point, e.g.the body; as such, they move within the framework of relativity. Reality knows neither ‘inside’ nor ‘outside’; it is ever full. But these relative concepts are helpful in our approach to the understanding of the total reality.

Thus we find that our knowledge of the manifold of experience the idam, also involves something else, namely, the unity behind the manifold. This unity behind the manifold, which is not perceptible to the senses, is indicated by the term adah meaning ‘that’, indicating something far away, unlike the ‘this’ of the sense experience. ‘This’ is the correlative of ‘that’; ‘this’ is the changeable aspect of reality; ‘that’ is its unchangeable aspect. If ‘this’ refers to something given in sense experience, ‘that’ refers to something transcendental, beyond the experience of the senses. To say ‘this’ therefore also implies at the same time something that is beyond ‘this’. This is an effect as such, it is visible and palpable; and behind it lies the cause, the invisible and the impalpable. Adah, ‘that’, represents the invisible behind the visible, the transcendental behind the empirical, a something that is beyond time and space. In religion this something is called ‘God’. In philosophy it is called tat or adah, That, Brahman, the ultimate Reality, the cause, the ground, and the goal of the universe.

So this verse first tells us that beyond and behind the manifested universe is the reality of Brahman, which is the fullness of pure Being; it then tells us about this world of becoming which, being nothing but Brahman, is also the ‘Full’. From the view of total Reality, it is all ‘fullness’ everywhere, in space-time as well as beyond space-time. Then the verse adds:

पूर्णस्य पूर्णमादाय पूर्णमेवाशिष्यते

pūrṇasya pūrṇamādāya pūrṇamevāśiṣyate

‘From the Fullness of Brahman has come the fullness of the universe, leaving alone Fullness as the remainder.’

What, then, is the point of view or level from which the sentiments of this verse proceed? It is that of the total Reality, the Absolute and the Infinite, in which as we have read earlier, the ‘within’ and the ‘without’ of things merge. The Upanishads call it as ocean of Sachchidānanda, the unity of absolute existence, absolute awareness, and absolute bliss. Itself beyond all distinctions of time and space, it yet manifests itself through all such distinctions. To the purified vision of the Upanishadic sages, this whole universe appeared as the fullness of Being, which was, which is, which shall ever be. In the Bhagavad-Gita (VII. 26) Krshna says:

वेदाहं समतीतानि वर्तमानानि चार्जुन ।
भविष्याणि च भूतानि मां तु वेद न कश्चन ॥

vedāhaṃ samatītāni vartamānāni cārjuna |
bhaviṣyāṇi ca bhūtāni māṃ tu veda na kaścana ||

‘I, O Arjuna, know the beings that are of the past, that are of the present, and that are to come in future; but Me no one knows.’

That fullness of the true Me, says Krshna, is beyond all these limited categories, such as space and time, cause and effect, and substance and attribute.

# Chinese Philosophical Pythagoreanism. Thought of the Day 14.0

The universe is just like a right-angled triangle of standard unit. The two right-angle sides oppose each other and yet also complement each other like quantum entanglement. They superpose and counterbalance each other in a system. They grow and decline, but identically equal to “1”. From this perspective, the Traditional Yin-and-Yang Double Fish Diagram and the core idea behind it is counterbalance and unity of opposites, a geometric expression of “being of beings” and an illustration of the ground of “being”, or the “being” in “being of beings”, the “one” in “one is all”. It roots in the Pythagorean Theorem and goes beyond the Pythagorean Theorem. It illustrates the idea in the Euler Equation:

e·e−iθ =(cosθ + isinθ (cosθ − isinθ) = cos2θ + sin2θ = 1),

and has opened the door of relativity, expressing the key idea of Fuzzy Set in modern mathematics developed in contemporary and modern times. “If each right-angled side multiple itself, the sum will be the square of hypotenuse. The extracted root is hypotenuse”. Thus, the xian (in Chinese meaning either “hypotenuse” or “profound”) is “profound” because we don’t understand the principle and add too much mysterious explanations to it.

This reminds us of the “Needham question”, why didn’t science rise in China? And the emotional sigh of Mr. Liang Shumin, “The Chinese culture is a pre-matured one of mankind.” In a material desire-pursuing physical world, metaphysics is so lonely and with- ering. As Heidegger pointed out, truth is hardly acceptable just because it’s too simple. Facing the rapidly-changing advancement of science, we need inquire with earnestness and reflect with self-practice, and ask ourselves whether we need “lead people to perfection” and “having known where to reset at the end, one will be able to determine the object of pursuit”, and try our best to “return to things themselves”; Facing the cultural flourishing, whether we should keep to the original intention of “speech” and come back to the original point of language.

Facing the complicated world, human beings need another renaissance to search, keep and return to the “1” to create “a community of common destiny”. The philosophical element of traditional Chinese culture and the dialectical unity of Marxism could make continuing contributions to the aim since whether ancient or modern, Chinese or foreign, arts or sciences, they must be “unified” in basic principle of the universe.

# Whitehead’s Non-Anthropocentric Quantum Field Ontology. Note Quote.

Whitehead builds also upon James’s claim that “The thought is itself the thinker”.

Either your experience is of no content, of no change, or it is of a perceptible amount of content or change. Your acquaintance with reality grows literally by buds or drops of perception. Intellectually and on reflection you can divide them into components, but as immediately given they come totally or not at all. — William James.