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)


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.

Gauge Fixity Towards Hyperbolicity: General Theory of Relativity and Superpotentials. Part 1.


Gravitational field is described by a pseudo-Riemannian metric g (with Lorentzian signature (1, m-1)) over the spacetime M of dimension dim(M) = m; in standard General Relativity, m = 4. The configuration bundle is thence the bundle of Lorentzian metrics over M, denoted by Lor(M) . The Lagrangian is second order and it is usually chosen to be the so-called Hilbert Lagrangian:

LH: J2Lor(m) → ∧om(M)

LH: LH(gαβ, Rαβ)ds = 1/2κ (R – 2∧)√g ds —– (1)


R = gαβ Rαβ denotes the scalar curvature, √g the square root of the absolute value of the metric determinant and ∧ is a real constant (called the cosmological constant). The coupling constant 1/2κ which is completely irrelevant until the gravitational field is not coupled to some other field, depends on conventions; in natural units, i.e. c = 1, h = 1, G = 1, dimension 4 and signature ( + , – , – , – ) one has κ = – 8π.

Field equations are the well known Einstein equations with cosmological constant

Rαβ – 1/2 Rgαβ = -∧gαβ —— (2)

Lagrangian momenta is defined by:

pαβ = ∂LH/∂gαβ = 1/2κ (Rαβ – 1/2(R – 2∧)gαβ)√g

Pαβ = ∂LH/∂Rαβ = 1/2κ gαβ√g —– (3)

Thus the covariance identity is the following:

dα(LHξα) = pαβ£ξgαβ + Pαβ£ξRαβ —– (4)

or equivalently,

α(LHξα) = pαβ£ξgαβ + PαβεξΓεαβ – δεβ£ξΓλαλ) —– (5)

where ∇ε denotes the covariant derivative with respect to the Levi-Civita connection of g. Thence we have a weak conservation law for the Hilbert Lagrangian

Div ε(LH, ξ) = W(LH, ξ) —– (6)

Conserved currents and work forms have respectively the following expressions:

ε(LH, ξ) = [Pαβ£ξΓεαβ – Pαε£ξΓλαλ – LHξε]dsε = √g/2κ(gαβgεσ – gσβgεα) ∇α£ξgβσdsε – √g/2κξεRdsε = √g/2κ[(3/2Rαλ – (R – 2∧)δαλλ + (gβγδαλ – gα(γδβ)λβγξλ]dsα —– (7)

W(LH, ξ) = √g/κ(Rαβ – 1/2(R – 2∧)gαβ)∇(αξβ)ds —– (8)

As any other natural theory, General Relativity allows superpotentials. In fact, the current can be recast into the form:

ε(LH, ξ) = ε'(LH, ξ) + Div U(LH, ξ) —– (9)

where we set

ε'(LH, ξ) = √g/κ(Rαβ – 1/2(R – 2∧)δαββ)dsα

U(LH, ξ) = 1/2κ ∇[βξα] √gdsαβ —– (10)

The superpotential (10) generalizes to an arbitrary vector field ξ, the well known Komar superpotential which is originally derived for timelike Killing vectors. Whenever spacetime is assumed to be asymptotically fiat, then the superpotential of Komar is known to produce upon integration at spatial infinity ∞ the correct value for angular momentum (e.g. for Kerr-Newman solutions) but just one half of the expected value of the mass. The classical prescriptions are in fact:

m = 2∫ U(LH, ∂t, g)

J = ∫ U(LH, ∂φ, g) —– (11)

For an asymptotically flat solution (e.g. the Kerr-Newman black hole solution) m coincides with the so-called ADM mass and J is the so-called (ADM) angular momentum. For the Kerr-Newman solution in polar coordinates (t, r, θ, φ) the vector fields ∂t and ∂φ are the Killing vectors which generate stationarity and axial symmetry, respectively. Thence, according to this prescription, U(LH, ∂φ) is the superpotential for J while 2U(LH, ∂t) is the superpotential for m. This is known as the anomalous factor problem for the Komar potential. To obtain the expected values for all conserved quantities from the same superpotential, one has to correct the superpotential (10) by some ad hoc additional boundary term. Equivalently and alternatively, one can deduce a corrected superpotential as the canonical superpotential for a corrected Lagrangian, which is in fact the first order Lagrangian for standard General Relativity. This can be done covariantly, provided that one introduces an extra connection Γ’αβμ. The need of a reference connection Γ’ should be also motivated by physical considerations, according to which the conserved quantities have no absolute meaning but they are intrinsically relative to an arbitrarily fixed vacuum level. The simplest choice consists, in fact, in fixing a background metric g (not necessarily of the correct Lorentzian signature) and assuming Γ’ to be the Levi-Civita connection of g. This is rather similar to the gauge fixing à la Hawking which allows to show that Einstein equations form in fact an essentially hyperbolic PDE system. Nothing prevents, however, from taking Γ’ to be any (in principle torsionless) connection on spacetime; also this corresponds to a gauge fixing towards hyperbolicity.

Now, using the term background for a field which enters a field theory in the same way as the metric enters Yang-Mills theory, we see that the background has to be fixed once for all and thence preserved, e.g. by symmetries and deformations. A background has no field equations since deformations fix it; it eventually destroys the naturality of a theory, since fixing the background results in allowing a smaller group of symmetries G ⊂ Diff(M). Accordingly, in truly natural field theories one should not consider background fields either if they are endowed with a physical meaning (as the metric in Yang-Mills theory does) or if they are not.

On the contrary we shall use the expression reference or reference background to denote an extra dynamical field which is not endowed with a direct physical meaning. As long as variational calculus is concerned, reference backgrounds behave in exactly the same way as other dynamical fields do. They obey field equations and they can be dragged along deformations and symmetries. It is important to stress that such a behavior has nothing to do with a direct physical meaning: even if a reference background obeys field equations this does not mean that it is observable, i.e. it can be measured in a laboratory. Of course, not any dynamical field can be treated as a reference background in the above sense. The Lagrangian has in fact to depend on reference backgrounds in a quite peculiar way, so that a reference background cannot interact with any other physical field, otherwise its effect would be observable in a laboratory….



If e0 ∈ R1+1 is a future-directed timelike unit vector, and if e1 is the unique spacelike unit vector with e0e1 = 0 that “points to the right,” then coordinates x0 and x1 on R1+1 are defined by x0(q) := qe0 and x1(q) := qe1. The partial differential operator

x : = ∂2x0 − ∂2x1

does not depend on the choice of e0.

The Fourier transform of the Klein-Gordon equation

(□ + m2)u = 0 —– (1)

where m > 0 is a given mass, is

(−p2 + m2)û(p) = 0 —– (2)

As a consequence, the support of û has to be a subset of the hyperbola Hm ⊂ R1+1 specified by the condition p2 = m2. One connected component of Hm consists of positive-energy vectors only; it is called the upper mass shell Hm+. The elements of Hm+ are the 4-momenta of classical relativistic point particles.

Denote by L1 the restricted Lorentz group, i.e., the connected component of the Lorentz group containing its unit element. In 1 + 1 dimensions, L1 coincides with the one-parameter Abelian group B(χ), χ ∈ R, of boosts. Hm+ is an orbit of L1 without fixed points. So if one chooses any point p′ ∈ Hm+, then there is, for each p ∈ Hm+, a unique χ(p) ∈ R with p = B(χ(p))p′. By construction, χ(B(ξ)p) = χ(p) + ξ, so the measure dχ on Hm+ is invariant under boosts and does note depend on the choice of p′.

For each p ∈ Hm+, the plane wave q ↦ e±ipq on R1+1 is a classical solution of the Klein-Gordon equation. The Klein-Gordon equation is linear, so if a+ and a are, say, integrable functions on Hm+, then

F(q) := ∫Hm+ (a+(p)e-ipq + a(p)eipq dχ(p) —– (3)

is a solution of the Klein-Gordon equation as well. If the functions a± are not integrable, the field F may still be well defined as a distribution. As an example, put a± ≡ (2π)−1, then

F(q) = (2π)−1 Hm+ (e-ipq + eipq) dχ(p) = π−1Hm+ cos(pq) dχ(p) =: Φ(q) —– (4)

and for a± ≡ ±(2πi)−1, F equals

F(q) = (2πi)−1Hm+ (e-ipq – eipq) dχ(p) = π−1Hm+ sin(pq) dχ(p) =: ∆(q) —– (5)

Quantum fields are obtained by “plugging” classical field equations and their solutions into the well-known second quantization procedure. This procedure replaces the complex (or, more generally speaking, finite-dimensional vector) field values by linear operators in an infinite-dimensional Hilbert space, namely, a Fock space. The Hilbert space of the hermitian scalar field is constructed from wave functions that are considered as the wave functions of one or several particles of mass m. The single-particle wave functions are the elements of the Hilbert space H1 := L2(Hm+, dχ). Put the vacuum (zero-particle) space H0 equal to C, define the vacuum vector Ω := 1 ∈ H0, and define the N-particle space HN as the Hilbert space of symmetric wave functions in L2((Hm+)N, dNχ), i.e., all wave functions ψ with

ψ(pπ(1) ···pπ(N)) = ψ(p1 ···pN)

∀ permutations π ∈ SN. The bosonic Fock space H is defined by

H := ⊕N∈N HN.

The subspace

D := ∪M∈N ⊕0≤M≤N HN is called a finite particle space.

The definition of the N-particle wave functions as symmetric functions endows the field with a Bose–Einstein statistics. To each wave function φ ∈ H1, assign a creation operator a+(φ) by

a+(φ)ψ := CNφ ⊗s ψ, ψ ∈ D,

where ⊗s denotes the symmetrized tensor product and where CN is a constant.

(a+(φ)ψ)(p1 ···pN) = CN/N ∑v φ(pν)ψ(pπ(1) ···p̂ν ···pπ(N)) —– (6)

where the hat symbol indicates omission of the argument. This defines a+(φ) as a linear operator on the finite-particle space D.

The adjoint operator a(φ) := a+(φ) is called an annihilation operator; it assigns to each ψ ∈ HN, N ≥ 1, the wave function a(φ)ψ ∈ HN−1 defined by

(a(φ)ψ)(p1 ···pN) := CN ∫Hm+ φ(p)ψ(p1 ···pN−1, p) dχ(p)

together with a(φ)Ω := 0, this suffices to specify a(φ) on D. Annihilation operators can also be defined for sharp momenta. Namely, one can define to each p ∈ Hm+ the annihilation operator a(p) assigning to

each ψ ∈ HN, N ≥ 1, the wave function a(p)ψ ∈ HN−1 given by

(a(p)ψ)(p1 ···pN−1) := Cψ(p, p1 ···pN−1), ψ ∈ HN,

and assigning 0 ∈ H to Ω. a(p) is, like a(φ), well defined on the finite-particle space D as an operator, but its hermitian adjoint is ill-defined as an operator, since the symmetric tensor product of a wave function by a delta function is no wave function.

Given any single-particle wave functions ψ, φ ∈ H1, the commutators [a(ψ), a(φ)] and [a+(ψ), a+(φ)] vanish by construction. It is customary to choose the constants CN in such a fashion that creation and annihilation operators exhibit the commutation relation

[a(φ), a+(ψ)] = ⟨φ, ψ⟩ —– (7)

which requires CN = N. With this choice, all creation and annihilation operators are unbounded, i.e., they are not continuous.

When defining the hermitian scalar field as an operator valued distribution, it must be taken into account that an annihilation operator a(φ) depends on its argument φ in an antilinear fashion. The dependence is, however, R-linear, and one can define the scalar field as a C-linear distribution in two steps.

For each real-valued test function φ on R1+1, define

Φ(φ) := a(φˆ|Hm+) + a+(φˆ|Hm+)

then one can define for an arbitrary complex-valued φ

Φ(φ) := Φ(Re(φ)) + iΦ(Im(φ))

Referring to (4), Φ is called the hermitian scalar field of mass m.

Thereafter, one could see

[Φ(q), Φ(q′)] = i∆(q − q′) —– (8)

Referring to (5), which is to be read as an equation of distributions. The distribution ∆ vanishes outside the light cone, i.e., ∆(q) = 0 if q2 < 0. Namely, the integrand in (5) is odd with respect to some p′ ∈ Hm+ if q is spacelike. Note that pq > 0 for all p ∈ Hm+ if q ∈ V+. The consequence of this is called microcausality: field operators located in spacelike separated regions commute (for the hermitian scalar field).

Highest Reality. Thought of the Day 70.0


यावचिन्त्यावात्मास्य शक्तिश्चैतौ परमार्थो भवतः॥१॥

Yāvacintyāvātmāsya śaktiścaitau paramārtho bhavataḥ

These two (etau), the Self (ātmā) and (ca) His (asya) Power (śaktiḥ) —who (yau) (are) inconceivable (acintyau)—, constitute (bhavataḥ) the Highest Reality (parama-arthaḥ)

The Self is the Core of all, and His Power has become all. I call the Core “the Self” for the sake of bringing more light instead of more darkness. If I had called Him “Śiva”, some people might consider Him as the well-known puranic Śiva who is a great ascetic living in a cave and whose main task consists in destroying the universe, etc. Other people would think that, as Viṣṇu is greater than Śiva, he should be the Core of all and not Śiva. In turn, there is also a tendency to regard Śiva like impersonal while Viṣṇu is personal. There is no end to spiritual foolishness indeed, because there is no difference between Śiva and Viṣṇu really. Anyway, other people could suggest that a better name would be Brahman, etc. In order not to fall into all that ignorant mess of names and viewpoints, I chose to assign the name “Self” to the Core of all. In the end, when spiritual enlightenment arrives, one’s own mind is withdrawn (as I will tell by an aphorism later on), and consequently there is nobody to think about if “This Core of all” is personal, impersonal, Śiva, Viṣṇu, Brahman, etc. Ego just collapses and This that remains is the Self as He essentially is.

He and His Power are completely inconceivable, i.e. beyond the mental sphere. The Play of names, viewpoints and such is performed by His Power, which is always so frisky. All in all, the constant question is always: “Is oneself completely free like the Self?”. If the answer is “Yes”, one has accomplished the goal of life. And if the answer is “No”, one must get rid of his own bondage somehow then. The Self and His Power constitute the Highest Reality. Once you can attain them, so to speak, you are completely free like Them both. The Self and His Power are “two” only in the sphere of words, because as a matter of fact they form one compact mass of Absolute Freedom and Bliss. Just as the sun can be divided into “core of the sun, surface of the sun, crown”, etc.

तयोरुभयोः स्वरूपं स्वातन्त्र्यानन्दात्मकैकघनत्वेनापि तत्सन्तताध्ययनाय वचोविषय एव द्विधाकृतम्

Tayorubhayoḥ svarūpaṁ svātantryānandātmakaikaghanatvenāpi tatsantatādhyayanāya vacoviṣaya eva dvidhākṛtam

Even though (api) the essential nature (sva-rūpam) of Them (tayoḥ) both (ubhayoḥ) (is) one compact mass (eka-ghanatvena) composed of (ātmaka) Absolute Freedom (svātantrya)(and) Bliss (ānanda), it is divided into two (dvidhā-kṛtam) —only (eva) in the sphere (viṣaye) of words (vacas)— for its close study (tad-santata-adhyayanāya)

The Self is Absolute Freedom and His Power is Bliss. Both form a compact mass (i.e. omnipresent). In other words, the Highest Reality is always “One without a second”, but, in the world of words It is divided into two for studying It in detail. When this division occurs, the act of coming to recognize or realize the Highest Reality is made easier. So, the very Highest Reality generates this division in the sphere of words as a help for the spiritual aspirants to realize It faster.

आत्मा प्रकाशात्मकशुद्धबोधोऽपि सोऽहमिति वचोविषये स्मृतः

Ātmā prakāśātmakaśuddhabodho’pi so’hamiti vacoviṣaye smṛtaḥ

Although (api) the Self (ātmā) (is) pure (śuddha) Consciousness (bodhaḥ) consisting of (ātmaka) Prakāśa or Light (prakāśa), He (saḥ) is called (smṛtaḥ) “I” (aham iti) in the sphere (viṣaye) of words (vacas)

The Self is pure Consciousness, viz. the Supreme Subject who is never an object. Therefore, He cannot be perceived in the form of “this” or “that”. He cannot even be delineated in thought by any means. Anyway, in the world of words, He is called “I” or also “real I” for the sake of showing that He is higher than the false “I” or ego.

Black Holes. Thought of the Day 23.0


The formation of black holes can be understood, at least partially, within the context of general relativity. According to general relativity the gravitational collapse leads to a spacetime singularity. But this spacetime singularity can not be adequately described within general relativity, because the equivalence principle of general relativity is not valid for spacetime singularities; therefore, general relativity does not give a complete description of black holes. The same problem exists with regard to the postulated initial singularity of the expanding cosmos. In these cases, quantum mechanics and quantum field theory also reach their limit; they are not applicable for highly curved spacetimes. For a certain curving parameter (the famous Planck scale), gravity has the same strength as the other interactions; then it is not possible to ignore gravity in the context of a quantum field theoretical description. So, there exists no theory which would be able to describe gravitational collapses or which could explain, why (although they are predicted by general relativity) they don’t happen, or why there is no spacetime singularity. And the real problems start, if one brings general relativity and quantum field theory together to describe black holes. Then it comes to rather strange forms of contradictions, and the mutual conceptual incompatibility of general relativity and quantum field theory becomes very clear:

Black holes are according to general relativity surrounded by an event horizon. Material objects and radiation can enter the black hole, but nothing inside its event horizon can leave this region, because the gravitational pull is strong enough to hold back even radiation; the escape velocity is greater than the speed of light. Not even photons can leave a black hole. Black holes have a mass; in the case of the Schwarzschild metrics, they have exclusively a mass. In the case of the Reissner-Nordström metrics, they have a mass and an electric charge; in case of the Kerr metrics, they have a mass and an angular momentum; and in case of the Kerr-Newman metrics, they have mass, electric charge and angular momentum. These are, according to the no-hair theorem, all the characteristics a black hole has at its disposal. Let’s restrict the argument in the following to the Reissner-Nordström metrics in which a black hole has only mass and electric charge. In the classical picture, the electric charge of a black hole becomes noticeable in form of a force exerted on an electrically charged probe outside its event horizon. In the quantum field theoretical picture, interactions are the result of the exchange of virtual interaction bosons, in case of an electric charge: virtual photons. But how can photons be exchanged between an electrically charged black hole and an electrically charged probe outside its event horizon, if no photon can leave a black hole – which can be considered a definition of a black hole? One could think, that virtual photons, mediating electrical interaction, are possibly able (in contrast to real photons, representing radiation) to leave the black hole. But why? There is no good reason and no good answer for that within our present theoretical framework. The same problem exists for the gravitational interaction, for the gravitational pull of the black hole exerted on massive objects outside its event horizon, if the gravitational force is understood as an exchange of gravitons between massive objects, as the quantum field theoretical picture in its extrapolation to gravity suggests. How could (virtual) gravitons leave a black hole at all?

There are three possible scenarios resulting from the incompatibility of our assumptions about the characteristics of a black hole, based on general relativity, and on the picture quantum field theory draws with regard to interactions:

(i) Black holes don’t exist in nature. They are a theoretical artifact, demonstrating the asymptotic inadequacy of Einstein’s general theory of relativity. Only a quantum theory of gravity will explain where the general relativistic predictions fail, and why.

(ii) Black holes exist, as predicted by general relativity, and they have a mass and, in some cases, an electric charge, both leading to physical effects outside the event horizon. Then, we would have to explain, how these effects are realized physically. The quantum field theoretical picture of interactions is either fundamentally wrong, or we would have to explain, why virtual photons behave completely different, with regard to black holes, from real radiation photons. Or the features of a black hole – mass, electric charge and angular momentum – would be features imprinted during its formation onto the spacetime surrounding the black hole or onto its event horizon. Then, interactions between a black hole and its environment would rather be interactions between the environment and the event horizon or even interactions within the environmental spacetime.

(iii) Black holes exist as the product of gravitational collapses, but they do not exert any effects on their environment. This is the craziest of all scenarios. For this scenario, general relativity would have to be fundamentally wrong. In contrast to the picture given by general relativity, black holes would have no physically effective features at all: no mass, no electric charge, no angular momentum, nothing. And after the formation of a black hole, there would be no spacetime curvature, because there remains no mass. (Or, the spacetime curvature has to result from other effects.) The mass and the electric charge of objects falling (casually) into a black hole would be irretrievably lost. They would simply disappear from the universe, when they pass the event horizon. Black holes would not exert any forces on massive or electrically charged objects in their environment. They would not pull any massive objects into their event horizon and increase thereby their mass. Moreover, their event horizon would mark a region causally disconnected with our universe: a region outside of our universe. Everything falling casually into the black hole, or thrown intentionally into this region, would disappear from the universe.

Hyperbolic Brownian Sheet, Parabolic and Elliptic Financials. (Didactic 3)


Financial and economic time series are often described to a first degree of approximation as random walks, following the precursory work of Bachelier and Samuelson. A random walk is the mathematical translation of the trajectory followed by a particle subjected to random velocity variations. The analogous physical system described by SPDE’s is a stochastic string. The length along the string is the time-to-maturity and the string configuration (its transverse deformation) gives the value of the forward rate f(t,x) at a given time for each time-to-maturity x. The set of admissible dynamics of the configuration of the string as a function of time depends on the structure of the SPDE. Let us for the time being restrict our attention to SPDE’s in which the highest derivative is second order. This second order derivative has a simple physical interpretation : the string is subjected to a tension, like a piano chord, that tends to bring it back to zero transverse deformation. This tension forces the “coupling” among different times-to-maturity so that the forward rate curve is at least continuous. In principle, the most general formulation would consider SPDE’s with terms of arbitrary derivative orders. However, it is easy to show that the tension term is the dominating restoring force, when present, for deformations of the string (forward rate curve) at long “wavelengths”, i.e. for slow variations along the time-to-maturity axis. Second order SPDE’s are thus generic in the sense of a systematic expansion.

In the framework of second order SPDE’s, we consider hyperbolic, parabolic and elliptic SPDE’s, to characterize the dynamics of the string along two directions : inertia or mass, and viscosity or subjection to drag forces. A string that has “inertia” or, equivalently, “mass” per unit length, along with the tension that keeps it continuous, is characterized by the class of hyperbolic SPDE’s. For these SPDE’s, the highest order derivative in time has the same order as the highest order derivative in distance along the string (time-to-maturity). As a consequence, hyperbolic SPDE’s present wave-like solutions, that can propagate as pulses with a “velocity”. In this class, we find the so-called “Brownian sheet” which is the direct generalization of Brownian motion to higher dimensions, that preserves continuity in time-to-maturity. The Brownian sheet is the surface spanned by the string configurations as time goes on. The Brownian sheet is however non-homogeneous in time-to-maturity.

If the string has no inertia, its dynamics are characterized by parabolic SPDE’s. These stochastic processes lead to smoother diffusion of shocks through time, along time-to-maturity. Finally, the third class of SPDE’s of second-order, namely elliptic partial differential equations. Elliptic SPDE’s give processes that are differentiable both in x and t. Therefore, in the strict limit of continuous trading, these stochastic processes correspond to locally riskless interest rates.

The general form of SPDE’s reads

A(t,x) ∂2f(t,x)/∂t2 + 2B(t,x) ∂2f(t,x)/∂t∂x + C(t,x) ∂2f(t,x)/∂x2 = F(t,x,f(t,x), ∂f(t,x)/∂t, ∂f(t,x)/∂x, S) —– (1)

where f (t, x) is the forward rate curve. S(t, x) is the “source” term that will be generally taken to be Gaussian white noise η(t, x) characterized by the covariance

Cov η(t, x), η(t′, x′) = δ(t − t′) δ(x − x′) —– (2)

where δ denotes the Dirac distribution. Equation (1) is the most general second-order SPDE in two variables. For arbitrary non-linear terms in F, the existence of solutions is not warranted and a case by case study must be performed. For the cases where F is linear, the solution f(t,x) exists and its uniqueness is warranted once “boundary” conditions are given, such as, for instance, the initial value of the function f(0,x) as well as any constraints on the particular form of equation (1).

Equation (1) is defined by its characteristics, which are curves in the (t, x) plane that come in two families of equation :

Adt = (B + √(B2 − AC))dx —– (3)

Adt = (B − √(B2 − AC))dx —– (4)

These characteristics are the geometrical loci of the propagation of the boundary conditions.

Three cases must be considered.

• When B2 > AC, the characteristics are real curves and the corresponding SPDE’s are called “hyperbolic”. For such hyperbolic SPDE’s, the natural coordinate system is formed from the two families of characteristics. Expressing (1) in terms of these two natural coordinates λ and μ, we get the “normal form” of hyperbolic SPDE’s :

2f/∂λ∂μ = P (λ,μ) ∂f/∂λ +Q (λ,μ) ∂f/∂μ + R (λ,μ)f + S(λ,μ) —– (5)

The special case P = Q = R = 0 with S(λ,μ) = η(λ,μ) corresponds to the so-called Brownian sheet, well studied in the mathematical literature as the 2D continuous generalization of the Brownian motion.

• When B2 = AC, there is only one family of characteristics, of equation

Adt = Bdx —– (6)

Expressing (1) in terms of the natural characteristic coordinate λ and keeping x, we get the “normal form” of parabolic SPDE’s :

2f/∂x2 = K (λ,μ)∂f/∂λ +L (λ,μ)∂f/∂x +M (λ,μ)f + S(λ,μ) —– (7)

The diffusion equation, well-known to be associated to the Black-Scholes option pricing model, is of this type. The main difference with the hyperbolic equations is that it is no more invariant with respect to time-reversal t → −t. Intuitively, this is due to the fact that the diffusion equation is not conservative, the information content (negentropy) continually decreases as time goes on.

• When B2 < AC, the characteristics are not real curves and the corresponding SPDE’s are called “elliptic”. The equations for the characteristics are complex conjugates of each other and we can get the “normal form” of elliptic SPDE’s by using the real and imaginary parts of these complex coordinates z = u ± iv :

2f/∂u2 + ∂2f/∂v2 = T ∂f/∂u + U ∂f/∂v + V f + S —– (8)

There is a deep connection between the solution of elliptic SPDE’s and analytic functions of complex variables.

Hyperbolic and parabolic SPDE’s provide processes reducing locally to standard Brownian motion at fixed time-to-maturity, while elliptic SPDE’s give locally riskless time evolutions. Basically, this stems from the fact that the “normal forms” of second-order hyperbolic and parabolic SPDE’s involve a first-order derivative in time, thus ensuring that the stochastic processes are locally Brownian in time. In contrast, the “normal form” of second-order elliptic SPDE’s involve a second- order derivative with respect to time, which is the cause for the differentiability of the process with respect to time. Any higher order SPDE will be Brownian-like in time if it remains of order one in its time derivatives (and higher-order in the derivatives with respect to x).

Killing Joke: Tata Power’s CGPL launches Bio-Diversity Club in Mundra. A Response

Let us stray into the some of the cold corners of the human heart. Its not that this blog hasn’t been doing that already, but, this time, it is with a difference, and that being some sort of activism that I was involved in, in one of the most ecologically rich marine diverse system I have had the chance to walk through. So, whats the cold corner of the thought referred to being here. Thanks to Joe, I chanced upon reading this piece, which on a cursory glance and to the one who hasn’t been privileged enough to walk the ecosystem, this might be good intentioned, but then road to hell is paved with good intentions and for the perpetrators of this large-scale eco-disaster, man-made and waiting to explode and go defunct, hell is the other people, just to echo Sartre. The post in question deals with Tata Power’s thermal-fired power plant in Mundra in Kutch (variously spelled as Kachchh in the Indian state of Gujarat), through its wholly-owned subsidiary, Coastal Gujarat Power Limited (CGPL), and how as part of Corporate Social Responsibility (CSR), the company has had the audacity to launch a bio-diversity club in Mundra to safeguard the rich ecology, after having first built one of India’s largest 4000 MW power plant on the coast and in the process wrecking an irreversible damage to the rich ecosystem the company now has suddenly woken up to safeguarding. So, it is a matter of flogging the dead horse, or even more tersely bringing back to life from dead through these exclusively hollowed CSR activities. I shall come to that in a while, but before that let us look at the background of the disaster in brief.

Why is it that the Tata Mundra Project keeps bouncing back to become a bone of contention for all across the spectrum; from the financiers to the beneficiaries, the owners, as well as the people of the area? Why is this project, likely to become the largest power generation unit in the country, being opposed nationally and globally? Obviously, there is more than meets the eye. So, what exactly went wrong? Let us take the first step through the power quagmire in the country. India has always had chronic problems in meeting the electricity needs of its’ peoples. At the beginning of the post-liberal era or the mid-1990s, the then Congress Government, headed by P.V. Narsimhan Rao, initiated a series of measures to address the already growing crisis. It promulgated a Mega Power Policy, whereby projects of more than 1,000 MW would be built to improve the electricity grid in the country. This would apply to projects anywhere in the country, except for the states of Jammu & Kashmir and North-East India, where the cap on generational capacity was reduced to 700 MW. What looked like a decent plan on paper, however, failed in practice, as the gap between supply and demand just kept increasing. It was ten years before the Government decided to amplify the Mega Power Projects (MPPs) by setting up Ultra Mega Power Projects (UMPPs) in a bid to overcome the shortfall. The prefixing of ‘Ultra’ meant a multiplication by a factor of 4 of what the MPPs were hitherto envisaged to generate. In other words, a generational capacity of 4,000 MW made the projects Ultra Mega. Fair enough.

The Tata Mundra Project (Coastal Gujarat Power Limited and CGPL hereafter), a wholly owned subsidiary of Tata Power, became the first UMPP that got approved by the Government of India in 2006-07 and was established in Mundra in the Kutch area of Gujarat. Kutch is an ecologically fragile region having a coastline dotted with mangroves, sand dunes, coral reefs, mudflats and a nest of some of the rarest marine species. The coalfired thermal plant adds to the vulnerability of the marine ecology and is built near the massive Mundra SEZ (Special Economic Zone). It also happens to be one amongst the many power generational units over the coastline of 70 km that together would produce 22,000 MW of power.


Photo Credit: Sanjeev Thareja

So, how did this project come about? As in the case of UMPPs, the proposal for this project was initially nurtured by the government-owned Power Finance Corporation of India (PFC) and after a competitive bidding process, Tata Power took over by offering the lowest levelized tariff of Rs. 2.26 (US $ 0.04) per KWh. The project consists of five units of 800 MW each and is priced at a whopping US $4.14 billion. The funding for the project comes from a consortium of Indian banks, led by the State Bank of India and contributions from other National Financial Institutions; like India Infrastructure Finance Company Limited, Housing and Urban Development Corporation Limited, Oriental Bank of Commerce, Vijaya Bank, State Bank of Bikaner and Jaipur, State Bank of Indore, State Bank of Hyderabad and State Bank of Travancore and also through External Commercial Borrowing (ECB). The ECB comprises of International Finance Corporation (IFC) – the private lending arm of the World Bank Group, Asian Development Bank, Exim Bank of Korea, Korea Export Insurance Corporation and the BNP Paribas. Of the whopping cost of the project, the financing from IFC and ADB is US $450 million each, which appears to be a mere chunk of the whole. However, it has a huge leverage point, since it is assumed that funding from these multilateral giants gets approval only if safeguards and guidelines laid down by them are met successfully.

Switching over to the financial impracticality of the project, the coal that fires the plant comes wholly from Indonesia. After a decision by the Indonesian Government to link mineral exports to market rates in September 2010 and taking effect in 2011, importing coal has become dearer. The immediate logical implication of this is a restructuring of the tariff rates from the one that helped bid for the project in the first place. That was exactly the route undertaken and based on some riders that the Central Electricity Regulatory Commission (CERC) extended, including sharing profits earned by Tata’s Indonesian mining companies, sacrificing one per cent return of equity (RoE) and lowering auxiliary consumption of 4.75 per cent, further brought down the effective compensatory rate to 47 paise (US $ 0.01) per unit. As a result of this compensation, the retail rate from the CGPL for consumers in the five procuring states of Gujarat, Rajasthan, Haryana, Punjab and Maharashtra is expected to rise by 0.4 – 1.8 per cent.

This might come as a relief, but is clearly not, since four of the five states benefiting from the electricity generated at the plant, plan to legally challenge the move by the regulator to let CGPL pass on increased fuel costs on power purchasers. Maharashtra, Gujarat, Haryana and Punjab have taken an in-principle decision to approach the Appellate Tribunal for Electricity (APE) against the order by the CERC. These states are pondering filing separate petitions against the compensation extended to CGPL. An official involved with a long-drawn legal and regulatory battle, on conditions of anonymity, said, “The states are approaching the ATE on a major issue of sanctity of the PPA (power purchase agreement) signed by these for 25 years. CERC’s order will ensure that financial condition of buyer utilities will further deteriorate.” In a very recent development, Tata Power even signed the option of selling 5 per cent of its stake in its Indonesian coal mine to the Bakrie Group at US $250 million to reduce its debt.


Photo Credit: Sanjeev Thareja

Since the Kutch coastline is marine rich, fishing becomes one of the major means of livelihood. Thus, the main inhabiting population of the coast happens to be a migratory fishing community locally called bunders. They live in fishing settlements for 8-9 months of the year when fisheries reach an economic peak and then go back inland to the villages for the remaining part of the year. Other economic means are more rural-economy oriented and involvement in salt making, animal rearing and cashcrop cultivation is common. Mundra is a blessed oasis, in an otherwise hot and parched area that borders the great desert, with groundwater fit for drinking and agricultural and horticultural options in the offing. The Project has undoubtedly disrupted the prevalent order with a promise too difficult to keep and the players involved too complacent to mend their ways.

CGPL is located next to another UMPP, the Adani Power Project and both of these are fueled by coal- the thorn in the bush to climate change polylogues all over the world. Tata Mundra is fed entirely on coal imported from Indonesia. With the world trying hard to come to grips with excessive hazard due to the burning of this fossil fuel, the ratiocinating of the Government of India seems to have gone on a tailspin, with a trail of such plants dotting the vast shoreline of the country and the mineralrich hinterlands. All of this in an attempt to solve the official line of the “demand versus supply” equation.

The northern coast of Kutch, where Mundra is located, has witnessed large-scale, rampant industrialization in the last decade. Adani port (the largest private-sector port of the country), Adani SEZ, OPG’s coal-fired thermal plant and metal forging units have already done more than enough to cause irreparable damages to the geography of the land and sea. They have torn the social fabric of the population and left their economic means in tatters. What should really be occupying the mind of the company has eluded it completely, for CGPL plans for an expansion of 1,600 MW to the existing capacity, disregarding the mitigation that it has orchestrated in the first place.

So, what is amply clear is the presence of resistance to such large-scale industrialization by ground-level and grassroots movements, who are trying not just to safeguard the rich-biodiversity, but at the same time also trying to safeguard their livelihood, which is largely marine-dependent and is now under the constant threat of shrinking to a point. These people, of the ethnic clan and minority religion are getting sandwiched between the devil and deep sea with their resources fast depleting. And, these were the ones who primarily owned up to their coexistence with the diversity the piece now speaks of safeguarding, for the former knew this damage would be irreversible and the only way to save the region was to halt this rampant industrialization. The fisherfolk took their call and fought and are still resilient in the face of diminishing returns.

Machimar Adhikar Sangharsh Sangathan (MASS) that transliterates into Association for the Struggle for Fish workers’ Rights is a local organization of the affected communities which filed a complaint with the Compliance Advisor Ombudsman (CAO) – an independent recourse mechanism for the IFC and Multilateral Investment Guarantee Agency (MIGA) of the World Bank Group – in June of 2011. The complaint outlined parameters where the IFC committed significant policy breaches and supervision failures and wanted the CAO to weigh in with its findings. After a two-year long rigorous process, CAO came out with its findings that validated the complainant’s case by admitting to policy breaches, supervision failures and leniencies on IFC’s part. As a result, there were lapses and impacts:

1. The Environment and Social Impact Assessment (ESIA) filed by CGPL was deficient and shockingly, even failed to identify certain communities as project-affected. 
2. A cumulative impact study was not carried out despite the presence of certain large-scale polluting industries in the vicinity. 
3. CGPL failed to conduct adequate, meaningful and informed consultation with the affected communities and even shied away from sharing information about the action and mitigation plans. 
4. There was a clear violation of the environmental clearance with large stretches of mangroves, dry-land forests and bio-diversity rich creeks meeting a destructive end during the construction of the outfall and inlet channels. 
5. CGPL U-turned from the initial deal to have a closed-cycle cooling system and switched over to a cheaper and more environmentally destructive once-through cooling system. Why CGPL turned the degrees is reason defying. 
6. Access roads for the fisher-folk and pastoralists to their respective fishing and grazing grounds were either blocked or diverted, forcing the villagers to unusually longer routes and impacting their finances as a result of increased transportation costs and delays. 
7. The project has accentuated a decline in fish catch, which has been recorded empirically ever since the CGPL became fully operational. The situation has also aggravated due to the adjacent Adani project. 
8. There is inadequacy in addressing the health and environmental impacts of ash contamination from the project. It has contaminated drying fish, salt and animal fodder in the area, giving rise to significant health concerns. Adding to the woes, ignoring the potential hazard of radioactivity from the coal ash pond has deteriorating health impacts. Due to such health hazards, the children and the elderly are most vulnerable to respiratory ailments and the gravity of the situation can be gauged by the fact that the two coal plants are together burning 28 million tons of coal every year. 
9. Turning over to the finances and cost-benefit side of CGPL. The company totally ignored cost overruns and a likely tariff increase in times to come, by either misrepresenting it or underestimating its bid. As a result, the financial burdens would conveniently be placed on the customers.

CAO confirmed the inadequacy on IFC’s part to consider in its risk assessment studies the seasonally resident fishing community; the majority members of which belong to a religious minority and are most susceptible to be affected by the project. This excluded a population of close to 25,000 from application of land acquisition standards, biodiversity conservationism and other relevant policies enacted for their protection as laid down in the Performance Standard 5 of the IFC.ii The audit report pointed that IFC was lackadaisical in fulfilling requirements to manage impacts on airshed and the marine environment. On a more specific level, CAO brought to light that IFC did not ensure that its client CGPL applied the 1998 WB guidelines for thermal power, which puts a cap on the net increase on emissions of particulates or sulphur dioxide within the airshed. In case of marine environment, CAO found that the IFC did not possess any robust baseline data and thus, the future impacts of the project were bound to be missing a crucial component. IFC did not assure itself of the plant’s seawater cooling system as complying with the applicable IFC Environmental, Health and Safety (EHS) Guidelines. This could be gauged from the fact that when CGPL bid for the project, they had a closed water cooling system in mind, which suddenly got switched over to a once-open cooling system, which is more hazardous, even if economically cheaper. The failure not to comply is significant, since the thermal plume from the project’s outfall channel will extend well into the shallow waters of the estuary, posing an existential risk to marine life and marine resources. The coast of Mundra is getting dense with industries that are highly polluting and a failure to conduct a cumulative study when a new industry is planned is a grave injustice, not only to the geology of the region, but also to the demography of the region. The latter is largely left ignorant of the hazards that accompany the erection of a new industry, which, incidentally are the cumulative causes of an alliance with other industries in the vicinity. A lender like the IFC, which is known for its safeguards and guidelines and is at least insistent on paper that a strict adherence to it is be followed, could not have deliberately chosen to ignore this grave risk. This was nothing but a felony committed by IFC, since it failed to undertake a cumulative impact assessment of the area. It even failed to advice and admonish its client CGPL that environmental and social risks emerging from the project’s proximity to the Mundra Port and the Adani Power Plant and SEZ should have been assessed by a third party, neutral in stature, which in turn would help devise mitigation measures. Therefore, a compounded assessment made by the CAO sternly suggested that IFC’s review and adoption of its client’s reports are not robust to ensure that the performance standards and supervision requirements are met.

So, how did the IFC react to the recourse mechanism’s findings? They reacted by proficiently dismissing and essentially rejecting them. They audaciously defended their decision to fund the project, their client CGPL and even passed over any remedial action plan. Since the CAO reports directly to the President of the WBG, the report with its findings was tabled before Dr. Jim Yong Kim; who sat over it in bureaucratic silence for close to a month. He eventually cleared the management response, thus acquitting the IFC of any wrong doing and putting a question mark over the need for the compliance mechanism. Dr. Kim – who with his constant rant of a sensible investment by the Bank that would give highest importance to climate change mitigation on one hand and a commitment to eradicate extreme poverty on the other – clearly failed to deliver on his promises. Three months after the CAO’s findings, ADB’s Compliance Review Panel (CRP) decided to investigate policy violations while financing the 4,000 Mw coal plant. ADB’s board of directors approved the recommendation of its accountability mechanism, the CRP for a full investigation. According to Joe Athialy, in its eligibility report, CRP said, “The CRP finds prima facie evidence of noncompliance with ADB policies and procedures, and prima facie evidence that this noncompliance with ADB policies has led to harm or is likely to lead to future harm. Given the evidence of noncompliance… the CRP concludes that the noncompliance is serious enough to warrant a full compliance review.” CRP found the following evidence of noncompliance: insufficient public consultations; the project-affected area is defined erroneously; CGPL discharges water at a higher temperature than is allowed by ADB standards; ADB’s air emission standards are not met; insufficient cumulative impact assessments; flawed social and environmental impact assessments; harmful effects of the cooling system on the environment and the fish harvest; inaccessibility of fishing grounds and effects of coal-dust emissions. The full investigation is underway, Joe added.

Even as the world took note of these damning and scathing reports by extending solidarity with the fisherfolk and their families, joined in by numerous Indian organizations, all of this seemed to have fell on deaf ears. The unanimous nature of their solidarity centered around community woes in accessing contaminated water and the loss of ecological resources at their disposal and suffering the impacts of air pollution. The protesting organizations appeared to be stunned by the decision that would have given a chance for the World Bank President to prove to the world that he, with his past rooted in public health advocacy and constant reminders of the commitment of the Bank to phase out of funding fossil fuels had not just frittered away.

Though a lot of solidarity was promised, what seems to be moving would only require a keen eye to detect, if at all there is even a semblance of it. A meeting with the higher-ups of IFC went on to commit that any actions plans that were being formulated would be diligently shared with the people affected, but eventually, all that came out from the office was what had come even before the project had been commissioned, thus proving the entirety of the exercise of an action plan as a non-starter. Building pressure from the ground-up appears to be the only strategy to work at the moment and this was vociferously expressed at the Spring Meeting in 2014 (+ how the fishermen sued the IFC in a development in the recent past). MASS came out with a four-point agenda, from which there was no question of budging. The petition mentioned above demands urgent actions to restore, rehabilitate and resettle, to provide adequate compensations, to acknowledges its lapses, to refuse the financing of any expansion and to make the IFC pull out of the project.

So far the IFC has hinted that it will not be financing the After being scanned by the compliance mechanism of the IFC, a complaint was also filed with the Asian Development Bank. Their recourse mechanism, the Compliance Review Panel (CRP), has taken the complaint seriously for an audit and is in the process of reviewing adherence to the safeguards and guidelines of the Asian multilateral giant. It is worth noting once more that even the ADB has an investment of US $450 million in Mundra.

This clearly shows that the CGPL finds itself in serious human rights’ violations of the people inhabiting the region. The CRP review process is currently on and their findings are awaited most eagerly. The really surprising question however, is why would the CGPL go in for an expansion of units when it is yet to sort out these urgent issues? What fuels this surprise further is the financial health of the project, which we turn attention to now. And to add salt to the injury, here is the eye-wash in the form of a club to safeguard bio-diversity. According to the piece linked in the beginning, the conservation plan has already been prepared and being implemented. Significant among them on which the work has already started are (a) White Napped tit ( Machlolophus nuchalis), a bird species . (b) Olax Nana which are the two important species of flora and Fauna respectively.



Further insult come to the fore with CGPL having initiated the Club with the objective of developing a people centered-approach for bio-diversity conservation for creating more awareness among the employees. I wonder where are the people in the spectrum here, those who have been hitherto protectors of the ecosystem and who are now pushed further and further to the peripheries. They are as usual absent from the whole scheme of this delusional propaganda and are paying the price for they understand, they comprehend the episteme. The club will work towards motivating individuals to participate in activities that help protect the environment, and this is precisely where shamelessness embraces its expiration.

But, the most dismal comment on this initiative came from none other than the progenitor of this initiative, one Mr. KK Sharma, ED and CEO, CGPL, who said,

Kutch region is very special from a biodiversity point of view as it harbours its own unique forms of desert flora and fauna. Some species of plants and animals are of high conservation significance, both at the national and international level. Therefore, we at Tata Power’s Coastal Gujarat Power Limited took up this initiative to set up a bio-diversity club that will implement conservation programs to ensure the safety of the various species found in and around our power plants. We are grateful to our employees for the hard work they have put in to ensure the success of this initiative. CGPL’s Bio-diversity Club demonstrates our unwavering support and commitment towards society as a whole.

The entirety of the article/post written before this comment proves his point to be utterly dross, and reminds me of the saying that paranoia feeds on the crumbs of reality, and thats the reason for this sudden wake-up call from a disaster-producing and inducing slumber. In conclusion, what is this, but a drivel of sorts, where everything from the beginning seems to be heading the wrong way and where any attempt to straighten things out only lands the CGPL between the devil and the deep sea, or between the blades of a scissor? Then, why the perseverance by CGPL to make further promises when even the ones it made in the beginning failed disastrously? Is it enough to expand generational capacity to address the deficiency between supply and demand, even if it takes burning fossil fuels and the dirty coal? There isn’t much of an alternative in the thoughts of growth-led development pundits. With the country poignantly poised at an economic downturn and living on a daily basis getting costlier and costlier, whatever energy is generated is beyond affordance for the millions of citizens anyway. Then why not choose for cost-effective renewable sources that at least don’t kill by smoke; for the buzzword today is “coal kills” and there is no denying that. For the people at Mundra, as am sure with people all over the coal and thermal-power belts, their lives are getting darker by the day and whatever electricity is produced at the plant at whatever gain or loss; for the former is a myth while the latter a reality, makes nothing brighter for them.