Rishabh Jha / International Journal of New Technologies in Science and Engineering Vol. 3, Issue. 7, July2016, ISSN 2349-0780
Inconsistencies between Thermodynamics and Relativity
Rishabh Jha, Department of Physics, Indian Institute of Technology Kanpur, India
Abstract: The fundamental laws of thermodynamics are contrasted with the laws of special and general theory of relativity. Various inconsistencies emerge as a result. Problems due to which these inconsistencies arise are highlighted. These inconsistencies have major consequences in the applicability of statistical mechanics in curved spacetime background. Finally the paper concludes with few important comments, a plausible method to resolve these inconsistencies and a summary of all the highlighted problems and inconsistencies.
Keywords: Thermodynamics, Special Relativity, General Relativity
I. INTRODUCTION Thermodynamics is a firmly established empirical science with a strong theoretical background provided by statistical mechanics. It is one of the most verified branches of Physics which has been developed over centuries. On the other hand, special and general theories of relativity (together they will be referred to as a single theory of relativity or simply relativity in this paper) came into existence much later than the advent of thermodynamics. They have been tested and verified over a shorter span of time but with breathtaking accuracy. Both branches are considered as pillars of Physics, both based on assumptions which have been contrasted in this paper. The paper starts with a very brief review of both theories to the level required in this paper. Then it comes to the core of the paper where the fundamentals of both theories are contrasted and inconsistencies found. In a small section on the application of statistical mechanics in curved spacetime background, the consequences are highlighted. Hence, a need for new kind of mathematics and insights are required to resolve them. The paper concludes with one such attempt which may be plausible.
II. REVIEW OF THERMODYNAMICS The following review of thermodynamics has been extensively used in this paper. These are all standard relations that can be seen in any standard textbook on thermodynamics (see [1]). The basic equation of thermodynamics is Euler relation given by:
TS = U + pV - µN (1) where T, S, U, p, V, µ and N refers to temperature, entropy, internal energy, pressure, volume, chemical coefficient and number of particles respectively.
The first and second laws of thermodynamics can be written together as:
TdS = dU + pdV - µdN (2)
Combining these two equations gives Gibbs-Duhem relation as follows:
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SdT – Vdp + Ndµ = 0 (3)
Equation pairs (1, 2) or (2, 3) can be considered to be the most basic equations of thermodynamics on which the entire subject is further built upon. But a crucial assumption goes in there. These are extensivity and homogeneity. Qualitatively, the concept of extensive and intensive functions simply refer to the fact that whether they depend on the bulk of matter or not respectively. Mathematically, intensive functions are homogeneous of degree zero while extensive functions are homogeneous of degree one. Hence any multivariable extensive functions can be defined as:
f(cx, cy, cz) = cf(x, y, z) (4) where c is any scalar constant.
Hence, we have the following functions as extensive in thermodynamics:
S = S(U, V, N) (5)
We further assume these functions to be mathematically well behaved, so that they can always be inverted to give the following extensive relations:
U = U(S, V, N) ; V = V(S, U, N) ; N = N(S, U, V) (6)
Equation (5) or any equation in (6) is known as the fundamental equation of state from which all equations of state can be derived. The fundamental equation of state contain within itself, all the required piece of thermodynamic information of that state.
III. REVIEW OF THEORY OF RELATIVITY The following review of relativity is far from complete and almost only those parts have been reviewed which are required for the purpose of this paper. These are all standard relations that can be seen in any standard textbook on relativity (see [2]). Our convention is that of mostly negative signature of metric, i.e., gµν = (+, -, -, -). Latin indices run over space coordinates (1, 2, 3) while Greek indices run over spacetime coordinates (0, 1, 2, 3). A. Special Theory of Relativity This was the first of theories given by Einstein which completely revolutionized the notion of space and time. The crux of the theory states that space and time are not independent concepts but actually interrelated ones and the concept of spacetime hence emerges out of this. Consider a frame of reference S′ moving at constant velocity u with respect to the frame of reference S along the x-axis of both S and S′. We will be working in natural units where c=G=ℏ=1. Here c, G and ℏ denote speed of light, Newton’s gravitational constant and Planck’s constant respectively. The coordinates of S be referred to as xµ = (t, x, y, z) and coordinates in S′ be x′µ = (t′, x′, y′, z′). Then we define two functions as follows:
β=√(1-u2) ; ϒ = 1/β (7)
It is assumed without the loss of generality that at t = t′ = 0, origins of S and S′ coincide. The
Available online @ www.ijntse.com 2 Rishabh Jha / International Journal of New Technologies in Science and Engineering Vol. 3, Issue. 7, July2016, ISSN 2349-0780 coordinates of S and S′ are related by Lorentz transformations as follows:
t′ = ϒ(t - ux) (8) x′ = ϒ(x-ut) (9) y = y′ (10) z = z′ (11)
The principle of relativity is postulated as following: the laws of physics are the same in all inertial frames of reference. In other words, it states that all laws must have mathematical forms that remain Lorentz invariant under the above set of transformations (equations (8) to (11)). This is the sacred principle on which the special theory of relativity is based along with the second postulate that the speed of light in vacuum has the same value c in all inertial frames of reference, irrespective of the source or anything else.
B. General Theory of Relativity This is the generalization of the theory of gravitation as formulated by Newton. This theory is considered by many as perhaps the most beautiful theory in all of Physics (see [3]). This theory also has the specialty of showing how geometry manifests itself so elegantly in physical forms. The core idea is the principle of equivalence which states that ([4]): “The outcome of any local experiment (gravitational or not) in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime.” and “The gravitational motion of a small test body depends only on its initial position in spacetime and velocity, and not on its constitution.” Gravitation arises due to the curvature of spacetime manifolds which in reverse affects the curvature of spacetime manifolds. This makes the theory nonlinear. All the information of spacetime is encoded in the metric gµν. The first derivative of this metric uniquely defines the Christoffel connections as follows:
α αβ Γ µν =0.5g (∂νgµβ + ∂µgβν − ∂βgνµ) (12) ν where ∂ν = ∂/x . One can always find a coordinate system where these connections vanish identically in flat spacetime background as in special theory of relativity.
Metric are assumed to be symmetric in both the indices and accordingly, Christoffel connections are symmetric in both the lower indices. Christoffel symbols fail to transform as tensors. Using these Christoffel connections, the Riemann curvature tensor is defined as (being second derivative of the metric tensor):
α α α µ α µ α R βγδ = ∂γΓ βδ − ∂δΓ βγ + Γ βδΓ µγ − Γ βγΓ µδ (13)
This satisfies the following relation:
Rαβµν = −Rβαµν = −Rαβνµ = Rµναβ (14)
This can be contracted using the metric tensor to get Ricci tensor as well as Ricci scalar (being a scalar, this quantity is invariant under any general coordinate transformations in general theory of relativity).
Then comes the concept of parallel transport. Unlike special relativity, vectors are not free to be
Available online @ www.ijntse.com 3 Rishabh Jha / International Journal of New Technologies in Science and Engineering Vol. 3, Issue. 7, July2016, ISSN 2349-0780 parallel transported from one point of spacetime to another. Curvature induces a correction term which prohibits free parallel transport of vectors. To begin with, the concept of covariant derivative as being a more general form of partial derivative is introduced. Here it will be illustrated only for the case of contravariant vectors. Consider a contravariant vector Aµ. Then covariant derivative is defined as:
µ µ β µ ∇νA = ∂νA + A Γ βν (15)
It can be seen directly that the first term in the normal partial derivative term while the second term is the correction due to the curvature of spacetime manifolds. The second term goes to zero identically in case of special theory of relativity, thus reducing the covariant derivative to partial derivative as it should be. This basically gives the derivative of the contravariant vector (in this case) along the tangent vectors of a manifold. This second non-trivial term (correction term due to curvature of spacetime manifolds) is responsible for preventing free parallel transport of vectors from one point to another in spacetime.
IV. CONTRASTING THERMODYNAMICS AND GENERAL THEORY OF RELATIVITY: LAW OF EXTENSIVITY Extensive properties of a system are those properties that depend on the bulk of the system unlike intensive properties that are independent of the bulk of system considered. Examples of intensive properties of a thermodynamic system are density, temperature, pressure, etc. Examples of intensive properties of a thermodynamic system are energy, mass, volume, etc.
We have noted above the relationship between an extensive function, intensive function and a homogeneous function. All extensive functions are homogeneous functions of degree one while all intensive functions are homogeneous functions of degree zero. Hence these are interrelated. We have also seen that this extensivity/homogeneity is a crucial assumption behind the fundamental equation of a thermodynamic state.
Law of extensivity states that extensive properties of a system are additive in nature. For example, if there are n-systems, each having an extensive property Ai (i running from 1 to n), then collectively for an isolated system containing all n-systems as sub-systems altogether, we have the combined extensive property A for the entire system given by:
Σi (Ai) = A (16)
So far, all good. But this assumption (based on common intuition) fails to hold exactly in the context of general relativistic scenarios (not so common in our intuition). This is already explicit in the case of extreme relativistic systems, i.e., black holes where the entropy of the black hole as given by Bekenstein-Hawking formula ([5], [6]) is not additive. The reason for this abnormal behavior is that the entropy depends on the surface area and not the volume.
In this paper, it will be shown that other extensive properties like energy E are also not exactly additive in nature. Exact additiveness holds only for a flat spacetime background but not in curved spacetime background which happens to be the case in general theory of relativity. On the face value of the problem posed, it is tempting to think as follows that energy E is a scalar and hence must be
Available online @ www.ijntse.com 4 Rishabh Jha / International Journal of New Technologies in Science and Engineering Vol. 3, Issue. 7, July2016, ISSN 2349-0780 invariant under any general coordinate transformation in general theory of relativity. Accordingly, additiveness must hold precisely even in the most generalized conditions. However tempting it may seem but it suffers with a major flaw that energy E is not a true scalar but a component (time component to be precise) of 4-vector momentum Pµ. Hence in order to comment on the transformation of E, the transformation of Pµ must be considered first.
Consider two spacetime points A and B where two thermodynamic systems (simply referred to as A and B) have been kept. Consider the spacetime difference between these two points to be ∆xµ. Let µ ν 4-momentum of A and B is P 1 and P 2 respectively. Let energy (time component of 4-momentum) of A and B is E1 and E2. Here subscripts 1 and 2 are just for the sake of labeling systems A and B ν respectively. In case of flat spacetime background, P 2 can be simply parallel shifted from B to A and then added directly whose time component would be then E1 + E2. But this is not the case in curved spacetime background.
ν In curved spacetime background, shifting P 2 from B to A causes a change in 4-vector induced by curvature as follows:
ν ν µ ∆P 2 = (∇µP 2).( ∆x ) (17) ν ν β ν µ => ∆P 2 = (∂µP 2 + P 2Γ βµ).( ∆x ) (18) ν ν µ β ν µ => ∆P 2 = (∂µP 2).( ∆x ) + (P 2Γ βµ).( ∆x ) (19)
The first term on the RHS of equation (19) is the term which holds in flat spacetime background and it is the second term of the RHS which creates all the difference. This is the correction due to curvature of spacetime manifolds. Hence, the new 4-momentum after shifting from B to A is:
ν ν ν P′ 2 = P 2 + ∆P 2 (20) ν ν ν µ β ν µ => P′ 2 = P 2 + (∂µP 2). ( ∆x ) + (P 2Γ βµ).( ∆x ) (21)
The first two terms on the RHS of equation (21) is the same as that in the case of flat spacetime background. It is the third term on the RHS which is the correction due to the curvature of spacetime manifolds.
Since we are working in natural units, hence the 0th component of the 4-momentum is the energy. 0 Initial energy of B: E2 = P 2 0 Final energy of B (after being shifted to A): E′2 = P′ 2 0 0 µ β 0 µ => E′2 = P 2 + (∂µP 2). ( ∆x ) + (P 2Γ βµ).( ∆x ) (22) using equation (21).
0 0 0 We know that P 2 = E2 and for a conservative system, ∂µP 2 = 0. Using same arguments, ∂µP 1 = 0. These are valid in both the cases of curved and flat spacetime backgrounds.
0 β 0 µ => E′2 = P 2 + (P 2Γ βµ).( ∆x ) (23)
Finally adding the two 4-momentums and taking the time-component gives the total energy as:
0 0 Etotal = P 1 + P′ 2 (24) β 0 µ => Etotal = E1 + E2 + (P 2Γ βµ).( ∆x ) (25.a)
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This is the final result where we can see that extensivity is just an approximation and the third term on the RHS of equation (25) is the correction due to curvature of spacetime manifolds. As expected, on a tangent manifold, where general theory of relativity reduces to special theory of relativity, we can ν always find a coordinate system for which affine connections Γ βµ vanish identically, hence reducing equation (25) to the standard known result of thermodynamics (E1+E2). In classical domain as well, the assumption of a flat space and an absolute concept of time are assumed. Hence in the classical domain too, the standard result E1+E2 is derived from equation (25).
If instead of moving system B to A, we would have kept B at its place and moved A from B, then following the same line of reasoning as followed above, the final result obtained would have been as follows:
β 0 µ E′total = E1 + E2 + (P 1Γ βµ).( ∆x ) (25.b)
Prime over Etotal is simply a label to differentiate it from Etotal in equation (25.a). Hence it can be seen that the value of total energy of the system actually depends on the fact that whether B has been moved to A or A has been moved to B (Etotal ≠ E′total). This is not quite unexpected as we know that energy in itself has got no physical meaning. It is the difference of energy that matters. Moreover, total energy of the system indeed depends on the spacetime location of the system. As an analogy, it can be provided that the total energy of a system measured at sea level and the same system measured at the peak of a hill will be definitely different. But the crucial point remains that extensivity is still an approximate assumption. From now onwards, equations (25.a) and (25.b) will be collectively denoted as equation (25) in this paper.
V. IMPACT ON STATISTICAL MECHANICS IN CURVED SPACETIME MANIFOLDS Thermodynamics in itself is an empirical science. Statistical mechanics gives a strong theoretical background to thermodynamics. But all standard textbooks read by the author deals with the fundamentals of statistical mechanics in flat spacetime background. This is actually important to emphasize the wonders of this pillar of Physics.
The study of statistical mechanics starts with the following assumption ([7]): Consider a system having total energy E. Then particles are assumed to be non-interacting. The total energy E is then equal to the sum of the energies ei of the individual particles, given by:
Σi(eini) = E (26) where ni denotes the number of particles having energy ei. Hence
Σi(ni) = N (27) where N denotes the total number of particles in the system.
But in a more general curved spacetime background, the concept of extensivity becomes an approximation. Hence the basic equations from where the statistical mechanics start themselves become an approximation in general theory of relativity. It is obvious, yet alarmingly important to highlight that all results of statistical mechanics based on these equations fail to be true in general
Available online @ www.ijntse.com 6 Rishabh Jha / International Journal of New Technologies in Science and Engineering Vol. 3, Issue. 7, July2016, ISSN 2349-0780 theory of relativity. All standard results become an approximation in the most general background. No doubt that whatever be the most general theory of statistical mechanics, it must boil down to the standard known statistical mechanics under suitable limits, just like relativistic Physics boils down to classical Physics in suitable limits.
VI. CONTRASTING THERMODYNAMICS AND SPECIAL THEORY OF RELATIVITY Taking the analyses to a tangent manifold for the sake of contrasting special theory of relativity and thermodynamics, we know that extensivity holds precisely on the tangent manifold. The reason is that general theory of relativity reduces to special theory of relativity on any tangent manifold of spacetime and affine connections vanish identically in special theory of relativity, hence from equation (25) it is clear that extensivity holds exactly on a tangent manifold.
Hence entropy S = S(U, V, N) is the fundamental equation of state in thermodynamics. All thermodynamic functions are assumed to be well-behaved. So this fundamental equation of state can be inverted always to get equation (6), thus showing U, V and N are also extensive variables just like S.
In special theory of relativity, we know that volume transforms as: V βV = V′ (28) => V(S, U, N) βV(S, U, N) = V′(S′, U′, N′) (29)
Using the definition of extensive function being a homogeneous function of degree one, we have:
βV(S, U, N) = V(βS, βU, βN) (30)
Instantly we see that Lorentz transformations (L.T.) take the following form in thermodynamics:
L.T.: V βV ; S βS ; N βN (31)
Considering these transformations, it turns out that all intensive variables like T, p, etc. are invariant under Lorentz transformations. But in practice, S is taken to be Lorentz invariant. This change of convention can always be achieved by attributing the transformation to temperature T as:
T βT (32) keeping S as Lorentz invariant. Since everywhere in thermodynamics, T and S always appear as TS, so it doesn’t matter which convention is being used because at the end, the following holds true always:
TS βTS (33)
What happens to U under Lorentz transformations? This standard result ([8]) can be calculated as follows. Let at u = 0 (velocity = 0), values of thermodynamic quantities be U0, p0 and V0. Consider a fluid having density ρ, pressure p and velocity u (speed being u). Then momentum density is:
ϱ = ρu + pu/c2 (34) => Total Momentum: P = [(U + pV)/c2].u (35) => Force: F = dP/dt (36)
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Hence rate of change of energy is:
dU/dt = F.u – p.dV/dt (37)
Using p=p0 and V=βV0, we get:
dU/dt = (dU/dt).(u2/c2) + (p.u2/c2).(dV/dt) + [(U+pV)/c2].u.(du/dt) – p.dV/dt (38) => [1-(u2/c2)].d(U+pV)/dt = [(U+pV)/c2].u.du/dt (39)
Integrating gives:
2 U + pV = (U0 + p0V0)/ √(1-u ) (40) -1 => U = β (U0 + p0V0) – pV (41)
Again using p=p0 and V= βV0, we finally get:
2 2 -1 U = (U0 + p0V0u /c ) β (42)
Hence under a Lorentz transformation, U transforms as:
-1 2 2 L.T.: U β (U + pV.u /c ) (43)
This taken together with equation (31) completes the formulation of Lorentz transformation of extensive variables in thermodynamics. Remember that under these set of transformations, all intensive variables remain Lorentz invariant. So this beautifully divides into categories of extensive variables (transforming as per equations (31) and (43)) and intensive variables (being Lorentz invariant).
Using equations (31) and (43) and accordingly, all intensive variables being Lorentz invariant, this finally leads to check whether Euler relation holds the test of being Lorentz invariant or not (which every equation must be on a tangent manifold of spacetime for it to be consistent with special theory of relativity). Euler relation is given by equation (1). Plugging in there, it is straight forward that Euler relation fails to be Lorentz invariant, thus violating the sacred demand of special theory of relativity that all equations must be Lorentz invariant for it to be consistent with special theory of relativity. The problem is also clear that it is actually the weird transformation of U that creates all the issue. Hence some kind of new insight and mathematics are required to solve this problem.
VII. WHAT’S MISSING? Few comments are in order. Starting sequentially, the result that the law of extensivity holds only approximately in a curved spacetime background has been highlighted by Thanu Padmanabhan in his paper [9] where he writes in the introductory section that this holds to “a high degree of accuracy” but he doesn’t go further to mathematically formulate his ideas. He specifically comments on the energy which has been taken as an example to illustrate the final result in this paper. But a great deal of difference lies in this fact that in this paper, energy has been taken to be a representative example representing the entire set of extensive thermodynamic quantities. Also this result has got further consequences. The key point which actually leads to this result is that energy is not a true scalar but a
Available online @ www.ijntse.com 8 Rishabh Jha / International Journal of New Technologies in Science and Engineering Vol. 3, Issue. 7, July2016, ISSN 2349-0780 component of a 4-vector. Hence one of the important consequences of this result when tracked backwards is that all extensive variables are not true scalars actually but a component of some 4-vectors. This crucial point will be used below to highlight a plausible solution resolving the issue of Euler relation failing to be Lorentz invariant on a tangent manifold of spacetime. It is imperative to stress the fact that thermodynamics is an empirical science. This is the strength of this field but as explained shortly, this is also the reason for its failure in a more general background. Since mostly all experiments are conducted on a tangent manifold, hence no local experiments can detect this correction term in equation (25). Further, statistical mechanics has been developed so as to explain thermodynamics and give it a firm theoretical basis. The impact on statistical mechanics in curved spacetime background clearly tells about the limitations of statistical mechanics. This is also understood because almost all experiments conducted in statistical mechanics or biophysics are local where extensivity holds up to an unbelievably great level of accuracy. But globally speaking, extensivity is just an approximation.
The Lorentz variance of thermodynamics is an equally shocking result. While this paper was being written down, it was brought to the author’s knowledge that a similar work has been done in [10]. It is important to comment on the convention of whether S needs to be kept Lorentz invariant or T, though finally results will be the same irrespective of the convention chosen as highlighted above. Keeping T Lorentz invariant and making S go to βS under a Lorentz transformation is philosophically satisfying as it neatly categorizes all thermodynamic variables as extensive variables (transforming as per equations (31) and (43)) and intensive variables (being Lorentz invariant). But Fulling–Davies–Unruh effect ([11], [12], [13]) is a very surprising result in quantum field theory that tells that (loosely speaking) an accelerated observer measures a “warm background” when compared to an inertial observer. This relative concept of temperature actually motivates to make use of the convention that S is Lorentz invariant and T becomes βT under Lorentz transformation. The whole problem arises due to the weird transformation of U under Lorentz transformations. Hence somewhere the answer lies at having a new insight to look at energy. A good set of references to start this discussion on U is given in [10]. Another consequence of this Lorentz variance of Euler relation is that in the context of a global background too, Euler relation fails to transform covariantly which is one of the sacred principles of general theory of relativity for any equation to be consistent with general theory of relativity. Though straight forward, but this consequence has not been noted in [10].
An important distinction exists between thermodynamic pressure and relativistic pressure which is generally used in stress-energy-momentum tensor Tβµ. The former is defined only in the thermodynamic limit of number of particles and volume going to infinity. Throughout this paper, thermodynamic quantities have been used which by definition exist only in thermodynamic limit. We have also seen that failure of extensivity implies (since it is an “if and only if relation”) that all extensive quantities are actually component of some 4-vectors. Accordingly one of the plausible solutions to resolve the problem of Lorentz variance of Euler relation is to find those 4-vectors whose component(s) is (are) extensive thermodynamic quantities (This is also evident from the fact that whatever corrections one can do in the standard Euler relation (equation (1)), Euler relation is never going to be Lorentz invariant as each time all modifications would yield some absurd results). Hence accordingly, Euler relation would then be generalized to some kind of tensor relation which will be covariant in general theory of relativity and Lorentz invariant in special theory of relativity. This standard Euler relation (equation (1)) would be obtained from the generalized Euler (tensor) relation as one of the components of the tensor equation. There must be then new relations apart from the standard Euler relation (equation (1)) corresponding to other components of the generalized Euler (tensor)
Available online @ www.ijntse.com 9 Rishabh Jha / International Journal of New Technologies in Science and Engineering Vol. 3, Issue. 7, July2016, ISSN 2349-0780 relation. Since in the classical limit and flat spacetime background (on a tangent manifold of spacetime), extensivity holds precisely, hence other relations so obtained would must reduce to triviality and the only meaningful remaining equation would must be the standard Euler relation (equation (1)). These require new mathematics (since the present mathematics is inconsistent among them) and insights. This situation reminds the author of a famous quote (possibly) by Carl Sagan that “Somewhere, something incredible is waiting to be known.”
Finally it is important to point out that though extensivity plays a major role in thermodynamics but for the first and second laws of thermodynamics to hold, the concavity property of entropy is of paramount importance rather the extensive properties of entropy.
VIII. SUMMARY This paper started with a review of thermodynamics and theories of relativity. Then by contrasting the fundamentals of these pillars of Physics, the following results were obtained.
Law of extensivity holds only approximately in the most general curved spacetime background. It becomes precise only in the limit of flat spacetime background (say on a tangent manifold of spacetime on which most experiments are conducted) as well as classical limits owing to the fact that affine connections vanish identically. o Consequently, foundational equations of statistical mechanics fail to hold in general theory of relativity.
The present mathematical formulation of thermodynamics and special theory of relativity are mathematically inconsistent because Euler relation fails to be Lorentz invariant on the tangent plane of spacetime (where general theory of relativity reduces to special theory of relativity). This inconsistency arises due to the weird transformation property of U under Lorentz transformations. This indicates that new insight(s) and mathematics are required to look at this entire problem as a whole and the interpretation of energy and its mathematical properties in particular. o Consequently, Euler relation also fails to be covariant under a general coordinate transformation in general theory of relativity making it mathematically inconsistent with general theory of relativity too.
Problems and inconsistencies have been highlighted in this paper with full mathematical support. The paper clearly points out that understanding thermodynamics and statistical mechanics in the context of both the theories of relativity are far from complete and is definitely an open problem. An attempt has been made to illustrate as why this problem is coming up at all under section (VII) of this paper. In that same section, a plausible way to resolve these inconsistencies has also been pointed out, though it is very sketchy and (of course) very far from complete. These require calm thinking and discussions which definitely need to be accompanied by new insight(s) and mathematics.
ACKNOWLEDGMENT I am highly thankful to my friend Shounak De for asking thought provoking questions which led me
Available online @ www.ijntse.com 10 Rishabh Jha / International Journal of New Technologies in Science and Engineering Vol. 3, Issue. 7, July2016, ISSN 2349-0780 to these ideas presented here and without whom this work wouldn't have been possible at this point of time.
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[1] Herbert Callen, “Thermodynamics and An Introduction to Thermostatistics,” Paperback Edition, June 2006. [2] Wolfgang Rindler, “Relativity: Special, General, and Cosmological,” Paperback Edition, April 2006. [3] S. Chandrasekhar, “The general theory of relativity: Why “It is probably the most beautiful of all existing theories”,” Journal of Astrophysics and Astronomy, 5(1):3-11, March 1984. [4] Wikipedia Reference: https://en.wikipedia.org/wiki/Equivalence_principle [5] J. M. Bardeen, B. Carter, S. W. Hawking, "The four laws of black hole mechanics". Communications in Mathematical Physics 31 (2): 161–170, 1973. [6] Jacob D. Bekenstein, "Black holes and entropy". Physical Review D 7 (8): 2333–2346, April 1973. [7] R. K. Pathria, Paul D. Beale, “Statistical Mechanics,” Paperback Edition, February 2011. [8] Richard C. Tolman, “Relativity, Thermodynamics and Cosmology (Dover Books on Physics),” Paperback Edition, November 1988. [9] T. Padmanabhan, “Stellar Dynamics and Chandra,” Current Science, Vol. 70, No.9, May 1996. [10] J. Dunning-Davies, “Extensivity and Relativistic Thermodynamics,” arXiv:0706.4020v1 [physics.gen-ph], June 2007. [11] S.A. Fulling, "Nonuniqueness of Canonical Field Quantization in Riemannian Space-Time," Physical Review D 7 (10): 2850, doi:10.1103/PhysRevD.7.2850, 1973. [12] P.C.W. Davies, "Scalar production in Schwarzschild and Rindler metrics," Journal of Physics A 8 (4): 609, doi:10.1088/0305-4470/8/4/022, 1975. [13] W.G. Unruh, "Notes on black-hole evaporation," Physical Review D 14 (4): 870, doi:10.1103/PhysRevD.14.870, 1976.
Rishabh Jha (Author):
He has received his professional training in Physics in the department of Physics, IIT Kanpur, India. His publication include "Kerr-Newmann Black Hole Thermodynamics, Quantum Geometry and Information Theory", International Journal of Research and Development in Physics (IJRDP), Volume 2, Issue.1, October, 2015. His research area is mainly focused on theoretical Physics. The topics include a vast spectrum ranging from black hole thermodynamics to quantum information. He has been a KVPY/INSPIRE fellow awarded by the Government of India. His other research works include quantum holonomies and phase transitions on which he is presently working. For any other information or details of his research work, kindly contact him at [email protected].
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