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Inconsistencies Between Thermodynamics and Relativity 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: Available online @ www.ijntse.com 1 Rishabh Jha / International Journal of New Technologies in Science and Engineering Vol. 3, Issue. 7, July2016, ISSN 2349-0780 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.
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