On thermodynamic equilibrium in a gravitational field

Balázs, N. L.; Dawson, Jef

doi:10.1016/0031-8914(65)90089-3

Cited by 14 publications

(14 citation statements)

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“…being proportional to T d loc , where d is the space-time dimension and T is the asymptotic Minkowski space temperature. T loc is the temperature measured by a standard local thermometer, such as a Carnot cycle [15,16]. This behavior is in agreement with the so-called Tolman-Ehrenfest effect which argues that in a system at thermal equilibrium in a stationary gravitational field, the temperature varies with the space-time metric according to the relation (1).…”

Section: Introductionsupporting

confidence: 60%

“…From a physical point of view, this universal behavior (which has also been derived using other approaches [15,16,22,23]) can be traced back to the requirement of thermal equilibrium in a gravitational field. Indeed, the emergence of a local temperature, and consequently a temperature gradient, is unavoidable in thermal equilibrium to prevent heat (which interacts with gravity) to flow from regions of higher to those of lower gravitational potential.…”

Section: Discussionmentioning

confidence: 69%

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QED plasma in a background of static gravitational fields

2014

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We obtain the effective Lagrangian of static gravitational fields interacting with a QED plasma at high temperature. Using the equivalence between the static hard thermal loops and those with zero external energy-momentum, we compute the effective Lagrangian up to two-loop order. We also obtain a non-perturbative contribution which arises from the sum of all infrared divergent ring-diagrams. From the gauge and Weyl symmetries of the theory, we deduce to all orders that this effective Lagrangian is equivalent to the pressure of a QED plasma in Minkowski space-time, with the global temperature replaced by the Tolman local temperature.Comment: Revised version to be published in Phys. Rev. D (9 pages, 4 figures

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Section: Introductionsupporting

confidence: 60%

Section: Discussionmentioning

confidence: 69%

QED plasma in a background of static gravitational fields

2014

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“…Relativistic effects make the gas warmer at the bottom and cooler at the top, by a correction proportional to c −2 , where c is the speed of light. This is the well known Tolman-Ehrenfest effect, discovered in the thirties [1,2] and later derived in a variety of different manners [3][4][5][6][7][8][9][10][11]. The temperatures T 1 and T 2 measured by the same thermometer at two altitudes h 1 and h 2 in a Newtonian potential Φ(h) are related by the Tolman law…”

Section: Non-uniform Equilibrium Temperaturementioning

confidence: 99%

Death and resurrection of the zeroth principle of thermodynamics

Haggard

Rovelli

2013

Phys. Rev. D

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“…Relativistic effects make the gas warmer at the bottom and cooler at the top, by a correction proportional to c −2 , where c is the speed of light. This is the well known Tolman-Ehrenfest effect, discovered in the thirties [1, 2] and later derived in a variety of different manners [3][4][5][6][7][8][9][10][11]. The temperatures T 1 and T 2 measured by the same thermometer at two altitudes h 1 and h 2 in a Newtonian potential Φ(h) are related by the Tolman lawThe general-covariant version of this law readswhere |ξ| is the norm of the timelike Killing field with respect to which equilibrium is established.…”

mentioning

confidence: 99%

Death and Resurrection of the Zeroth Principle of Thermodynamics

Haggard

Rovelli

2013

Int. J. Mod. Phys. D

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The zeroth principle of thermodynamics in the form "temperature is uniform at equilibrium" is notoriously violated in relativistic gravity. Temperature uniformity is often derived from the maximization of the total number of microstates of two interacting systems under energy exchanges. Here we discuss a generalized version of this derivation, based on informational notions, which remains valid in the general context. The result is based on the observation that the time taken by any system to move to a distinguishable (nearly orthogonal) quantum state is a universal quantity that depends solely on the temperature. At equilibrium the net information flow between two systems must vanish, and this happens when two systems transit the same number of distinguishable states in the course of their interaction. I. NON-UNIFORM EQUILIBRIUM TEMPERATUREAccording to non-relativistic thermodynamics, a thermometer (say, a line of mercury in a glass tube), moved up and down a column of gas at equilibrium in a constant gravitational field, measures a uniform temperature. But this prediction is wrong. Relativistic effects make the gas warmer at the bottom and cooler at the top, by a correction proportional to c −2 , where c is the speed of light. This is the well known Tolman-Ehrenfest effect, discovered in the thirties [1, 2] and later derived in a variety of different manners [3][4][5][6][7][8][9][10][11]. The temperatures T 1 and T 2 measured by the same thermometer at two altitudes h 1 and h 2 in a Newtonian potential Φ(h) are related by the Tolman lawThe general-covariant version of this law readswhere |ξ| is the norm of the timelike Killing field with respect to which equilibrium is established. A violation of the uniformity of temperature seems counterintuitive at first, especially if one has in mind a definition of "temperature" as a label of the equivalence classes of all systems in equilibrium with one another. In a relativistic context a physical thermometer does not measure this label and we must therefore distinguish two notions: (i) a quantity T o defined as this label (proportional to the constant in (2)), and (ii) the temperature T measured by a standard thermometer.In the micro-canonical framework the entropy S(E) is the logarithm of the number of microstates N (E) that have energy E and T can be identified with the inverse of the derivative of S(E),where k is the Boltzmann constant. The fact that two systems in equilibrium have the same T can be derived by maximizing the total number of states N = N 1 N 2 under an energy transfer dE between the two. This gives easily T 1 = T 2 . In the presence of relativistic gravity, this derivation fails because conservation of energy becomes tricky: intuitively speaking, the energy dE reaching the upper system is smaller than the one leaving the lower system because "energy weighs". Is there a more general statistical argument that governs equilibrium in a relativistic context? Can the Tolman law be derived from a principle generalizing the maximization of the number of micro...

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On thermodynamic equilibrium in a gravitational field

Cited by 14 publications

References 1 publication

QED plasma in a background of static gravitational fields

QED plasma in a background of static gravitational fields

Death and resurrection of the zeroth principle of thermodynamics

Death and Resurrection of the Zeroth Principle of Thermodynamics

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