Tolman-like temperature gradients in stationary spacetimes
Jessica Santiago · Matt Visser
Abstract and summary · read the original at the source
In one page
Heat runs downhill in gravity. Richard Tolman showed in 1930 that a fluid sitting still in a static gravitational field, in perfect thermal equilibrium, does not have one temperature — it reads hotter deeper in the well, by exactly the redshift factor. Jessica Santiago and Matt Visser at Victoria University of Wellington point out that this famous result carries a hidden ingredient. A heat bath is not specified by a temperature alone; it needs a temperature and a rest frame. In a static spacetime there is only one natural rest frame, so nobody notices the choice being made. Around a rotating body there are several natural choices, and they give different answers, which means Buchdahl’s 1949 extension of Tolman is only part of the story. Their result is one equation covering all the cases: the temperature gradient is minus the four-acceleration of the fluid, whatever the spacetime and whatever the material. Pick a free-falling frame and the gradient vanishes. Pick the zero-angular-momentum frame around a spinning black hole and the Hawking temperature redshifts sensibly all the way out.
Why it matters hereChapter 4 is about taking the metric seriously as the thing that sets what instruments read, and this paper is a clean demonstration of that: the same equilibrium fluid, in the same spacetime, reports different temperature gradients depending on which frame you call the rest frame — and the metric, through the lapse function, tells you the number. It reaches chapter 2 through its worked example, which is how to carry the Hawking temperature of a rotating black hole out from a horizon where the local value diverges to a spatial infinity where it is finite.
What it claims
01Tolman in 1930, with Tolman and Ehrenfest, showed that a fluid in thermal equilibrium in a static spacetime carries a relativistic temperature gradient: the locally measured temperature is a position-independent constant divided by the norm of the static Killing vector, so temperature multiplied by the square root of minus the time-time metric component is constant across the fluid.Section I, Equations 1.1 and 1.5; Section III, Equation 3.1
Settled physics02For a photon gas in internal equilibrium the relativistic Euler equation reduces to the statement that the four-acceleration equals minus the gradient of the logarithm of the temperature, and because a thermal equilibrium gradient cannot depend on the substance or the state of matter, that relation extends to any system in internal thermal equilibrium — so any accelerating thermal bath shows a temperature gradient, whether the spacetime is Minkowski, Schwarzschild or Kerr-Newman, and Einstein’s equations are not needed to derive it.Section II, Equations 2.1 to 2.5
Published and peer-reviewed03Buchdahl’s 1949 extension of Tolman rests on a purely kinematic result valid for any Killing flow, that the four-acceleration is the gradient of the logarithm of the Killing vector norm — but it is incomplete, because specifying a heat bath requires a four-velocity as well as a temperature, and in stationary non-static spacetimes several different natural four-velocity fields can be defined, where in static spacetimes the normal flow and the Killing flow can be made to coincide.Section I; Section IV, Equations 4.1 to 4.5
Published and peer-reviewed04The other natural choice in a stationary spacetime is the normal flow, the fluid four-velocity proportional to the gradient of the time coordinate, which is automatically free of vorticity and gives a four-acceleration equal to the gradient of the logarithm of the ADM lapse function, hence a temperature equal to a constant divided by the lapse — the same as the Killing-flow answer in static block-diagonal spacetimes and typically different otherwise.Section V, Equations 5.1 and 5.2; Section VIII, Conclusions
Published and peer-reviewed05For an axially symmetric black hole the normal flow is the zero-angular-momentum-observer flow, which is definitely not a Killing flow, and it gives a redshifted temperature proportional to the square root of minus the inverse time-time metric component that stays well behaved from just above the horizon out to spatial infinity — a plausible way to define the redshifted Hawking temperature for Kerr and Kerr-Newman, where the locally measured Hawking temperature diverges at the horizon and is finite far away.Section VII B, ZAMO normal flow, Equations 7.8 and 7.9
Published and peer-reviewed06Choosing a free-fall normal flow makes the Tolman temperature gradient vanish altogether, and the choice of coordinates — Boyer-Lindquist against Doran, for instance — does not change the physics but guides which four-velocity is physically appropriate for the heat bath, so the existence and size of a Tolman gradient cannot be separated from that choice.Section VII B, closing paragraph; Section VIII, Conclusions
Published and peer-reviewed
Read it · abstract
Abstract
It is (or should be) well known that specification of a heat bath requires both a temperature and a 4-velocity, the rest frame of the heat bath. In static spacetimes there is a very natural and unique candidate for the 4-velocity of the heat bath, the normalized timelike Killing vector. However in stationary non-static spacetimes the situation is considerably more subtle, and several different “natural” 4-velocity fields suitable for characterizing the rest frame of a heat bath can be defined — thus Buchdahl’s 1949 analysis for the Tolman temperature gradient in a stationary spacetime is only part of the story. In particular, the heat bath most suitable for describing the Hawking radiation from a rotating black hole is best described in terms of a gradient flow normal to the spacelike hypersurfaces, not in terms of Killing vectors.
The way in
https://doi.org/10.1103/PhysRevD.98.064001Published in Physical Review D volume 98, article 064001 (2018) by Jessica Santiago and Matt Visser of the School of Mathematics and Statistics at Victoria University of Wellington, dated 6 July 2018 and submitted to arXiv as 1807.02915 on 9 July 2018. The arXiv posting carries arXiv’s non-exclusive distribution licence and the published article the APS default licence, neither of which is a Creative Commons licence, so this page carries the summary, the claims and the authors’ own abstract and sends the reader to the source. Jessica Santiago’s work was supported by a Victoria University of Wellington PhD Scholarship.
How to cite it
Jessica Santiago, Matt Visser (2018) Tolman-like temperature gradients in stationary spacetimes. doi:10.1103/PhysRevD.98.064001
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