Müller I. A history of thermodynamics. The doctrine of energy and entropy (Müller I. A history of thermodynamics. The doctrine of energy and entropy.pdf), страница 68

10.1. A kind of density distribution inthe ultimate white dwarfThe last step makes use of the differential equation in the formȡ1 d § 2 d 1 (U/A) 2/3¨r AL2r 2 dr ¨©dr·¸.¸¹Obviously, degeneration of the electron gas has played a decisive role in theforgoing analysis. It is less clear that the relativistic square root in the equation forp is essential for the result. However, it is! Without that relativistic contributionthere is no mass limit.Insert 10.2The usual interpretation of the Chandrasekhar limit is that the electron gas cannotwithstand the gravitational pull of bigger masses than 1.4 MƁ .

It is assumed thatunder great pressure the electrons are pushed into the protons of the iron nuclei toform neutrons. The star thus becomes a neutron star, with a truly enormous mass15density: 10 times the already large density of a white dwarf. Neutron stars havetheir own mass limit – 3.2 MƁ – according to a theory presented by J. RobertOppenheimer (1904–1967) in 1939. If a star is bigger than that, – and does not getrid of the excess mass by nova- or supernova-explosions – it collapses into a blackhole, at least according to current wisdom.

There seems to be no conceivablemechanism to stop the collapse. It is tempting to pursue the matter further in thisbook. However, there is a touch of science fiction in the subject and I desist, – withregret.Chandrasekhar has left his mark in several fields of physics. In hisautobiography he says that he was … motivated, principally, by a questafter perspectives…compatible with my taste, abilities and temperament.Stellar dynamics was the subject of only the first such quest.

Othersfollowed: x Brownian motion, x radiative transfer, x hydrodynamic stability,x relativistic astrophysics, x mathematical theory of black holes. WheneverMaximum Characteristic Speed299he found that he understood the subject, he published one of his highlyreadable books, – in his words: a coherent account with order, form, andstructure. Thus he has left behind an admirable library of monographs forstudents and teachers alike. His work on white dwarfs, but also his lifelongexemplary dedication to science, was rewarded with the Nobel prize inphysics in 1983, fifty years after he discovered the Chandrasekhar limit.The maximal mass of a white dwarf is notalone in having been named afterChandrasekhar.

There is also the NASA X-rayobservatory which is called Chandrasekharobservatory, and a minor planet,– one of about15000 – which was named Chandra in 1958.Fig. 10.2. Subrahmanyan ChandrasekharMaximum Characteristic SpeedAfter Jüttner there was a period of stagnation in the development of relativistic thermodynamics. To be sure, there was some interest, and in 1957John Lighton Synge (1897–1995) streamlined Jüttner’s results in a neatsmall book12 which, however, did not significantly add to previous results.Also Eckart provided a relativistic version of thermodynamics of irreversible processes,13 in which he improved Fourier’s law of heat conduction byaccounting for the inertia of energy, cf. Chap.

8. However, his differentialequation for temperature was still parabolic so that the paradox of heatconduction persisted. Understandably that paradox has irritated relativistsmore than it did non-relativistic physicists. After all, if no atom, ormolecule can move faster than the speed of light, heat conduction should12J.L Synge: “The Relativistic Gas.” North Holland, Amsterdam (1957).13C.

Eckart: “The thermodynamic of irreversible processes III: Relativistic theory of thesimple fluid.” loc. cit.30010 Relativistic Thermodynamicsnot be infinitely fast. This problem was the original motive for Müller todevelop extended thermodynamics, cf. Chap. 8, and its relativistic version.14Shortly afterwards, Israel15 published a very similar theory and, eventually,it was shown by Boillat and Ruggeri16 that extended thermodynamics ofinfinitely many moments predicts the speed of light for heat conduction.Thus the paradox was resolved; the field is conclusively explained byMüller in a recent review article.17A decisive step forward in the general theory was done by N.A.Chernikov in 1964 18 when he formulated a relativistic Boltzmann equation.Let us consider this now.Boltzmann-Chernikov EquationI have already mentioned the elegant four-dimensional formulation which isnow standard in relativity.

It was introduced by Hermann Minkowski(1864–1909). Minkowski had taught Einstein in Zürich and later he becamethe most eager student of Einstein’s paper on special relativity. He suggested that the theory of relativity makes it possible to take time into accountas a kind of fourth dimension and he introduced the distance ds betweentwo events at different places and different times19ds 2g cAB dx c A dx c Bc 2 dt c2 (dx c1 ) 2 (dx c2 ) 2 (dx c3 ) 2 .in a Lorentz framewith coordinates ct´,x´14aI. Müller: “Zur Ausbreitungsgeschwindigkeit ...” Dissertation (1966) loc. cit.A streamlined version of relativistic extended thermodynamics may be found in:I-Shih Liu, I. Müller, T.

Ruggeri: “Relativistic thermodynamics of gases.” Annals ofPhysics 169 (1986).15 W. Israel: “Nonstationary irreversible thermodynamics: A causal relativistic theory.”Annals of Physics 100 (1976).16 G. Boillat, T. Ruggeri: “Moment equations in the kinetic theory of gases and wavevelocities.” (1997) loc.cit.17 I. Müller: “Speeds of propagation in classical and relativistic extended thermodynamics.”http:/www.livingreviews.org/Articles/Volume2/1999-1mueller.18 N.A.

Chernikov: “The relativistic gas in the gravitational field.” Acta Physica Polonica 23(1964).N.A. Chernikov: “Equilibrium distribution of the relativistic gas.” Acto Physica Polonica26 (1964).N.A. Chernikov: “Microscopic foundation of relativistic hydrodynamics.” Acta PhysicaPolonica 27 (1964).19 H. Minkowski: “Raum und Zeit.” [Space and time] Address delivered at the 80thAssembly of German Natural Scientists and Physicists, at Cologne. September 21st, 1908.The address has been translated into English and is reprinted in “The Principle ofRelativity.

A collection of original memoirs on the special and general theory ofrelativity.” Dover Publications pp. 75–91Boltzmann-Chernikov Equation301In this manner the tensor gƍAB, whose invariance defines the Lorentzframes, may be interpreted as a metric tensor of space-time. Its componentsin a arbitrary frame xA = xA(xƍ B) can be calculated fromg ABwx cC wx c Dc .g CDwx A wx BIn particular, for a rotating frame – on a carousel (say) – with coordinates(ct,r,ș,z) given bytƍ = t, xƍ 1 = r cos(ș + Ȧt), xƍ 2 = r sin(ș + Ȧt), xƍ 3 = zthe metric tensor readsg ABȦr§ Ȧ2 r 2·0 0 ¸¨1 2cc¨¸0100 ¸.¨¨ Ȧr0 r2 0 ¸¨¨¸¸c0001©¹The metric tensor has some significance, because it allows us to write theequation of motion of a free particle, whose orbit is parametrized by IJ, inthe formd2 x BdIJ 2A BB dx dx , ī ACdIJ dIJwhereBī AC1 BD § wg DA wg DC wg AC ·¨¨¸.g2wx Awx D ¸¹© wx CBIndeed, in a Lorentz frame, with ī AC= 0, the solution of this equation isa motion in a straight line with constant velocity, which is the definingfeature of an inertial frame.

The parameter IJ is usually chosen as the propertime of the moving particle, i.e. the time read off from a clock in themomentarily co-moving Lorentz frame. With that, the equation of motionmay be written in the formdp BdIJ1 B A Bī AC p p ,µcwherepAdx AdIJis the four-momentum of the particle as before.The equation of motion represents the equation of a geodesic in space-time. This isa nice feature, much beloved by theoretical physicists, because it supports theirpredilection for a specious geometrical interpretation of the theory of relativity.

Thenotion was useful for Einstein, when he developed the theory of general relativity;but most often it is used to confuse laymen with talk about curved space, etc.30210 Relativistic ThermodynamicsOf course, nobody will try to solve the equation of the geodesic in its general formin order to calculate the orbit of a free particle. It is so much easier to solve it in aLorentz frame and transform the straight line obtained there to an arbitrary frame.The relativistic – non-quantum – formulation of the Boltzmann equationwas derived in a series of three remarkable papers by N.A. Chernikov. It isan integro-differential equation for the relativistic distribution functionF(xA,pa) which readspAwFwFd ī ABp A pBwx Awp d³ (F(pcC)F(qcC ) F(p C )F(q C )h  pq ! dedQ.Comparison with the classical Boltzmann equation, cf.

Chap. 4, easilyidentifies the individual terms. I do not go into that, other than saying thatx the term with ī represents the acceleration of a particle between twocollisions,20 andx the collision term on the right hand side vanishes for the MaxwellJüttner distribution because of conservation of the energy and momentum vector pA in the collision.Chernikov uses the equation for the formulation of equations of transferfor moments of the distribution function and he concentrates on 13moments, which is rather artificial for a relativistic theory; it is moreappropriate to include the dynamic pressure and thus come up with a theoryof 14 moments.21 But we shall not pursue this question here, because sofar – apart from the finite characteristic speeds – the multi-moment theoryhas not provided any suggestive results that go beyond Eckart’s reformulation of the Fourier law, see Chap.