Thermodynamics of relativistic rotating perfect fluids

ANNALS

OF PHYSICS:

76, 301-332 (1973)

Thermodynamics

of Relativistic

G. Racah Institute

of

HORWITZ

AND

Rotating

J.

Fluids

KATZ

Physics, The Hebrew University Received

Perfect

of

Jerusalem, Jerusalem,

Israel

October 21, 1971

A new formulation of thermodynamics for special and general relativistic rotating perfect fluids is developed. Both isolated systems and portions of isolated systems electrically uncharged or charged are treated. Exploiting the symmetry of motion of stationary axisymmetric fluids, the global thermodynamic functions, including total energy and spin, are defined as free scalars, represented by hypersurface integrals of conserved vectors. Local equilibrium parameters such as local temperature and chemical potential are scalar functions. There also exist global equilibrium parameters, global temperature and global chemical potential, which are free scalars. The connection between local and global conditions of thermodynamic equilibrium is made clear and explicit. Thermodynamic potentials are introduced in the context of treating open systems in a relativistically invariant way.

A formulation of thermodynamics in a form compatible with special relativity was developed many years ago by Planck [I] and Einstein [2] and later for general relativity by Tolman and Ehrenfest [3]. Many extensions, generalizations and a variety of different derivations [4] have been proposed over the years. In recent years, there has also appeared a serious challenge to the Planck results [5]. The present work is not directly concerned with this controversy, which deals with the appropriate “transformation laws” for temperature, heat and other thermodynamic quantities. In the context of our relatively simple treatment of much more complex systems than those treated in the above-mentioned polemics, the controversy appears rather superfluous, a semantic rather than a physical disagreement. What is, in fact, the nature of the problem of applying relativity to equilibrium thermodynamics? The central problem of relativity, both special and general, is to analyze how various physical quantities will appear to different observers and to transcribe relations to a form valid for any member of some given class of observers. Thus one commonly seeks some scalar, vector or tensor definition for physical parameters as well as a covariant form for all equations valid under Lorentz transformations for special relativity and general coordinate transformations in general relativity. 301 Copyright All rights

0 1973 by Academic Press, Inc. of reproduction in any form reserved.

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A system in thermodynamic equilibrium, at rest in a given inertial frame, and which remains at rest in this frame while undergoing various processes is the subject of conventional thermodynamics. How to reformulate the definitions of the various quantities and the laws of thermodynamics for an arbitrary inertial observer, having a constant velocity relative to the system, is the problem of (special) relativistic thermodynamics. Classically, thermodynamic equilibrium exists not only for systems moving with uniform velocity (i.e., with the center of mass at rest in some inertial frame), but for systems all parts of which also possess a common constant angular velocity about an axis [6]. How does one generalize the thermodynamics for such a system to an arbitrary observer? For rotating systems there is the additional complication of dealing with nonuniform local behavior and of relating local to global quantities, local to global thermodynamic equations. There are, in a sense, two different approaches possible to the relativistic formulation: (i) One can seek an appropriate definition for all relevant thermodynamic quantities as measure by a moving observer. Thus a definition of internal energy, work, heat, etc., as measured by the moving observer would arise, as well as the appearance of additional variables related to momentum. Thus, for example, internal energy in the rest frame will be associated with the four vector energymomentum [7] for an isolated system, enthalpy with the four vector enthalpymomentum [S] for a confined system. One then seeks covariant expressions for the first and second laws of thermodynamics, scalar or four-vector expressions, from which definitions and transformation laws for intensive parameters, notably temperature, are defined. Unfortunately, there is a substantial amount of arbitrariness as to which definition of heat and work to apply to the moving observer and this leads to lack of uniqueness [9] of the results for the transformation of both extensive and intensive parameters. Furthermore, as a result of the nonequivalence of simultaneity for different frames, some rather grotesque physical descriptions arise [lo]. (ii) An alternative approach is possible which does not have these ambiguities; it is closer to the approach adopted by the general relativist. One can seek to define the thermodynamic variables as quantities which will not transform under a given class of coordinate transformations. The thermodynamic laws are thus given in terms of scalar quantities, and are form invariant. For example, the energy can be defined as a scalar quantity (which is, of course, equal to the rest frame energy) and hence will have the same value for all inertial 0bservers.l For 1Of course,to measurethis energy, the movingobserverwould have either to add up the conventionalenergydensities at differenttimesor usea differentdefinitionof densitywhichhe could measuresimultaneously.

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static uniform systems, global scalar expressions suffice. However, for nonuniform systems- whether the origin of the nonuniformity is rotation or long-range forces-- one also needs local scalar expressions in addition to global ones. In the latter case, global scalars correspond to the standard definitions as measured in the center of mass rest frame, while local scalars correspond to the quantities measured in the local comoving frame. Thus, the temperature according to this approach will always be a scalar. For the uniform case, it is a free scalar, i.e., a number, while for the nonuniform case the local temperature is a scalar function varying from point to point. The relations connecting the local temperatures at different points (for the rotating systems at points with different velocities) is not a transformation law, but rather an expression for the position variation of the locally measured temperature.2 It is the second approach which is the basis of the present work. Since the application to static uniform systems is relatively trivial, and there is considerable interest in relativistic rotating systems, we shall present our development for rotating systems. Both locally and globally results will be formulated in terms of scalar quantities in the affine sense, thus being valid in arbitrary coordinate systems.

In general, the rotating

systems are nonuniform

and the description

has two

levels-local and global. The local temperature is found to be uniform in the nonrelativistic limit on the one hand and to correspond to the appropriate limit of the ,general relativistic theory on the other, i.e., a gravitational force corresponding to centrifugal force. Loca.1 thermodynamics is usually assumed known, as in hydrodynamic descriptions [I I], and, in effect, corresponds to standard thermodynamics written in density form3 or for volumes small enough to be locally uniform. The criteria for the validity of such form of local thermodynamics raise some interesting and important questions,4 but for the purposes of the present work we shall assume them to hold. 2 In this respect it is worthwhile to note that the results derived by Planck-Einstein, for special relativity,, differ intrinsically from those of Tolman-Ehrenfest for general relativity. The former results is a transformation law, the variation for different observers; the latter result on the other hand is an expression of the variation of the local temperature for a system with varying gravitational field as a function of the local field, when in an overall state of thermodynamic equilibrium. 3 Notice that even though we are dealing here with special relativity, the only basis to say that we would expect a priori in a local comoving frame-which is after all an accelerated frameto have conventional results holding would be on the basis of the strong equivalence principle [12]. ’ A commonly stated criterion for the validity of local thermodynamics for nonuniform systems is that the mean free path for all excitations be small compared with a typical dimension over which the gradients of densities are negligible. This kind of description is not necessarily adequate.

First of all, it is really only meaningfulas a low densityapproximation(for the excitations). Second, if there is a very long-range correlation,

it may be completely inadequate. A diagonal

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The basis for the global form is less clear. One can obtain the global expression by assuming that the entropy variation depends on variations of relevant constants of the motion. For the rotating fluid these are expected to be the energy A4 (inclusive of the rest mass) the spin J, the number of particles N and the volume V. For a self-bound system, however, the volume is not an independent variable. There are conventional ways of connecting global to local thermodynamic relations, at least in special coordinates. The problem of relating global to local expressions is distinctly nontrivial, as we shall discuss further below, and has not hitherto been solved in a general way. The method which is developed in the present work is to make the connection between global and local quantities by exploiting the symmetries of the motion. Global scalars are related to conserved vector densities defined with the aid of Killing vector fields in a manner related to the conserved vector densities introduced by Komar [14] in general relativity. Global thermodynamics is expressed in terms of scalar relations. Energy, spin, particle number, volume are constants of the motion related to conserved vector densities. Conservation is related, however, to the symmetry of the motion rather than to dynamical relations; to emphasize this we shall designate them as kinematical conservation relations. Thus conservation is not based on relations like the vanishing of the divergence of the stress-energy tensor T 11”)which implies the equations of motion of the matter. Instead, it is based on the vanishing of the divergence of an energy vector density and a spin vector density defined in terms of timelike and spacelike Killing vector fields respectively. The vanishing of their divergences follows as an identity due to the invariance of the systems to time translations and rotations about the axis of symmetry. This lack of dependence on equations of motion is intrinsically more satisfactory than the more common methods of treating relativistic thermodynamics, since, although equilibrium of forces is a necessary condition for thermodynamic equilibrium, the thermodynamic description should not be expected to relate directly to the forces if they are balanced. This paper proceeds in four divisions: Part Part dynamics Part Part

I: Isolated perfect fluids in rotation; II: Open systems in rotation-treatment of part of a closed system; thermodynamic III: Isolated charged fluids in rotation; IV: Isolated fluids in general relativity.

In part I we first indicate the problems

involved

of the equilibrium potentials;

in making

thermo-

the connection

mean field theory does not lead to any breakdown of local thermodynamics provided that the gradients of the field are negligible over large enough regions. An off-diagonal mean field could lead to a nonlocal theory. As a concrete example leading to a nonlocal equation of state we note the case of the Klein-Gordon geon [13].

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between global and local thermodynamics. Then, introducing Killing fields associated with the symmetry of motion, the conserved energy and spin vector densities and differential forms are introduced. This yields hypersurface independent definitions of energy and spin which are thus free scalars. The thermodynamics is then formulated globally in scalar form and the local form emerges naturally from this. Or reciprocally, we could build up our global form based on the assumedlocal form for the entropy. Part II treats the thermodynamics of rotating fluids open with respect to one or more of the conserved quantities. The results are derived by dividing an isolated system into one small part regarded as open, with the rest being considered as a reservoir. The motivation is to find equilibrium conditions for subsystemstoo large to be regarded as uniform, but nonethelessnot closed. Results are derived for systemsin contact with volume reservoirs (having constant surface pressure), and for a system defined by delineating a specific volume. For these open systems it becomes clear that the global (constant) equilibrium parameters, global temperature F, global chemical potential ,& become the relevant parameters whose zero variation corresponds to equilibrium. Thus, for example, constant global temperature T and not constant local temperature T becomes the condition of no heat flow. Thermodynamic potentials are then discussedmore generally and some further examples are introduced including the Massieu functions related to the Helmholtz and Gibbs free energies. In part III the same approach is extended to treat a charged rotating fluid. Only metastable equilibrium is possible for such a system, but we treat rotating system both for didactic reasonsand for the reason that one may have situations with a low enough rate of radiation to make our thermodynamic treatment adequate for reasonably long times. The vector potential is related to the Killing vectors and the correspondingly conserved vector densities of energy and spin associatedwith the electromagnetic field are introduced. Energy and spin densities for the electromagnetic field are not uniquely defined, and we examine two cases. Locally,, the contribution to the entropy variation due to the electromagnetic field vanishes up to a divergence, which for isolated nonradiating systems vanishes rapidly enough not to contribute to the total entropy. The result is a local thermodynamics which is independent of the electromagnetic field as conventionally assumed.The global form is also evaluated. In part IV we show the slight formal modification necessaryto extend the method to general relativistic systems.Formally the results are very similar to those found for the electromagnetic field. There are, however, important conceptual differences. The symmetry becomesthe symmetry of spacerather than that of motion; energy and spin take on purely geometric definitions. As a result, thermodynamic equilibrium implies Einstein’s equations and thus, due to the Bianchi identities, the equations of motion, in contrast to the special relativistic case.

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AND KATZ

I. AN ISOLATED ROTATING

I-l.

PERFECT FLUID

The Problem of Relativistic Formulation for Rotating Fluids

Let us consider a perfect elastic fluid made up of particles of one type and rotating about an axis; generalization to more complex systems would appear to be straightforward; in particular, charged fluids are treated below. A basic assumption we make is that there is a valid local thermodynamics. By assumption, conditions obtain wherein there exist regions whose dimensions are small enough to be uniform and yet large enough to be macroscopic and hence have welldefined local equilibrium. Long range forces, if present, would have to be constant over the dimensions of such regions. The stress-energy tensor Tuy (A, p, v, p, u = 0, 1,2,3) is divergenceless since the system is isolated, 3YTu* = 0 3

(1)

(a, means a/&‘), and this leads to the existence of a conserved four-vector PA, the energy-momentum. There is similarly a conserved angular momentum tensor and a conserved particle number. There is associated with each of the above-named conserved quantities a number, the total energy M and the total spin J, in addition to the total particle number N. These quantities can be related to integrals in a special coordinate system over quantities which can reasonably be regarded separately as energy, spin and particle number densities. The preferred coordinate system is one with the origin at the center of mass and the x3, or z axis, the axis of rotation. Then, M = j, Too dV,

J = jy(Toly

- xTo2) dV,

N = j

P

n dV;

(2)

n being the number of particles per unit volume and P any volume large enough to contain all the matter. The first and second laws of thermodynamics are then given by the following relations for quasistatic processes [15]: SQ = 6M--8J-

rs;SN,

(3)

Q being the heat transfer and TSS>SQ=6M--8J-$8N,

(4)

S being the entropy (the equality holding for a reversible process), Q being the

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constant angular velocity, ,Ci the chemical potential and T a global temperature.5 These relations would follow in the preferred coordinate system with the constancy of the parameters Q, T and i;, as an extremal condition on the entropy for M, J and iV constant. How should one generalize (2) to a general coordinate system? What form does (4) take? Do we relate M to the four-vector PA, with PO = M in the rest frame, or do we take the scalars (PAPA)‘/” or u,PA [18] ? Similar questions arise with respect to J, except that the relevant quantity is a second rank tensor. Should Eq. (4) be written as a scalar, vector or a tensor relation? On the other hand, a conventional local description is widely used [l 1, 191, whereby local densities of energy o(x), and pressure p(x) exist as well as entropy density s(x). The local thermodynamics is given by the equation

TW4 = @a/n> + PWn);

(5)

regardless of rotation, charge and possible boundary constraints. What is the connection between this local expression (5) and the global formula (4) with the equality sign ? In general terms, we can raise three questions to point out some of the nontrivial aspects which arise in relating local and global relativistic thermodynamics. (i) The global form (4) uses constants of motion deduced from the dynamical equations; the local expression (5) is in no way related to dynamics. (ii) N, M and J are associated locally with conserved tensors, respectively, of rank 1, 2 and 3; Eq. (5) contains only scalar functions. (iii) When fields are present, like with charged matter, M and J include contributions from the field; Eq. (5) holds with or without charge, with or without gravitation or any other field whose source is or is not in the matter itself. 5 The question of the interpretation of this free scalar as a temperature has been a matter of some controversy. The terminology of a global temperature is occasioned by its role as the parameter characterizing equilibrium of a system with respect of variation of the total energy. T is ultimately related to the local equilibrium parameter T [cf. Eq. (31)]. Giles 1151 has, for example, objected to characterising T as a temperature. The general relativistic thermodynamics is in this respect completely analogous. In that case, Tolman does not address himself at all to the interpretation of the constant, and is only concerned with the local temperature. Einstein (16) considered the analog of T as the temperature in a process which involved heat transfer from noncontiguous regions. On the other hand Balasz [I71 objected to the interpretation of T as a temperature. As we show clearly below for open systems, P has an interpretation of a temperature on the basis of one of the most basic criteria for a definition of temperature: equality of the parameter T for two contiguous open systems is the condition for no heat flow between them. This is completely relevant if the energy is transferred in such a manner as to leave the angular velocity constant.

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In what follows, we shall first recapitulate and apply a mathematical device which permits the establishment of the connection between global and local expressions in a very convenient fashion. I-2. Symmetry

of Motion and Killing Fields

Most treatments of relativistic thermodynamics begin with dynamical considerations like those presented in the beginning of Section I-l, though in classical thermodynamics one knows that they are superfluous for describing time-independent states. If the forces are balanced, a stationary situation results and we should expect that we no more need to know the forces to describe the equilibrium for relativistic systems then we do for classical ones. Let us now proceed to describe a method of finding conserved quantities based on symmetries of motion to describe thermodynamic equilibrium. The approach evidently can be extended to treat more general time independent situations, such as steady state heat flow. The classical result that a rotating system in thermodynamic equilibrium can only translate uniformly and rotate as a whole is expected to hold also in relativity. There are two symmetries associated with such motion: one is associated with translation in time and one with rotation of coordinates about the axis of rotation. Since we are considering special relativity here, we express our results in coordinates which are orthogonal everywhere although, as will be evident, our results are valid, except for trivial changes, in arbitrary coordinate systems. In addition to the stress-energy tensor of the fluid,

where 7’“” GEdiag( 1, - 1, - 1, - 1) and u” is the velocity field u2 = q&4w

= zlA14A = 1.

We also introduce the particle number density vector n* s nub

(8)

with the total number N being N=

s,nAdZA.

The matter occupies a finite portion of space; .Z is a space-like hypersurface including all of the matter. There is a coordinate independent manner by which we can characterize the fact that we have a stationary, axisymmetric system. We note that this is the great simplifying feature involved in introducing the symmetries of motion rather

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PERFECT FLUIDS

than of space. The fluid functions CT,p, u” and n should admit two Killing fields,6 a time-like one f”, 5” E &JA > 0, and a space-like one $, $q,\ < 0. This means that field functions should be form-invariant for the one parameter (a or b) group of motions defined by the equations dxA = f” da and dxA = qAdb. These displacements belong to the group of motion of the Minkowski space if they satisfy Killing’s equations (see Appendix): a,t, + 45, = a,9, + b

= 0.

(10)

As a result, the derivatives of the scalar functions of the fluid, in the t and r) directions should be zero:

0 = pano = +ano = paAP = qaAp = . .. .

(11)

and the velocity field, which has to satisfy (see Appendix) the equations

uya,p - syavd = uuavrlh- T~a,uA= 0 should thus be in the (5, T)-plane and there should, moreover, invariant scalar fields c(x) and Q(x), such that

(12)

exist two form

VAE iy E f” + @A. Denoting (x”) ($) = (0, --y, local angular functions only

(13)

= (t, x, y, z) the “standard coordinates” for which 5” = SoA and x, 0), corresponding to the fluid rotating about the z axis with velocity Q(+); all the scalar functions in these coordinates are of z and p = (x2 + y 2) l12. Note also that in standard coordinates, 5 c (u”)-’ = (1 - p2522)V

(14)

pQ being the local velocity of the rotating fluid. Let us now construct two vector densities, the energy density vector ma z TV”?

(15)

G -T$.~v;

(16)

and the spin density vector jA

both lying in the (5, T)-plane. From (6), (11) and (12) or (13) it follows that a,id = aaja = 0,

(17)

irrespective of whether ayTuvis zero or not.7 B A short review on Killing fields and their associated geometric properties is given in the appendix. 7 &ma:= i&jA = 0 corresponds to the subset of nondynamical or trivially satisfied equations among a,Tpy = 0 which follows from symmetry.

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It is now convenient to introduce differential forms A = mA dZ,, ,

f

= j” dZA ,

.N = nA dZ,, .

(18)

These quantities are scalars and represent, respectively, the energy content, the spin content and the particle number contained in the small comoving volume element V, V- = uA dzl, .

(19)

This is a very convenient way to represent functionally results for small but macroscopic uniform regions. In fact, as a result of (17) and, because of (11) and of (13) and as a result of a& = 0.

(20)

syToodV,

(21)

the following numbers M=

and

//A’=

J = j- y = j, (Toly - xTo2) dV, 2

(22)

N=JEJlr

(23)

are C-independent quantities and may be evaluated in arbitrary coordinates (by introducing appropriately the determinant of the metric under the sign of integration). ml, j” and n* satisfy some sort of a strong conservation law [20] existing independently of whether the dynamical equations are satisfied or not. The conservation laws (17) and (20) are, however, related to symmetries of motion of the fluid. The space-time itself admits .$ and 7 as two of the ten Killing vectors associated with the PoincarC group of motion [21]. We finally note, as is easy to check, that .M - Q$

= ovA d&

= [oV

= CC?,

(24)

d being the internal energy (plus the rest-mass) contained in the volume -Y-. I-3. Thermodynamics

of Isolated Rotating Fluids

The elastic simple fluid we are considering is rotating about the z axis; the pressure goes to zero at the boundary. It is to be recalled that for an isolate system the entropy is independent of the volume occupied by the system. Thus S is a function of M, J and N. Each of these four quantities is a Z-integral of a local

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scalar differential form. In the absence of fields, the general local structure leading to (4) for quasistatic reversible variations, is of the form SY = (SJH - Q(x) S$ - P(x) Sdv)/?;(x)

(25)

in which F(X), Q(x) and p(x) are scalar functions and 9

= suA dzl,

= sA dZA .

(26)

In fact, it is now easily seen from (25) that S is an extremum, for a constrained variation (aA4 = 6J = 6N = 0) leaving - -- the system isolated; introducing the (free scalar) Lagrangian multipliers l/T, Q/T and p/T, we have

0 = ss- [(G) SM- ($) SJ- (f) m] 1 jJg-+L =[S~(r-+)“.“-

j,(++)sM] W’)

Thus as condition for extremum we have T = T, Q = D and p = p, T, Q, and ,C are numbers. Then, in view of the constancy of these parameters, we recover the global expression of the first and second laws of thermodynamics. Let us now show how (25) is connected with (5). From (13), (15) and (16) one immediately gets the following relation: bC--QSs$

= oY8T;dZA,

(28)

providing the variations preserves the symmetries St” = 67” = 0 (coordinates are by definition unchanged in the process). With the help of (24), (28) may be written equivalently Sd - 9 S$ = {(SCLY + p W),

(29)

and with this relation, (25) becomes equivalent to SY = (l/T)@& + p W) - (/Ii/T) SM.

This expression is already reminiscent define a local temperature T(x)

and a local chemical potential

of conventional = F//~(X)

thermodynamics

(30) if we (31)

HORWITZ

312

AND KATZ

Note also that 9, b, JV are all proportional to V. If we now assume,as in conventional thermodynamics, that 9 is a homogeneousfirst order in b, -/- and JV, then the value of the local chemical potential is thereby determined: p = (0 + P -

Ts>/n,

(33)

and hence (30) may be written in terms of specific value variations as follows:

T Wn) = %dn> + p W/n>,

(34)

which is the local form (5).

II. ROTATING II-l.

PERFECT FLUIDS-OPEN

SYSTEMS

Introductory

There are a number of motivations for studying systemswhich are not closed; we shall enumerate some of them: (i) In many applications of interest the thermodynamic equilibrium is only approximate, the approximation being valid only over a fine part of a larger system; these partial systemsare not closed and yet may be too large to be considered uniform. (ii) There is a significant clarification of the interpretation of the “global” parameters i=, p and external pressures& , j& ,... the pressuresat the boundary surfaces of the open system. The relevant parameters determining equilibrium between regions are the global parameters rather than the local ones. As an example, we note that it is not the vanishing of the gradient of the local temperature T which determines the absence of heat flow, but the absenceof a gradient in the global temperature T; similarly it is the behavior of the global and not the local chemical potential which determines particle flow. We continue to treat caseswhere the local thermodynamics is assumedto hold. What we seek to do is to find the appropriate thermodynamic functions for such open systemsas are extremal as a condition for equilibrium for systemsin contact with appropriate reservoirs, i.e., with various specified constraints. By the use of the kinematically conserved vector densities, we are able to readily apply essentially standard methods of deriving results for open systems, taking them to be a small part of closed systems.The closed system will be regarded as a composite system comprising the open system and the residual system which will be regarded as a reservoir. The implication of the residual system being regarded as a reservoir has two aspects: (a) any conserved quantities (for uniform

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systems.-extensive variables) lost to the system are transferred to the residual system, (b) the reservoir is taken to be large enough so that those thermodynamic parameters complimentary to the conserved quantities transferred, the equilibrium parameters T, fi, etc., are not changed by the process. The open system is, however, taken Iarge enough to be nonuniform, since under our assumptions otherwise we could simply use local thermodynamics. The .thermodynamics of open rotating systems will be developed for systems having the symmetry of the motion, to lift this restriction complicates matters enormously. Thus, we define our open system to be axisymmetric and, of course, stationary. As a device to separate the composite system into subregions, it is more convenient in relativistic notation to define the volumes by step functions introduced into the definitions of the various matter functions. This can be done in an invariant manner allowing us to integrate over arbitrary space-like hypersurfaces as previously for the isolated fluid. 11-2. ConJined Systems with Fixed Pressures at their Surfaces A rotating confined system represents, perhaps, the simplest example possessing many of the features of nonclosed systems. By a confined system we shall consider a system contained within one or more isobaric walls. The results for such a system will be derived by separating the closed system into two by means of isobaric surfaces. For simplicity we shall describe results for a single isobaric surface, but the generalization to more than one surface is quite obvious. The composite system consisting of wall plus confined system is closed; the inner part will represent our confined system, the boundary plus outer part we shall regard as a wall attached to a volume reservoir. We seek to find the appropriate thermodynamic potential, given in terms of the properties of the confined system alone which is extremal for equilibrium as well as the conditions for thermodynamic equilibrium. Having derived this in a manner independent of the details of the wall, we can then express general variations for a confined system. Let U:Sfirst consider some particular isobaric surface, denoted /l corresponding in cylindrical coordinates to p = pO(z) for z, < z < z2 ; fl is, of course, cylindrically symmetric. To make the separation of the composite system in an invariant fashion we introduce the step functions fi and f. which are, respectively, unity (zero) inside and zero outside (inside). fi and f. are scalar functions with zero derivatives in the f and 7 directions:

paAfi = paAfo = qaAfi = gaAfo = 0.

(35)

Actually we could have f. be unity only between d and the external boundary, but as the pressure is zero there we need not consider this boundary explicitly.

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In the standard coordinates, for example,

.fi = 4% - PM(Z - 4 - @ - 41,

(36)

where e(x) is the standard step function, 1 for x > 0, and 0 for x < 0. All of the matter functions are now redefined to include the step functions and this is indicated by an index i or o accordingly. Thus,

and it is noted that the total stress-energy tensor is independent of the separation. In the same way, we shall define all vector densities in the two separate regions: moA = fomh,

etc.

(39)

All of these vector densities are divergenceless because of (17) and (35): aAm$ = aAjiA = i&Q = aAmoA= aAjoA = a,noA = 0.

(40)

Thus, Mi

c

Jzfid,

Ji = I =fif,

etc.

(41)

are Z-independent free scalars, as long as 2 extends to spatial infinity, or at least beyond the region of definition. To determine the conditions of equilibrium, the boundary (1 must be regarded as closed with respect to transfer of mass, spin and particles, but consider that we can displace the surface rl keeping the boundary pressure constant, i.e., we consider variations which preserve the symmetry. The entropy of the composite system is extremal (maximum) for Mi + 44, = constant, Jt + Jo = const. and Ni + N, = const. yielding the constant global equilibrium parameters l/T, Q/T and p/ilT, respectively: SS = SS, + SSO= (SMi + SMo)/T - (SJi + 6J,) II/T - (SNi + 6No) F/T.

But a direct calculation

(42)

leads to (43)

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and similarly for 6S, . S& and 8f0 are delta-functions thus be written

315

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and the integral in (43) may

where pi is the pressure on the inside of A, PO the pressure on the outside of fl, and Vi (VJ the inner (outer) volume as measured by the nonrotating observer at rest on the axis of rotation. This term represents the work performed on the inner and outer subsystems, respectively. The p 6V term in (30), for example, or rather to be more specific its integral does not represent an external work on a region; it is rather an effect on the energy due to change in velocity (cf. the definition (19)). Under the conditions of the constraining wall, SM, = &Ii = 6M, = *.* = SN, = 0, and with Vi + V, = cons& the two expressions of type (43) reduce to ssi + (j&/T) sv, = ss< - (&/T)

svi = 0

(45)

which thus [cf. (43) and (44)] implies as a condition of equilibrium that pi = jjO ; the pressure inside the boundary A must equal the pressure outside or at the wall for equilibrium. Referring back to (40), there was not any condition on the boundary pressures for conservation; the equality of pressures then arises purely as an equilibrium condition, not as a result of conservation. The change in entropy of the confined system is equal to the work performed by the volume reservoir. Thus we have found in terms of parameters of the confined system alone ssi - (Pi/T) s vi = 0,

(46)

and since for the variation in contact with the volume reservoir, pi = j = const., we define the thermodynamic potential Y, for which SY = &Is - (jp)

V] = 0

(47)

(we have suppressed the index i and are henceforth dealing with the confined system). The thermodynamic potential Y is a special case of the entropy transforms denoted in the notation recommended by Callen [22] Massieu functions Y = S[ji/T]. ?P is a function of M, J, N and FiTand for a general variation of !P we have SY = (l/T) 6M - (Q/T) 6J - (ii/T) 6N -

V @/IT).

(48)

Turning to the connection with the local forms, it will be evident that Y has some but not all of the properties desirable for describing both global and local 595/7W-2

316

HORWITZ

AND

KATZ

properties from a confined system. !P can be related to a conserved vector density p = [s - (p/T)] df

(49)

and a related differential form (50) From (50) and (43), the differential forms of left and right members of (48) are equal; thus s$b = (l/T) SJZ - (Q/T) S$ - (F/T) SJV - [Y- S(F/T)

(51)

and from (30), s+ = (2% + p sv - p SJv)/T - 2y- s(g).

(52)

This expression has at least one property that can be regarded as an unsatisfactory feature. The variation of C,!Jis not altogether local; it depends on variations of F//T which are defined on the surface. It is to be noted that # does not correspond to the local Massieu form (9 - pV/T), which while integrable and conserved, does not have the extremal property globally for fixed pressure on the surface. 11-3. Thermodynamic

description of an open system

The type of constraints of most general interest, both for physical application and for statistical mechanical calculations, is that of an open system, defined by given boundaries of separation which constitute walls nonrestrictive to M, J and N. Again, purely for simplicity of notation, we take our open system to be the inside portion of a closed system; to take a symmetric shell is a trivial generalization. On the other hand, it is highly desirable, in terms of the method, to take the dividing surface to be one which preserves the symmetry of the regions; we again separate the regions by step functions fi and f0 , the former unity inside the open system, and the latter unity outside in the volume, heat and angular momentum reservoir. The total entropy variation, the composite system being closed, is ss = S& + ss, = 0,

(53)

SS, = (l/T) SM, - (o/T) SJ, - (p//T) 6ZV,,

(54)

but, with (43),

since our subsystems are defined in terms of a tied S(Mi + 44,) = S(Ji + Jo) = S(N, + No) = 0; hence,

boundary

SS = S& - [(l/T) SM< - (Q/T) SJi - (,2/T) SN,] = 0.

surface. But (55)

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For the extremum, with l/T, D/F, ,E/;ITconstant, we can define the thermodynamic potential for the open system (dropping henceforth the subscript i) x7 = S[S - (M - m - /%N)/T] = 0.

(56)

The general variation of 17 is then

L! is a function of I/T, ~?/Tand ,E/ilTand a functional off (Sf, it is to be recalled, is a delta function at the surface). If 17 is known, for example, from statistical mechanics, we would have the relations an/a(l/T)

= A4,

ahya(ji/T) = -N,

aIqa(Q/T)

= -J, (58)

ahyaf(x) = cpv-.

If the boundary surface is also an isobaric surface, 17 also becomes a function of I’, instead of a functional off: alI/aV = ji/r .Z7,in contrast with Y, has completely satisfactory local as well as global properties. Introducing the conserved vector density (cf. (24) and (33)) d = [s - (u - pn)/T]fu^

= fpu”/T

(59)

L7 = s, TTT,

(60)

and the differential form T = w’ dZ,, ,

the latter being a hypersurface independent constant of the motion. Comparing right an.d left members of (57), using (59), a comparison with the above results confirms that the differential forms are equal: s7r = [Jzz S(l/T)

- f S(!z/T) - Jlr s(p/T)]f+

(pY/s=)

85

(61)

+ (pY-/T)

Sj- (62)

S(p/T)lf + (pV/T) SJ

(63)

Then rearranging 6s~ == [S(A - 03)/T

- (SJ&’ - 0 S$)/T]f-

.A’- &T/T)

and comparing with (24) and (29) gives 6~ = [8 6(1/T) - (p/T) B’- - Jlr

Thus Lr corresponds exactly to the negative of the grand potential divided by the temperature and (63) is the form of the local volume variation plus the surface

318

HORWITZ

AND

KATZ

term, dependent on the pressure at the surface. U is just the Massieu function n = S[l/F, o/T, ,,Ci/T], while 7~is the local Massieu form 17 = sP[l/T, pL/7J. II-4

Thermodynamic

Potentials for Rotating Fluids

In the previous sections two examples of thermodynamic potentials have been derived for rotating relatistic fluids. The examples already suffice to indicate the caution necessary in making a formal definition of the thermodynamic potential, whether based on global or local definitions. Not all thermodynamic potentials defined for uniform systems are convenient for non-uniform systems; some of the criteria which we propose as desirable are the following: (i) The appropriate integral (global) thermodynamic function should be extremal with respect to chosen variables. (ii) The global quantity should be defined in terms of local kinematically conserved vector densities and be directly additive. (iii) A desirable, though not essential, property is that the differential form corresponding to the thermodynamic potential should comprise a volume contribution which is strictly local plus a term at the surface. To clarify the above points we bring one counterexample to each of the above points. As we mentioned previously, a conserved local enthalpy form can be defined: Its integral is a constant. But it is not an appropriate function which is extremal for fixed pressure at the surface. The function above was satisfactory in this respect, but it is deficient in the sense of (iii). Its differential form has a volume term which is not local. The point of (ii) is a limitation to the introduction of all energy (in contrast with the entropy) representations [22]. There is a definite preference for entropy and Massieu functions instead of the internal energy and its Legendre transforms. Thus, if we were to build up a global result from the energy form d rather than the entropy, the good global variable is given instead by the integral of {S, which is equal to M - QJ. How could we predict this in advance? Also the local variation of 58 is not particularly meaningful. Without trying to be complete we record two more convenient thermodynamic potentials. The first is related to the Helmholtz potential S” FZ S[l/T,

Q/T] = S - (A4 - .nJ)/T = 1, (9 - 8/T) f = f, - SAT,

where we introduced the form 9 corresponding the volume V: SS* = M6(1/T)

- JS(D/T)

to the Helmholtz

- (,2/T) 6N + &T

(64)

free energy of

8f

(65)

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319

FLUIDS

and, locally, -S(fF/T)

= [c? 6(1/T) - (p/T) 6V - (P/T)

SJ-If + [(Pi - 4/u

v

(66)

With S” a function of l/T, a/T, N and a functional off,

as*/a(l/Tj = 44, asyaiv = p/T,

as*ja@T) = 4; as*jaf= piy.

(67)

The corresponding local results are

-appyav

-appya(i/T>= 8, -app)/aJr/‘ We can similarly

= P/T, VW

= -pp.

define the Massieu function related to the Gibbs function

S** cc S[l/T, = j, [y

o/T, p//T] = (S - [M - OJ + jWJ/T) (69) -

(8 + F+W’lf

-

jr -(g/T)f,

where the global variation SS** = A4 8(1/F) - Ja(.!f?/T) - (ii/T) SN - VS(j/T). The local variation

(70)

is

-S(fg/T)

= 18 W/T)

+ (P/T) W” - (v/T) hv

- W W~>lf - b.Jlr)lTl W

(71)

Again if S** is a given function of its natural variables l/T, Q/T, N and p/i=,

as**/a(l/T) = M, as**/alv = -(ji/T),

as**la@jT) = -J, as**/a(pp) = 4.

(72)

The corresponding local relations are ---B(2?/T)/8(I/T)

-appya~

= 6,

= --cLI~,

-a(S’/T)/W

= p/T,

-a(~pya(~/T~

= v-c.

(73)

This last relation is, of course, not a good local relation, just as was the analogous case for Y.

320

HORWITZ

AND KATZ

In summary, we have defined these additional thermodynamic potentials formally, but it is clear that they could be derived in a manner analogous to the functions of II-2 and of 11-3. Moller [23] has discussed thermodynamic potentials for uniform systems in special relativity. The obvious advantages of scalar definitions throughout in relation to the question we raised in part I for isolated systems are completely relevant here. The power of this form of resolution of the difficulties is evident.

III. III-l.

AN ISOLATED CHARGED ROTATINGFLUID

Introductory

The thermodynamics of systems which include fields, electromagnetic as well as gravitational, have certain features in common in which they differ from the simple fluids treated above. In the first place the localization of the energy is distinctly more tenuous; this manifests itself in a degree of freedom in the choice of the energy and spin densities and in the interpretation of the divergence terms added to the energy which vanish globally due to asymptotic conditions. A second feature is that the local thermodynamics does not depend at all on the field quantities. The results in part III are formulated for a rotating perfect fluid consisting of charged particles with a common charge e per particle. A charged rotating system will be, at most, in a metastable equilibrium, but both for illustrative and formal purposes and because there is an interest in systems approximately in equilibrium since they radiate at sufficiently slow rates for the radiation to be negligible for relatively long time intervals. Local thermodynamics, when valid, for systems with an electromagnetic field present is identical with that of a system without the electromagnetic field. Thus relabeling our previous result with a subscript m, material energy M, and material spin J,, in contrast with the electromagnetic quantities which will be labelled with a subscript e, the total variation of the local entropy form is written

or equivalently (34). These results can be expected to be valid provided that the regions small enough to be uniform both with respect to rotational effects and electric and magnetic fields are yet large enough to have good thermodynamic properties, namely local equilibrium and negligible surface contributions, the only long range terms that persist being, in effect, mean field terms and they do not contribute to local thermodynamics.

RELATIVISTIC

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PERFECT

FLUIDS

321

The global expression likewise has a structure formally independent of the field. Thus, T SS = SM,,, - a SJ, -

s, &+&c) S./V.

(75)

But here, at least, one must be cautious as Cpm.is no longer constant; and in terms of the electromagnetic contributions to the electrochemical potential, to be derived below, pa = edAh, (76) it is rather 5&i

+ P-Le>= tL

(77)

which is constant. One could, in principle, either build up the global expression for the entropy and its variations by adding up the local entropy forms and its variations or proceed in the converse direction. There are some difficulties in proceeding from the local to the global forms, especially for the charged fluid case. A basic problem for the present case is that since the local expression is independent of the electromagnetic field, how does one add up field independent energies, etc., to produce a global entropy which while intrinsically also independent of the field is yet based on conservation of total energy and total spin, i.e., including contributions of field and material quantities. If we wish to proceed from global expressions, we proceed on the following principle which we have used before and which is so clearly expounded in Giles [ 151. The entropy has to be a function of the constants of motion M, J and N. Equilibrium corresponds to an extremum of the entropy for constant M, J and N. The condition for equilibrium yields a constant global temperature T, angular velocity 0 and global electrochemical potential ,L Hence, ~&.3=6M-~8J-,&V,

(78)

but M f M,,, and J # J, . We must first find a way to set up a scalar expression for the electromagnetic energy and spin M,=M-Mm,

J,=J-J,,,.

Then, if we can fmd conserved energy and spin forms J%‘, and je such that

(79)

322

HORWITZ

then, by comparing

AND

KATZ

(75) and (80), this implies that

or the integrand is zero up to a divergence term and up to factors which vanish when Maxwell’s equations are satisfied. 111-2. Scalar Energy and Spin for the Electromagnetic

Fields

The stress-energy tensor is the sum of the matter tensor TK and the field tensor Tr; separately the matter and field tensors are not divergenceless, only their sum is. On the other hand, the material energy and spin densities-corresponding exactly to the quantities defined above in Eqs. (15) and (16) - remain divergenceless even in the presence of the field as the symmetries remain satisfied, i.e., remain kinematically conserved quantities. In deriving the corresponding kinematically conserved vector densities for the electromagnetic field, we again exploit properties of symmetry of motion. Let us introduce the symmetric gauge-invariant stress-energy tensor for the electromagnetic field

AA being the vector potential. The electric and magnetic fields are, by general considerations, stationary and axisymmetric. A large class of gauges are consistent with the vector potential also having the same symmetry. In cylindrical coordinates A will be independent of time and of the azimuthal angle 91. A major point for the introduction of the Killing fields is that this symmetry can be exploited in a coordinate independent fashion. The application of Killing fields to represent the symmetries is more complicated in the present case: the basic idea, however, remains the same. If we perform an elementary displacement in a symmetry direction, then the corresponding quantity remains unchanged. The differential form of this statement for A”, which is a vector field is the same as for Us, (12) namely,

In cylindrical

AVa”p - pa”AA = 0,

(84)

A”a,?f - ya”A”

(85)

= 0.

coordinates these will again correspond to a,JA = 0 and a,AA = 0.

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PERFECT

323

FLUIDS

Proceeding by analogy with the material case, (15) and (16), we define the forms Je E Fii; d& = c$*Tehy dZl, = $(E2 + B”) dV, #; = JeA dz& = --q”T;”

d& = v A (E A B) dV,

(86) (87)

the right hand member being the result in rectangular coordinates and is in accord with the standard definitions. Do these in some reasonable sense properly represent vector densities of energy and spin and if so under what conditions ? In contrast to the material case, the forms (86) and (87) are not conserved on the basis of symmetry conditions (84) and (85) alone, as may be easily checked. In fact, using (84), (85) and (lo), we have first

a,@,

Fn”f” = a,(A”f”)

=

4~”

= 6x,

= G%T”)

(88) (89)

where @ represents a scalar electric potential and x a scalar magnetic potential. These relations are valid in arbitrary coordinates. Then, it follows that ,$“Ti” + @O” = $F’t”

+ @iA i- a”(F”“@),

(90)

T”T& + xOA = $F$”

+ xi” + a,,(l@‘X),

(91)

in which iA G enun

(92)

OA G a,FAv + iA

(93)

is the electric current and

(OA = 0 represents, of course, the two sets of Maxwell’s equations corresponding to 0 = div E + en = O” and 0 = (rot B)” + ik = Ok). The form of definitions of energy and spin vector densities (86) and (87) are not the most appropriate and we shall proceed with two alternative ones. To illustrate the nonuniqueness we shall discuss two alternative pairs of definitions in the following discussion. Using (90) and (91), we define the energy and spin densities, respectively, m, ’ = Tt”E” + QiO” jeA = -[Ti”q”

a”(@F”“) = 5”F2/4 + @pi”,

+ x0” -

a”(xF’“)]

= -[q’F2/4

+ xi”].

(94) (95)

From (20), (84), (85) and (92), it follows that these vector densities are conserved.

324

HORWITZ

AND

KATZ

We define a second pair of energy and spin vector densities, respectively, ” = T:“f” - a”(@F’“) m e-

= m,” - @OA,

(96)

and jiA = --[T~“T”

- 3,&F”‘)]

= j,’ - x0”.

(97)

These quantities, in contrast to m,” and jeA are only conserved quantities with the use of 0” = 0. The subtraction of the terms like a”(@FI\“) in both pairs of definitions is made for later convenience in finding the conditions on the variation which lead to a net zero contribution of the electromagnetic quantities to the entropy. Formulation in terms of Ei,” andJ,” would lead to difficulties in this matter. The integrals over both pairs of densities are hypersurface independent free scalars:

The integrals involving A%‘@’and ye’ differ from 2, and A only by the addition of the superpotentials which do not contribute to the integrals. The equality of these integrals, respectively, to M, and J, is thus independent of Maxwell’s equations; but their conservation properties depend on Maxwell’s equations. In contrast with this is the case of the A, and ze which are conserved forms independent of Maxwell’s equations; the integrals are correspondingly hypersurface independent quantities. On the other hand the identification of the integrals with the total energy and total spin, respectively, requires the use of Maxwell’s equations. 1113. Thermodynamics

of Isolated Charged Fluids

Returning to our global equation for the entropy variation in terms of scalars of material and field quantities, we can insert our expressions for Af, and J, , taking advantage of conservation to obtain the results in terms of variations of forms. From (93)-(95) and (13), we have &A’, - i2 62, = [zP(6F2/4) + u” S(A”P)] dZA

but,

UYiA= UAjV 9

and eu”A” = i”A”/n.

RELATIVISTIC

ROTATING

PERFECT

325

FLUIDS

Thus taking (101) and (102) into account in (loo), we may write after some simple algebraic manipulation (103) in which1 9 = v~~~(F“” 6A,) da& = ~@““l SA,) dV(Z = 1, 2,3).

(104)

For nonradiating fields-the case we are concerned with here-9 is thus a spatial divergence which goes to zero for I + co as r4; the Z-integral of 9 is thus zero. We may now decompose (78) as follows: T&S-

[6Mm-~8Jm-j

(105)

z

Comparing with (103) we see that this result is consistent with (75) if Maxwell’s equations (0” = 0) are satisfied. Thus the global form of (78) which depends on the constants of motion corresponding to total energy and total spin, is verified. From (105) we can obtain the local expression T&Y

= (Sd2’m - 0 6$, - (pm k/V) + (&dl, - Q Sye - &ue S/V) + 3’.

(106)

9 is some as yet undetermined divergence whose integral is zero. To establish the nece.ssary form of local variation from the global is a problem which we have yet to solve. However, we shall be satisfied for the present purposes that there exists a choice which is consistent with the results of local thermodynamics, which we are assuming to hold. Thus using (103) and (29) we find SY = (S8 + p a%‘- - /J &V-)/T + (0” d& + g + 9)/T.

(107)

Hence, i.f we choose 9 = -9’ and assume Maxwell’s equations, we have obtained the standard local results. Let us now proceed in an analogous manner with A@‘~’and ye’ (cf. (98) and (99)), then &de’ - Q Sye’ -

&L~ &A’” = [(do”

Thus again for the global form-the ~6S-8i14,+~6Jm+

- ~“0~) 6A, - v”Ay 60”] d& - 9.

9 integral not contributing-we jI;pm8M=

-

jzv’A,80”dZ~.

(108)

have (109)

Thus in this case we require, in addition to the Maxwell equations, that the variation of one Maxwell equation 60° = 0 when the hypersurface .Z is t = 0.

HORWITZ AND KATZ

326

In the global form (109) we have dropped the terms proportional to OA, since they are already necessarily satisfied in order to have conservation. We obtain the local form from this global expression again in the manner analogous to which we obtained Eq. (107), substituting the identity for the conserved matter and electromagnetic field forms 69 = (66 + p ST+‘-- p tW)/T

- (u”A, 60A d&)/T.

(110)

The terms in OA which are included in the identity (108) used to obtain the electromagnetic field terms have not been included in (110) since to obtain this local form we already had to take 0” = 0. The local variation of entropy is then consistent with the local thermodynamics under the following conditions: (i) (ii)

Maxwell’s equations are satisfied: 0” = 0. One linear combination of the variations

of Maxwell’s

equations is

zero: 80AdJYA = 0. This result is apparently a more restrictive type of solution than above, since it involves an additional condition; here both sets of Maxwell’s equations have to be satisfied (to obtain conservation) in addition to the condition on the variations SO”.

IV. AN ISOLATED PERFECT FLUID IN ITS OWN GRAVITATIONAL FIELD-GENERAL

IV-l.

RELATIVISTIC TREATMENT

General Remarks

The thermodynamics of a rotating perfect fluid in general relativity has been studied by Bardeen [24]. We have, however, much to gain in simplicity and clarity by using the techniques developed in the preceeding sections. We shall try to avoid as far as possible the specific technicalities of general relativity. The formal aspects have already been described elsewhere [25] in a different context. We shall here describe the results on a more or less qualitative basis, intended for nonspecialists in this particular field. Analogy with what has been done above will permit us to skip part of the mathematics. We should add that conserved vectors for energy and spin were introduced in general relativity by Komar [14]. As we have already emphasised more then once, the preceding description has been set up in a coordinate independent manner. Using arbitrary coordinates will not change the formalism except for minor points. For instance, every conserved vector density is to be multiplied by the square root of the absolute value of the determinant of the local metric associated with the coordinates. The perfect fluid, however, continues to be described by a(x), p(x) and u”(x).

RELATIVISTIC

ROTATING

PERFECT

FLUIDS

327

In going over to general relativity we note that the presence of gravitation first of all makes it impossible to find back coordinates which are orthogonal everywhere like in special relativity and secondly implies that the local metric of the space has a physical meaning, i.e., the ten-component gravitational field g,,(x). IV-2. Energy, Spin and Geometry In general relativity where matter and fields are the sources of the gravitation, we expect the gravitational field to share the same properties of symmetry. Thus for a stationary axisymmetric fluid, the metric g,, should have the same symmetry, i.e., (see Appendix) it should admit the two Killing fields [ and 7: (111) (112)

If we were to proceed by analogy, we would attempt to introduce the following vector densities, which are indeed conserved, as energy and spin vectors: AEns pp

(113)

j” s -py

(114)

in which TVA is the matter tensor (6), with 7”” replaced by g”“; (^) indicates the necessary multiplication by (-g)‘/”

E (-det gUJ112,

(115)

which includes the gravitational contribution. However, a first order calculation in the static case (no rotation) shows that since

(-g)‘/” N 1 + u,

(116)

where U is the (nondimensional) gravitational potential, a Z-integral of ritA thus takes the energy of the gravitational field into account twice. To find the correct results we take instead the following kinematically conserved vector densities: ?fzgA= (2P”A - syq .p,

(117)

jg,” E -(F$

(118)

- &,5?) 7y,

$,A and jg,” are the Komar energy and spin conserved vectors whose Z-integrals are equal to the mass and spin only if Einstein’s equations of the field g,, are satisfied. The ten equations will be represented by O,, = 0 without further details. It is, however, a characteristic property of these equations that they imply the

328

HORWITZ

AND

KATZ

dynamical equations of the matter. Thus, in sharp contrast with what we found in special relativity, the dynamics must be taken into account from the beginning. If we had taken the present definition for energy- and spin-conserved vectors instead of those of (15) and (16), the situation would have been the same in special relativity, the Z-integrals of the corresponding 112,~and jBn would be equal to the total mass and spin if the dynamical equations are satisfied. IV-3. General Relativistic

Thermodynamics

We may now proceed by analogy with what was done with the Maxwell field except that in general relativity, in contrast to special relativity, we cannot separate matter and field contributions. In a way the situation is actually simpler since gravitation manifests itself only through the metric while our formulation is coordinate independent anyway and this makes the situation look the same as in special relativity. Starting from the local law form (34) which is again assumed valid for local thermodynamics in general relativity, we have, of course, also Eq. (29), that is, T&Y=6&+ptW-/~tr.6N,

(119)

where Y, JV, d, V are scalar forms, i.e., (-g) II2 has been inserted in the right places. Thus multiplying by 5 defined in (13), we have again as in (30) TsY+psJv

= {(sa+pw).

(120)

The relation of the right hand member of this equation with the variation of energy and spin forms A?~ and & is, however, different from (29) and on the contrary closely analogous to (108) for the electromagnetic field, namely,

in which gU is again a divergence which at spatial infinity goes like r4. There are two other terms related to Einstein’s equations that have their analog in charged fluids in special relativity. Thus, if Einstein’s equations are satisfied,

o,, = 0

(122)

and if the varied field and matter functions satisfy the restriction vu 8(8,? - SvA&‘) dZA = 0,

(123)

RELATIVISTIC

ROTATING

329

PERFECT FLUIDS

whose geometrical meaning has been discussed in [25] (they are specific to gauge field theories), then, as in special relativity, we may write

and the conditions of thermodynamic equilibrium of the isolated fluid are again T = T, fi = B and p = tYi. Thus under these conditions, since the integral over C vanishes, we again recover the expression for the entropy variation (cf. (4)) of the form

also for general relativity; this being expressible in terms of conserved forms.8 The distribution of local temperature and chemical potentia1 are then found to be [24] T = T&‘(l - ~2)~l/2 026) and p = pe-“(l

- J+J~)-~/~,

(127)

where U may be interpreted as the general relativistic gravitational potential and w is the local velocity of the fluid as measured by a free falling observer. Such an observer is rotating relative to a free falling observer at infinity. In the limit of no rotation (w = 0) we get the Tolman [26] and Klein [27] results and in the limit of no gravitation one gets back the special relativistic results (31) and (32).

APPENDIX:

ON GROUPS OF MOTION

AND KILLING

FIELDS

Consider the space-time described in arbitrary coordinates. (What follows is, in fact, valid for n-dimensional Riemanian manifolds.) Minkowski’s metric is replaced at each point by the local metric g,, of the local tangent vector space. 8 Alternatively, we could have also derived our results beginning with this form, involving a global variation for F, a and p as conditions for an extremum with constrained variations. Then the local properties sufficient to yield this global extremum would be (making use of the equality of the right members of Eqs. (120) and (121)) (i) extremal property of the local thermodynamics

6~+pSY-&v-=o, (ii) the satisfying of Einstein’s equations, (iii) zero variation of Einstein equations.

330

HORWITZ

AND

KATZ

Consider a small displacement described by one real parameter a: 2 = Xh + 6x” = XA + P(x) sa.

(128)

fP is a vector field. These infinitesimal transformations generate a one parameter group of displacements and this group is called a “group of motion” if length and angles remain invariant in the displacement. Consider, in particular, the square of the length of a small vector dx” in the local tangent space ds2 = g,” dx” dx”.

(129)

S(ds2) = (Sg,“) dx@ dx” + g,“(S dx”) dx” + g,” dx@(S dx”) = 0.

(130)

One has a group of motion if

but Sg,, =

6~"

a, g,, = OAang,,

Sa,

(131)

and 6 dxA = d 6~" = Sa d@ = Sa dx” a”@.

(132)

Thus (130) is true for any vector dx”, provided that [using (131) and (132)] 8” aAh + g,, a,eA + g,, a,el = 0.

(133)

These are the differential equations 0” should satisfy in order for (128) to generate a one parameter group of motion. Equations (133) are known as Killing’s equations and a field satisfying Killing’s equations is a Killing field. If coordinates may be chosen so that g,, = 7UVeverywhere as in special relativity, then a,,g,, = 0 and with 0, = g,,& equations (133) may be written a,6 + a,e, = 0,

(134)

which are of the form of Eqs. (10). It is quite easy to show that a necessary and sufficient condition for the space to admit a one-parameter group of motion is that there exist a coordinate system for which all the components g,, are independent of one coordinate. If one has two independent Killing fields-a time-like one tA and a space-like one $coordinates exist in which g,,, is independent of two coordinates: the time-like one x0 and let us say x1. If x1 is periodic x1 = p: (modulo 277); then the situation is a stationary cylindrically symmetric metric and the coordinates in which g,, is independent of x0 and y are cylindrical coordinates.

RELATIVISTIC

ROTATING

PERFECT FLUIDS

331

Suppose a Killing field ~9~exists, satisfying Eqs. (133); then a one-parameter group of motion exists and any scalar field like p(x) is invariant along a trajectory of motion: Sp = 0 or 8” aAp = 0, (135) which is of the form of Eqs. (11). If a vector field exists like uA,then 6(u, dxA) = 0 and, using (133), this is equivalent, as may be easily calculated, to & 8”UA- 1P’ap = 0,

(136)

which is of the form of Eq. (12). One can in the samemanner calculate the differential conditions on any tensor or tensor density, due to the existence of the group of motion or if one prefers to the symmetry of the space.

ACKNOWLEDGMENTS It is a pleasure to thank S. Alexander and G. Rakavy for a number of valuable conversations and especially for raising some questions which helped clarify some important issues in the present work. REFERENCES 1. 2. 3. 4.

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