PHYSICAL REVIEW D VOLUME 41, NUMBER 6 15 MARCH 1990

Hamiltonian theory of self-gravitating perfect fluid and a methodof effective deparametrization of Einstein's theory of gravitation

Jerzy KijowskiInstituteJor Theoretical Physics, Polish Academy oj Sciences, Al. Lotników 32/46,02-668 Warszawa, Poland

Adam SmólskiInstitute oj Mathematics, Technical University, Pl. Jednosci Robotniczej 1,00-661 Warszawa, Poland

Agnieszka Górnicka *Department oj Physics, University oj Warsaw, Hoza 69, 00-681 Warszawa, Poland

(Received 16 May 1989)

The Hamiltonian forll).ulation of the theory of a gravitational field interacting with a perfect fluidis considered. There is a natural gauge related to the mechanical and thermodynamical propertiesof the fluid, which enables us to describe 2 degrees of freedom of the gravitational field and 4 de­grees of freedom of the fluid (together with 6 conjugate momenta) by nonconstrained data (g,P)

whereg is a 3-dimensional metric and P is the corresponding Arnowitt-Deser-Misner momentum.The Hamiltonian of the theory, numerically equal to the entropy of the fluid, generates uniquely theevolution of the data. The Hamiltonian vanishes on the data satisfying the vacuum constraint equa­tions and tends to infinity elsewhere as the amount of the matter tends to zero. In this way the vac­uum theory with constraints is obtained as alimiting case of a "deep potential well" theory.

at each point of ~ and at each instant oftime t=xo. Fi­nally, the parameters can be interpreted as the lapse func­tion

(6)

(5)

© 1990 The American Physical Society

The freedom in the choice of lapse and shift correspondsto the gauge freedom of the theory with respect to thegroup of space-time diffeomorphisms.

In the theory of constrained Hamiltonian systems it isusually possible to replace constraints by a "deep poten­tial well" without changing essentially the behavior of thesystem (one could even argue that all the constrained sys­tems known in classical physics are idealized situationswith a deep potential well3). The goal of this paper is toshow that general relativity coupled to a perfect fluid isindeed a theory with constraints (2) and (3) replaced by apotential well. We show that the phase space of bothgravitational and thermomechanical degrees of freedom(2+4 per point) can be described by the same mutuallyconjugate objects (pkl,gkl) as in the vacuum case. Thezero on the right-hand side of Eqs. (2) and (3) is replacedby the corresponding components of the matter energy­momentum tensor. However, the equations can nolonger be considered as constraints. They enable us tocalculate uniquely the lapse and the shift in terms of thedata (pkl,gkl)' Finally, the time evolution of the systemis uniquely generated by a regular, nonconstrained Ham­iltonian H=H(pkl,gkl)' Therefore, our formulation iswell adapted to numerical simulations.

lN=---=VlgOOI

and the shift vector

N - NI-k -gkl -gkO'

187541

(4)

K =8rrG we denote the gravitational constant) is noto.miquelydefined unless the parameters (N,Nk) are fixed

I. INTRODUCTION

R (g) __ l_(pklp -.lP2)=O (2)detg ki 2

.P denotes again the trace of the tensor P ki) and

Kkl is the second fundament al form of ~ (extrinsic curva­tUTe)and K=Kklgkl. Four of ten Einstein equations donot contain time derivatives of p ki and gkl:

where a vertical bar denotes the covariant derivative withrespect to the metric connection generated by g. They;;an be treated as Hamiltonian constraints for the general­~ed Hamiltonian system (pkl,gkl)' There are 4 (perpoint) "Lagrange multipliers" N and Nk, canonically~onjugate to the four constraints. As for the usual con­~..rained Hamiltonian systems, the dynamics of the datapkl,gkl) resulting from the remaining six Einstein equa­

::--ons

The Hamiltonian formulation of general relativityl,2may be sketched as follows. Let ~ be a 3-dimensionalinitial-value surface. Cauchy data for the gravitationalfield are described by the Riemannian metric gkl on ~ (la­tin indices run from l to 3) and by the so-calledArnowitt-Deser-Misner (ADM) momentum pkl, where

----~~=---_.-~=-

KIJOWSKI, SMÓLSKI, AND GÓRNICKA 41

(16)

(14)

(12)

(13)

(11)

-= 1L=-Vg-e(V) .

V

Here g= Idetg/1vl= IgOOI(-I)(detgkl) is the absolute valueof the determinant of the 4-dimensional metric tensorwith the signature (-, + , + , +), and V is the specificvolume (per mole) of the fluid in its rest frame. It is con­venient to introduce the rest matter density (moles pervolume): p= l/V. In this way we have

j=h ;;;1;~dx a /\ dx{3/\ dx y

l=_h_€/1a{3yr-Ir-2r-3€ dxv /\dxA/\dxa (17)

3! ~a~{3~y/1VAa '

(the factor dx/1 is to be omitted in the ILthterm). The for­mulas (17) and (18) enable us to express j/1 in term s ofderivatives of the fluid configuration Sa:

Using the notation ;~=a;a /ax/1, we have

which enables us to- assign to each vo1ume D in Z thenumber n (D) of moles (or the number of particles) of thematter contained in D:

n (D):= fr. (15)D

The pull back of r from Z to X via the configuration ; is adifferential three-form (Le., it is a vector density in thephysical space-time X):

j= r-a-.J (dxo /\dx l/\dx2/\dx3)ax/1

_~_1_"/1 d v/\d A/\d a- 3' - 1/21 €/1VAaX X X. (g)

3

= ~ r(-l)/1dxO/\ ... /\dx/1/\ .". /\dx3 (18)/1=0

where €/1a{3ydenotes the completely antisymmetric tensordensity (Levi-Civita symbol; we use the convention ofMisner, Thorne, and Wheeler2). The vector density j canbe spanned with respect to basic three-forms:

The quantity p (or V) has to be expressed in terms of un­known functions ;a and their derivatives. This is possiblebecause Z is equipped with the volume structure (scalardensity or a differential three-form)

;a(x). The equations will be derived from the variationalformula

where the Lagrangian L is equal to minus the rest energydensity of the fluid:

(8)

(10)

de=-pdV+TdS,

S=S(e,V) ,

e =e( V,S)

II. RELATIVISTIC HYDRODYNAMICSAS A LAGRANGIAN FIELD THEORY

where p is the pressure and T is the absolute temperature.In our paper we also need another description of thefluid, given by the so-called "entropy picture." To get itwe divide Eq. (8)by T:

dS= ~de + ~dV . (9)

Here, the entropy S has to be expressed in terms of themolar internal energy and the molar volume,

and Eq. (9) gives the values of remaining two parametersT and p. Given a function (10) we construct a noncon­strained Hamiltonian H = H(pkl,gkl) such that the evolu­tion equations derived from H are precisely the Einstein­Euler equations for the self-gravitating perfect fluidwhose mech anical and thermodynamical properties aredescribed by (10).

(internal energy per mole expressed as a function of themolar volume V and the molar entropy S). The function(7) uniquely determines the properties of the fluid, ac­cording to the Gibbs relation4

In the present section we show how to derive the rela­tivistic Euler equations for a barotropic perfect fluid fromthe first-order variational formula. 5 This formulation ofthe hydrodynamics is essential for our purpose. We as­sume therefore that the energy is a function of thespecific volume V only and formula (8) reduces tode = -p dV. In the present section the space-time metricgis given a priori.

Consider an abstract 3-dimensional "matter space" Zequipped with an appropriate geometric structure whichwill be specified later. Points of Z correspond to particlesof the matter. A configuration of the fluid is describedcompletely if for each point x = (x /1) of the physicalspace-time X a particle z = ;(x) whose world line passesthrough x is specified. The configuration can thus berepresented as a mapping ;: X ---+2. Given a coordinatesystem (za), a = 1,2,3, the mapping is described by threefunctions za=;a(x). Physicallaws governing the evolu­tion of the system will be formulated in term s of partialdifferential equations for the three unknown functions

1876

The Hamiltonian vanishes on constraints (2) and (3)and-as the amount of the matter tends to zero-tends toinfinity outside the constraints. This shows that ourdescription of the dynamics may also be used for numeri­cal analysis of the vacuum equations, since the con­straints can be considered as an "infinitely deep potentialwell." Replacing it by a deep, but finite well does notchange substantially the dynamics.

The specific form of the Hamiltonian function dependson the specific Gibbs functional equation of the liquid:

41 HAMILTONIAN THEORY OF SELF-GRAVITATING PERFECT ... 1877

Because af the Noether theorem, variational formula (11)implies conservation laws

where VfL means the covariant derivative with respect tothe space-time metric. Because of the continuity equa­tion (20) only three among the four equations (28) are in­dependent. They are equivalent to three variational equa­tions:6

r= -h"/i EfLa/3r;;;~~

;6,· .. ,;1=h ( - 1)fLdetI ~6, , ~~

;~, ,;~1\

f.l

(19)

VfLT~=O (28)

since dr =0 as a differential four-form in the three­dimensional material space Z. In coordinate languagethis means that the divergence of the current vanishes:

(again, the f.lth column of the 3X4 matrix ;~ is to beomitted; this is indicated by {tJ. The current j is con­served by virtue of the definition

dj= [_a-r]dXOl\dXll\dX2I\dX3=0. (21)axfL

Decomposing r as a product of a four-dimensionalvolume density (g)I/2, normalized velocity vector field ofthe fluid u fL (u fLufL= -1), and a rest-frame matter densi­ty p,

(29)

1Ta=-vip(e+ Vp )UOUk(;-I)~

=-j017Uk(;-I)~ ,

The above formulation of the fluid mechanics as a La­grangian field theory leads in a natural way to the HamiI­tonian formulation5 (later used also by Kiinzle and Nes­ter7). Momenta canonically conjugate to configurations;a are given by 1Ta=P~, where

fL= 8L

Pa 8;~'Equation (26) implies that

(20)dj=d;*r=;*dr=O

we obtain the formula

Finally, using (8) we obtain

where (;-I)~ is a 3X 3 matrix inverse to (;%) and17 = (e + Vp) is the enthalpy of the fluid. The abovecanonical structure is equivalent to the description basedon the so-called Clebsch variables,8 although it is simplerand physically more natural.

Because of the reparametrization freedom of the ma­terial space Z the theory may be further reduced.Indeed, any reparametrization of Z which leaves invari­ant the volume structure (14) is physicaliy equivalent tothe previous one. The space of equivalence c1assescan beparametrized by four parameters (e.g., r or p and v kl.The Poisson brackets between those quantities can be im­mediately calculated9 from the canonical structure car­ried by the variables (;a,1Ta). Some authors introducethose brackets in an axiomatic way as a "noncanonical"Hamiltonian formalism. 10

The parameters za=;a(x) can be considered as "poten­tials" for hydrodynamics: we express physical quantities(p and ufL) in terms of the first derivatives of the poten­tials in such a way that the continuity equation is au­tomatically satisfied. The analogous ansatz in electro­dynamics consists in expressing physical quantities (elec~tromagnetic tensor F fLV) in terms of the potential A fL insuch a way that the first pair of Maxwell equations is au­tomatically satisfied. First-order field equations for phys­ical quantities (Euler equations in hydrodynamics and thesecond pair of Maxwell equations in electrodynamics) be­come variational second-order equations for the poten­tials. In both theories the ansatz is not unique or "uniqueup to gauge transformations." In hydrodynamics gaugetransformations correspond to unimodular (Le., withoutchanging the volume form r) transformations of thematter space Z.

(22)

(23)

(26)

(27)

ar ra=J.v81l-J'fL8V . (24)a;~:'1.. . I.. I..

The easiest way to prove that the variational formula (11)is equivalent to Euler's equations of motion of the fluid isto calculate the canonical energy-momentum tensor afthe above field theory:6

T~ = aL ;~ -8~L . (25)a;~"-

Because offormulas (13), (23), and (24) we have

aL a_ aL JE- ar a

ara ;1.. - a a 'v ara ;1..:'fL P J :'fL

_ [ de] ar a- e - V dV Uv a;~;A

=-vi [pe- ;~ ](8~+UfLUA)'

1 ,,/ ,".vp= vi V -rJ gfLV .

Formulas (19) and (23) give us the necessary expression ofp in terms of ;~. The reader may easily check that thefollawing fundament al identity holds:

41

(38)

(34)

(33)

(32)

° o 3kT=x -x --.2mc2

We identify the param eter T with the proper time ret ar­dation multiplied by an arbitrary dimensionless constante:

1

t==e(xO-t)= 13xOT , (39)

where the constant {3=2mc2/3ek has dimensions oftem­perature. Hence,

13 dr =T (40)dxo

similarly as in form ula (30). We interpret therefore the"material time" (up to a multiplicative constant e) as the"proper time retardation" due to the chaotic motion ofthe partic1es. This phenomenonenables us to construct(at least theoreticalIy) a "radium thermometer." We in­ject a drop of radioactive radium into the fluid. Because

molecules move chaoticalIy around the theoretical worldlines of the fluid (lines tangent to the vector field u /l) andthe mean kinetic energy of this motion with respect tothe rest frame is equal (for low temperature) totkT=mu2/2. Because of this motion the proper time tfor the particles is retarded with respect to the physicaltime xO calculated along u/l. For velocities v much smalI­er than the velocity of light c this retardation can be cal­culated from the form ula

- 1 -L=-Vg-j(V,T)=-Vgpj(1/p,T), (31)V

(where 13 is a positive constant), together with the choiceof minus the free energy [fi V, T)=e - TS] as a La­grangian of the theory

leads to correct equations. To prove this statement wecalculate again the energy-momentum tensor

T~= :~ r;f-8~L ,where a=O, 1,3. Using the thermodynamical identity

dj=-pdV-SdT

we obtain

(35)

(36)

(37)

(30)

KIJOWSKI, SMÓLSKI, AND GÓRNICKA

aL {-Q __ V= [ [j. _ VaJ ]~ + 13 aj auu {-o ] ar {-QaS-~~A - g aV ar p aT ar ~u as-~~A

=[(j+V:p)u +S(8u+uuu )13{-O]ar{-Qv v V ~u. as-~~A

= - v~[(j+ vp )(8~+U/lUA )+Sf3~u/l(8~ +u UUA)] .

IV. MICROSCOPIC INTERPRETATIONOF THE "MATERIAL TIME" T

Moreover,

Suppose that the liquid is composed of molecules withmass m. If the temperature of the fluid equals T, the

FinalIy, we obtain formula (27) for T~ with internal ener­gy defined by e =j + TS. Again, the Noether theoremimplies the energy-momentum conservation V/lT~ =0,equivalent to four independent variational equations

For a=O this gives the entropy conservation

0= 8L = -13a (Sj/l)= -13pu/la S .8~ /l /l

It is easy to check that four equations (36) are equivalentto the system of three equations of momentum conserva­tion [three independent equations among (28)] plus entro­py conservation (37).

In the above formulation the phenomenological con­stant 13 may be chosen arbitrarily. The choice 13= l isalso possible. It gives (time X temperature) for the di­mension of the new potential r=zo. As we shalI see inthe next section, it is better to choose the dimension of 13

equal to the temperature and to measure r in units oftime.

III. THERMODYNAMICS

To describe thermal properties of the fluid we need oneIllore potential. 11 We add therefore a new dimensionr=zO to the matter space Z. This way we obtain the 4­dimensional matter space-time Z. with r playing role of a"material time." To describe the configuration of thefluid we need now an additional function zO=S-°(x/l).The physical interpretation of the new variable will begiven in the next section. However, a purely phenomeno­logical point of view is aIso possible. VIe prove that thefolIowing ansatz for the temperature,

1878

41 HAMILTONIAN THEORY OF SELF-GRA VITA TING PERFECT ... 1879

V. SELF-GRA VITATING FLUID

(42)

(49)

(50)

(47)T={3 1 -{3-----ly -goo NV1-n2

Our gauge conditians imply s{;a=O, tk=O for k=1,2,3,and to= 1. Th.e formula (41) reads Pb ={3Sj!l={3ShSg.These observations enable us to rewrite the Hamiltonianformula (42) as

and

where the quantity a = J Sh is equal to the total entropyof the system. Now, the quantity Jf = H - (3a plays therole of the Hamiltonian of the system described bycanonical variabies (pkl,gkl) and the evolution is uniquelydetermined by the Hamiltonian.

For the sake of simplicity let us limit ourselves to thecase of spatially compact space-times. In this case the to­tal gravitational energy vanishes identically2 (H=O) andthe global Hamiltonian is determined by the total entropyof the system: Jf = - {3a . The quantity U = (3hS playsthe role of minus the Hamiltonian density on the Cauchysurface 2:. To express the Hamiltonian in terms ofcanonical parameters (pkl,gkl) we consider the constraintequations corresponding to components Too and Tg ofthe energy-momentum tensor (27):

X(g,p)=_l_ [R(g) l_(pklp _1-P2)]167TG detg ki 2

2 .

= _N_p_e_+_p_N_'_N_i= _p_e_+_p_n_2N2-NiNi 1-n2

_ 1 IYk(g,P)- 87TGhPk II

V detg NNk(pe +p)

h N2-NiNi

=_ -vctetg Nk ~

h N 1-n2

We stress that Y is a covector field since Pk li I is a covec­tor density and h is a scalar density. For a given funda­mental equation (10) of the fluid the pressure p is given asa function of the parameters (e, V). Therefore, the sixequations (44), (47), (49), and (50) can be solved withrespect to the following six variabies: the lapse functionN, the shift vector Ni, and the two thermodynamical pa­rameters (e, V). In this way we are able to express thesesix parameters as functions of five purely geometric vari­ables: twa scalars X,Z:=h2/detg and a covector Yk.The simplest way to solve the above system consists incalculating the square of the length of the vector Yk :

fying the material time with the time coordinate xO im­plies

(41)

(45)

(44)p=h Y~=h V~-h ~Vg Vdetg Vdetg

where N is the lapse function and Ni is the shift vector.Moreover,

of chaotic motion of the particles the lifetime of radiumgets lengthened proportionally. to the temperature of thefluid. Therefore, measuring the lifetime we measure thetemperature.

(the overdot denotes the "time" derivative), which simplymeans that the time derivatives of canonical parametersare equal to the variational derivatives af the Hamiltoni­an with respect to the conjugate parameters. Because ofthe invariance of the theory with respect to space-timediffeomorphisms, we are allowed to impose the following"gauge conditions": {;a(x/l)=xa. The three conditionscorresponding to a = 1,2,3 mean that the coordinates x k

are comoving with the fluid (constant on the fluid worldlines). In this gauge we have {;%=S% and, according to(1),we have

j!l = - h vi €/la[3y{;~{;~~

= -hvi €/l123=hSb . (43)

Using (23) we obtain

The Hamiltonian field equations giving the time deriva­tives of the Cauchy data in terms of the functiona1derivatives of the Hamiltonian can be briefty written as2

-SH=- -l-fpkISg -g' Spkl+fp' °sra-,?aspO167TG ki ki a ~ ~ a

The theory of self-gravitating fluids will be based onthe Lagrangian L = L grav +L mat> where L grav is theEinstein-Hilbert Lagrangian for the gravitational fieldand Lmat is as in Sec. III. In Hamiltonian formulation,the complete Cauchy data for the theory consists of Cau­chy data for both gravitational and hydrothermodynami­cal field: (pkl,gkl,{;a,p~), where p~ =aL /a{;~are momen­ta canonically conjugate to hydrothermodynamical po­tentials.5 Using Eqs. (30), (31), and (33), we show that themomentum canonically conjugate to (l is equal (modulothe factor (3) to the entropy current:

where V detg =y detgkl denotes, as usual, the three­dimensional volume density on the Cauchy surface}:=fxo=constJ and

1V-· -.n=- N'NJg··N IJ'

(46) (51)

Because of formula (30), for a=O gauge condition identi- Equation (44) implies

(64)

41

(63)

(60)

(59)

(61)

(62)

p'(e', V')=8p(e, V)=8p [ ~ ' V' ISimilarly

S'(e', V')=S(e, V)=S [ ~, V' ]

To prove that the free Einstein theory corresponds to alimiting case when the Hamiltonian tends to infinity out­side the constraint subspace [X=O, Y=Oj, assume thatwe change the mass of particles of the fluid: m' =8m,where e is a real number (finally, we aregoing to pass tothe limit 8~O). For a given kinematical state of the par­ticles we have therefore T'=8T.Moreover, we assumethat the total molar energy changes in the same way:e'=8e (this assumption is compatible with the fact thatboth the rest energy mc2 and the kinetic energy fkTchange in the same way). On the other hand, we do notchange the density (particle number) of the fluid: p'=p.Hence, V' = V. Consequently, according to (8) we havep'=8p and S'=S. If p'=p'(e', V') gives the pressure asa function of energy and volume for the new liquid and ifp =p(e, V) is the corresponding function for the old one,we have

VI. CONSTRAINT MANIFOLD AS A BOTTOMOF A "POTENTIAL WELL"

To parametrize the dynamics we use the "material time"'T given by the formula (39) with the corresponding valueof 8. This way we keep the parameter {3 unchanged([3'=(3)during the entire operation.

Consider now Eq. (49) for the new fluid (Le., with p, e,and p replaced by corresponding primed quantities). Di­viding both sides by 8 and using (61) we obtain on theright-hand side the same function of parameters(e' la, V') as we have had for the cold fluid. The sameobservation is valid for Eq. (51) divided by a2• On theother hand, Eq. (52) is the same for both fluids. We con­clude that

where e =e(X, Y,Z) and V= V(X, Y,Z) are correspond­ing solutions for the first fluid. Using (62), (63), and (64)we obtain the following expression for the Hamiltoniandensity of the second fluid:

and

(57)

(52)

(55)

(54)

(58)

(53)

KIJOWSKI, SMÓLSKI, AND GÓRNICKA

[XI ]F(X,O,Z)=S VZ' VZ '

e=VX-VYVV2Z-1,

VZvYp(e, V)= , ~ X ,

where the function p(e, V) is determined by the stateequation (9):

S=S(e(X,Y,Z),V(X,Y,Z»=:F(X,Y,Z). (56)

Practically (see Sec. VII) Eq. (54) is very often highlynonlinear and cannot be solved analytically. To calculatethe function F(X, Y,Z) (i.e., the Hamiltonian) it is how­ever sufficient to observe that it satisfies the followingHamilton-Jacobi equation:

[ aF ]2 _ 4 aF aF =Oax ayaZ .

where the function S=S(e, V) on the right-hand side isprecisely the one which defines the fundamental equation(lO). It is possible to find a thermodynamical conditionfor the function (58) which prevents the occurrence ofcaustics (see Appendix C). If the fluid satisfies this condi­tion the Hamilton-Jacobi equation can be solved globallyfor y> O, providing the Hamiltonian of the coupledmatter and field system. The Hamiltonian is a functionof X (depending algebraically on pkl and on derivatives ofgkl up to the second order), Y (derivatives ofboth pkl andgkl up to the first order), and Z (algebraic dependence ongkl)' Hamiltonian form ula (48) implies the following evo­lution equations with U(X, Y,Z )=(3hF(X, Y,Z):

p= as las.av ae

Inserting (53) into (54) we obtain finally a single nonlinearequation for the param eter V= V(X, Y,Z) which has tobe solved. Having solved this equation we can express allthe thermodynamical parameters together with the lapseand the shift in terms of geometrie quantities (X, Y, Z),Le., in terms of canonical variabIes pkl and gkl' More­over, we can find the entropy density

Inserting the above value into (49) and (51) we obtain thefollowing nonlinear equations for twa thermodynamicalparameters (e, V):

The proof is given in Appendix A. Observe that thethree-dimensional space of parameters (X, Y,Z) has apseudo-Riemannian structure with the signature(+,+,-) and the characteristics of (57) are straightlines. The initial-value condition for Eq. (57) is given bythe following observation: for y=o Eq. (51) impliesn =0. Hence, Eq. (49) reduces to X=e IV and Eq. (44)reduces to 1/V=vZ. Finally, we have e(X,O,Z)=XlvZ, V(X,O,Z)=I/VZ,and

1880

41 HAMILTONIAN THEORY OF SELF-GRAVITATINGPERFECT ... 1881

U'(X, Y,Z)=h{3F'(X, Y,Z)=h{3S'(e', V')=h{3S [ ~, V' ) =h{3S [e [~, ~'Z)' V [~, ~,Z ) )

=h{3F [ ~, ~,Z ) = U [ ~, ~,Z )(65)

VII. EXAMPLES

Example 1. A monoatomic ideal gas is described bythe fundament al equation

where m is the rest mass per mole. It follows that

p ='2:..- e -m (67)3 V

and Eq. (54) reads

It is easy to see that U' satisfies (57) if U does. We provein Appendix B that for XFO and Y+O the value of theright-hand side tends to infinity as e goes to zero. Thisproves that indeed, the constraint manifold {X=O, Y=OJcan be considered as a bottom of "a very deep potentialwell." The dynamics of the free gravitational field canthus be approximated by the nonconstrained dynamics ofthe self-gravitating fluid. For e« 1 the degrees of free­dom transversal to constraints become "fast degrees offreedom" and decouple practically from the dynamicsalong the constraints. Starting the numerical simulationfrom the bottom of the well the system will remain for along time in the vicinity of the bottom.

(71)S=f(eJ.lV) ,

(73)

where f is an increasing concave (f' > 0,1" < O) functionand J1 is a positive constant. Moreover, we assume[(J1+1)xf"(x)+J1I'(x)]<O, which is necessary for (71)to be concave (thermodynamical stability condition).This class includes the ultrarelativistic ideal gas (J1 =t)and thephoton gas [S=f(e3Va)I/4, J1=3]. An interest­ing class of fluids corresponds to J1= 1, where the pres­sure is equal to the energy density pe. This makes theenergy-momentum tensor proportional to the metric.Einstein equations contain therefore a "dynamicalcosmological constant" (e.g., if S=Ve V we haveT=2Ve IV and the constant is equal to T2).

For a general fluid described by (71) we have1 ep=-- (72)

J1 V

and Eq. (54) can be solved:

[ )1/2 11/2

X2 __ 2_yZ+X X2- 4J1 YZJ1+ 1 (J1+ 1)2

V= 1-----------­2Z(X2-yZ)

cal systems that are described by a fundament al equation

(66)S =R ln(e -m)3/2V ,

V4Z(X2- YZ)-.±V3mXZ + V2(.±YZ -X2+-±.m2Z)5 s 2S

+fVmX-fs-(m2+y)=0. (70)

APPENDIXA

The function F(X, Y,Z) is given again by (56), i.e.,

F(X, Y,Z)= f{[ V(X, Y,Z)X

-VYYV2(X,Y,Z)Z-1]J.lV(X,y,z)j ,

(74)

(75)

Take the function

with V(X,Y,Z) given by (73). For S=VeV the expres­sion (74) can be somewhat simplified:

[ ]1/2F(X y Z)= X(Yl-YZIX2+1)

, , 2Z

(69)

(68)2e-m = VzvY -X.3 V YV2Z-l

'2:..- [x- vY YV2Z-l-~)= VzvY -x3 V V YV2Z-l '

which can be rearranged into

Inserting (53) into (68) we obtain

As the roots of a fourth-order algebraic equation aregiven by a rather lengthy formula we not e only that thereexists a unique solution satisfying the conditionV~ l!vZ. The entropy density is given by Eq. (56).

Equation (70) becomes much simpler when the restmass can be neglected. For such a ultrarelativistic idealgas, however, the fundament al equation is a special caseof a more general class that is described in the next exam­ple.

Example 2. Let us consider a class of thermodynami-

F(X, Y,Z)=S(e(X, Y,Z), V(X, Y,Z)) .

At each point (X, Y,Z) the system of equations

aF = as ~ + as aVax ae ax aV ax '

aF = as ~ + as aVay ae ay aV ay ,

aF = as ~ + as aVaz ae az aV az '

(Al)

(A2)

(A3)

(A4)

41

(AB)

(AI5)

(AI7)

112aF

az

Z aF _yaFaz ay

v=

aF aF aFPx= ax' Py= ay' Pz= az .

This proves that aF faX, aF /aY, and aF jaZ are con­

stant on the characteristic lines of (57). The equationsgenerated by (A14) are linear. Therefore, the lines can beparam etri ze d as follows:

The "Hamiltonian" (A14) do es not depend on the "posi­tions." Hence, the "momenta" are constant on charac­teristic lin es. Of course,

Finally, taking (A3) and (A4) as independent equationswe obtain

The solvability of the system (A2)-(A4) implies that atleast two formulas among (AlI), (A12), and (AB) aresatisfied. Equating the right-hand sides of any two ofthem we obtain (57).

Conversely, we will prove that (57) is sufficient for Fbeing obtained from Eq. (Al). The function satisfying(57) is constant on the characteristic lines generated bythe following "Hamiltonian":

where s denotes the parameter along the line andXo, Zo, Px> Pz are initial conditions on the surface

!Y=O}. Let us calculate the value of the (mutuallyequivalent) expressions under the square root in the for­mulas (AlI), (AI2), and (AB). The value equalsl/Zo > O. This enables us to define the functionv= V(X, Y,Z) by any of the formulas (AIl)-(AB).Moreover, define e=e(X, Y,Z) by the formula (53)[again, it is easy to prove the inequality (V2Z -1) > O].The reader may easily check that both V and e are alsoconstant on the characteristic lines. Hence,

(A5)

(A6)

(A7)

(AlI)

(A 10)

(AI2)

KIJOWSKI, SMÓLSKI, AND GÓRNICKA

112

1/2

=F[ e(X, Y,Z) l )V(X,y,Z)'O, V2(X,y,Z) =S(e(X,y,Z),V(X,y,Z)),

[~~r

2[~n-4Y[~;]'

asav

v=

4 [~~ rV= I--~-~ ~--42 [ ~~ r - y [ ~~ r

-[ e(X(O),O,Z(O)) O 1 )F(X, y,Z)=F(X(O),O,Z(O))-F V(X(O),O,Z(O))' , V2(X(O),0,Z(O))

aF ae aF ae-----ayaX ax ayae av ae av '-----ax ayay ax

aF av aF av-----as = ax a y a y axae ae av ae av-----

ax ay ayax

Using (55) we have

as aF ae aF ae- -----av ayax ax ay

p = as - aF av aF a v- -------ae ax ay ayax

1882

can be uniquely solved with respect to (aS /ae) and(aS /av). This proves that there must be two linearly in­dependent equations among the above three. Suppose,those independent equations are (A2) and (A3). Hence,

Using Eqs. (53) and (54) we can eliminate the derivativesof e(X, y,Z):

~=- av +v (A8)ax axP ,

~=_ av _ VV2Z-1 (A9)ay ayP 2V'Y

Inserting (A8) and (A9) into (A7) and using (54) we finallyobtain

or, equivalent1y,

Assuming that (A2) and (A4) are independent we obtainin a similar way the result

HAMILTONIAN THEORY OP SELF-GRAVITATING PERFECT ...

APPENDIXC

respands to vanishing af the heat capacity cv, which canbe excluded for physically reasonable systems.

(Cl)

(C2)

1883

=1+AY, (C3)

_y ajazo

l+2jY ajazo

l-Y ajaxo'

2jY~axo,

==det

X(Xo,Zo)=Xo-Yj(Xo,Zo) ,

Z(Xo,Zo)=ZO+Yj2(XO'ZO) .

The corresponding Jacobian is equal to

J=det a(X,Z)a(Xo,Zo)

We define a mapping 4>: (Xo,Zo)---+(X,Z). To eachpoint (Xo,Zo) on the plain Y=O we assign the intersec­tion point (X,Z) of the plain Y=const*O with thecharacteristic !ine of Eq. (57) starting from (Xo,Zo). Be­cause of Eq. (AI6) the value of the parameter s corre­sponding to Y=const equals s = - y 14Pz. Denotingj=PxI2Pz, where px=aF lax, pz=aF laz on the ini­tial plain Y=O, the mapping 4> can be written as

where we have denoted

APPENDIXB

The entropy is necessarily an increasing and concavefunction of e and V, sa when e gaes to zero, S tends toinfinity or to a finite constant. The latter possibility cor-

where the function S=S(e, V) has been defined by theformula (58), Le.,

S(e, V)=F [ ~,O, ~2] . (AI8)

A physical interpretation can be given only to those solu­tions af (57) which correspond to physically admissibleentropy functions (AI8) taken asinitial conditionF(X,O,Z).

From Appendix A, Eqs. (AlIHA13) we conclude that

v[ ~, ;,Z ]=V(X,Y,Z) (B l)

and from (53) we have

[X y ] ee 8'(ji'Z =e(X, Y,Z) . (B2)

Thus, according to (65),

U'(X, Y,Z)=h{3S [; (X, Y,Z), V(X, Y,Z) ]. (B3)

41

A=2j aj _ aFazo axo

y=o

(C4)

Using (58)we express the derivatives of F in terms of thermodynamical quantities of the fluid:

[ aF ] = V as = ~, [aF] = _1- y2 [ as e + as V] = _1- ~ (e +p V) ,ax y=o ae T az y=o 2 ae aV 2 T .

[ a2F] _ v2 a2s [a2F] _ l y3 [ a2s + y a2s + l ]ax2 y=o - ae2' axaz y=o - - 2" e ae2 aeav T '

[ a2F] = l.~(e + V)+ 1-V4 [e2 a2s +2eY a2s + V2 a2s ]az2 y=o 4 T P 4 ae2 aeav av2

(C5)

(C6)

where 77 = e +p V denotes the enthalpy function. The expression in parentheses can be further simplified:

·._-----~--. -

41

(C7)

(C9)

(ClO)

we get

3A=--25(e-m )2

and for fluids described by S= f(eIlV),

A ~ [ II'~0' rl'lI' -I) .We see that for f-L> 1 the coefficient A is positive. Thishappens, in particular, for the ultrarelativistic gas(m =0) and for the photon gas. For f-L= 1 we have A =0and (57) can be solved globally, also in nonphysical sectorY<O.

Berlin, 1979).6J. Kijowski, B. Pawlik, and W. M. Tulczyjew, BulI. Acad. Po­

lon. Sci. Ser. Sci. Math. Astron. Phys. 27, 163 (1979).7H. P. Kiinzle and J. M. Nester, J. Math. Phys. 25, 1009 (1984).8y. E. Zakharov and E. A. Kuznetsov, S~y. Sd. Rev. C 4, 167

(1984).9D. Lewis, J. Marsden, and R. Montgomery, Physica 18D, 391

(1986).!Op. J. Morrison and J. M. Greene, Phys. Rev. Lett. 45, 790

(1980);D. D. Holm, Physica 17D, 1 (1985); D. D. Holm, Lec­tures given at the Centro Matematico Estivo, Session on Re1­ativistic Fluid Dynamics, Noto, Sicily, 1987 (unpublished).

llJ. Kijowski and W. M. Tulczyjew, University of Calgary re­port, 1981 (unpublished); Mem. Acad. Sci. Torino serie V,6-7,3-17 (1982-83).

(C8)

KIJOWSKI, SMÓLSKI, AND GÓRNICKA

=_p[ap] +[~]3e v av e

Finally we get

A=~-3[~+V2[:~ tl= -3 [ -V2 [~] ]~ ~ av2 s '

where ~=e +pV denotes the enthalpy function. Observethat only the half-space y> O is physically meaningful.Equation (C3) implies that A > O is the necessary andsufficient condition for the global existence of the func­tion F within this half-space. The condition is satisfied inall the examples considered in Sec. VII. For a perfect gas

'Present address: lnstitute for Theoretica1 Physics, Po1ishAcademy ofSciences, Al. Lobników 32/46, 02-668 Warsawa,Poland.

IR. Arnowitt, S. Deser, and e. W. Misner, in Gravitation, Anlntroduction to Current Research, edited by L. Witten (Wi1ey,New York, 1962).

2e. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation(Freeman, San Francisco, 1973).

3V. I. Amo1'd, Mathematical Methods oj Classical Mechanics(trans1ated from the Russian by K. Vogtmann and A. Wein­stein) (Graduate Texts in Mathematical Physics, V01. 60)(Springer, Berlin, 1978),p. 75.

4K. Huang, Statistical Mechanics (Wiley, New York, 1963),Chap. 1;see a1soRef. 2.

5J. Kijowski and W. M. Tulczyjew, A Symplectic FrameworkJorField Theories (Lecture Notes in Physics, Vol. 107) (Springer,

1884