IL NUOVO C I ~ E N T O VoL. 91 B, N. 1 11 Oennaio 1986

Relativistic Neutral Superfluids. Exact Solutions (*).

G. :P~_PI~I a n d M. W E i s s

Department o/ -Physics and Astronos~y, University of lr Regina, Sasls. $4S 0A2 Canada

(riocvuto 1'11 Set tembre 1985)

S u m m a r y . - - Relat ivist ic superfluids are of great in~erest in astrophysics and cosmology. Itowever, only a fraction of all the known exact, perfect- fluid solutions of Einstein equations also represents superfluids. Those found so far are given in this paper . They all correspond to neut ra l superfluids and are characterized, with a few exceptions, b y equations of s ta te P = c~@-~ const, wi th 0 < ~ < 1 and 0 proport ional ~o the square of the number density. Some exact, superfluid solutions corre- sponding to inflat ionary universes are also given ~nd classified.

PACS. 04.20.Jb. - Solutions to equations. PACS. 98.80.Dr. - Relat ivist ic cosmology. PACS. 95.30.8f. - Rela t iv i ty and gravi tat ion.

1 . - I n t r o d u c t i o n .

The r e l e v a n c e of s u p e r f l u i d i t y a n d , m o r e in g e n e r a l , of B o s e - E i n s t e i n con-

d e n s a t i o n t o a r e a s of g r e a t c u r r e n t i n t e r e s t such as a s t r o p h y s i c s , c o s m o l o g y

a n d p a r t i c l e p h y s i c s h a s b e e n r e p e a t e d l y s t r e s s e d in t h e l i t e r a t u r e . I t is , for

i n s t a n c e , w i d e l y t h o u g h t tha~ a n e l e c t r o n s u p e r f l u i d e x i s t s in t h e i n t e r i o r of

w h i t e d w a r f s (1) a n d t h a t n e u t r o n a n d p r o t o n supe r f l u id s a r e p r e s e n t in n e u t r o n

s t a r s (1). T h e cores of t h e l a t t e r ones , i t is s p e c u l a t e d , m a y e v e n c o n t a i n p i o n

a n d q u a r k c o n d e n s a t e s (1).

(*) Research supported by the Natura l Sciences and Engineering Research Council of Canada. (1) G. BAY~ and C. P]~TtIICK: Annu. t~ev. Astron. Astrophys., 17, 415 (1979).

31

~ 2 G. P A P I l i I a n d M. w ~ I s S

In cosmology the effects of pho ton (~) and gravi ton (a) superfluids have been studied in connect ion with the e~rly universe. In addit ion, the Higgs fields in metas tab le false vacua of inf lat ionary cosmologies (~,~) correspond to superfluids satisfying the ve ry special equat ion of s ta te P ~ - ~. P h o to n condensation, wi th consequent generat ion of strings, ~lso occurs in Weyl- Dirse universes when the cosmological cons tant A is posi t ive (~).

Similar considerations apply to part icle physics in which s y m m e t r y breaking, a more common designation for those phase t ransformat ions by which l=Iiggs fields acquire nonvanishing expecta t ion values, pl~ys a major role in a t t empt s to describe quark confinement and to nni fy the v~rious interact ions (7).

In some of the ment ioned problems the superfluid phase at ta ins relut ivist ic energies because of the high part icle densities involved. For cosmological spplicutions and, to ~ lesser extent , for some of the as t rophysical ones, a general relativist ic t r ea tmen t of the superfluids is desirable. Superfluidi ty is here unders tood to correspond to the phenomenologieal , macroscopic theories of London (s) and Landau ("), as re formula ted relutivist icsl ly b y l%OTm~-(.o) and, more recent ly, by I s g ~ (n). These relat ivist ic generalizations are, however, well grounded in a microscopic descript ion of the fluid (~,~a).

Since the energy-momentum tensors of perfect fluids ~nd superfluids coincide formally, the general-relativistic s tudy of the la t te r ones results some- what facil i tated. One can, in fac~, t ake advantage of the r a the r impressive number of known exact , perfect-fluid solutions of Eins te in equat ions (~). Some

(2) V.A. KuzMI~ and IV[. E. S~A~OSH~IKOV: Phys. Lett. A, 69, 462 (1979). (s) D.J. GI~oss, M. J. P E ~ and L. G. YAFFE: Phys. l~ev. D, 25, 330 (1982); Y. KI- xVDI, T. ~/[o~IYA an4 It. TSV~:AH~a~A: Phys. Bey. D, 29, 2220 (1984). (4) A. II. GUT~: Phys. t~ev. D, 23, 347 (1981). (5) See also A.D. LIN~)]~: in The Very Early Universe, edited by G.W. GIBBO~S, S.W. HAWKINS and S. T. C. SIKLOS (Cambridge University Press, Cambridge, 1983). (~) G. PAPI~I: Phys. Lett. A, 107,. 26 (1985). (7) See, for instance, T.D. L]~]~: Particle Physics and Int~'oduction to ~Field Theovy (Harwood Academic Publishers, Chur, 1981). (s) F. LonDon: Super]luids, Vol. 1 and 2 (John Wiley and Sons, New York, N. Y., 1950). (9) See, for instance, L. D. LA~DAV and E. 1Vf. LIFSmTZ: Flq~d Mechanics (Pergamon Press, I~ew York, N.Y., 1982). (~0) F. R O T ~ : Helv. Phys. Acta, 41, 591 (1968). (11) W. ISlCA~L: Phys. Lett. A, 86, 79 (1981). (12) 1VL-C. L]~vNo, G. PAPI~I and R. G. RYSTE1)HANICK: Can. J. Phys., 49, 2754 (1971); ~.-C. LEVNG: 1Vuovo Cimento B, 7, 220 (1972). (la) W. ISrAeL: in Quantum Gravity, Proceedings o/ The I I I Moscow Seminar, edited by 1~. A. ~r and V.P. F~OLnOV (World Scientific, Singapore, 1986). (~) See, for instance, ~he most comprehensive treatise Exact Solutions o/ Einstein's ~ield Equations, edited by D. K ~ , H. S ~ ' ~ I , IV[. I~IcC~LL~ and E. H]~RLT (VEB Deutscher Verlag der Wissenschaften, Berlin, 1980).

~]~LATIVISTIC N:EUT:RAL SUPERFL~D'IDS. EXACT SOLUTIONS ~

of these solutions have con t r ibu ted decisively to the deve lopmen t of mode rn cosmology (1~).

Of course superfluids, even in the phenomenologic~l fo rmula t ion adop t ed here, are no t jus t per fec t fluids, bu t obey addi t iona l res t r ic t ions (~0,~). Sim- i larly one m ~ y expect superfluid cosmologies to have character is t ics of the i r own. Some of our recent work in th is direct ion (~6) h~s, for instance, shown tha t , under sui table condit ions, quant iza t ion of circulat ion or of flux, for a charged superfluid (~), do indeed represen t dis t inct ive features . We have also found (~7) t h a t a t least two conformal ly flat solutions of the S tephan i t y p e (~), the inter ior Schwarzschild solution ~nd its general izat ion (~4), m ~ y exhibi t u l t rabar ic and super luminal behuviour .

As a first step toward the as t rophys ica l applicat ions, we have also deter- mined some exact solutions for a superfluid flowing in a rigid solid (1s). He re solid r ig id i ty and mot ion de te rmine the met r ic to wi thin an ~ rb i t r a ry constant , while the proper t ies and mot ion of the superfluid ~ffect the s t ruc ture of the solid, even though solid and fluid are assumed r be noninteract ing.

I n this pape r we wish to comple te our work, which refers exclusively to neu t ra l superfluids, b y giving in sect. 4 and 5 the details of the solutions pre- sented in ref. (~,~s). I n sect. 2 we outl ine the theo ry of the re la t iv is t ic super- fluid, while sect. 3 contains a br ief in t roduct ion of the me thod of Infeld-Schild- Taube r (~%~o) on which the solutions of sect. 4, 5 and 6 are b~sed. Section 6 contains a n u m b e r of exac t solutions for universes wi th tension (P ~ - ~) and the i r cosmological classification. As a l ready ment ioned , these ~re essen- t ia l ly inf la t ionary universes. Final ly , we have collected in the appendix all those known exact , perfect-f luid solutions which also describe superfluids ac- cording to the cr i ter ia set out in sect. 2.

2. - The f u n d a m e n t a l equat ions .

/~0THE~ (~0) ~nd la te ly ISRAEL (n) h~ve g iven equiva len t cov~r iant fo rmu- lat ions of supcrfluidity. Fol lowing these authors , we wri te the energy- m o m e n t u m tensor for a neu t ra l superfluid in the fo rm

(15) See, for instance, A. K. RAYCHAVDHVnI: Theoretical Cosmology (Clarendon Press, Oxford, 1979). (16) G. PAPINI and !VI. W~ISS: Phys. JLett. A, 89, 329 (1982). (17) CT. PAPINI and M. WEIss: JLett. Nuovo Cimento, 44, 83 (1895). (is) G. P~_PINI and iV[. WEISS: J. Math. Phys. (N. Y.), 26, 1028 (1985). (1~) L. INFIELD and A. SCHILD: Phys. Rev. 68, 250 (1945); 70, 410 (1946). (~o) G.E. T~VBEI~: J. Math. Phys. (s :Y.), 8, 118 (1967); Gen. Rel. Gray., 3, 17 (1972); I I Advanced Seminar o/the International School o/Relativistic Astrophysics, Eriee, 1976, Tel Aviv University preprint 565-76.

3 - I I N u o v o ~ i m e n t o B .

34 G. rAeINI and M. WEISS

where g, n , t " and u~ represent chemical potent ia l , par t ic le number densi ty, pressure and veloci ty of the superfluid, respectively, and

(2.2) u u u ~ : 1 .

The energy dens i ty ~ is re la ted t o / t , n a n d / ~ b y the equat ion

(2 . s ) /~n = ~ + .P ,

where we have assumed t h a t t he en t ropy of the superfluid vanishes (~1), and b y

(2.4) Q;c, = #n;~,

(2.5) P;~ = n#:~,

which follows by different iat ion f rom (2.3) and the definit ion of the chemical po ten t ia l

(2 .6) ~ = ~ .

Superfluidi ty is in t roduced by means of the equat ions

( /~u.) , , - - ( # u . ) , . = 0 , (2.7)

while

(2.8) (nu~);z = 0

expresses par t ic le conservat ion. In sect. 3 to 6 we consider Eins te in equat ions wi th a eosmologicai t e r m

~Bg~'~ - - A g l , , = - - 8 z T g ~ (2.9) / ~ _ I

and transform them to conformally fiat space-time.

3 . - The Infeld-Sehild-Tauber m e t h o d .

Under a eonformal t r ans fo rmat ion

(3.1) ~..(z) = /~(z )~ . , ,

(~1) See also D. R. TILLEY and J. TILL:EY: Super]luidity a n d S u p e r c o n d u c t i v i t y (John Wiley and Sons, New York, N.Y., 1974).

~ELA~VIST~C NE~J~AL sVP~vLVIDS. ~xxc~ SOLW~ONS 3S

w h e r e Vz~ ~ (1, - - 1, - - 1, - - 1), eqs . (2.9) t r a n s f o r m t o

(3.2) _ ~ f l . ~ . ~ - ~ + ~ o ~ . ~ - ~ + ~ . ~ . ~ - ~ _ 2 ~ . ~ - ~ - A ~ ~ =

= - 8 z ~ f l - ~ ( # n u , u ~ - - P w ~ ) �9

I n d e r i v i n g (3.2) we h a v e a s s u m e d %hut (3.1) i n d u c e s t h e fo l l owing t r a n s f o r -

m a t i o n s (**):

F o l l o w i n g I~FELD-Somz,D (~9) a n d TAVBER (80) we i n t r o d u c e t h e v a r i a b l e s

X 1 X 2 X 3 35 4 (3.4) p ~ - - , q - - ~ - - , z ~ - - , v ~ - -

8 8 8 8

a n d t h e i r d e r i v a t i v e s p~ ~ q~, z~ a n d v~ w i t h r e s p e c t t o x~. T h e y a r e a l l o r t h o g o n a l

to t h e v e c t o r

X6~ (3.5) s ~ = S , ~ - ~ - , s~s ~ = 1

8

a n d h a v e t h e fo l l owing p r o p e r t i e s :

(3.6) p~, q~ =- - - s - ~ p q , p~, z~ ~ - - s - ~ p z , p~, v~ = - - s - ~ p v ,

q a z o~ ~ - - 8 - 2 q Z ~ g ~ v c~ ~ - - 8 - ~ q v ~ ZaV c~ ~ - - 8 - ~ Z V .

T h e f o u r - v e l o c i t y is n o w r e w r i t t e n in t h e f o r m

(3.7) u z - f s z + g v s + h p , + lq~ + m z z ,

w h e r e f , g , h~ l a n d m a r e e a c h f u n c t i o n of t h e v a r i a b l e s s, v, p , q , z. I n t h i s

w o r k we choose for s i m p l i c i t y h ----- t = m = 0 a n d c o n s i d e r fl a f u n c t i o n of s

a n d v on ly . T h e n

(3.s) fi,, fl ,~ + t~v~,

w h e r e t h e p r i m e a n d t h e d o t r ep re sea~ d e r i v a t i v e s r e l a t i v e t o s a n d v~ r e spec -

(22) In this connection see also T. FULTON, F. ROHRLICH and L. W I ~ w ~ : R e v . M o d .

P h y s . , 34, 442 (1962).

36

t ively.

(3.9)

~, PAPINI 3~lld ~I. WEISS

Similarly for fl,~ we obta in

~,,~ = (~"- ~ ' + ~ ~ 8 , ) ~ + t ~ + (8 ' - ~18)

�9 (s,,s~ + v~s~) + (s-~fl ' - s-~8~)V,~.

B y subst i tu t ing eqs. (3.4) to (3.9) into (3.2) and equat ing the coefficients of s~,~, v~,s~§ v~s, , v ,v~ and ~,, on bo th sides o~ the result ing equat ions, we obta in

(3.1o) 4/~'~ - 2fl~" + 28-~flfl'- 28-~8~ = 8 ~ I ~ ,

(3.11) 48f l ' - - 2fl8' § 2s-~f18 = 8wzn]g ,

(3.12) 48 ~ 2~N = s ~ g ~ ,

(3.13) 4s-~fit~,_ 4s-2v~fl _ ~,2 _ s -~(1_ v ~) 82 § 2fl/~" § 2s--~(1- v ~ ) / ~ ' =

- ~ ) . -- 8 ~ ( - - . p - - A

Similarly eq. (2.7) yields

(3.14)

while (2.2) and (2.8) give

(3.15)

and

(3.16)

f4 § ~ / - ~ g ' - ~'g = o,

] ~ §

nil _~_ ~gs - -2 ( l __ ~2) .J_ ~[]l_~ 38--1f _~_ ~8--2(1__ V2 ) - - 3gs_~v] = O.

Final ly f rom eqs. (2.4) and (2.5) we obta in

(3 .17) e 'Z - 4q~' = ~ ( ~ n ' - 3n /~ ' ) ,

(3.18) ~Z - 4o8 = ~ ( Z ~ - 3~8),

(3 .19 ) P ' fl - 4 P f l ' -~ n ( f l # ' - /~fl') ,

(3.20) P ~ - 4 ~ 8 = ~ ( ~ - ~8) .

4. - E x a c t c o n f o r m a l l y f iat so lut ions . T h e case u~ ~ s~.

We first consider a fluid flow ~or which ] ~ 1 and g -~ 0. ]~quation (3.15) is satisfied automatical ly, while eq. (3.16) implies

(4.1) n(s , v) = s-~G(v) ,

:RELATIVISTIC NEUTRAL SUPEI~FLUIDS.

where G(v) is that

(4.2)

~XACT SOLVm~O~S 37

an unknown funct ion. I t immedia te ly follows f rom eq. (3.11)

fl = s - ~ ( v ) ,

where ~(v) must be determined. Equa t ion (3.14) indicates t h a t / t is a funct ion of s only. On using (4.2) in (3.12), we obta in

(4.3)

which can be in tegra ted to

(4.4)

2 T ~ - T ~ = o ,

T = - (d~v q- d~)-~,

where dl and d2 are in tegra t ion constants . I f t h en follows tha t

(4.5) fl = s-l(d~v + d~)-l.

l~rom eqs. (3.10), (4.1) and (4.5) we get

(4.6) - - d ~ s - ~ ( d l v q- ~4) -3 = 4Jr#G(v).

S ince / , is a funct ion of s only, we see f rom (4.6) t h a t

01 (4.7) ~ = - , 8

where e~ is a constant~ and t ha t

(4.8) n = 4~c~s~(d l v q - d~) 3 �9

Using (4.5), eq. (3.13) gives for P the expression

(4.9) p = _ 3d~ - - d~ q - 2 d l d ~ v q - A

8zcs4(dl v q - d~) ~ ,

which satisfies (3.19) and (3.20). When we solve (2.3) for s, we obta in

3d~ - 3dl + A

By direct subst i tut ion, one can easily see t ha t (4.10) satisfies (3.17) and (3.18).

3 ~ G. FAPII~I and M. W:EISS

Since

(4.11) fl~ q 8z~ '

( 4 . 1 2 ) n - - d 2 f13 4zc~

a r e c o n s t a n t e x p r e s s i o n s , while

(4.13) # _ d~ c~(d~v + d2) fl # - d~ -

is v -dependen t , we m u s t h a v e d~ = 0. The c o n d i t i o n A = 2d~ t h e n follows f r o m (2.6). The c o n f o r m a l l y fiat so lu t ion is, t he r e fo re (23),

el -- 1 1 (4.14) fl = 1 (A/2)1/2 ' ~ - - , n - - _P - - - - @. s s 2~e~As 3 ' 4~s4A

I t is also is ea sy t o ve r i fy t h a t a d d i t i o n a l so lu t ions can be o b t a i n e d b y rep lac ing v in t u r n wi th p , q or z in all equa t ions . These solut ions do aga in reduce t o (4.14) because of t h e cond i t i on dx = 0.

5. - E x a c t c o n f o r m a l l y fiat so lut ions . T h e case u~ = jo.

The choices / = v, g = s a n d / = - - v , g = - - s wh ich sa t i s fy (3.15) cor-

r e s p o n d to a f r a m e of re fe rence in which t h e superf lu id is a t res t . F r o m (3.14) we ge t

(5.1) s y - - v/2 = 0 , \ ,

which has a separab le so lu t ion of t h e f o r m

(5.2) /~ = As~v~,

where A a n d 1 are a r b i t r a r y cons tan t s . E q u a t i o n (3.16) becomes

1 - - v 2 (5.3) v n ' § - - i~ = 0

8

(~ Solution i) of ref. (le) satisfies the weaker conditions (3.17)-(3.20) only.

R E L A T I V I S T I C N]~UTRAL S U P E R F L U I D S . :EXACT SOLUTIONS

und has the separable solution

(5.4) n = B s ~ ( 1 - - v~) ~/~ ,

where B and ~ are constants . By writ ing fl as

(5.5) fl = ill(s)fl~(v)

and using (5.2) and (5.4) in {3.12), we obta in

(5.6) v ~ ( 1 - v ~ ) ~ fl~

where ~ is constant . F r o m (5.6) we immediu te ly get

(5.7) fl~ = 2 ~ - ~ ( ~ A B ) I / 2 s ~+v+m2 .

B y using (5.5) and (5.7) in (3.4) we obta in for g -}- 2 + 4 ~ 0 and ~ V= 0

27~ ~ v a+~ (1-- v~) ~/2. (5.8) &fi~ - ~ + ;t +

Subst i tu t ing (5.8) and (5.7) into (3.10)~ we obta in for ~ + )l + 2 va 0

1 11/~ (5.9) /3, = 2 V (a~ + 2 + 2)(~ - / 2 + 4)J vr -- v ,)~/ , .

Inser t ing (5.9) into (5.8) yields a-----0. Thus

1 11" (sv)(~+"'/" (5.10) fl = fl , fi2 = 4(zAB) ~/2 ii + 2)(2 + QJ

for ~ r -- 2 and -- 4. Express ion (5.10) also satisfies (5.6), as expected. (3.13) the pressure follows

- - A B ]~ - - 2 3 2 7 ~ A 2 B 2 A (5.11) zO 2 2 + 4 (sv)~ -- (~ + 2) ~ (] + 4) 2

and sutisfies eqs. (3.19) and (3.20). Similarly f rom (2.3) we obta in

3 A B ( 2 -F 2) 3 2 z A m B i A (5.12) e -- 2(~ + 4) (sv)~ + (;~ + 2) ~(2 + 4) ~ (svp~+,,

39

From

4 0 G. PAPINI and ~ . W~ISS

which obviously satisfies eqs. (3.17) and (3.18). A solution we have recent ly g iven (2a) follows f rom (5.10)-(5.12) for ~ = 2.

W h e n A ~ 0, (5.11) and (5.12) yield the equa t ion of s ta te

2 - - ~ (5.13) .P - - 3(~ + 2) e

Iqo exac t separable solutions exis t for ] 2 = �89 g2 _ s~/2(1_ v~) and ]3 ~_ 0, g~ ~ s~ / (1 - v 2) and at the same t i m e nonvan i sh ing /x and n. The solutions. found in this section and the previous one correspond to ~ r i e d m a n n universes . For the solution of sect. 4 the expans ion O, shear a,~, vor t i c i ty (o~ and ac- celerat ion as all van ish (~5.~). The corresponding universe is thus comple te ly isotropic and homogeneous .

l~or the second solution we aga in find a~, ~ oJ~, ~ a , = 0, bu t

(5.14) 0 = 3(~ § 2) 3/2 (~ ~- 4) ~/2 t(_z+4)/2. 8(~AB) 112

The expans ion is then slowing down if the decelerat ion p a r a m e t e r Q satisfies the inequa l i ty

(5A5)

This occurs if

(5.16)

1(00) Q = - O % 3 ~ - 0 ~ > o .

t > L4(JrAB)~/2 (~ _~ 4)~/2j .

Since the fluid mot ion is shear free, i r ro ta t iona l and geodetic and the energy m o m e n t u m is t h a t of a per fec t fluid, i t follows f rom l~aychaudhur i ' s t h e o r e m (36)

t h a t the met r ic 4eseribes a comple te ly isotropic and homogeneous universe .

6 . - E x a c t s o l u t i o n s w i t h _P ~ - - - ~ .

When P = - ~, E ins te in ' s equat ions become independent of us, while eq. (2.8) no longer applies, as requi red of an inf la t ionary universe (4). The me t r i c corresponds to a De-Si t te r universe , bu t , while us would r ema in c o r n -

24) This is solution if) of ref. (16). (~) G.F. 1~. ELLIS: l~elativistic cosmology, in Proc. S.I.E., Course XLVII (Academic Press, New York, N.Y. , 1971), p. 112. (26) A.K. I~AYCHA~I)H~I: Phys. l~ev., 98, 1123 (1955).

RELATIVISTIC lq~EIYTRAL SUP:E~FLZIIDS. :EXACT SOLITTIO:NS 41

pletely mldetermined if the fluid were perfect, it must here satisfy the remaining

equation (2.7). The cosmological parameters do of course depend on u~. We present here a number of solutions t h a t can be found for two functiollal

forms of fl and several forms of u~. We also give the corresponding cosmologicul parameters .

C a s e A ) . fi ~-- f l (s) , u , = ](s, v ) s , -~ g(s , v ) v ~ .

fl and P can be immedia te ly determined f rom eqs. (3.10)-(3.13). We find

(6.1)

an.d

(6.2)

1

C 1 -~- C282

p _ 16cl e2 -- A 1 8 z (e~ + c~s~) ~ '

where e~ and e~ are coustnnts. The calculation of # depellds on the funct ional form of uz:

i) ] = 1 , g--~0. We obtain

B (6.3) /t = -- ,

8 where B is cons tant .

The cosmological parameters are

(6.4) ~ = ~,----- a , - = 0 ,

ii) ] = v , g ~ - - s . We get

0 = 3(c~. -- c~s~)

8

(6.5)

and

(6.6)

# = B ( s v ) ~

a ~ = a)gv ~-- O ~ ag = O ~ O = - - 6 c ~ s v .

Consideration of the deceleration parameters Q indicates tha t

1 - - 2e2 t 2 (6.7) Q -

2c2t 2

and the expansion slows down when t 2 < 1/2c2 if o~ > O, or speeds up if e2 < 0 for any value of t. This universe is homogeneous and isotropic.

iii) ] ~--O, g = s / ( 1 - - v ~ ) ~ l ~. Equat ion (3.14) yields

B(v) ( 6 . 8 ) # = ~ - - ,

8

4 2 G. PAPINI all~ 1VI. WEISS

where B(v) is an a r b i t r a r y func t ion . W e also ob ta in

(6.9) (rz~ - - - - 3(1 - - v2) az~ v~,v~ 3 s ( 1 - - v2) I/2 r]"~ c~ + c~s~ s(1--%~)" ~ s#s,

(-ogv = O , a~ - - s ( c l + e2s 2) s ~ , 0 = s ( 1 - - v2) 1/2

iv) f = l i v e , g = s / [2(1- -v2)] 1]~. I n this case (3.1~) gives

(6.1o) # = B s ~-~ exp [~ sin -~ v ] ,

while t he cosmological pa r ame te r s are

(6.11)

V - - q )

[ V 2 ( 1 - v2)~/2s 3V2(I- v 2)

2sv v ] 1 -~- - . _ ~ V ~ V ~ - - - - . _ - - - - - ~ - - ,

3V/2 (1 - V2) 3/2 3~ /2S(1 - - V~) 1l~ J Cl + e2s 2

o

C2S ~ - C~ ~ ~ 1 - e~ 82

(9~v ~ 6 .

Case B) . fl = T(v) / s , u , = ](s, v) s . + g(s, v) v . .

F r o m (3.10)-(3.13) we o b t a i n

(6.12) # C3

87)

and

(6.13) t ) 3c~ § Ac~ 1

- - 8 ~ '0 4 8 4 "

As for case A) we cons ider t h e fo l lowing poss ibi l i t ies :

i) ] = 1. I n th i s case we aga in f ind t h e chemica l p o t e n t i a l g iven b y (6.3), while

(6.14) a,v = ~o/~* = 0 , a~ = 0 , 0 = O.

This describes an i so t rop ic and h o m o g e n e o u s universe .

]:r NEUTRAL SUFERFLUIDS. EXACT SOLUTIONS 4 3

ii) ] ---- v, g ~ s. As p rev ious ly , tt is g iven b y (6.5), whi le

- - 3 (6.15) a ~ ~ m~ --~ 0 , a~ ~ O , 0 ----

c3

Thus Q = - - 1 a nd t he expans ion is speeding up. Again , th is is an i so t rop ic a n d h o m o g e n e o u s un iverse .

iii) ] = O, g ~ s / ( 1 - - v ~ ) u*. The chemica l p o t e n t i a l is g iven b y (6.8). T h e n

(6.16) a ~ ~- - - [ 3 (1 - - v2) 3/2 v~v~ sv ~(1-- v2),2 s , s ~ -

v 2 - - 3 c o ~ - - 0 , a , = 0 , 0 - -

(~3(1 - - v2) l ] 2

3s (1 - - v2) ~/~ ~ '

iv) f = 1/V~, g = s / [ 2 ( l - v~)]*~ 2. # is g iven b y (6.10), while

( 6 . , 7 )

C 3 [ q) V

sv V,2 (1__ v2)1/2 s 3%/~(1__ V2)

2sv v ]

+ 3x/~(1_ v~)~f 2 v , , v ~ - 3 V ~ s ( l _ v 2 ) ~ 2 ~,,~ ,

- - ( 1 - - v 2 ) ~ / 2 1 v 3 - 3 ~ o ~ = O , a ~ - - s~ - - 2 v v ~ , 0 =

7 . - C o n c l u d i n g r e m a r k s .

I t is i n t e res t ing to obse rve t ha t , in Mmos t ~11 solut ions cons ide red (,e,37.35),

e q u a t i o n s of s t a te ex i s t ~nd ~re of the f o r m P : 2~ + a, where O < ~ < 1 ~nd a

(27) R. TABENSKY: Some exact solutions in relativist/ie Hydrodynazaies, Ph.D. Thesis, University of Berkeley, Cal. (1972). (2s) H. ST~PHA~r Commun. Math. Phys., 4, 137 (1967); Exact Solutions of Einstein's .Field Equations, edited by D. K_RA~n, H. ST:EI~ M. ~r and E. HERLT (VEB Deutscher Verlag tier Wissenschaften, Berlin, 1980). (39) j . A . ALLNVTT: Gen. Rel. Gray., 13, 1017 (1981). (30) p. LET~LIEI~ and R. TABENSKY: J. Math. Phys. (N. Y.), 16, 8 (1975). (31) F. WAINW-21OHT: Commun. Math. Phys. , 17, 42 (1970). (a2) A. BA~N~S: J. Phys. A, 5, 374 (1972). (33) j . IBANEZ and J.M. SANz: J. Math. Phys., 23, 1364 (1982). (34) A . H . TAUB: Plane symmetric similarity solutions ]or sel]-gravitatiqw fluids, in General Relativity, Papers in Honor o] J . .L. Synge, edited by L. 0'RAIF]~A~TAIGH (Clarendon Press, Oxford, 1972). (2~) A . F . DA TEIXEII~t, I. WOLK and !~. iYL So~: J. Phys. A, 10, 1679 (1977).

~4 G. PAPINI an~ M. WEISS

constant . Only for the solution (A.61)-(A.63) of ref. (~) we h~ve found a = -- 2, while a = -- 1 for the inflution~ry solutions of sect. 6.

We h~ve ~lso found tha t generally ~ oc n 2 with the exception of the solu- t ion (A.102), (A.103) of ref. (3~) for which (A.105) gives ~ oc n 3/4, the solution of sect. 5 which yields ~ oc exp [2(~ + 4)/3(~ + 2)] in the par t icular case A = 0,

the Stephani solution (A.6), (A.7), which requires addit ional assumptions to specify the ~rbitrury functions it eoatMns (~s), und the generalized interior Schwnrzsehild solution (A.108), (A.109) for which ~ = coast (~). These results

ure consistent with the equation ~ = ~(n) given by ISl~AE~ (~a). As discussed in ref, (~), the solutions (A.108), (A.6) ~nd thu t of sect. 5

c~n lead to ul trub~rici ty and superlumin~li ty (~). Throughout this work we h~ve ~ssumed, with l~o~m~ und ISICAEL~ tha t

the en t ropy S of the superfluid vunishes. A different opinion is held in this respect by some uuthors (a~). The in t roduct ion of the en t ropy S as an uddition~l p~rumeter mukes the solution of Einstein equations somewhat easier and we

h~ve indeed found exact solutions (an) with S r 0. The solutions of sect. 6 show whut ~ vur ie ty of influtiollury eosmologies

one can have, even when the fluid veloci ty is restr icted by conditions (2.7) und (3.7) with h = 1 = w = 0. ~ o r e inflat ionary solutions c~n be e~sily ob-

ruined from the results of the ~ppendix, which ~re based on u v~riety of co-

ordinates and symmetries. I t should finally be ment ioned tha t some of the equations given in sect. 2

ure equivalent to the relativistic form of the Luuduu-Ginzburg equation. This fact bus recently been used to identify the (~ utomic g~uge ~) in theories of the Weyl-Dir~e type (a~), These appear be t te r suited thun the E i n s t e i n - ~ x w e l l

theory for the t r ea tment of the charged superfluid, as they cun be shown to incorporate superconductivi ty. We ulso expect the method of sect. 3 to be

purt icularly useful in the s tudy of the churged cuse.

(36) S.A. BLZID~AI~ and ~. A. RZIDER~AN: Phys. Rev., 170, 1176 (1968); ~I. RVDEn- ~A~: Phys. l~ev., 172, 1286 (1968); S. A. BLIY])~A~ and IV[. A. RyI)Enl~A~ : Phys. t~ev. D, 1, 3243 (1970); S.A. BLVD~A~: Equations of state of qHtradense relativistic matter, in Proe. S.I.J~., Course LIV (Academic Press, New York, N, Y., 1972). (3~) W.G. DIXON: Arch. Ration. Mech. Anal., 80, 159 (1982); I.M. KI~ALATNIKOV and V. V. LEB]~DEV: Phys. J~ett. A, 91, 70 (1982); V. V. L]~BED~V and I. IV[. KHALAT- l~I~OV: Sov. Phys. JETP, 56, 923 (1982). (as) /V[. WEISS: On charged and neutral superflnids in a conformally flat space-time, Ph.D. Thesis, University of Regina (1985). (a9) D. GanG0nAS~ an4 G. PAPINI: Phys. Lett. A, 82, 67 (1981); G. PAPINI: Phys. Lett. A, 91, 105 (1982); Nuovo Cimento B, 70, 113 (1982); Proceedings of the X X Orbis Seientiae Conference in honour of P. A. M. Dirac, Miami, 1983, High Energy Physics, edited by B. Kvnsv~o~Lv, S.L. MI~TZ and L. P~nL~VTTE~ (Plenum Press, New York, N. Y., 1985); Proceedings of Sir A. Eddingtong Centerany Conference, Nagpur, 1984, l~elativistic Astrophysics, Vol. 4, edited by V. DESABBATA and T.M. K~A])E (World Scientific, Singapore, 1984).

:RELATIVISTIC N]ET~T/~A.L STuTP:EI~FL'U'IDS. EXACT SOLZJTIO:NS 4 5

~ P P E S T D I X

Other exact neutral superfluid solutions.

Severa l solut ions in t he h t e r a t u r e also descr ibe n e u t r a l superf luids. W e h a v e l is ted be low those t h a t we h a v e so f a r f o u n d in t h e course of our research .

i) T a b e n s k y ' s so lu t ion (~7), r e p r e s e n t e d b y the m e t r i c

(A.1) ds~ : _ t ( d t ~ - - d x 2 __ d y 2 - - dz ~) ,

y ields , in t h e c o m o v i n g f r ame , the equa t i on of s t a t e

3 1 (A.2) "P---- ~ - - 4 t 8"

l~rom (2.3), (2.6) a n d (A.2) we f ind

(A.3) ~ = a n 2 ,

where a is cons t an t . F r o m (A.2), we also see t h a t

(A .4 : ) ~ = \ 4 a ] t ~1--~ "

T h u s

(A.5) # - - 2an = ~ / ~

a n d eq. (2.7) becomes an iden t i ty .

ii) A genera l i za t ion of t he R o b e r t s o n - W a l k e r so lu t ion w i t h nonze ro ex- p a n s i o n has been g iven b y S T E ~ m ~ I in t h e f o r m (28)

(A.6) 1 r 3V4 l ~ + +

where

(A.7)

v- - - vo(t) +

3C-'(t)

~4 o

C~(t) - - O~(t) /9 4Vo(t) (Ix - - xo(t)] 2 ~- [y - - yo(t)] 2 ~- [z - - zo(t)]2},

2 C C . 4 V

P = - e + .~oV,~ '

a n d t h e func t ions V o ( t ) , O( t ) , C ( t ) , Xo( t ) , yo( t ) a n d Zo(t) a l e a r b i t r a r y . ~o is t h e

,~6 G. PAI:'INI a n d M. w ~ s s

gravitat ional constant. In the comoving frame u~ becomes

- - o - - 3 V , ~ ,o (A.8) u~, = ~ / - - g o o ~ , - OV c5~,

and from (2.3) and (A.7) we obtain

2CC,~V (A.9) n --

~oV,~

From (A.7) we get

(A.10) 2CC,4

~ - ~o(~,)~ [ v , ~ v ~ - vv,o~]

which substi tuted into (2.5) yields

(A.11) ~ _ V,~ V,,~ # V V,~

Equat ion (A.11) gives in turn

V (A.12) ~ = v(v , ~, t) -~--.

From (2.7) one then sees tha t G can be a function of t ime only. Expression (A.9) becomes

(A.13) n(t) - - 2VV,, ~oG(t)

and satisfies eq. (2.4). F rom eqs. (2.5) we also obtain

V~ C,~ Ca4 (A.I~) 3 V -- C + C,,

(A.15) V , 4 V,~ V V,4 '

( & . 1 6 ) V , 4 _ V,4~ V V,~

G,~ G '

Equat ion (A.14) integrates to

(A.17) C(t) Ca(t)

G(t) -- V3(x' y' z, t )F(x, y, z) ,

RnLATIVlSTIC ~ V ~ A L SVP~nFnVlI)S. ~XAeT SOLVTIO~S 47

where E is an unknown funct ion of x, y, z alone. Thus eq. (A.17) requires the V given in (A.7) to be a factorizable funct ion of the space co-ordinates x, y, z. We consider two possibilities:

a) Xo, Yo and zo are constants and

(A.ls) vX(O -- ~ c ~ ( t ) - ,

Then b y subs t i tu t ing the result ing expression for V(t ) where a is a constant . into (A.17) we obtain

(A.19)

where a is a constant , and

(A.20)

F ( x , y, z) : a { l + a [ (x - - xo) 2 + (y - - yo) "~ + ( z - Zo)-~]} 3 ,

Equa t ions (A.15) and (A.16) are also satisfied. :By compar ing (A.]8) and (A.20), we obta in

CC,4 (A.21) r Voo"

We then find f rom (&.12) and (A.21)

(]C,~ ( i .22) /~ - V o , , V~o ,

while (A.13) gives

2v] (&.23) n - -

~o

b) Equa t ion (A.17) is also satisfied if

(A.24) r - - O~( t ) 9

I n this case V ( x , y, z, t) : Vo(t) and F(x, y, z) = A == consk We have now

C(t) C.dt) CC ~ 2 A (A.25) G(t) - - A V e ( t ) ' # - - 2" , n • - - V~ ,

A Vo Vo.4 Xo

while _P and ~ are as in (A.7).

4 ~ G. PAPINI and M. WEISS

iii) A l l n u t t ' s so lu t ion (29) is r e p r e s e n t e d b y t h e m e t r i c

dx 2 [/i + t ~ ~/2 dye_ F ( i . 2 6 ) dss - - 1 - - ex s -[- xSt(1 -~ t2)l/s L ~ ]

l] + t~\ ~Is ]

w h e r e m a n d e are real constants~ a n d b y

X 2 d t s , i-}- t s

(A.27)

(A.2S)

(A.29)

- - 6c 1- m s I P-- §

~o 4Xo x 2 t S ( l - t s)

6c ] - - m 2 1

u~ = x( i + t~ ) -v s~ .

F r o m (2.3) a n d (A.13) we also ge t

( 1 - - m s) 1 1 3 (A.30) n - -

2~o t s (1 -~- t s) X ~ ~t

- (i- m s) (A.3I) -P~ = 2XotS( 1 + tS)x 3 , P t =

(1 - - mS)( - 1 - - 2t s)

2uoxSt3(1 + t~) 2

which s u b s t i t u t e d in to (2.5) y ie ld

(A . . 32 ) ~ g( t ) x

where g(t) m u s t st i l l b e de t e rmined . Us ing (A.32) in (2.5), we ge t

(A.33) - - (1 + 2t 2) ~(t)

t ( ] + t s) g( t ) '

where t h e do t r ep resen t s d i f fe ren t ia t ion wi th r e spec t to t. i n t eg ra t e s to

(A.3~)

for c o n s t a n t A. Thus~

( i . 3 5 )

and , f r o m (A.30), we ge t

A g(t) -- t(1 + ts) 1/~'

A 1 # - - t ( 1 + t s ) l l s x

E q u a t i o n (A.33)

1 - - m 1 ] (A.36) n - -

2uoA t(1 + t s ) 1/2 X,

I ~ E L A T I V I S T I C N E U T R A L S T J P E R F L U I ] : ) S . E X A C T S O L ~ J T I O I ~ S

E q u a t i o n (A.28) can now be r ewr i t t en in the f o r m

6v A2~o (A.37) 0 : - - ~ - - - - - n 2 ,

~ o : [ - - m s

f r o m which (A.35) ~gain follows. E q u a t i o n s (2.4) and (2.7) arc satisfied.

iv) T a b e n s k y ' s second met r i c (~7) in co rne r ing co-ordin~tes is

(A.38) ds 2 _ exp [a2z2/2] (dt2 _ dz2) _ az(dx2 ~- dy2) , V~z

where a is a cons tan t . The equa t ion of s ta te is

1 ra 2 z 23 ( 39) p = =

F r o m (A.39), (2.3) a nd (2.6), we ob ta in

(AA0) 0 = en2,

where e is an in t eg ra t ion cons tan t . Thus

1 n = ~ c a(az)l/4 exp [a2z~/4] (A.41)

and

(A.~2) / ~ - ~n - - 2on = ~/9~a(az)l l 4 exp [-- a~z2/4] .

I n the eomov ing f r ame

(A.43) exp [a 2 z2/4]

U 0 - - ( a Z ) 1 /4 '

49

(A.44) ds 2 ~ exp [-- (22 cosh%)/2P] (dt. ~ _ t~ dz~) _ t eosh z(dx 2 -F dY 2) ~/ t cosh z

for c o n s t a n t 2 and b y

(A.45) p = ~ = ~- t - m (cosh z) 1/2 exp t ~ .

4 - I I N u o v o Gimen to B .

hence (2.7) reduces to a single equa t ion which is also satisfied b y (A.42).

v) TA]~ENSr~u (27) also f o u n d an explosive perfect - f lu id solut ion corre- spond ing to an isotropic, i nhomogeneous universe. I n co rne r ing co-ord ina tes the solut ion is g iven b y

50

In this case (2.3) and (2.6) yield

( i A 6 ) ~ = eln ~

where v~ is an integrat ion constant . F r o m (A.46) we obtain

n : ~ t - m 6 ( c o s h z ) ~ l " exp [ ~ c~ z] V2v~ 4t~ J

(A.47)

and

( i .4S)

Since

4t ~ J "

G. PAPINI and. l~I. WEISS

(A.53)

We also have

. [ ( 3 ) ] and eq. (2.7) is satisfied.

vii) LETELIEI~ and TABENS~ (so) also found cylindrically symmetric , perfect-fluid solutions with ~ - ~ E(/ t ) . In comoving co-ordinates (and pos-

exp [-- ~ (cosh 2 z)/4t2] o (A.49) u~ = tl/~(cos h z) l] 4 (~lt ,

eq. (2.7) is also satisfied.

vi) The plane-symmetr ic perfect-fluid solution found b y L E ~ : E ~ and T ~ E ~ S ~ Y (~0) is represented in comoving co-ordinates b y

. (• ds~ = N-; exp -- E § g E -~ e �9

1

1 [ ( (A.51) P = q = ~ i E exp /E-~ E -1 a ,

where E = E(z) is an arb i t rary function. In the usual way we find

[(

R E L A T I V I S T I C :N'EUTI~A:L S '0"PERFL~I]:)S.

i t i v e s ign) , w e h ~ v e

(A.55)

E X A C T S O L U T I O N S 5 1

(A.56)

W e a l so o b t a i n

(A.57)

(A .5s )

(A.59)

~tnd

P = ~ o = ~ E ~ - 3 / 2 e x p s - I ~ .

~ C3~b 2 ,

n - - ~ E R -3/4 exp E § ~ .E -~ a/2 , ~2c~

3 s ~/2]

(A.60) u( ,~ RaI4-E-~ exp [ - - ( E + 3.E-~)(~/2] .

E q u a t i o n (2.7) is a g a i n sa t i s f i ed .

(A.61)

(A.62)

(A.63)

w h e r e

~ n d

(A.6~)

vi i i ) I n W a i n w r i g h t ' s (~) p e r f e c t - f l u i d s o l u t i o n

ds 2 = _ ~ (dxl) 2 - -

(dx2)2 2 �9 cos (ax 1) dx ~ dx 3 § 4u - - - - - cos (ax ~) dx 2 dx 4 § 2 dx 3 dx 4 -- 2u (dx') 2 ) a

1 P = A - - ~ a 2 + Po cos [(0.)2/2 (x~)] ,

119 ] = ~ ~ a o - - 3 A - - P0 cos [(2) 3/2 (xa)] ,

a 2 = r

A = c o s m o l o g i c a l c o n s t a n t ,

Po--~ eons~

1 1 u(x~) = ~ a2 + -~ po cos [(2)~/2 (x3) ] .

~ 2 G. P A P I N I 211(]. l~I. WEISS

W e find

(A.65)

and

(A.66)

Since

(A.67)

2 "]s/s tt = e a s + 5p0 cos [(2) ~/s (x~)]

z [ 2 ] - . ~ n = ~ a s + 5 po cos [(2)~/s (x~)] .

1 [1 s 1 ]~/2 u~ = ~ ~ a + ~ po cos [(2)3~(x~)] ~ ,

eq. (2.7) is satisfied for # given b y (A.65). :For this solution 0 and ~ are given b y

po cos [ (2) s/~ (x3)] (A_.68) 0 -~ 3{~a ~ �88 cos [(2)8/S(xS)]} '

203 (A .69 ) a ~ ~ - - 3

ix) BA~]~S ( s s ) found perfect-f luid solutions described b y

(A.7o)

(A.71)

ds s - - [.N(z)]S dx 2 ~- .F(x) d~ s ~- dz s - - x sdt s ,

p = 3 A s i ( a ) bNS(z) 1 2 x s

(A.72)

where

(A.73)

.~ _ 3 A S l ( a ) b.N'S(z) l 2 x s

.~ = ax2 -}- b ln x ~- o ,

I';' [ a sin ( V ~ + B)

N ( z ) = I A z @ B

[A sinh ( ~ @ b)

for a =/= 0~

for a = 0,

for a > 0,

for a = 0,

for a < 0,

where A and B are constants . W e discuss the three possibilities separately .

a) :For a ~ 0 we have

- - b ( A z + B ) s (A.74) ~ + .P ~ x 2

RELATIVISTIC NEZYTRA]U SUPERFLUIDS. EXACT SOLUTIONS ~ 3

:F rom P , . = n/*,~ a n d (A.74) we o b t a i n

(A .75)

On us ing

(A.76)

we g e t

(&.77)

# = e(x)(Az + B)mL

o'(x) ~ 1

0

Thus , for a c o n s t a n t , we o b t a i n

( • ~ = - (Az + X

- - b (A.79) n = - - (Az + B) 2-~1"~ .

~X

T h e e q u a t i o n

b r ,q(}~ A ( A z - / B ) =/un~ =

X 2

impl ies A = 1. T h u s

(A.81) # = x (z @ B ) ,

- - b ( A . ~ z ) n -= - -

0CX

I t fo l lows f r o m

(z + B ) .

(A.s3) u~ = V~oo ~ z + B 0o

a n d (A.81) t h a t (2.7) is sa~isfiecl.

b) ~ o r a > 0, we g e t

(A.84) ds ~ =

(&.85) P =

1 [1 ] A 2 sin 2 (v/a~ + B) ~ 4x~ + / 7 ( x ) dg~ -~ dz 2 - - x 2 dt ~ ,

b I A~ sin~ ( V ~ @ B) ' 3A 2 I(a) -- -~ -~

(A.86) b A~ sin~ ( ~ f ~ + B ) e = - - 3 A 2 I ( a ) - ~

5~ G. PAPII~I ~nd M. w~Iss

On using (A.85) we get

(A.87) P'~ -- 2x -- A 2 b v/~ sin (X/~z - / B ) cos (~/~z + B)

V~ - - b A s

~t~ , o - - XS

which integrates to

(A.88) # = F ( x ) sin ( V ~ -k B) .

sin s (~/~z -~ B)/~'-2~ V

Again on using (A.85) we obtain

(A.89) F ( x ) = - ,

where ~ is an integration constant. Thus

(A.90) # ~-- - sin (%/-a~ -k B ) , X

_ b A s (A.911 n -- sin (Vfh~ -k B ) .

~ X

Equat ion (2.7) is then satisfied for /z given by (A.90) and

X

(A.92) u~ : A sin (v~h~ -k B) 5~"

c) For a < 0 we have

(A.93) P ---- 3 A 2 I ( a ) : --AbS sinhS ( V ~ a z ~- B) 2 X ~

(A.94) ~ : - - 3 A S I ( a ) b A s sinh s ( % / ~ -k B) 2 X 2

Proceeding as in a) and b ) , we find

sinh ( ~ + B) ( A . 9 5 ) ~ = ,

X

- - b A 2 sinh (%/- -~ ~- B) (A.96) n --

Equat ion (2.7) is satisfied by (A.95) and

X (A.97) u ~ : - - A sinh ( ~ / ~ a z ~ B) by.

RIilI,ATIVISTIC NEUTRAL SIIP:ERFLUIDS. :EXACT SOLI_ITIOlgS

X) F o r t he so lu t ion of I~A~EZ a n d SAgz (38)

5 5

(A.98)

ds ~ = - - r 2 dt 2 + r 2 dO 2 + r 2 s in ~ 0 d~o ~ + 2 ( 1 - 2er2) -~ dr 2 ,

1 . P = ~ - - 6 e , Q = ~ r r ~ @ 3 c ,

where e is ~ cons t an t . F r o m

(A.99)

we o b t a i n

~e

(A.100) Q = 3 c + B n 2

a n d

(A.101) # = 2 B n - -

E q u a t i o n s (2.3)-(2.7) are also satisfied.

xi) T:~VB (84) a n d DA TEIXEn~+:, WOLK a n d S o ~ (ss) f o u n d a p l a n e - s y m - m e t r i c s ta t ic , perfec t - f lu id so lu t ion w i th m e t r i c

(A.102) Z

ds 2 : z 2 ( d x ~ -~- dy ~) + ~ dz 2 - - exp [ 2 v ] d t 2 ,

where

(A.103) 2 z P , , X o P Z ~ + p - - 1 - - ~ , 2',o = - - xo ~ z ~ , (~ -[- P ) v,~ = - - P ~ .

F o r P = ~/3, t h e y also f o u n d

(A.:L04)

P = po(36z 2 - - 12z 7 + z 12) ,

=_ XoPo (216 - - 108z 5 @ 18z 1~ - - z 15) 5

where xo a n d Po are cons tan t s . P roceed ing as above , we o b t a i n

(A.106) # = ~ -

56 C-. PAPINI a n d M. W~ISS

w h i l e

(~.107) u0 ~- e x p Iv] ~ c P -~/~ .

c a n d 2 a r e c o n s t a n t s . E q u a t i o n s ( 2 .3 ) - ( 2 .7 ) a r e t h u s s a t i s f i e d .

x i i ) T h e g e n e r a l i z e d i n t e r i o r S e h w a r z s e h i l d m e t r i c is g i v e n i n r e f . (1~):

d r ~ - - ~ d t ~ ( A . 1 0 8 ) ds~ 1 - - C ~ r ~ -~ r~ (dO~ + sin~ 0 dq02) - - ~ ~ .

t t e r e

(A..109)

u~ = rf~(t) s i n 0 s i n 9 ~- rf~(t) s i n 0 cos ~ + r/a(t) cos 0 +

+ / d r ) ( 1 - C~r~) "~ - c - ~ ,

3 C ~ 2 C - - - - c o n s t , P ~ - - - ~ - / - - - u ~,

~0 ~0

w h e r e 11, ]3, f3 a n d ] , a r e a r b i t r a r y f u n c t i o n s of t. W e f i n d

2 C A ( ~ k . l l O ) n - - - , # = - - ,

;4o # ~ 4 ~4

w h e r e A is a c o n s t a n t . A l l r e m a i n i n g e q u a t i o n s a r e s a t i s f i e d .

�9 R I A S S U 1 W T O (*)

I f luidi re la t ivis~ici sono di g r a n d e in t e re s se i a as t rof is iea e in eosmologia . C o m u n q u e solo u n a p a r t e di t u t t e le soluzioni e s a t t e e pe r u n fluido p e r f e t t o 4el le equaz ion i di E i n s t e i n r a p p r e s e n t a p u r e i superf iuidi . Quelle t rova~e f ine a d e r a sono da t e i a ques to lavoro . T u t t e eo r r i spondono ai super f lu id i n e u t r i e sono e a r a t t e r i z z a t e con poehe eeeezioni d~ equaz ion i di s t a t e P = cr 0 + eons t con 0 < ~ < 1 e ~ p r o p o r z i o n a l e al qua - d r a t o del la dens i t~ n u m e r i e a . Si d a n n o e si elassif ieano a leune soluzioni e s a t t e pe r super - f luidi ohe eo r r i spondono a tmive r s i inf laz ionar i .

(*) Traduz ione a eura della l~edazione.

PeYmTtmaCTC~a~ HefiTpaAbHa~l cBepTeKyqaR atH~KOCTb. ToqHh[e pemenH~.

Pe3IOMe (*). ~ PeHgTHBnCTCKHe cBepTeKy~ne ~ n ~ 0 C T H Hpe~cTaB~t-IOT 60Ylt,nIO~ ~ T e p e c B acxpoqbnaaKe rl K0CMOH0rna. O~naKo, T0nBK0 qaCTb n3 BCeX HaBeCT~LX TOHH]bIX pe- mer~r~ ypaBaenm] ~ T e ~ J ~ a ~lmt a~eazn ,no~ Xn~K0CTr~ n p a r o ~ m , i ~ t cBepxTeKyqnx ~rd~aK0CTe~. B aT0~ pa6oTe HpHB0~HTC~t peme~Ha, ~oay~eHnble paHee, ~ZL~ cBepxTeKyHI~IX XC_g~0CTe~. 3Tr~ pemeHHg OTHOCgTCa K He~Tpan~H~lM cBepXTeKyqHM ~H~KOCT~tM H xa- pa~TepH3yIOTC~ y p a ~ H e r m ~ r i COCTOm~n~ 29 ~_~ ~ + eons t , r~e 0 < cr < 1 H ~ n p o n o p - ~ o r m m , rm ~ a ~ p a r y mmT~OCTn. T a ~ e npn~o~aTC~ n r,~accn(t)rm~pyrorcz neKoTop~,le T0tlHbIe cBepxTeKy~ae perrreHag, C00TBeTCTByIOIIUHe p a c n ~ a p ~ o n ~ M C a BcenemmrM.

(*) Hepeae3eno peOatcttue~t.