Instability of higher dimensional de Sitter space

V01ume 191, num6er 3

PHY51C5 LE77ER5 8

11 June 1987

1N57A81L17Y 0 F H 1 6 H E R D 1 M E N 5 1 0 N A L DE 5171"ER 5 P A C E P.F. 6 0 N 2 A L E 2 - D f A 2 Department 0f App11edMathemat1c5 and 7he0ret1ca1Phy51c5, Un1ver51ty0f Cam6r1d9e, 511ver5treet, Cam6r1d9e C83 9EW, UK Rece1ved 16 Fe6ruary 1987

7he u5e 0f the euc11deanf0rmu1at10n 0f 4uantum 9rav1tya110w5u5 t0 5h0w that the d-d1men510na1de 51tter 5pacet1me 15a150 un5ta61e t0 the f0rmat10n 0f d-d1men510na161ackh01e5.

1. Recent1y, Myer5 and Perry [ 1 ] and Accetta and 61e15er [2] have 1nve5t19ated the pr0pert1e5 0f h19herd1men510na1 61ack h01e5. 1t 15 0ur c0ncern 1n th15 fetter t0 5ee whether the f0rmat10n 0f 5uch 61ack h01e5 1nduce5 an 1n5ta6111ty 1n the de 51tter 5pacet1me, 5uch a5 appear5 t0 happen 1n the f0ur-d1men510na1 ca5e [ 3,4]. H19her-d1men510na1 de 51tter 5pace 15 c1a551ca11y 5ta61e t0 the L1f5h1t2-Kha1atn1k0v pertur6at10n5 [ 5 ]. We u5e then the euc11dean f0rmu1at10n 0f 4uantum 9rav1ty, wh1ch a110w5 u5 t0 5h0w that h19herd1men510na1 de 51tter 5pace 15 5em1c1a551ca11y un5ta61e t0 the f0rmat10n 0f 61ack h01e5. 2. d-d1men510na1 de 51tter 5pacet1me mu5t 6e 1dent1f1ed [6] a5 a max1ma11y 5ymmetr1c, c0n5tant ne9at1ve-5pacet1me-curvature (p051t1ve R1cc1 5ca1ar) 501ut10n 0f the vacuum E1n5te1n e4uat10n5, Ra6 = A9a6, w1th c05m01091ca1 c0n5tant A > 0. 7h15 5pace can 6e v15ua112ed [7,8] a5 a ( d + 1)-hyper60101d, d

-X20 + E X2~=(d-1)/A,

(1)

a=1

where we def1ne H = [ A ( d - 1 )] ~/2. 7he hyper60101d 15 em6edded 1n E a+~ w1th metr1c d

d52=-dX2+

2 dX~,

(2)

a=1

and ha5 t0p0109y R X5 a-~ and 1nvar1ance 9r0up 5 0 ( d , 1 ). 7 h e d-d1men510na1 de 51tter metr1c 15 the metr1c 1nduced 1n th15 em6edd1n9. 1n c00rd1nate5 t e ( - 0 0 , 00);~./a~1, ~/1d~2,... , ~12~(0 , 7~); ~ff1e(0, 2~r), def1ned 6y

Xd=H -~ c05h(Ht) 51n ~d-~ 51n ~d-2... ×51n ~u2 c05 Vt ,

Xd-t =H -1 c05h(Ht) 51n ~/~/d--1 51n ~d-2... X51n ~2 51n ~

,

Xa•2 =H-1 c05h(Ht) 51n ~¢-~ 51n ~a-2... c05 ~ 2 ,

X, = H - 1 c05h(Ht) c05 4/a- 1 , X0 = H - L 51nh(Ht),

(3)

th15 metr1c 15 wr1tten a5 d 5 2 = - d t 2 + H -2 c05h2(Ht) d122•t ,

(4)

where d122• t 15 the metr1c 0n a ( d - 1 )-5phere. Metr1c (4) 15 a K = + 1 d-d1men510na1 R06ert50n- Wa1ker metr1c, wh05e 5pat1a1 5ect10n5 are ( d - 1 )-5phere5 0f rad1u5 H -~ c05h(Ht). 7he5e c00rd1nate5 c0ver ent1re1y the d-d1men510na1 de 51tter 5pace, wh1ch w0u1d f1r5t c0ntract unt11 t = 0 and expand thereafter t0 1nf1n1ty. 1n 0rder t0 5et the d-d1men510na1 de 51tter metr1c 1n 5tat1c f0rm, we 1ntr0duce the c00rd1nate5 t• e ( - 00, 00); r e ( 0 , H-1);~ud~2, ~//d--3,~, ~6¢2e(0, •); ~6¢1e( 0, 2~r), wh1ch are def1ned 6y [7]

0370-2693/87/$ 03.50 • E15ev1er 5c1ence Pu6115her5 8.V. (N0rth-H011and Phy51c5 Pu6115h1n9 D1v1510n)

263

V01ume 191, n u m 6 e r 3

PHY51C5 LE77ER5 8

where an 0verhead d0t den0te5 t1me der1vat1ve and a pr1me den0te5 der1vat1ve w1th re5pect t0 r. Fr0m e45. (8) we 5et f0r a d-d1men510na1 5chwar25ch11d-de 51tter metr1c

X d = H -~ 51n ~//d~1 51n ~d~2... 51n ~//2 C05 ~/1 , X d ~ = H -1 51n ~//d- 1 51n ~/,/d-2

"•"

11 J u n e 1987

51n ¢/2 51n ~1 ,

Xd•2 = H -~ 51n ~d-1 51n ~d-2... c05 ~2 , d5 2 = -

)(3 = H -1 51n ~d-1 51n ~/d-2 c05 ~ffd 3 X2 = H - 1 51n 4/d- 1 c05 ~ud~2

,

+

(5)

5ett1n9 r = H -~ 51n ~,ud~1, the metr1c can 6e wr1tten a5

d5 2 = - (1 --H2r 2) d t •2 + (1 --H2r2)-1dr2

dt2

Ar 2~.) • dr d - 1J

2 +r e

dE22•2

where d0}~2 15 the metr1c 0n a ( d - 2)-5phere. 7he5e c00rd1nate5 c0ver 0n1y the p0rt10n 0 f the 5pace w1th X1 > 0 and Y~a=2 X ] < H -2, 1.e. the re910n 1n51de the part1c1e and event h0r120n5 0f an 065erver m0v1n9 a10n9 r = 0. Metr1c (6) 5h0w5 an apparent h0r120n at r = [ ( d - 1 )/A] 1/2. We 06ta1n n0w the d-d1men510na1 9enera112at10n 0f the u5ua1 5chwar25ch11d-de 51tter metr1c fr0m the vacuum E1n5te1n e4uat10n5 6y u51n9 the an5at2

where F ( d ) = 8 n [ ( d - 2 ) A d • 2 ] 1, wh1ch 15 deduced fr0m def1n1n9 the ma55 0f an 1501ated 5y5tem 1n d d1men510n5 6y 51mp1y 9enera1121n9 t0 h19her d1men510n5 the 5tandard a5ympt0t1c ana1y515 0f 501ut10n5 t0 the E1n5te1n e4uat10n5 1n f0ur d1men510n5 [1]. 501ut10n (9) c0mp1ete5 the 0ne 06ta1ned 6y 7an9her11n1 [9 ], wh0 d15cu55ed the ca5e 0f a h19her-d1men510na1 5chwar25ch11d 61ack h01e.

2 + r 2 d9-22•2

.

(7)

3. We 1nve5t19ate n0w the ex15tence 0f p055161e 5em1c1a551ca11n5ta6111t1e5 1n a h19her-d1men510na1 de 51tter 5pace. We 5ha11 u5e the euc11dean appr0ach t0 4 u a n t u m 9rav1ty. 7h15 appr0ach de5cr16e5 pr0ce55e5 1n 4 u a n t u m f1e1d the0ry wh1ch are c1a551ca11yf0r61dden and, there6y, m19ht d15c0ver e55ent1a11y 5em1c1a551ca1 1n5ta6111t1e5 1n 5pace5 that are c1a551ca11y 5ta61e. 7 h e d-d1men510na1 ver510n 0f the F a d e e v - P 0 p 0 v euc11dean f0rm 0f the funct10na1 1nte9ra1 f0r 9rav1ty [ 10 ] can 6e wr1tten a5

2=JD9a~

exp( --1d + 9au9e f1x1n9 term5),

8 y den0t1n9, re5pect1ve1y, t, r, V/d-2, •.... ~a-,, ..... ,9tt 6 y X ° , X 1 , X 2.... , X " , . . . , X a - ~ , w e 5 e t t h e c 0 m p 0 nent5 0f the curvature ten50r:

where 1d 15 the d-d1men510na1 euc11dean act10n

R00 = - 1"f2,~1~2 + ~1JJ~+ ~

1a= - (16n6d) J (R-2A)91~2 d d x .

+ ~v~e ~-a - ~v~2~e~-a + • ( d - 2 ) r -1 v~e "-a , R11 = ~{e a-" + ~J.2 e~ " - ~21)ea-" - •v••

-•v ~2+~v~2~+~(d-2)r-12~ , R==-(d-3)(e-~-

1)+~re-a(2~-v~),

R33 =R22 51n 2 ~d-2,

Rjj=R22 1-1 51n2 ~ d - . , (8)

j--1

n=2

264

,

(6)

,

~ dt 2 + e a d r

26mF(d) rd•3

d-1]

(9)

X 0 = H -1 c05 4/d•1 51nh(Ht•) .

d 5 2 ------ e

1

Ar 2 "~

26mF(d) rd 3

,

X1 = H -1 c05 ~//d- 1 c05h(Ht•) ,

+r2 d12 } 2

1

(10)

(1 1 )

We ad0pt a5 60undary c0nd1t10n [ 3 ] that the 5pace 6e a5ympt0t1ca11y d-d1men510na1 de 51tter, 1.e., we 100k f0r 9e0de51ca11y c0mp1ete and n0n51n9u1ar 501ut10n5 t0 the E1n5te1n e4uat10n5 Ra6=9a6A, w1th A > 0, wh1ch are the d-d1men510na1 ver510n 0f the 9rav1tat10na1 1n5tant0n. Fr0m (1 1 ) we 9et

1d = - ( d - 2 )A V(d) (16rr6d)

--1

,

1n wh1ch V(a) 15 the v01ume 0f the d-1n5tant0n. We c0n51der f1r5t the 1n5tant0n a550c1ated w1th the

V01ume 191, num6er 3

PHY51C5 LE77ER5 8

d-d1men510na1 E1n5te1n metr1c, wh1ch 15 06ta1ned 6y ana1yt1ca11y c0nt1nu1n9 (t•--,1r) 501ut10n (6). 7he 51n9u1ar1ty at r----[(d-- 1 )A - 1] 1/2 15 0n1y an apparent c00rd1nate 51n9u1ar1ty 1f r 15 1dent1f1ed w1th per10d 22t[(d-- 1 )A-1] 1/2.7he de 51tter 6reen funct10n5 w111 6e theref0re per10d1c 1n 1ma91nary t1me and th15 feature 1ead5, 6y ana109y w1th para11e1 ca1cu1at10n5 1n the 61ack h01e [ 11 ] and f0ur-d1men510na1 de 51tter [ 12] ca5e5, t0 the c0nc1u510n that a h19her-d1men510na1 de 51tter 5pace 6ehave5 a5 th0u9h 1t ha5 an 1ntr1n51c temperature 7 = (2n) - 1[ ( d - 1 )A -•] -1/2.

(12)

We 5ee then that, f0r a 91ven va1ue 0f the c05m01091ca1 c0n5tant, the num6er 0f d1men510n5 0f 5pace tend5 t0 decrea5e the de 51tter temperature. 1n 0rder t0 5ee 1fthe E1n5te1n metr1c 15 a 10ca1 m1n1mum 0r c0rre5p0nd5 t0 an 1n5ta6111ty 0fthe de 51tter 5pace, 0ne ha5 t0 determ1ne the e19enva1ue5 0f the d1fferent1a1 0perat0r [ 13 ] 6a6ca

• a6 - - -

•

[] ~cd - - 2Rac6a~a6 = 2(~cd

(13 )

(where (~a6 15 a metr1c pertur6at10n hav1n9 d 2 - Y/-11 1 de9ree5 0f freed0m) 6y u51n9 the techn14ue5 rev1ewed 1n ref. [ 13 ]. 1f a11 the e19enva1ue5 2 are p051t1ve 0r 2er0, the E1n5te1n metr1c w111 c0rre5p0nd t0 a 10ca1 m1n1mum, 6ut 1t w111 c0nta1n an 1n5ta6111ty whenever 50me 0f the5e e19enva1ue5 are ne9at1ve. 1t 15 p055161e, h0wever, t0 c1rcumvent the 5u6t1et1e5 0f the ca1cu1at10n 6y u51n9 an 1nd1rect pr0cedure. 1t 15 rea11y p055161e t0 5h0w that a11 the e19enva1ue5 are p051t1ve f0r the ana1yt1c c0nt1nuat10n 0f (6) when the metr1c 0n the (d-2)-5phere, d~Q2~2, 15 expre55161e a5 the K~h1er metr1c [ 14] c0rre5p0nd1n9 t0 a tw0-5phere. 1n 0rder t0 5ee that th1515 actua11y the ca5e, we 1ntr0duce the c0mp1ex tran5f0rmat10n

2=2 tan ~Vd~2 exp(1f df2d~3 ) ,

(14)

fr0m wh1ch we 06ta1n d9~2,~2 = (1 + ~92) -2 d2 d9,

(15)

and the a550c1ated 1¢d1h1er p0tent1a1 K = 2 109(1 + ~29) . 7h15 1n5tant0n 15 then 5ta61e [ 3 ].

(16)

11 June 1987

An0ther p055161e metr1c wh1ch w0u1d 5at15fy the a60ve 60undary c0nd1t10n 15 the euc11dean c0nt1nuat10n 0f (9). 5uch a metr1c 5h0w5 h0r120n5 51tuated 6etween + [ ( d - 1 )A -1 ] ~/2 and + [26mF(d)] 1/d-3. F0r [2mF(d)]2/d-3A= [ ( d - 2 ) / ( d - 1)] 2/d-3 a11 the r00t5 6ec0me the 5ame and there6y a de9enerate h0r120n 15 f0rmed. 1n th15 5pec1a1 ca5e f0r the euc11dean c0nt1nuat10n 0f (9) we can 06ta1n a d-d1men510na1 Nar1a1 metr1c [ 15 ] d52 = ( 1 - - X 2 A ) d•r2 + ( 1 - - X 2 A ) -1 dX 2

+A - 1 dff2~2 . 7he 5u65t1tut10n x=2-1/2 c05 2, r =A-1/2p 1ead5 then t0 d52 =A - 1(51n22 d 2 p + d 2 2 + d~a~2) ,

(17)

wh1ch ha5 a t0p0109y 52×5 a-2, each 5phere hav1n9 a rad1u5 A - 1/2. 1n mer91n9 the h0r120n5 we have a110wed a 51mu1tane0u5 rem0va1 0f a11 the 51n9u1ar1t1e5 50 that, 1n pr1nc1p1e, the 1n5tant0n c0u1d 5eem 5ta61e. 70 5ee that th15 15 n0t actua11y the ca5e 6ut 0n1y a c0n5e4uence 0fa p00r c00rd1nate ch01ce, we wr1te (17) 1n the f0rm 0f a c0mp1ex K~1h1er 11ne e1ement, d25= (1 + ~w~) -2 dw dv~ + (1 + ~29)-2 d2 d9,

(18)

w1th w = 2 exp(1p) tan •2 and 2 a5 91ven 6y (14). 7he K11h1er p0tent1a1 f0r 52× 5 d-2 6ec0me5 thu5 the 5ame a5 that f0r 5 × 52. Hence, 0n 5 X 5 d- 2 there w111 6e an e19enfunct10n 0f the 0perat0r 6 w1th ne9at1ve e19enva1ue, wh1ch 1mp11e5 the d-d1men510na1 ver510n 0f the 1n5ta6111ty f0und 1n f0ur-d1men510na1 de 51tter 5pace [3,4]. 1 w0u1d 11ke thank 5.W. Hawk1n9 f0r h05p1ta11ty 1n DAM7P and f0r 5u99e5t1n9 t0 me the 5u6ject 0fth15 w0rk, and J.J. Ha111we11 f0r very en119hten1n9 d15cu5510n5.

Reference5 [ 1 ] R.C. Myer5 and M.J. Perry, Ann. Phy5. 172 (1986) 304. [2] F.5. Accetta and M. 61e15er, 7herm0dynam1c5 0f h19her d1men510na161ack h01e5, Ferm11a6 prepr1nt (1986). [3] P. 61n5par9 and M.J. Perry, Nuc1. Phy5. 8 222 (1983) 245.

265

V01ume 191, num6er 3

PHY51C5 LE77ER5 8

[4] M.J. Perry, 1n: 7he very ear1y un1ver5e, ed5.6.W. 61660n5, 5.W. Hawk1n9 and 5.7.C. 51k105 (Cam6r1d9e U.P., Cam6r1d9e, 1983 ). [ 5 ] P.F. 60n2~11e2-D1a2, D A M 7 P prepr1nt ( 1987 ). [6] D. Ratra, Phy5. Rev. D 31 (1985) 1931. [7] E. 5chr6d1n9er, Expand1n9 un1ver5e (Cam6r1d9e U.P., Cam6r1d9e, 1965). [ 8 ] 5.W. Hawk1n9 and 6.F.R. E1115,7he 1ar9e 5ca1e 5tructure 0f 5pace-t1me (Cam6r1d9e U.P., Cam6r1d9e, 1973). [9] F.R. 7an9her11n1, Nu0v0 C1ment0 27 (1963) 636. [10] L.D. Fadeev and V.N. P0p0v, 50v. Phy5.-U5p. 16 (1974) 777.

266

11 June 1987

[ 11 ] 6.W. 61660n5 and M.J. Perry, Pr0c. R. 50c. A 358 (1978) 467. [ 12 ] 6.W. 61660n5 and 5.W. Hawk1n9, Phy5. Rev. D 15 ( 1977 ) 2738. [ 13] D.J. 6r055 and M.J. Perry, Phy5. Rev. D 25 (182) 330. [ 14] L. A1vare2-6aum6 and D.2. Freedman, 1n: Un1f1cat10n 0f the fundamenta1 part1c1e 1nteract10n5, ed5.5. Ferrara, J. E1115 and P. van N1euwenhu12en (P1enum, New Y0rk, 1980). [ 15 ] H. Nar1a1, 5c1. Rep. 70h0ku Un1v. 35 ( 1951 ) 62.