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The scalar wave equation on static de Sitter and anti-de Sitter spaces

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1989 Class. Quantum Grav. 6 893

(http://iopscience.iop.org/0264-9381/6/6/013)

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Class. Quantum Grav. 6 (1989) 893-900. Printed in the U K

The scalar wave equation on static de Sitter and anti-de Sitter spaces

D Polarskit Raymond and Beverley Sackler Faculty of Exact Sciences, School of Physics and Astronomy, Tel-Aviv University, 69978 Tel-Aviv, Israel

Received 24 March 1988, in final form 21 October 1988

Abstract. Exact solutions to the scalar wave equation are given on a de Sitter (dS) and anti-de Sitter (Ads) background with static metric, for arbitrary dimensions d. In particular in the case of dS space the boundary conditions at the horizon are in agreement with the Hawking effect and the solutions form a complete set of physical modes.

1. Introduction

Spaces satisfying Einstein’s equations with a cosmological constant have attracted considerable attention during recent years in the context of cosmology and supergravity theories [l, 21. A lot of work has been done in quantisation on a d s background with global non-static coordinates [3]. In this work we give a method of solving the scalar wave equation on the static d s and A d s spaces in the general and not necessarily massless conformally coupled case. This is especially interesting in the case of static dS space which has a horizon and the Hawking radiation that goes with it.

2. Static metric on ds and A d s spaces

As is well known, d s and A d s spaces are maximally symmetric spaces. d s space has the group of isometries SO(4, 1) and can be viewed as the four-dimensional hyperboloid

(x’))’ - (XI))’ - (x)’))’ - (x’))’ - (x‘))’ = -a2 (2.1)

embedded in a five-dimensional flat space with metric

Too= 1 q i i = - l for i=1, . . . , 4 qu = 0 when i # j . (2.1‘)

On the other hand, Ads space has the group of isometries S0(3,2) and can be viewed as the four-dimensional hyperboloid

(x0))’+ (XI))’ - (x*))’- (~’1)’- (x‘))’ = a)’ (2.2)

t Supported in part by the Israeli Academy of Sciences.

0264-9381/89/060893 + 08$02.50 @ 1989 IOP Publishing Ltd 893

894 D Polarski

embedded in the five-dimensional flat space with metric

Too= 7711 = 1 7722 = 7733 = 7744 = -1 T,, = 0 when i # j . (2.3)

This is why both spaces can be described by polyspherical coordinates [4]. We will show that the static metrics on these spaces are just special cases of such coordinates. Indeed, let us consider the following parametrisation of Ads space

xo= a cosh (1, sin r

x’ = a cosh (1, cos r

x2 = a sinh CC, sin 8 sin 4

x3 = a sinh (1, sin 0 cos 4

x4=as inh(1 ,cos8

which gives us the metric

a2[cosh2 (1, dT2-d(1,2-sinh2 (1,(d82+sin2 8 d4’)].

It is easy to see that the change of coordinates

t = a r r = a sinh (1,

brings the metric (2.5) into the static Ads metric

( l + r 2 / a 2 ) dt2-(1+r2/a2)-1 drZ-r2(d02+sin2 ed4’).

Analogously, we can parametrise ds space as follows

xo= a cos (1, sinh T

X I = a cos (1, cosh 7

x2 = a sin (1, sin e sin 4 x3 = a sin (1, sin 8 cos 4 x 4 = a sin (1, cos 8

and this will yield the metric

a2[cos2 (1, dr2-d(1,’-sin2 (1,(d02+sin2 6 d4’)].

Again, defining new coordinates

t = a7 r = a sin (1,

we get the static d s metric

(1-r2/a2) df2-(1-r2/a2)-1 dr2-r2(d8’+sin2 t9 d+*).

3. Scalar field in a static A d s space

Let us consider a scalar field satisfying the equation

0 CP + m2CP + 5R CP = 0

(2.9)

(2.10)

(2.11)

(3.1)

Scalar wave equation on static d s and Ads spaces 895

where the Laplace-Beltrami operator

corresponding to the metric ( 2 . 5 ) is given by

1/a2[cosh-’ (I, d:-sinh-2 (I, cosh-’ (I, d+(sinh2 (I, cosh (I, d,)-sinh-’ (I,L2]. ( 3 . 2 )

m is the mass of the field, [ is some coupling constant and R is the Ricci scalar given by the constant - 1 2 1 ~ ’ . We can therefore write equation ( 3 . 1 ) as follows

cl@ = - A l a 2 @ (3 .3)

with A / a 2 = m 2 + [ R . Introducing the ansatz

= exp(-iwa7)Rwf((I,) y,,(e, 4 ) ( 3 . 4 ) we get the ‘radial equation’

( sinh-’ (I, cosh-’ (I, d,(sinh2 (I, cosh (I, d + ) + m - m - ( 3 . 5 )

This equation is of the form

d,(coshP (I, sinh” t/J a+)+ r ( r + p - 1 ) Z(l+q - 1 ) - - n( n + p + q ) ) R,’ = 0 (3 .5’) ( coshP IC, sinh“ (I, cosh2 (I, sinh’ (I,

w i t h p = l , q = 2 , r = a w , A = n ( n + 3 ) . As the Laplace operator is a Casimir of the s o ( 3 , 2 ) algebra, different values of A,

or equivalently n, define irreducible representations of S 0 ( 3 , 2 ) . The eigenfunctions will be homogeneous functions of degree n in the embedding coordinates x defined in 0 2 [ 4 ] . The exact solutions of (3 .5‘) are given by [ 4 ]

Z - n + r 1 - n - r - p + l q + l , 1+-; tanh2(I, 2 2

Rwll = tanh’ (I, coshn (LF - ( 2 ’

R,‘, = cOth“+’-’ (I, coshn (I,

. -I--. 2 9 q - 3 , 2 tanh2 (I,). r - n - Z - q + l p + I + q + n + r

, I -

Hence, equation (3 .5) has the exact solutions

1 - n + a w 1 - n - a w R,’, = tanh’ (I, coshn (I, F ; I + ; ; tanh’t/J) ( 3 . 6 )

aw - n - I - 1 2

3 + 1 + n + aw 2 9 1 - Rw12 = coth‘+’ (I, coshn (I, F (

We note, en passant, that if the space is d-dimensional, we have to put q = d - 2. Only the solutions ( 3 . 6 ) are regular at the origin, a fact important for physical applications. These solutions agree with those given in [ l , 51. Let us note that a massless conformally coupled scalar field corresponds to A = -2 or n = -1 , -2 . Two complete sets of solutions can be found by allowing aw to take discrete integer values for given 1 and n while the boundary conditions corresponding to each set are analysed in [ l ] .

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4. Scalar field in a static d s space

We now turn our attention to a scalar field in a d s space satisfying the equation

Cl@ = Ala2@ (4.1)

where again Ala2= -m2-112/a2. Introducing the ansatz

we get the radial equation

(i2a2w2 [ ( / + I ) )I sin-2 IC, cos-' J, a,(sin2 IC, cos 4 a,) - 2+.2-~ R,' = o

COS J, sin J,

which is of the form

8,(cosp J, sin4 J, 8,) r( r + p - 1) I ( 1 + q - 1) sin2 J, + (4.4) cosp J, sin4 J, -( cos2 J,

w i t h p = l , q = 2 , r= iaw and A=n(n+3) . Again the numbers n define irreducible representations of SO(4,l) and the solutions

are homogeneous functions of degree n in the embedding coordinates. Equation (4.4) has the solutions [4]

(4.5) I - n + r I - n - r - p + l q + 1

; I+-; -tan2 J, 2 2

RwI1 = tan' J, COS" 4 F -

-CwI2 = cot4+'-l J, COS" J, F(

( 2 '

. -1--. 2 , q-3 , 2 -tan2 +) r - n - l - q S 1 p + l + q + n + r 9 1- 2

and so the solutions of equation (4.3) are given by

(I-n;iaw, I-n-iaw ; l+i; -tan2 J,

2 R,', =tan' J, COS" J, F (4.6)

; - l++; -tan2 J, ) . (4.7) iaw - n - 1 - 1

2 3 + I + n +iaw

2 ' 1 - Rw12 = cot'+' J, COS" J, F(

As in the preceding case, these solutions are generalised to the case of a &dimensional space by putting q = d -2. We are interested in solutions which are regular at the origin and so we keep only the solutions (4.6). Applying (2.10) and using the transfor- mation formula

F ( a , b ; c; z ) = (1 -z)-"F

we get in the static metric (2.11) the regular solutions

1 - n, + iaw 1 + n, + 3 + iaw 2

where

Scalar wave equation on static d s and Ads spaces 897

Taking into account (4.9) it is clear that the solutions Rwl+ and RUI- coincide as should be the case because they are both regular solutions of the same differential equation. Also, interchanging a and b in (4.8), it is easy to see that RWl is an even function of w. The massless conformally coupled case, with R / 6 = 2 / a 2 , corresponds to the values n = -1 and n = -2 [ 6 ] . Static ds space is especially interesting as it exhibits a horizon at r = a. It is therefore very interesting to investigate the asymptotic behaviour of the regular solutions near the horizon. Using the transformation formula (15.3.6) of [7], we get the asymptotic behaviour

(4.10) A(1 - r2/a2)iaw/2+ B(1 - r 2 / a 2 ) - i a w / 2

where

1+3 + n -iaw)( 1 - n;iaw)]-’ A = r( 1 +$)r( -iaw)

(4.1 1 )

In view of (4.9) and of the properties of functions it is easy to show that

A* = B. (4.12)

In the coordinates r* defined by

a a + r 1 - r2 /a2 2 a - r

r* = - In - d r dr* = (4.13)

the radial equation looks like a scattering equation in quantum mechanics with a potential at finite distance from the origin. Near the horizon, this equation describes free wavepackets, and it is easily checked that the asymptotic behaviour (4.10) is now

(4.14)

The physical meaning of (4.10) or equivalently (4.14) is clear. At the horizon we have outgoing as well as ingoing waves and there is as much information leaving as entering the space. The solutions have a logarithmic singularity on the horizon and infinitely many oscillations pile up near the horizon, a situation analogous to that of a black hole [SI. Also these boundary conditions are those required in order to build a complete set of solutions for w > 0.

CC A exp( -iwr*) + B exp(ior*).

5. The Hawking effect in d s space

Let us now consider the Hawking radiation in d s space. The static metric possesses a horizon and covers only part of the d s space. We also have coordinates which cover all of ds space (see appendix) yielding the metric

ds2 = a2[dT2 - cosh’ 7{dx’ + sin’ x do’}]. (5 .1 )

QSlm = E(~)Y,~ , (x , e,+). (5.2)

ysrm = sin’x c~+_:(cos x) X,(e, +) (5.3)

In these coordinates, the scalar eigenmodes will have the form

The function Yslm are spherical harmonics on S3, given by

898 D Polarski

where CL?: is a Gegenbauer polynomial in cos ,y of degree s - 1

1 - s 1 - s 1 2 2 2

Cj?:(cos x) ~ c o s s - ' x F -, -+-; l+i; tan2 x (5.4)

where s=O,1 ,2 , . . . and 1=0,1, . . . , s. Furthermore, F , ( T ) is a solution of

{ 1/ cosh3 T a/3 . r ( cosh3 T 3 / 8 7 ) + s( s + 2)/cosh2 T} Fs ( 7) = A F, = n ( n + 3)F,

with A given by (4.1). The solutions to this equation are given by

Any particular combination of u1 and u2 will define specific states of the scalar field. We note that u,(O) = u;(O) = 1, u2(0) = u l ( 0 ) = 0.

Let us now consider the Bogolyubov coefficients connecting both sets of eigenmodes and hence also the corresponding creation and annihilation operators

asw/l~2mlm2 = (@sllml 3 @p,12m2)

P s w / l / z m , m z = -(@sllml 3 @ 2 / 2 m 2 ) = as -p , / I /2mlm2.

The scalar product (a,, Q2) is given by

(Q1, Q 2 ) = -i I {@ldp@$ -8pQ1@f}& dS'"

where the integral can be taken on any spacelike hypersurface. depend on 8, 4 through Y,,, the Bogolyubov coefficients will with the same 1 and m. We will therefore write aSwlm and Pswlm the subscript m. We then have

@'swim = dw{aswlm@wlm + P s w l m @ . t l - m ) I and equivalently

@ w l m = E { a Y $ l m @ s l m - P s w l m Q ? - m } * S

As is well known, the relation

a L m = eXp(2.rraw)Pt,lm

(5.5)

(5.5')

(5.6)

As both eigenmodes connect eigenmodes and eventually drop

(5.7a)

(5.7b)

(5.8)

implies the presence of a Hawking radiation from the horizon. Alternatively, one can say that the global vacuum will contain w particles with a thermal distribution corre- sponding to the temperature 1/(2rka) .

Instead of looking at eigenmodes, we can look at wavepackets built out of eigen- modes [8]. An ingoing wavepacket near the horizon at t -- 0 will reach the observer at r = 0 at a very late time. Such a wavepacket Ojnlm can be written as

( J + 1 ) &

@jn, = I/& I,. dw e x p ( i 2 ~ n w / ~ ) @ , , ( r = a ) (5.9)

Scalar wave equation on static d s and A d s spaces 899

and

@ j n l = C { a $ n i @ s i - P s j n i @ ' z l (5.10)

where the definition of aSjfli ( P S j n l ) is obvious from (5.9). E is a small positive number while n must be very large and positive in view of (4.14). Calculating the scalar products in the global coordinates and choosing the r = 0 hypersurface, (4.10) yields

S

Rwl( T = 0, x 2. ~ / 2 ) = A(COS x ) ~ ~ ~ + A*(COS x)-~"" ~ A ( T / ~ - X ) ' " " + A " ( T / ~ - X ) - ' ~ " (5.11)

as

r 2 / a 2 = cosh2 r sin' x. (5.12)

The calculations are further analogous to those of [9]. The main contribution to a.Tjnl will come from integration of aswf with

as,,OCiw exp(aw~/2)~(iaw)k-'""[c~Sic~a(k)-'] where Fs = c lul + c2u2, k is exponentially large, k = exp(2m/e) . As must be the case, the contribution to asjni from the outgoing wavepacket vanishes because of the opposite sign in the power of k. Relation (4.8) is finally derived using (5.5'). The crucial point here in order to get this result is the asymptotic behaviour of the w eigenmodes near the horizon. As we have shown, this asymptotic behaviour is essentially independent of the mass and the coupling to the scalar curvature R.

6. Conclusion

A general method is given to solve the equation of motion for a scalar field, not necessarily massless conformally coupled, in static ds as well as static Ads space. The results agree with those already known in the literature. In the case of ds space, the asymptotic behaviour near the horizon is investigated and is shown to imply the presence of the Hawking radiation [lo]. Also, as is well known, these boundary conditions are those required in order to have a complete set of solutions.

Acknowledgment

A Casher is gratefully acknowledged for useful discussions.

Appendix

For the sake of completeness, we give the global coordinates for ds space. They are defined by

xo= a sinh r

x' = a cosh T COS ,y

x2 = a cosh r sin ,y cos 8

x3 = a cosh r sin ,y sin 0 cos 4 x4 = a cosh r sin ,y sin 0 sin 4.

900 D Polarski

References

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Guth A 1981 Phys. Reo. D 23 347 Linde A 1982 Phys. Lett. 116B 335

[3] Tagirov E A 1973 Ann. Phys., N Y 76 561 [4] Vilenkin N Ja 1977 Special Functions and the Theory of Group Representations (Amsterdam: North-

[5] Avis S J, Isham C J and Storey D 1978 Phys. Rev. D 18 3565 [6] Daksh Lohiya and Panchapakesan N 1978 J. Phys. A: Math. Gen. 11 1963 [7] Abramowitz M and Stegun I A (ed) 1965 Handbook of Mathematical Functions (New York: Dover) [8] Hawking S W 1975 Commun. Math. Phys. 43 199 [9] Lapedes A S 1978 J. Math. Phys. 19 2289

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[lo] Gibbons G Wand Hawking S W 1977 Phys. Reo. D 15 2738