Fields of point sources near the cosmological event horizon

Volume 132, number 8,9

PHYSICS LETTERS A

24 October 1988

FIELDS OF POINT SOURCES NEAR THE COSMOLOGICAL EVENT HORIZON D.V. GAL’TSOV and M.Yu. MOROZOV Department of Theoretical Physics, MoscowSeate University, 119 899 Moscow, USSR Received 13 July 1988; accepted for publication 11 August 1988 Communicated by J.P. Vigier

The fields of different spins generated by point sources localized at the origin of the static coordinate system in the de Sitter space—time are constructed in an explicit form. It is shown that the massless scalar field with arbitrary coupling, non-conformal vector field and gravitational perturbations is singular at the cosmological event horizon, in contrast to the conformal fields, which remain regular there. The connection with the cosmic no-hair conjecture and the cosmological blackhole problem is discussed.

The invention of the inflationary model [1] of the early universe has stimulated investigation of the classical and quantum dynamics of scalar and grayitational perturbations of the de Sitter space—time [2—4].Of particular interest in this context is the behavior of the minimally coupled massless scalar field. For such a field there is no de Sitter-invariant vacuum state [41, which leads to growth with time of its vacuum fluctuations. This effect is considered as

ds2= (1 —x2)dt2— (1 —x2) ‘dr2 —

r2 (do2 + sin2O d~’2) (1) where x= r/,~,r~ = (3/A)1”2 is the radius of the cosmological event horizon, A the cosmological constant. Let at the origin of this reference system, r= 0, —

be localized a point source T 0=

one of the possible reasons for switching off the de Sitter expansion regime. However, it must be emphasized, that the minimally coupled massless scalar field, already on a classical level, possesses some unphysical features which must be taken in mind in the discussion of the inflation. As was shown by Chernikov and Tagirov [5], the wave packet composed by this field does not move along the geodesic. In de Sitter space—time the non-conformal scalar field behaves rather as massive, than as massless. The purpose of the present article is to show, that there exists another interesting “pathology” of a non-conformal scalar field in the static de Sitter space—time which seems to have a connection with the so-called no-hair theorems implied by existence in this space—time of an event horizon. Considered a static coordinate parametrization of a part of the de Sitter hyperboloid

,

Ô( r)

~

(2)

4itr ofa massless scalar field 0 with an arbitrary coupling to the gravitational field. The scalar field, generated by this source, obeys the equation ~

(3)

where R = 1 2/r~is the Ricci scalar. We seek for the spherically symmetric static solution of this equation, 0(r). In this case eq. (3) simplifies to the following ordinary differential equation —

((lX2)~

+2(x’2x)~

_l2~)0 (4)

.

=

—

r~x

Its solution can be expressed through the hypergeometric function as follows Ø(r)=

~2F1U+~öo,

~_~

i;r~/r~),

(5) 387

Volume 132, number 8,9

PHYSICS LETTERS A

where ~ = (1 i~)1/2 In the limit of flat space—time A—~0(r~—~oc) (5) becomes equal to Ø=Q/r which is nothing but the usual Coulomb-like field of a point source. Now we investigate the behaviorof 0(r) for finite A in the vicinity of the cosmological event ho—

rizon, i.e. at x—4 1. Using the well-known formulas for the hypergeometric function one can easily show that, provided ~ the solution (5) logarithmically diverges as x—*l: Q 2), (6) 0 C0 in(1 x where we have denoted C 0 =,J~[P( ~+ ~ô0)f’(~ ~) ] (7) ~,

— —

—

—

—‘.

It can be shown that this divergence leads to the physical singularity of the stress—energy tensor ofthe scalar field 2~)O(~0~) + (2~—~)gppØ.~Øa Tpv = (1 — ~ +2~~~0DO.(8) —

24 October 1988

event horizon is even stronger, the leading term being (1—x2~1ln(l—x2).In contrast to the case of the minimally coupled field, the conformal scalar field (~=~)remains finite at the cosmological event horizon, since in this case the constant C 0 vanishes, and the next term of the expansion near x= 1 gives 2

T~=

(12) 6r~ In order to tothe see that this divergence can be attributed non-conformal nature ofindeed the field, we have considered also the case of the massless vector field A~with general (possibly non-conformal) cou-~-~-.

pling to the gravitational field. In this case the field equations can be written in the Lorentz gaugeA’~.a= 0 as ~ (13) where a is an additional coupling parameter, j~is some source. The standard case of the conformally invariant (electromagnetic) field corresponds to

Indeed, the derivative 0,,- is linearly divergent at the horizon

a= 1, but an additional a-coupling is also admissible (and can arise indeed when generalisation to curved space—time is made in the equation for the 4-poten-

0.

2) (9) 0 (1 x and, hence, the non-zero components of the stress— energy tensor behave at x—~1 as

tial ~ rather than in the equation for the field tensor

~

j=

r~C

—

—

Let us take the point-like source term (point charge) localized at the origin r= 0 (14) and seek for the spherically symmetric solution 4~,,2ô(r)~

2 2Q T~——-C 0[l—2~ln(l—x)](l—x)— 2

2

2

~=

A~=A°(r)ô~.Fromeq.(l3)weget

‘

(10)

2T~=T~C2o(l_4~)(1—x2)_L

For the minimally coupled field, mic terms vanish, arid we have

I

0, the logarith(11)

T~= T ~= T~= T r (1 x For the metric (1) these mixed components of the stress-tensor coincide with physical tetrad components and, hence, the linear divergence at the cos—

=

—

47tr~j°.

(15)

.

2) -‘ —

((l_x2)~+2(x_1_3x)~_3(i+a))A0

The solution ofthis equation can also be expressed through the hypergeometric function,

—

A°(r)=~

2/r~), (16) 2F1(~+~ô1, ~ r where the constant d 1 = [1 ~jl + a) 11 In the case of conformal coupling, ô1 =~,and the hypergeomet. nc function ~—~öI;

—

mological event horizon x= 1 is the physical singularity of the stress-tensor. For ~0 (and ~ the divergence of 0 at the 388

/2~

Volume 132, number 8,9

PHYSICS LE1’TERS A

2) = (1 —x2) , (17) 2F1 (~~ ~ ~ö~ ~ x so for the covariant component of the 4-potential we get —‘

—

A

2)=e/r.

(18)

0=A°(l—x Such a behavior, while giving the divergence of the quadratic scalar A~A at x-+ 1, must be considered as non-singular, since it does not lead to any divergence of the physically observed field quantities (the same divergence of 4,4 at the horizon takes place for the black hole case in a gauge where Ao(cc) =0). The situation for the non-conformal coupling a,~1 is quite different. In this case A°has a logarithmic divergence at x-+ 1 which leads to divergence of the physical component of the field tensor, —~r2/r~)

F,~=—~ r 2e —

—~

C~ln(l —x2),

24 October 1988

static gravitational perturbations is to start with the general spherical static ansatz ds2=e~dt2—e2 dr2—r2(c102+sin29dQ,2) ,

for which the corresponding non-zero components of the Einstein tensor are

G8=—e—5(l/r2—2’/r)+l/r2, G~=—e5(l/r2+ v’ /r) + 1 /r2, G~= = ~e—~[v” + ~ —

~ (ii’

—)~‘

)— ~ v’1’]

(23) (a prime denotes partial derivative over r). If we linearize the Einstein equations with the A term near the de Sitter background, e~=e~°(l+vI), &=e50(l+2.I) , (24) where eP0=(l_x2),

(19)

(22)

(25)

eA0=(l_x2)i,

we obtain for v 1 and

r~ where

‘~. I

the following set ofequations

(rAie~°)’ =8i~T8,

C~ V~[1’(

+~ôI)f’(~~i5l)]

(20)

.

e~°

[v’l—(v’O+~)2I]=0, (26) r where the source term corresponds to the point mass —

The only non-zero Newman—Penrose scalar singular at the cosmological event horizon,

~

~I

is also

=~F~,~(1Pn~’+m*Mmi’) ~_4Ciln(l_x2)

r,-

Mlocalized at the origin, (21)

and, hence, physical components of the stress-tensor diverge too. Thus we that the in the case a vector field the situation is see exactly same: theofconformally invariant field is not singular, while the non-conformal field is singular at the cosmological horizon. Somewhat more tricky isa the case of gravitational perturbations generatedby point source in the static de Sitter space—time. Since gravitational perturbations obey non-conformal equations, one can expect for them analogous divergence at the horizon. This is indeed the case, but now we can proceed further and explain how this divergence is removed in the corresponding exact solution of the A-modified Emstein equations. The simplest way to treat spherically symmetric

(27)

T0o_~~d(r). 4itr

The solution of eqs. (26) is elementary, 2M 2)]’ , (28) r~ = V1 = [x(1x or, in terms ofmetric perturbations, ~ =g~)+ h~ 9, 2M 2)]~ (29) r,h8 = _hc_ [x(l —x This quantity is clearly singular at the cosmological event horizon. However, now we can easily see how this divergence can be removed. Consider the wellknown Schwarzschild—de Sitter solution ‘~l

—

—

.

—

ds2=

d12—

—

~-

dr2—r2(d02+sin2O d~i2),

(30)

where 389

PHYSICS LETTERS A

Volume 132. number 8,9

24 October 1988

(31) A = r2 r4/r,-2 2Mr. If we expand A/r2 in a formal series of powers of M, we obtain up to first order

construct a spherical static massive field, falling down at spatial infinity, leads to singularity of this field at the event horizon. The problems considered above of non-conformal fields generated by point sources

zl/r2= (1 —x2) [1—(2M/r,-x) (1 —x2)

inside the event horizon in the static de Sitter space— time are nothing but cosmological counterparts of this black hole problem. The term “no-hair theorems” was applied recently to de Sitter cosmology in a different sense than here, rather as the statement that de Sitter space—time is an attractor for cosmological solutions ofthe A-modified Einstein equations generated by a large variety of initial data [7,8]. While it was claimed [7], that there was an analogy between black hole no-hair theorems and the cosmic no-hair conjecture in this sense, it is clear from our consideration that much closer to the corresponding black hole situation is just our problem of the field generated by some source inside the cosmological event horizon. Nevertheless in both contexts the cosmic no-hair property has a unique origin: the existence of an event horizon in the de Sitter space—time.

—

—

—]

.

(32)

Substituting this expression into eq. (30) we obtain g,,,- =g~)+ hg,- where g~,°) is the static de Sitter metnc and the non-zero components of ha,- are just the same as given by our previously found formula (29). It is not surprising, since in the context of the full non-linear Einstein equations the point mass presumably generates the Schwarzschild solution rather than a newtonian potential. In the presence ofthe A term the corresponding exact solution will be given by the Schwarzschild—de Sitter metric (30), the formal expansion of which in terms of M will lead to corrections exhibiting singular behavior at the cosmological event horizon r= r,-. It is clear that, in the context of the exact solution (30), the point r= r~is not singular, but, instead, the position of an event horizon itself is moved to the point r÷~~r~—M

(33)

(this formula is valid up to he lowest order in M/r~, provided AM2 << 1). The pathological behavior of non-conformal massiess fields at the cosmological event horizon can be attributed to the fact that in the de Sitter space— time these fields become effectively massive. The problems considered above are analogous to the problem of the existence ofthe regular hair of a black hole generated by some source inside the event horizon. As was shown by Beckenstein, a black hole can not have regular massive hair [6]. An attempt to

390

References [1] A.H. Guth, Phys. Rev. D 23 (1981) 347. 12] B. Allen, Phys. Rev. D 32 (1985) 3136. [3] B. Allen, Phys. Rev. D 34 (1986) 3670. [4] B. Allen and A. Folacci, Phys. Rev. D 35 (1987) 3771. [5] N.A. Chernikov and E.A. Tagirov, Ann. Inst. H. Poincaré A 9 (1968) 109. [6] D. Bekenstein, Phys. Rev. D 5 (i972) 1239, 2403. 17] W. Boucher and G. Gibbons, in: The very early universe, eds. G. Gibbons, S.W. Hawking and S.T.C. Sikios (Cambridge Univ. Press, Cambridge, 1983). [8lJ.D. Barrow, Phys. Lett. B 187 (1987) 12.