General Relativity and Gravitation, Vol. ~8, No. 8, 1996

Thermal Conditions for Scalar Bosons in a Curved

Spacetime

Carlos E. Laciana I

Received January 1~, 1996

The conditions that allow us to consider the vacuum expectation value of the energy-momentum tensor as a statistical average, at some particular temperature, are given. When the mean value of created particles is stationary, a planckian distribution for the field modes is obtained. In the massless approximation, the temperature dependence is like that corresponding to a radiation-dominated Friedmann-like model.

1. I N T R O D U C T I O N

In the semiclassical approximat ion the vacuum expectat ion value (VEV) of the energy-momentum tensor (EMT) can be used as a source of the Einstein equations, in the form

a.~ = -SrV(T..)reg. (1)

This approximat ion is useful as an asymptot ic value of more complete theories where the gravitat ional field is also quantized, when the t ime is larger t han thc Planck time. The mean value of T ~ can be obtained using the expression

(out, O] T~, [0, in) (Tt~v) -- (out, 0 ] O, in) (2)

1 Instituto de Astronomia y Ffsica del Espacio, Casilla de Correo 67, Sucursal 28, 1428 Buenos Aires, Argentina. E-mail: laciana@iafe.uba.ar

1013

0001-7701/96/0800-1013509.50/0 ~) 1996 Plenum Publishing Corporation

1014 Lac iana

which was introduced in [1]. We can relate eq. (2) to the v~.v with respect to some particular "in" or "out" vacua (see Ref. 2), i.e.

(T~v) = (in, 012bu~ [0, in) + i ~-~AijT~v(r r i , j

= (out,01T~, 10,out) + i ~--~'V~T,~(r r (3)

where {r t.J {r and (tout} U {r are the basis of solutions defined for the Cauchy surface labeled by "in" and "out" respectively. A~j and V~j are functions of the Bogoliubov coefficients that determine the transfor- mation between the two vacua. It is also useful to take into account that the expectation values can be expanded in terms of the vacuum polar- ization, which contains the local infinities, that are removed by the usual regularization methods (see Ref. 3) independently of the vacuum defini- tion used, and a term which is function of the created particles, due to the interaction of the field with the curved geometry. This last term is of course null in fiat spacetime, but in curved space it is in general infinite when it is calculated perturbatively. For instance in the particular case of fields minimally coupled to gravity, the infinity coming from the particle creation term cannot be removed with the usual regularization methods for an arbitrary vacuum definition (see Ref. 4). However it is easy to see [5] that when particle creation has a planckian distribution, which is lost when perturbative expansions are performed, the contribution to the EMT is finite and moreover coincides with the standard cosmology which uses perfect fluid classical sources in the Einstein equations. We can ask whether it is reasonable to get a planckian distribution for the mean value of created particles. The answer is affirmative because it is obvious that the actual background radiation of the Universe has a black body distribu- tion with temperature approximately equal to 3~ (see Ref. 6). In another case, for example in the black holes studied by Hawking [7], a planckian distribution is predicted. Also the distribution seen by an accelerated ob- server (Rindler observer) is planckian-like, and therefore by generalization of the equivalence principle we hope that for some Bogoliubov transforma- tions we obtain a thermal spectrum. This result is shown in the present work for a scalar field minimally coupled to a Robertson-Walker metric. The expression obtained for the vEv of the EMT is compared with the one coming from the statistical average of T ~ at a given temperature, which is obtained using Thermo Field Dynamics (TFD) theory [9]. In a previous work [10] it was proved that a field in a curved geometry presents a form similar to that of the field in a thermal bath. In TFD the interaction with

Thermal Conditions for Scalar Bosons in Curved Spacetime 1015

the bath is half-filled by the so-called "tilde" modes, which is more natural than the interaction between the classical background and the quantum field. In the present work from, the comparison mentioned before, suffi- cient conditions on the Bogoliubov transformation are obtained, in order to get the same functional form for vEv of the V.MT and the statistical average obtained from TFD.

2. TIME D E P E N D E N T BOGOLIUBOV TRANSFORMATION

In a pioneering work [11] Parker developed a consistent method to obtain the annhilation-creation operators at each time in curved space. The operators defined at different Cauchy surfaces are related by time- dependent Bogoliubov transformations. Parker found in [11] the differ- ential equation satisfied by the Bogoliubov coefficients. This equation is another way of writing the field equation. In order to resolve that equation it is necessary to give some initial conditions. The Bogoliubov coefficients give us the mean value of created particles between an initial time to and a present time t. In order to make the paper more comprehensive, in this section we repeat in brief the formulation given by Parker in [11].

The action for a minimally coupled scalar matter field is

s = ~ v ~ d4x(O.~O"~ - m2~ ~) (4)

and the metric is the spatially flat Robertson-Walker, given by

ds 2 = dt 2 - a ( t )2 (dx 2 + dy 2 + dz2). (5)

Therefore the variation of the action gives the field equation

( v , 0 " + m2)~ = 0 (6)

(with # = 0, 1, 2, 3). The metric given by eq. (5) represents an unbounded spacetime.

Therefore in a Fourier representation of ~ integrals appear, and in or- der to make the calculation easier, we can use a discretization in the same form as in [11], i.e. we can introduce the periodic boundary condition ~o(x + n L , t ) = ~(x, t), where n is a vector with integer Cartesian compo- nents and L a longitude which goes to infinity at the end of the calculation. (Really in our cas~ it is not necessary because the results used do not de- pend on L.) Then we can introduce the set of functions {r U {r defined by

, ( ) Ck(x) = ( L a ( t ) ) 3 / 2 V ~ - ~ expi k x - W ( k , t ' ) d t ' , (7)

1 0 1 6 L a c i a n a

with W an arbitrary real function of k = [k[ and t. The field can be expanded in the form

~0(X, t) = E [ak(t)(~k(X)+ a*k(t)C*k(X ) ], k

(8)

where the operators ak(t) and ak, (t) are time dependent and satisfy the commutation relations

[ak(t),ak,(t)] = O, [atk(t) ,atk,( t)] ---- O, [ak(t) ,aik,(t)] = ~k,k'. (9)

The operators act on the vacuum I0, t) in the form

ak(t) 10, t) = 0, d k ( t ) I0, t) = I lk , t ) .

Moreover we define the operators

Ak : = ak(t = t l) with t > t l _> to,

Ak t : = akt ( t = tl). (10)

Also we define the vacuum 10) := 10, t = tl) on which the operators defined by eq. (10) act. These operators are related to the time dependent ones by the following Bogoliubov transformation:

ak(t) = cz(k, t)*Ak +/5(k, t ) A t _ k ,

a-k*(t) = a(k, t )A*_k + ~(k, t)*Ak �9

Clearly we use the conditions

cz(k, tl) = 1, f~(k, t l) = O.

(11a)

(llb)

(12)

We can rewrite the field ~o(x, t) in terms of the new operators in the form

tp(x,/:) = E [AktX/k(X) + A*ktX/*k(X)]' (13) k

with g!k(X) and ~ ( x ) solutions of eq. (6). We can make the separation

tI/k(X ) = h(k, t) exp ik.x (14)

with

i [a(k, t)*e -i Ito w~t' h(k,t)= (()) ( (La-(3/22Wk, t))l/2

+ fl(k, t)*eiI~o Wdr

Thermal Conditions for Scalar Bosons in Curved Spacetime 1017

Replacing eq. (14) in eq. (6) we have

(oLe~ I Wde + ~e-~ S Wde)M -- W , . ~ I Wdr + ~e-~ S Wde) W ~c~e

+ 2iW(&e ~ I Wdr _ ~e- i I wdt') + &e i I Wde + ~e- i I Wdt' __ 0 (15)

with

1(~____) ~ 1(~____)2__ 9 (__~) 2 3 ( ~ ) * _ W 2 . M := - 3 + ~ ~ - 5 + J (18)

In the following we will call w 2 = k2/a 2 + m 2. From eqs. (12) and (13), Parker [11] proves that the Bogoliubov coef-

ficients satisfy the equation

~ = - & exp (2i [ t W(k , t ')dt ') . (17) \ Jto

From the functional form (14), for which the invariance of the Klein- Gordon product holds, we obtain

lal 2 -I/~[ 2 = 1. (18)

Using eq. (11) we note that nk := I~kl 2 = {0l aikEl,~)akEt)I 0) is the mean value of particles created between tl and t. Equation (18) allows us to write (as in Ref. 11)

a(k, t) = e -i7~ (k,t) cosh O(k, t), (19a)

/~(k, t) = e iT~(k't) sinh 0(k, t). (19b)

7~, 7a and 0 are functions of k and t that must satisfy the initial conditions given by eq. (12) and the field equation (15). As we can see from eq. (19a) the number of particles is only a function of 8, i.e. nk = sinh2O.

Replacing eqs. (19) and (17) in eq. (15) and splitting in real and imaginary parts, the following system of equations is obtained:

(1 + tanh 0 c o s r ) M + 2W-~ = 0, (20a)

M sin F + 2W/} = 0, (20b)

with r := "ra + 7~ - 2 f~t ~ Wdt'. The solutionw to eqs. (20) give us the set of time-dependent Bogoli-

ubov transformations compatible with the Robertson-Walker metric. The stationary particle creation is a particular case with ~} = 0, and therefore from eq. (20b) we have

M = 0. (21)

1018 L a c i a n a

3. VEV OF EMT AS A THERMODYNAMICAL OBJECT

In [4] the analogy with the Bogoliubov transformation which relates states at different temperature with those at different Cauchy surfaces was shown. Here we will exploit this analogy in relation to the vEv of EMT. First of all we calculate the classical EMT in curved spacetime in the usual form (see Ref. 2), i.e. as

2 5S T . . - j_~ ~g..

We obtain the expression

T. . = �89 ~o.} - �88 ~ . } + �88162 ~} (22)

where {, } is the anticommutator and r := 0v~o. We can now calculate the EMT operator, replacing eq. (13) in eq. (22)

and obtaining

T.v = E { AkAk' D.v[ TWk(x), ff~k' (x)] + Ak Atk, D.v[~k(X), ff~*k' (X)] k k '

+ AtkAk, D.. [q?*k(X), qk' (x) ] + AtkAtk, Dvv [**k(X), ~*k' (X) ]

+ {k ~ k'} }, (23)

with {k ~-~ k'} indicating that the last term of eq. (23) is equal to the first replacing k by k ~. The differential operator D . . is defined by

D . . [9, r := 1 0 . 9 0 . r - lg~,.O~'~o 0,.r + lm2g..~or (24)

We can now calculate the vEv of Tu- using the vacuum ]0), and obtain

(01%. I0) = 2 ~ a e {D..[~k(=), **k(=)] }. (25)

Replacing now eqs. (11) and (13) we also have

�9 k(X) = ~*r +/~*r (26)

Thermal Conditions for Scalar Bosons in Curved Spacetime 1019

Putt ing this last equation in eq. (25) yields

(01T~ 10> = 2 E R e { (1 + 21fllg)D~[@k, ~k*] + 2a*f~D.~,[$k, ~-k] k

1 1 + ~ ( l~12)'E~v[~k, Ck*] + ~ (~*,0)'E~v[~k, ~-k]

1 l (6o~6ov -- -~ g~v) [ ( ldxl2 + l[Jl2) l(bk[2

with

(27)

E~v[~b, ~o] := ~b(6ova~o + 6o~Ov~o - g~v(o). (28)

For simplicity we can consider the case in which the particle creation is stationary. Then ]~ = & = 0, and therefore we have

(017 . I0> = {(1 + 21 12)D. [ k, + -kl }. (29) k

As we will see the difference between the VEV given by eq. (29) and a statistfcal average is the "interference" term, for the modes k and - k , of eq. (29).

A simple way to calculate the statistical mean value of some physical observable is by means of the thermo field dynamics (TFD) formulation (see Ref. 12). In this formulation the idea is to take into account the interaction between the system and the thermal bath by means of quantum fluctuations of the bath (see also Ref. 9), which are called "tilde" fields. These fields are auxiliary because they are not measurable. Those fields only act in the thermalization of the system modes. The tilde modes are associated with tilde creation-annhilation operators which act on the vacuum of the thermal bath I0). The space of states, in this formulation is extended in order to include the tilde and the states of the system, i.e.

We will call the total vacuum IiO> := 10>| Moreover we introduce a T parameter in order to identify the temperature of the state, i.e. [0, T> and a Bogoliubov transformation that relates the vacuum at zero temperature with the one at temperature T:

ak(T) = Akcosh O(k, T) - At k sinh 0(k, T). (30)

1020 L a c i a n a

In order to obtain the zero temperature operators, 8(k, T = 0) = 0 is necessary.

Then we have the creation-annhilation bosonic fields Ak and Ark which operate in the form

AkllO) ---- O, AtkllO) ---- [[lk), etc.

with the commutation relation

[Ak, A r k '] ---- ~k,k' �9

The tilde operators, represent the quantum effect of the reservoir and satisfy

fi-kllO) = O, Atk[lO ) = IIik), etc..

[~/~k, A tk ' ] = ~k,k' �9

The thermal operators ak(T), ark(T), ak(T), a*k(T) satisfy

ak(T)HO, T ) = O, atk(T)HO, T) = Hlk,T), etc..

[ak(T), ark, (T)] ---- r ,

and analogously the tilde operators. The tilde and no-tilde operators are mutually commutative.

With the new Fock space we can calculate the statistical mean value of any physical observable. For example if we have the physical operator /], the mean value can be obtained by means of

(A) = (0, TIIAII0, T).

We will calculate now the statistical mean value of the energy-mo- mentum operator Tu~. In order to do that we can represent the operator as a functional of the thermal modes r and r in a way analogous to that in curved spacetime:

CUr -~ ~/-:~(ak(T)ak, (T)D~,[r r ] kk'

+ ak(T)atk , (T)D,~,[r r ]

+ atk(T)ak ' (T)D~,~,[r r ]

§ atk(T)atk,(T)D~v[r r

+ { k ~ - * k ' } } . (31)

T h e r m a l C o n d i t i o n s for Sca lar B o s o n s in C u r v e d S p a c e t i m e 1021

It is easy to prove (see Ref. 13) using the transformation given by eq. (30) and the inverse that

(0[[ akt(T)ak(T) [[0) = (0, T[[ AktAk []0, T)

and in analogous form with ak(T)ak(T) and ak(T)akt(T). Therefore we can state

<Tuu) = <011Tu~[ak(T), akt(T), ak, (T), ak, t(T) ] II0 >

and then the following expression is obtained:

(0]] Tu~ ]]0) = 2~--~Re {(1 + 2nk)Du~,[r CkT*] }, (32) k

where n k = (011 akt(T)ak(T) II0).

As we can see from eq. (32) the Ck modes will be equivalent to the thermal ones if the interference term in eq. (29) is null.

The difference from the "genuine thermality" has the same cause as that in the case of the Rindler vs. Minkowski observers, analyzed in [14]. It is related to the fact that (0lak(g)ak,(t)[0) ~ 0. In [14] the thermality is restored by assuming that the observation is restricted to a finite space- time region, which is introduced mathematically by means of the wave packets (see also Ref. 8). In our case, instead of using wave packets we will exploit the fact that the interference term oscillates. First we calculate each component of that term:

1 Re {2c~*f~D00[r C-k] } = ~ cosh Osinh O 1r215

( 1 W 3H)2 w 2

Re (2c~*f~D0~[r C-k] } = - R e {~ei'Y~kr , (34)

1 Re {2c~*f~Djj [r C-k] } = ~ cosh 0 sinh 01r - a2w2)cos F. (35)

There are no contributions from the term given by eq. (34), because when the summation ~-~k=_oo +~176 is performed, a self-cancellation is produced.

1022 L a c i a n a

The terms given by eqs. (33) and (35), when the limit to the continuum is taken (~-~k --* (L /2 r ) 3 f d3k), are proportional to the integrals:

Ioo = ~o~176 [l (W + 3H)2+ J - W2] cosF

- (---~ + 3 H ) Wsin F}cosh 8 sinh 0,

/o I j j = k2dk(2kj 2 - a2w2)cos 1-'cosh 0 sinh 0

(36)

(37)

[where we used d3k = 47rk2dk, and F as in eqs. (20)]. The particle creation is not modified when 7 changes, as we can see from eq. (19). Then we can propose a ~/phase with the functional form

= , k . (38)

If we take # --+ co and introduce an ultraviolet cutoff (which is less restric- tive than the wave packet and can be justified because k/a is the physical momentum associated to the k mode, as is shown in Ref. 11) in the in- tegrals (36) and (37) we can then apply the Riemann-Lebesgue theorem, which is as follows: "If f is a real function absolutly integrable in the in-

terval [a, b], then lim~-~oof:f(t) sin(Tt + ~)dt = 0 with ~i a real constant". Clearly this is also true when we have "cos" instead of "sin". By means of this theorem we can eliminate the interference term in eq. (29), and it turns out to be equivalent to the statistical mean value given by eq. (32).

4. THERMAL SPECTRUM CONDITION

In order ~o obtain thermal distribution for the k modes, we need to apply an extremun condition to a thermodynamical potential. This potential will be a function of the energy and the entropy of the bosonic gas in curved space-time. The energy can be calculated by means of the metric hamiltonian /:/ in the form

E = <Ol IO>,

where

[t=/a3d3x oo

Thermal Condit ions for Scalar Bosons in Curved Spacet ime 1023

in the discretized formulation is f d3x = L 3. For eq. (29) when the oscil- latory term is dropped, we have

with

k (39)

1 3 H ) 2 + w 2] (40) ~k ~--- ~"~ [ ~ ( ' ~ ~- ~ - W 2

the energy by mode. As the energy for the system studied has the form of a set of decoupled quantum oscillators, it can be considered as a bosonic gas in flat spacetime. Therefore we can use the expression for the entropy used in Minkowski space. In order to do that we can write the operator (see Ref. 12) as

t'[ = - ~ { a ~ ( t ) a k ( t ) 1 0 g sinh2~ - ak( t )a~( t ) logcosh2~}. (41) k

The expectation value of the operator given by eq. (41) is the entropy of a bosonic gas (Ref. 15, eq. 54.6, p.147):

K=(OiR[O)=--~-~{nklognk--(l+nk)log(l+nk)}. (42) k

In a way analogous to [12] we will obtain an extremum, with respect to the Bogoliubov angle 8, of the thermodynamical potential

A = - T K + E,

where T is a Lagrangian multiplier that can be interpreted as the temper- ature. Then performing the variation

~A/Se = 0

the following equation is obtained:

sinh 2 T log cosh 2-----~ + e = 0.

Therefore we get a planckian spectrum

1 nk = e~k/T _ 1 (43)

1024 L a c i a n a

(with units such that h = c = kB = 1). It is interesting to note that we can eliminate the interference term in

the hamiltonian, without using the phase condition given by eq. (38), by means of the criterion known as hamiltonian diagonalization. As we can see from eq. (33) that condition is

- 3 H W

W 2 = tO 2 .

(44a)

(44b)

This condition is, moreover, compatible with the dynamic equation for W, given by eq. (21). When eq. (44) holds at the observation time, D00[r C-k] ---- 0 is assured. Of course the condition (44) has a mean- ing when the calculation is "in-out ' , with the parameter t fixed at the observation time. Then we can think of condition (44) as Cauchy data on W and l/d. So when condition (44) is satisfied the energy by mode is [as we can see from eq. (40)].

= (45)

In particular if we suppose that the rest energy can be dropped against the kinetic one, i.e. k2/a 2 >:> m 2, we can take ~k ~ k/a, and obtain

1 (46)

n k ~ e k / a T - - 1 "

Moving from eq. (39) to the continuum set of modes, the particle creation contribution to the energy density is given by

1 fo~176 p = 2~r2a4 k3nkdk. (47)

Replacing eq. (46) in eq. (47) and performing the integral, we have

1 4 foo k,3 7r2 p = ~ T Jo ek-; ~- l dk' = "-~ T 4, (48)

which is the expression obtained from the standard phenomenological radiation-dominated Friedmann cosmology (Ref. 16, p.91).

T h e r m a l Cond i t i ons for Scalar Bosons in C u r v e d S p a c e t i m e 1025

5. CONCLUSIONS

The v s v of the EMT is different to the statistical mean value, even for the stat ionary particle creation. In the last case the difference is an interference te rm between the modes k and - k , which can be eliminated by a condition on the phase of the Bogoliubov transformation. At the hamiltonian level the coincidence with the statistical value is satisfied by means of the criterion known as diagonalization of the hamiltonian condi- tion (which for the scalar field is identical to the condition of minimization of the energy). By means of this condition also the mean value (Too), defined by Ut iyama-De Wit t [1], and the v~.v of the GUT coincide [see eq. (3)].

The coincidence between the v sv and the statistical mean value is another view of the analogy between the Bogoliubov transformation that connects the different vacua in curved spacetime and the one that relates vacua to different tempera ture in TFD.

From the extremal conditions on the thermodynamical potential, we obtained a thermal spectrum. The fact that particle creation turns out to be thermal avoids the existence of infinities coming from this effect, in the source of the Einstein equations. This is true also when the hamiltonian diagonalization conditions are used. In general those conditions produce inconsistences with the adiabatic regularization [4,17]. All these factors suggested utilization of the usual covariant regularization techniques, in order to eliminate the infinities coming from the vacuum polarization term, together with some thermodynamic conditions that "regularized" the cre- ation particle term. Tha t approach has allowed us to take into account the back-reaction effect in an easy way, without using numerical calculation (see Ref. 5).

ACKNOWLED GEMENTS

This work was supported by the European Community DGXII and by the Depar tamento de Ffsica de la Facultad de Ciencias Exactas y Naturales de Buenos Aires.

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