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PHYSICAL REVIEW 0 VOLUME 40, NUMBER 6 15 SEPTEMBER 1989
observer-dependent quantum vacua in curved space. II
Mario A. Castagnino
and Instituto de Fisica de Rosario, Consej o Nacional de Investigaciones Cientigcas y Tecnicas U—niversidad Nacional de Rosario,AU. Pellegrini 250, 2000 Rosario, Argentina
Jorge B.SztrajmanInstituto de Astronomra y Fssica del Espacio, Casilla de Correo 67, Suc. 28, Ciudad Un&versitaria, 1428 Buenos Aires, Argentina
and Area de Fssico-Matematica, Ciclo Basico Comun, Universidad de Buenos Aires, Ciudad Universitaria,Pabellon II, 1428 Buenos Aires, Argentina
(Received 12 May 1987; revised manuscript received 30 May 1989)
An observer-dependent Hamiltonian is introduced in order to describe massless spin-1 particles in
curved space-times. The vacuum state is defined by means of Hamiltonian diagonalization andminimization, which turns out to be equivalent criteria. This method works in an arbitrarygeometry, although a condition on the fluid of observers is required. Computations give the vacuacommonly accepted in the literature.
I. INTRODUCTION
It is well known that a "vacuum" is a notion thatstrongly depends on observers; the best-studied case isthe one of Rindler observers: they perceive particles in aquantum state considered as the vacuum for Minkowskiobservers. '* A way to obtain this dependence wasshown, in the first part of this work, for the case of thescalar field by introducing a new Hamiltonian, invariantunder coordinate transformations, but dependent on theobservers, and then applying the Hamiltonian minimiza-tion and diagonalization. Hamiltonian diagonalizationhas been used in the literature in order to define the vacu-um state when a field in a curved space is studied, butwithout introducing the observer in the definition of theHamiltonian. '
Furthermore, in a previous paper we introduced aconformal formalism to study the electromagnetic field incurved space and we defined a "conformal derivative"that depends on an auxiliary timelike vector field, whichwe believe could be a field related to the observers* sys-tem.
In this paper we join the concepts of Refs. 3 and 6,generalizing the results of Ref. 6 to the massless vectorfie1d case and preserving the conforma1 invariance of thetheory. It is well known that this problem of radiationdetected in accelerated frames is well studied in the spin-0 case. On the contrary, the really interesting physicalproblem, the spin-1 massless case, is not so well studied(probably because of the technical difficulties that wesolve in this paper). Thus, we believe it is interesting toobtain a complete solution of the second one.
In Sec. II we review the conformal transformations andthe conforma1 derivative. In Sec. III we review the con-cept of reference system and introduce the concept ofnatural time and the one of the adapted chart. In Sec.IV we state the temporal gauge and the field equations tostudy the electromagnetic field. In Sec. V a conformally
invariant inner product is introduced in order to ortho-normalize the solutions of the field equations. In Sec. VIwe define the new conformal Hamiltonian explicitly de-pending on the Quid of observers. Section VII is devotedto obtaining the vacuum associated with the Quid via theenergy minimization and diagonalization, and we showthat the vacuum obtained is the same using bothmethods. In Sec. VIII we solve the problem of the elim-ination of the unphysical photons. In Sec. IX the pro-posed formalism is applied to particular cases and it isshown that the vacua obtained are the same as thosecommonly accepted in the literature.
II. CO%FORMAL DERIVATIVE
Let V4 be an arbitrary Riemannian space-time and letg„be its metric tensor, with principal diagonal(1,—1, —1, —1) in its diagonal form. We suppose V4 tobe of C class and globally hyperbolic. Greek indicesrun from 0 to 3 whereas latin indices run from 1 to 3.
We define a conformal transformation in V4 as
gpv~gpv =Mx)gpv (2.1)
A, (x) being a positive-definite arbitrary function.The contravariant components of the metric tensor
transform as
g" ~g" =A, '(x)g" (2.2)
dx"=dx", dxp =A, dxp, g =k g
d'g —A. d'g, dc7p —k dip
From (2.1) and (2.2) we can obtain the transformationlaws for all geometrical objects of the manifO1:
r P.p=rv. p+ ,'(5P.Vp 1ni+5PPV.-ink g.pVP in'), —
R =A, '(R —3A, 'V"V„ink, ——,'A. 'V" in' V„ink, ),
(2.3)
40 1876 1989 The American Physical Society
40 OBSERVER-DEPENDENT QUANTUM VACUA IN CURVED SPACE. II 1877
where I " is the Riemannian connection, R the curva-ture scalar, g =det(g„), dpi the volume element, anddo.„the surface element of V4.
We shall assume that the massless fields on V4 trans-form under conformal transformations as
(2.4)
A tensor (spinor) g will be called a Weyl tensor of weight
p if its covariant components transform like (2.4).In order to assure the conformal covariance of the
theory we need the action S to be a Weyl scalar of weightzero:
S=S . (2.5)
This implies that the Lagrangian density X transformslike
i(%+V„F"),
U„ that appears in (2.13) is the unitary vector tangent tothe Quid of observers at each point, covering space-time.In this way, Q (and then D ) becomes observer dependent.
We propose for the action the expression
S= JX(Q,D„Q)dr) (2.14)
obtained by replacing V„by D„ in X. Even if this actionwill be, in general, observer dependent, it could turn outto be observer independent for some special Lagrangian.This will be the case for the Lagrangians that we shalluse. In order to satisfy condition (2.5), it is necessary tochoose the Weyl weight of the field f appropriately. If
is a tensor field of range s and weight p:)"s
(2.15)
then
where F" is an arbitrary vector function that vanishesquickly enough at infinity. Since the transformation lawof V„g is difFerent from the one of a Weyl field, we intro-duce a conformal derivative D„[in fact, the gauge deriva-tive of the conformal group (Ref. 7)] with the property
(D„P„.. .„) =A, D„P„.. .„
then the kinetic term of the Lagrangian
(2.16)
(D„g) =At'D„Q, , (2.7) (2.17)
where p is the weight of f.In order to define D„we introduce the conformal con
nection Cl'„as
transforms as
(2.18)
Ct'„= I t'„,+ ,'(ot'„Q, + ot'—,Q„—Q~g„, ), (2.8) and, comparing with (2.6),
where Q„ is an auxiliary vector field whose conformal be-havior is
p =(s —1)/2 . (2.19)
Q„=Q„—V„ink, .
Then, from (2.3), (2.8), and (2.9) it follows that
C ~„,=Cl'„
(2.9)
(2.10)
For example, for the scalar field (s =0) p = ——,', and for
the vector field (s = 1)p =0, etc.
III. REFERENCE SYSTEM
Now, if V„ is a Weyl vector of weight p, its conformalderivative is defined to be
D, V„=a„V„—C~„,V, +pQ, V„. (2.1 1)
D g 0 (2.12)
Clearly, Q„ is not a Weyl vector. However, Q„can beexpressed by means of a unitary Weyl vector of weight —,',U„, as
Q„=ZU V U„——', U„V U" . (2.13)
In principle U„ is an arbitrary field. However, in Ref.6 we have introduced the assumption that this vectorfield is related, somehow, to the physical observer. As weshall see, this assumption is proved below: the vector field
The generalization to Weyl tensors of arbitrary range isstraightforward.
The operator D„has the ordinary properties of V'„
[e.g., D„(Pg)=PD„g+(D„P)gj, and, in addition, the
very useful one
As in Ref. 3 we shall define a reference observer systemin general relativity as a Quid of observers moving freely,such that their space-time paths would cover all space-time. Each observer has a clock, which measures x, acontinuous increasing arbitrary function of proper time.If each observer is labeled with three numbers x',x,x,where x'=const on each fluid line, then (x,x',x,x ) isa particular geometrical chart.
If the Quid is curl free, it has global spacelike hyper-surfaces orthogonal to the lines of the Quid. In this paperwe shall call natural time t, a quantity that is constant oneach of these hypersurfaces.
In this way, the time t appears naturally associated tothe fiuid. A chart defined by (t,x',x,x ) will be calledthe adapted chart. The natural time t may be used tocharacterize the lines of the fluid as x"=x"(t). The vec-tor v"=dx "/dt is tangent to the lines of Quid. ' BesidesU"=v~/~v"
~is the vector that appears in (2.13), because
only with this choice all the formalism turns out to be sa-tisfactory. This fact proves our assumption of Ref. 6about U". When we use an adapted chart we have
1878 MARIO A. CASTAGNINO AND JORGE B. SZTRAJMAN 40
googp~= 0 Vij
(3.1)D„(V&W„*—W~*V.)=(le&—
g )a„
X[+—g ( V"W" —W&*v )]v"=(1,0,0,0) U"=(goo ', 0,0,0) . (3.2) (5.3)
IV. TEMPORAL GAUGE
We shall study the electromagnetic field in the so-called temporal gauge. ' The Lagrangian density is
and then, the last term in (5.2) disappears, applyingGauss's theorem as usual. Hence, without any loss ofgenerality, we can write (after a redefinition of the con-stants)
with
+IJ~4 pv (4.1) (v, w„),= —f [x,(v D.w„' —W„'D.V )
+A2( V,D"W„" —W,'D„V")]do
F„.=D„A.—D.A„=a„A.—a.A„. (4.2) (5.4)
Thus, even if we use the conformal derivative, the actionturns out to be observer independent.
The temporal gauge condition is
(4.3)
A very important condition to be satisfied by (, )z isthe X invariance, i.e., (, )z =(, )z=(, ). This re-quirement is equivalent to asking that the conformaldivergence of the expression under the integral in (5.4)should vanish:
and, in the adapted chart, it is simply
Ao=p . (4.4)
D [A, ,(V"D W„* —WqD, V~)
+A,2( V,D"W„' —W*D"V„)]=0 . (5.5)
In this way, the degree of freedom Ao disappears fromX and the observer will see no temporal photons. Thefield equations are obtained in a conformally covariantway by means of D"D„A;—D"D;A„=O (5.6)
Developing (5.5) and using the field equation in theadapted chart
and we obtain
BXaA,
D F»=q FI &=0P P (4.6)
and the commutation relation
D D"A —D "D A =S" A
where S" is a symmetric tensor, (5.5) yields
(A, +A, )(V D'D'W,* —W*D'D "V )=0
(5.7)
(5.8)
V. INNER PRODUCT
We need an inner product in the space of solutions ofthe field equations (4.6), in order to orthonormalize suchsolutions. The most general expression, bilinear in thefields and their conformal derivatives is
(V„,W") = i f [A, ,(v—"D W„* —W"'D V„)
+A,2( V"D„W„'—W"*D„V )
+A, 3( V D„W"'—W'D„V")]do
(5.1)
where X is a Cauchy surface, V„and 8'„are complexsolutions of the field equations, and A, „A,2, A, 3 are con-stants to be determined. Note that the conformal invari-ance of (, )z is assured by construction.
The expression (5.1) can be reduced by taking into ac-count the identity
From (5.8), it is clear that, as V„and W„remain arbi-trary, we must have
A, )+A2=0 . (5.9)
In particular, in the adapted chart, where V = Vo= W = Wo =0, (5.10) simply becomes
( Y",W„)= i f (V'W—,"0 —W; V' )go~ '~~dX, (5.11)
where, O means the derivative with respect to the naturaltime t; i.e., in the adapted chart we reobtain the usualinner product.
VI. HAMILTQNIAN
Therefore, we simply take A, &= —12=1 and define theinner product as
( v", w„)= i f[(—v".D W„* —W„'D, V")
—(V D"W„' —W„*D"V„)]do
(5.10)
V„D"8',* —8'„D"V = V D„S'"*—8' DpV"
+g»( V„W", —W„*V,) (5.2)The energy-momentum tensor is obtained, as usual,
from the action as
but T„„=2(—g) ' 5S/5g" (6.1)
OBSERVER-DEPENDENT QUANTUM VACUA IN CURVED SPACE. II
In this way, T„results in a Weyl tensor of weight —1:TOO IY io j0+ 4gooy y +li Fmj (6.7)
The Hamiltonian is defined via '
H& —= T„v"d2 = T„v"U d 2
and since 0"=v"and dX =XdX, it follows that
(6.2)
(6.3)
Expression (6.7) can be written using the covariantderivative
~~built with the metric tensor y,j induced on X:
TQQ 2 y Ai, OAj, Q+ &goo(g g g g ) Ai il Aj1m
(6.8)
H~=H~ . (6 4) Therefore,
The inclusion of v" in the definition of H& means thatHz depends on the Quid of observers. However, H& is in-variant under a geometrical chart change (but it is notunder an observer's system change. )
If v" is a Killing vector in the region bounded by twoCauchy surfaces X and X', then H& =H&. (Ref. 10).However, in general, H& depends on X.
In the adapted chart, the Hamiltonian reads
Now, using the identity
(yijylm yimylj )g 1/2A
—[(yijy lm yimylj )g1/2 A
y'j(y' —goo' Fi; )i Aj (6.10)Hx= f Toogoo
' dX . (6.5)and the field equations
Hx= ,' f-d&[goo '"y"A;oA, ,o
+g ' (y'y' y—' y')A, i, Al ] . (6.9)
From (6.1) and (4.1) we obtain
(6.6)
(y' goo'"+a)i =goo '"[A;,00+-,'[»(y/goo)], QA;, 0
ymi, oAi, oI (6.11)
and, in the adapted chart, where y =det(y;j ), the last term in (6.9) can be written as
( yijy
lm imy
lj)
1/2 A '[ A;00+ —,'[»(y/goo)] QA;0 —y' y;QA, QI A,
+[(yijylm yimylj)g 1/2A A ] (6.12)
——,'[ln(y/goo)], oA;, oA,
+y' y;,oAl, oA, I . (6.13)
Note that in (6.13) only temporal derivatives appear, justas in the scalar case.
Replacing (6.12) in (6.9) and using Gauss's theorem, wefinally obtain
Hx= —,' f dXgoo ' y' [A;OAjo —
A;OOA
VII. THE VACUUM
A, (x)= g [ak(k)p;(O, A )+atk(k)p, '(k, j1,)]k, k
(7.1)
and the Hamiltonian turns out to be
Let [P;(O, A, ),P,*(k, i(, )] be a basis of solutions of fieldequations (4.6). As in Ref. 6, A, is an index related to thepolarization mode. The field can be expanded in thisbasis as
H~= —,' g f dXgoo '/ y'j(ai(k)ak (k')[P;0(O, A. )P 0(k', A, ') —P;00(k, l, )$.(k', X')
k, k', k, A,'
——,' [ln(y/goo)] 0$; „-(k,l)gj(k', l')+y™y;oitll 0(k, A )P (O', 1,') I
+akt(k)ak. (k') [p;0(k, l)$. 0(k', A, ') —p;00(O, A, )p (O', A.')
——,'[ln(y/goo)] 0$,'0(k, A, )P (O', 1,')
+y y 0/1'0(k i()P (k A )I )+H.c. (7.2)
In addition, the basis [P;(O, A. ),P,*. (k, i, ) I must be ortho-normal in the inner product (, ) defined in Sec. V:
As we know, each basis of solutions has an associatedvacuum state ~0), defined by
(y"(O, A, ),y„(k', A, ')) = —5 .5„„,(y~(k, z), y„*(k,z ))=0,
~hi'~kk'
(7.3)
a (k)~0) =0, Vk, i, . (7.4)
Therefore, choosing a vacuum ~0) is equivalent to choos-ing the basis [P, (O, A, ), P; (k, A, )].
In analogy with the scalar case, we shall consider the
1880 MARIO A. CASTAGNINO AND JORGE B. SZTRAJMAN
cases where the natural time can be separated from thespace variables in the field equations. It is possible toseparate the variable t in Eq. (6.11) when the componentsof the metric tensor can be written as
The set IS;(k, A, ) I is a basis of solutions of the spatial sideof the field equations and it may be orthonormalized on Xaccording to
goo =r(t)h (x), y; =f (t) A; (x) .
In such a case we write
$;(k, A, ) = Tkk(t)S;(k, A;x) , .
(7.5) f d X g' y'JS (k, k)SJ*(k', A, ') =5k' 5„„
and then (7.8) yields
(7.9)
[1n(y/goo)] 0=[in(f /r)] 0,y' ym, ,0=5,'(»fl, o .
Orthonormality conditions (7.3) turn out to be
(7.7)
It is easy to see that if (7.5) holds, the factors[ln(y/goo)] 0 and y™y~;0that appear in (7.2) only de-pend on t. In fact, they are
(TkkTk*k 0 TkkTkk, o)lx t (7.10)
Furthermore, since the separation constant of the fieldequation is real (see the Appendix), the result is that ifS;(k, A, ) is a solution, S (k, A, ) is a solution also (for thesame value of the separation constant). Thus, we can set
i (TkkTk'k', 0 Tk k'Tkk, o)lx f d&goo '"y"
XS;(k, A, )Si*(k', A, ')
(7.8)
S;(—k, A, )—=S;*(k,A, )
and then (7.9) reads
f d&goo ' y'~S;(k, A)SJ(k', A, ')=5kk5k
(7.11)
(7.12)
'(TkkTk/. ,0 T/ k Tkk, o)lx f d&goo
XS;(k,A, )S~(k', X') =0 .Introducing (7.6) in (7.2) and on account of (7.9) and(7.12) we obtain
Hz= —,' g (ak(k)ak( —k)I Tkk, o Tkk, ooTkk+ —,[ln(r/f)], OTkk, oTkkIx
+ak(k)ak(k) j I Tkk, ol—
Tkk, ppTkk+ —,'[»(«f)],OTkk, oTkkI z)+H c. (7.13)
But we can use the fieM equations for Tk& in order to ex-press Tk& in terms of Tk& and Tk& o..
that is to say,
Tkk 0Q= C/,—Tkk —
—,' [»(f«)],OTkk, o (7.14)
rTk A. ,o Ck Tk A,f (7.17)
where Ck is the separation constant. Finally, for Hz weobtain
and on account of condition (7.10), we obtain the follow-ing Cauchy data (choosing Tkk to be real )0):
rHz =
—,' g ak(k)ak( k) Tkk 0
—Ck—T„k—kA,
Tkklz=(1/&2)[f (~)/lck lr(~)]'",T k, ol =( —/&2)[I& I ( )/f( )]'" (7.18)
+—,' g a&(k)ak(k) I Tkk, ol Ck I Tkkl
kA,
where ~ is the natural time corresponding to X.The quantity hk is
+H. c. (7.15)
A. Hamiltonian diagonalization
We want Hz to take the diagonal form
The functions S;(k, A, ) are not important in order to selectthe vacuum, because they do not appear in (7.15) (Ref. 3).Now, to select a vacuum is equivalent to selecting theCauchy data Tkklx and Tkk Olx. We study the followingtwo criteria to fix these data.
j..e.,
hk I Tkk, , o I z Ckr(r)
= [Ick lr(r)/f (r)]'",
Hz =—,' g [akt(k)ak(k)+ak(k)ak(k)]
kA,
X [Ic„lr(r)/f(r)]'" .
B. Hamiltonian minimization
(7.19)
(7.20)
Hx =—,' g hk[a&(k)ak(k)+a&(k)ak(k)],
kA,
(7.16) In this case the data are such that (0;XIHzl0;X)™turns out to be minimized, where IO; X ) is the associat-
OBSERVER-DEPENDENT QUANTUM VACUA IN CURVED SPACE. II 1881
ed vacuum. From (7.15)
&OIHz IO& =I Tkk, ol Ckr I Tkkl &Olak(k)ak(k)IO& .
(7.21)
D"Foq=&h (I/&h E')1; .
Besides, the spatial part of the field equations is
h (i/h 3 ' S„)„=C„S,(x)
(8.2)
(8.3)
Therefore, in order to obtain the Cauchy data, we mustminimize the quantity
where Sl; =5; I—SI;.
Multiplying by T(t) and deriving with respect to t, weobtain
A=I Tkk, ol' —Ck —
ITkkl' (7.22) (~/h ~'-X ) =h '"C-„E, , (8.4)
B«, I Tkk, olz andI Tkklz are not independent; they are re-
lated by (7.10), which implies (choosing Tkk to be real)
where XI; =E;&
—EI;, or
=h-'"C E'llm
(8.5)—1
ITkk, olz=2Tkklz sin «g Tkk„olx
(7.23) and applying lli on (8.5) and taking into account that %is antisymmetric, we find that
Then
=(2T„I,»n «gT«ol, )-' —C, («f),T,', I, .
It is clear that, in order to minimize A, it must be
Ck(h 'i E') =0llI'
Hence, with the assumption that Ck&0, we have
(h—1/2Ei) —0 .
(8.6)
(8.7)
ArgTkk olz= —~/2 (7.25)
[the value +m/2 is forbidden by (7.23)]. Then, fromd A/dTkk I
z=0, we have
Tkklz= [ f (~)/4r—(r)ck]"" (7.26)
Since f and r can be chosen to be real, it is clear that Ckmust be negative. Thus,
Tk k I Z= —[f( r ) / I C. I
r ( & ) I'"1
2
and, from (7.23) and (7.25),
(7.27)
(7.28)
Therefore, the Cauchy data that minimize Hz are thesame that diagonalize it. That is to say,
0.y )D —I
0.y ) M (7.29)
VIII. GAUSS'S LAW
Of course, the solution Tk& that satisfies the required con-ditions on X may not satisfy these conditions on other hy-persurface X. In such a case lo;X)%IO;X') and there isparticle creation
i.e., we obtain Gauss s law again but no longitudinal pho-tons.
On the other hand, if Ck =0, condition (8.7) does notappear, thus it is clear that longitudinal photons are asso-ciated with the condition Ck =0 and transversal photonsare associated with the condition Ck&0. We could im-
pose Gauss's law as a supplementary condition, but thenthe canonical commutation relations result is incon-sistent. Furthermore, the existence of longitudinalmodes (Ck =0) is not compatible with the Hamiltoniandiagonalization. In fact, in the case Ck =0, Eq, (7.17) im-plies Tkk o=o and then the condition (7.10) result is ab-surd (O=i).
B. Elimination of the unphysical photons
In order to cope with this problem we shall preserve alongitudinal nondiagonal part in the Hamiltonian whichwill be eliminated a posteriori by means of an appropriatelimit.
In the case C„=o, and with the metric (7.5), the tem-poral factor of Eq. (7.6) is
T(t) =a Iv'r /f dt +b, (8.8)
where a and b are fixed numbers which are related bymeans of orthonormality condition (7.10), which yield
A. The problem of the longitudinal photonsIm(ba *)= ,'&f (r)/r (r)— (8.9)
When we work in the temporal gauge, the degree offreedom Ao disappears from X; so only three field equa-tions are obtained and Gauss's law D"FO„=O does notappear. In this way, longitudinal (unphysical) photonscould appear.
Developing D"Fo„we obtainb =( I/2a)v f (r)/r(v') (8.10)
and as the longitudinal part of Hz is
and taking a = —ia with a real (with no loss of generali-ty, because we can make it imaginary with an irrelevantconstant phase factor), it results in
D"Fo„=+goo[(goo) ' E'](~, , (8.1)
where E; =Fo, is the electric field. In the case of metric(7.5),
g aL(k)aI (k)Tkr, o IXk
from (8.8) we have
(8.11)
1882 MARIO A. CASTAGNINO AND JORGE B. SZTRAJMAN
I-tt = i—(a/2)V r(r)/f(r) g at (k)at (k) .X
k
(8.12)
Thus, the longitudinal photons will be removed by takingthe limit a~0 at the end of the calculation.
IX. EXAMPLES
(9.1)
and then, from (7.14) and (7.18) it follows that
A. o" is a Killing vector in JV* (neighborhood of X}
In this case, the metric is independent' of t. Thus, wehave
field. '" Of course, this procedure is possible because theBogolubov coefBcients' are gauge invariant as we shallprove. In fact, let Ip„(k),P&(k)] and IP„(k),P„'(k)I betwo bases of solutions related to the same geometricalchart and corresponding to the same gauge; k and k holdfor all the indices necessary to specify the modes P„andP„, respectively. The gauge condition can be written as
JVQ„(k)=JVQ„(k ) =0, (9.9)
where JV is an operator that specifies the gauge. In addi-tion we assume that JV is a linear operator (this is the casefor the gauges we are interested in). The transformationbetween both bases is
~k A, ,OO+ k ~kk
1Tl, i. lx 7k'. ,olz= —i&k/2,
(9.2)
(9.3)Pq(k) =g [a(k, k )P„(k )+P(k, k )P „'(k )], (9.10)
which means that
T (t) =(2k) ' e (9.4)
Hz =—,' g k[ai (k)a&(k)+a&~(k)a&(k))z .
kA,
(9.5)
This case includes Minkowski observers and Rindler ob-servers in flat space-time, comoving observers inSchwarzschild's metric, " in the static Einsteinmetric, ' ' and also the Killing observers in de Sitterspace.
B. Rindler observers
with —k being the separation constant. The transversepart of the Hamiltonian is
JV'P„'(k) =JV'P „'(k ) =0 . (9.11)
Clearly, Pz(k} is related to P&(k) by a gauge transforma-tion
Pq(k) =P„(k)+B„M(k) (9.12)
and analogously for P „'(k ) and P„(k ),
P„'(k)=P„(k)+B„M(k) . (9.13)
In order to find the relation between M(k) and M(k ) wepostulate
where a(k, k ) and p(k, k ) are the Bogolubov coeScients.Now we introduce another gauge condition character-
ized by the operator JV' and both solutions P„'(k) and
P „'(k ) in this gauge, i.e. ,
We consider the flat space-time in Rindler coordinatesas an important case of the former example. Units arechosen such that G =iti=c=k (Boltzmann's constant)= 1. All quantities are therefore dimensionless. The rela-tion between Rindler (vi, g,y, z) and Minkowski (t, x,y, z)coordinates is given by means of
M(k)=g [a(k, k)M(k)+P(k, k)M*(k)] .
So, we obtain
P„'(k)=+[a(k, k )P „'(k )+P(k, k )P „'*(k ) ] .
(9.14)
(9.15)
t =+gsinhrt, x =+gcoshil, (9.6)
dg =g d viz —d ii2 —dy2 —dz 2 (9.7)
with the top sign for t, x in R+ (x ) ltl) and the lowersign for t, x in R (
—x ) ltl };and the metric results in
The linearity of JV' implies, from Eq. (9.15), that if P „'(k )
satisfies the gauge condition (9.11) then P„'(k) does too.So we see that conjecture (9.14) implies that both solu-tions satisfy the same gauge condition and the gauge in-variance of the Bogolubov coeScients is now obvious.
In Ref. 14 it is shown that, in the Lorentz gauge, theRindler modes can be written as
The components of the metric connection, different from0, are
(9.8)
2, =e,b V"P ( a, b, . . .= rt, g),A~ =ej&V'"P (j,k, . . . =y, z),
(9.16)
In order to obtian the relation between the Rindler andMinkowski modes we must compare both modes in thesame gauge. This work is more easily achieved in theLorentz gauge, where the electromagnetic Rindler modescan be expressed by means of the solutions of the scalar
where e,b and e&k are the antisymmetric two-dimensionaltensors given by e„&=g and e, = 1, g and P being mass-less scalar Aelds:
(9.17)
So we find two independent Rindler modes:
OBSERVER-DEPENDENT QUANTUM VACUA IN CURVED SPACE. II 1883
$„(I,Q, k)=[2~(Qm)' I (ifl)] '(0, 0, —n„n )e '""+' "K,„(kg),p„(2,Q, k)=[2~k(Q~)'"I"(iQ)] '(/BE, „, in' 'J:,„,0,0)e 'n&+'"", (9.18)
(9.19)
where Q & 0 and k =(k~, k, ) label the modes, x =(y, z), k = ~k~, and n=k/k; E;n(k g) are the modified Bessel functionsof imaginary order. These modes are orthonormal in the inner product:
(f~,g„)= i—f (g "V' f„* f„*V—~„)dcr
as is Usual for the Lorentz gauge. If X is a hyperplane with g =const, the inner product can be written
(f"g„)= —i J [(g"~g„* f„*d—~„)+2k '(f „'gg g„fP—]4 'dt's dz (9.20)
The independent Minkowski modes in the Lorentz gauge and in the Rindler chart are chosen as
f„(1,k„,k) = [2(2m. )'co] 'i'(0, 0, n„n—~ )e
f„(2,k, k) = [2(2m)oi] .'~ (co/k)(gk„co ' cosh' —g sinhri, k„ro ' sinhg —cosh', 0,0)e
(9.21)
where co=—(k +k )'~ . Now we can compute the Bogolubov coefficients that connect modes (9.18) and (9.21) in accordwith
a„(Q,k, k„,k')= —(P"(m, Q, k), f„(n,k„,k')),P„(II,k, k, k') = (P"*(m,Q, k), f„(n,k„,k')),
where n, m =1,2. After calculations we get
aii=a22=(2n. ) '(0/co)'~ I (iD)k' (co k, )' —e ~ 5(k' —k), aiz=azi=0,
and
(9.22)
(9.23)
P»=P2z=(2m. ) '(0/co)'~ I ( iA)k—' (co —k„)' e " 5(k'+k), P,2=P2 =0 . (9.24)
This is the same result obtained by Takagi for scalar fields and it indicates that if the field is in the state ~OM ) (that is,it is devoid of usual Minkowski-space particles) then, from Eqs. (9.23) and (9.24), it may be deduced that a Rindler ob-server will detect
—(e2mn 1)—1 (9.25)
particles in mode Q, k. This is the Planck spectrum for radiation at temperature To =1/2m, however, the temperatureT as seen by the accelerated observer with acceleration a is given by the Tolman relation'
T= To/+goo= 1/2~$
and as we know that coordinate g is related to the acceleration as a = I /g, it follows that
T=a/2n .
(9.26)
(9.27)
Candelas and Deutsch' calculated the vacuum stress for the electromagnetic field as seen for an accelerated detector inMinkowski vacuum and they got a thermal spectrum, though not quite Planckian, with temperature a /2n:
T„=(a /m )I dQII d(A)(e —1) 'diag(1, ——,', —
—,', ——,'), (9.28)
0
where d(Q) is the density of states of the detector
d(Q)=1+0 (9.29)
that, when the adapted charts are used, the metric ten-sors of both space-times g„and g„' are conformally re-lated:
So, it is important to note that the particle spectrum(9.25) is exactly Planckian while the stress tensor is not; itis distorted by the density of states (9.29) (Ref. 17).
C. Conformal transplantation
Let us consider two different space-times such that ineach of them there would be an observer system (bothspace-times may eventually be the same). Let us suppose
g„'„(x)=A(x)g„(x) .
Clearly, we can consider thatl
gpv =gatv
and then, from (6.4), we get
~x[4';]=~&[4;]
(9.30)
(9.31)
(9.32)
I884 MARIO A. CASTAGNINO AND 3ORGE B. SZTRAJMAN 40
Therefore, if (P;(k, A, );P,*(k,A, )) is the basis that performsthe Hamiltonian diagonalization in the metric g„, it alsodiagonalizes the Hamiltonian in the metric g„' . Then,the well-known concept of the conformal vacuum can bedefined for the case of massless spin-1 fields, in the sameway as we did for the case of spin-0 fields.
Tki '(t)=i/a(r')la(r) exp ik J dt'/a(t') Tkz'(t) .
(9.43)
Since Tki' '(t) does not appear in Eq. (9.43), it is clearthat particle creation does not occur: i.e.,
iO)( ') —iO)( ) (9.44)
D. Geodesic Quid in a Robertson-Walker universe
We shall now study the conformal vacuum in the veryimportant case of Robertson-Walker universe. In theadapted chart, the line element is
ds =dt a(t—)[dy +p (y)(d8 + sin Odg )]; (9.33)
i.e.,
g~=l, y, =a (t)A; (y, 0,$), (9.34)
where
The functions r (x ) and f (x ) introduced in (8.5) are
r(xo)=1, f(x )=a (t),and, from (7.14), the field equation for Tt, i(t) is
Tki oo+(a o/a)T„i„o (C„ /a)T &k—=0 .
If we introduce the conformal time g as
t~ri: drj=a (t}dt
(9.35)
(9.36)
(9.37)
as it is well known, with this transformation, metric.(9.33) can be written in a conformal way to the Aat orstatic universe metrics.
Then Eq. (9.36) takes the form
d T„i./d O' C„T„i=0—and calling —Ck =k, we have
d2Tk~)d&'+k2T«p .
Cauchy data (7.27) and (7.28) become
T„„,= i/a (r)/2k, T„„,= —i v'k/2a(r) .
The solution of Eq. (9.39) satisfying (9.40) is
(9.38)
(9.39)
(9.40)
sing, 0 g 2m spatially closed (IC =1),r(y) = y, 0 & g & ao spatially Aat (K =0),
sinhy, 0&g& ~ spatially hyperbolic (K = —1).
In fact, we have only one conformal vacuum obtained byconformal transplantation from the vacuum of flatspace-time or from the one of a static universe.
X. CONCLUSIGN
We have developed a massless spin-1 particle modelwhich is observer dependent. This dependence was ob-tained introducing a Hamiltonian invariant under coordi-nate changes but observer dependent. The aim of thisdependence is to obtain quantum vacua associated withthe observer system as is commonly accepted in the litera-ture. ' ' The associated vacuum was obtained by meansof the Hamiltonian minimization or diagonalizationmethod. In addition, as is usual for massless fields, wehave preserved the conformal invariance of the theory.
Almost all the calculations have been made in the casewhere the metric has the form (7.5), but we think thatthis is not essential; the aim of this condition is only tosimplify the calculations. In fact, the separation of vari-ables is not an essential condition for the Hamiltonian di-agonalization. It only makes the calculations easier. Infact, Eq. (9.32) shows that we can give examples wherevariables do not separate, but diagonalization still works.Qn the contrary, the use of a curl-free Quid is certainlyessential, because it makes possible the notion of time.The particular cases we have developed show that the ob-tained vacua are similar to the ones that can be found inthe literature.
The results of this paper agree with those recently stat-ed for spin-0 particles.
Finally, we wish to point out that other requirements,which were not studied in this paper, must be satisfied inorder to obtain a good vacuum: it must render the theoryrenormalizable. Therefore, as usual, in order to have agood vacuum two conditions are necessary; a global anda local condition: i.e., (a) diagonalization and minimiza-tion of the Hamiltonian (global condition); (b) renormal-izability (local condition).
APPENDIX
Tki„'(t)=v a(r)/2k exp ik I dt'la(t'—)r
(9.41)
We shaH demonstrate that the separation constant ofthe field equation is real. To begin with we write the fieldequations in the adapted chart:
The solution associated with another surface X' (withtime r') will be
A, oo+ ,'tl ()'/goo)~, o-A, o )" )',oAI, o—
1/2(
1/2 Im~ ) (A 1)
TPi '(t)=i/a(r')/2k exp ik I dt'/a (t'—)1'
(9.42) We can separate variables when the metric tensor has theform
and the corresponding Bogolubov transformation is goo=r(t)h(x), y;i=f(t)A; (x) . (A2)
OBSERVER-DEPENDENT QUANTUM VACUA IN CURVED SPACE. II 1885
In such a case we write
A, (t, x) = T(t)S, (x)
and (Al) takes the form
[h( )]' I[h( )]' A' S„J„
(A3)
=[f(t)/r(t)]' T '(t)I[f (t)lr(t)]' To(t)J oS(x),
straightforward calculation gives
&pu;, u;) —(u, , pu, &
—f dXI&h A'iA' [(u, , —ui, )u*T
(—u,*i u—I', )u; ]]~~
(A6)
and as the integral of Eq. (A6) vanishes by using Gauss'stheorem, we find that
(A4) &pu, , v;) =(u;, pu, &, (A7)
where SI;—=S; &
—SI, . Now we shall prove that the opera-tor p de6ned as
which means that p is a Hermitian operator. Now, from(A4) we obtain the equation for S, (x), which has the formof an eigenvalue equation:
pE; =+h (+h A ™Si (A5) pE, =CS, (C=const) . (AS)
becomes Hermitian in inner product (5.11). In fact, aThe reality of C follows from the Hermiticity of p as usu-al.
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