CC BY
47
UDC 530.1; 539.1
VACUUM FLUCTUATIONS AS QUANTUM PROBES IN FRWL SPACE-TIMES
S. Zerbini
Department of Theoretical Physics, Trento University, TIFPA-INFN Trento, via Sommarivel460, 3S123 Trento, Italy.
E-mail: zerbiniQscience. unitn.it
Vacuum fluctuations related a massless conformally coupled scalar field in Friedman-Robertson-Walker-Lemaitre (FRWL) space-times are investigated. Point-slitting regularization is used and a specific renormalization proposal is discussed. Applications to generic black holes and FRWL form of the de Sitter space-time are presented.
Keywords: vacuum fluctuations, black holes, Hawking temperature.
1 Introduction
How to probe the properties of the Quantum Vacuum in relativistic quantum field theory (QFT)? It is well known that this is a difficult task, because one is dealing with relativistic systems having an infinite number of degrees of freedom. The possibility we shall discuss here is to study the vacuum expectation value (^2(x)) =< 0|^2(x)|0 > associated with quantum field ^(x). We will refer to it as "vacuum fluctuation". It contains physical informations on the quantum vacuum, and it is simpler than the vacuum expectation value of the stress-energy tensor.
In QFT, ^(x) is an operator valued distribution, thus (^2(x)) ill-defined quantity, and regularization and renormalization procedures are required.
One may use zeta-function regularization fl, 2], however, since in cosmology one is dealing with FRWL space-times and hyperbolic operators, point-splitting regularization is more appropriate. Given the off-diagonal Wightman function
W(x, x') =< 0|^(x)^(x')|0 > ,
L = -d2 - V2 + M2
(2)
M being the mass and t imaginary time with period p. A direct calculation within zeta-function regularization [3,4] gives for the renormalized fluctuation associated with this massive free field
M ^ -i(nßM) M2 /M2, ..
M2
M=0
and an arbitrary mass scale is present: bad thermometer.
However, in the massless case, there IS a drastic simplification and the result is
Wx)2>R
1
12ß2
T2 12
(4)
(1)
No ambiguity is present, simple reading for the temperature T: good thermometer [5].
It is also instructive to present a point-splitting regularization derivation. Making use of the images method and KMS condition, it follows that thermal Wigthman function is periodic of period P in the imaginary time t, and reads
point-splitting regularization means x' = x + e, e a small cut-off, and the removal of cut-off consists in taking the coincidence limit x' ^ x.
2 Fluctuation as Quantum thermometer
Fluctuations may give informations on thermal nature of the quantum vacuum. In order to illustrate the issue, let us consider a free massive scalar field defined on the Minkowsky manifold M4, make a Wick rotation to imaginary compactified time with period p. As a result, one is dealing with a massive free quantum
field in thermal equilibrium at temperature T = —, and
P
the relevant operator is defined on Si x R3 and reads
Wß (x,x') = ¿2
1
|x - x'|2 + (t - T' + nß)2
(5)
The term n = 0 is the only singular term when x ^ x', and coincides with the Minkowski contribution. Renormalization prescription : subtract this term. Thus, for x ^ x', we get again
< $(x)2 >=
1
1
n2
1
2n2ß2 — n2
' n=l
12ß2
T! 12
(6)
3 Spherically symmetric dynamical black holes
Fluctuations associated with massless scalar fields work well as good thermometers in finite temperature
QFT in Minkowski space-time. What about the use of fluctuation as thermometers in presence of gravity?
Static black holes and FRWL space-times in cosmology are example of Spherically Symmetric Space-times (SSS), and both may have an unified descriptions [6]. With regard to this, first let us briefly review the Kodama-Hayward formalism. A generic dynamical SSS may be defined by the following metric
ds2 = Yij(x^dxW + , i, j = 0,1
where the two dimensional normal metric is
¿72 = Yj (xi)dxidxj ,
4 Fluctuations in FRWL space-times
The main idea: use the fluctuation related to a suitable quantum probe, which has to be a good thermometer. The probe: a conformally coupled massless scalar field. Its Lagrangian reads
(7)
L = (-2d W - 1 R4>2
(15)
(8)
with xi associated coordinates. Furthermore, the quantity R(xi) is the areal radius, a scalar field in the normal 2-dimensional metric. Dynamical or apparent trapping horizons are determined by
YijdiRHdjRh = 0 .
(9)
R Ricci scalar curvature.
It is convenient to re-write the flat FRWL spacetime making us of the conformal time n dn = —>
a
ds2 = -dt2 + a2(t)dx2 = a2(n)(-d^2 + dx2) . (16)
Recall that the Wightmann function is
W(x,x') =< 0|^(x)^(x')|0 >. Now the field ^(x) is "free" and admits the expansion
Another important invariant quantity is the Hayward ¿(x) = ^ /k( x)ag + h.c. surface gravity kh = -(AyR)h (see for example [7,8]), %
which is a generalization of the Killing surface gravity.
(17)
There is also conserved Kodama vector, Ki
eij 3Rj
generalization of the Killing vector. Relevant for us Yre the so called Kodama observers, defined by condition R = RQ.
First Example: 4-dimensional static Schwarzschild BH with x = (t, r) as coordinates in normal space.
where fk(x) are the associated modes. The related Conformal Vacuum is defined by
ak|0 >=0. As a result
W(x,x') = £ /k(x)/fci (x'),
(18)
(19)
ds2
-V (r)dt2 +
dr2 V (r)
+ r2dQ2 = d72 + r2dQ2 (10)
where the modes functions /^(x) satisfy conformally invariant equations
with
V(r) = l - G =1. (11)
r
Areal radius R = r, the event horizon V(rH) = 0,
VH 1
i.e. rH = 2M the surface gravity kh = =
Kodama vector is the Killing vector K = (1,0,0,0). Kodama observer: r = r0: constant areal radius.
A dynamical SSS example: flat FRWL space-time, relevant in cosmology
6 J
/k (x) = 0 .
(20)
In this case, solutions of this equation are simple and characterize the Conformal Vacuum:
/k(x) =
a — ink
„,—ik•;
k ^Vka(n) '
k = |k|.
(21)
ds2 = -dt2 + a2 (t)(dr2 + r2dQ2):
(12)
The related Wightmann function W(x, x') can be evaluated, and this covariant bi-scalar distribution reads
Areal radius is R = a(t)r, physical radial distance, w(x x')
1
1
H(t) = — is the Hubble parameter. There is a a
dynamical trapping horizon (Hubble radius) 1
4n2a(n)a(n') |x - x'|2 - |n - n' - i£|2
Rh
h (t)-
The Kodama vector is K = (1, -H(t)R, 0,0).
(13)
(14)
It is convenient to make use of the proper time t as evolution parameter. As a result, the proper-time-parametrized Wightman function is
11
Kodama observer: R = R0 constant areal radius,
, Ro namely r = ——.
a( t)
W (x(t ),x'(r ')) = 2 2 where the invariant distance is defined by a2 (t, t') = o(t)o(t') (x(t) - x(t2 .
(22)
5 Point-splitting regularization
Due to the isotropy of flat FRWL space-time, one may restrict only to radial time-like trajectories, namely x(t) = (^(t),r(T)).
Putting t' = t + £, with £ small and t = ■dt-, and
dT
introducing the four-acceleration along the trajectory
x x'
A2
t
Vt2 -1
+ Wt2 -1
one obtains
a2(r,e) = -£2 - — ( , ) 12
A2 +H2+2t2dtH
As a result, for small £ the Wightman function is
W (t,E) =--+
( , ) 4n2 e2 +48n2
6 Renormalization
A2 + H2 + 2t0dtH
a(t) = eHoi, H(t) = Ho constant.
For Kodama observers with R = R0, the conformai time can be evaluated in terms of proper time t
n(T )
H1o *
(24)
e4+O(e6). (25)
and the dS invariant distance is
a2(s) = --(1 - R2H2)--2
4H0
sinh
H0
2^1 - RqH2
(30)
(31)
This is an example of stationary space-time: it depends only on the difference s = t — t'.
General formula (27) leads to the fluctuation for dS
+ O(e2).
(^2(x))j
HoRo
48n2 V (1 - ROH2)
+ H02
(32)
the first term in the bracket being the acceleration of
R0
In Minkowski space-time, one has H(t) = 0, and for inertial trajectories, one has vanishing acceleration, namely
Wm(T,e) = -¿2¿2 .
1
(^2(x))« = 48n2 [A2 + H2 + 2*0dtH] .
T = A Tu = 2n .
ds2
-dt2 + e'
2Hot
dx
(4>2(x))i
1
T 2
T gh
12(1 - RqHO)
(33)
(26)
Tgh temperature
Hn is the well known Gibbons-Hawking
being a kinematical red-shift
Choice of the renormalization prescription: subtract this contribution. As a consequence, the renormalized vacuum fluctuation reads [61
(27)
(i — R2H2)
factor. One is dealing with a good thermometer.
8 de Sitter space as black holes
The de Sitter space-time admits also static patch
Some remarks are in order. The first term is depending on the trajectory through A and hence t. The second one is depending on the dynamical space-time through H
vanishing for stationary space-times.
H=0
Thus one has only two cases: A
the related Unruh temperature is
ds2 = -(1 - H02r2)dtQ +
dr2
+ r°dQ°
1 - HQr2
(34)
(28)
On the other hand, if A = 0, namely inertial observer, one has (^2(x))R = 0, and the Minkowski renormalized result is obtained.
7 de Sitter FRWL space-time
A very important example of FRWL space-time is de Sitter one, its flat FRWL metric being
(29)
ts is the time coordinate, and the areal radius is R = r.
The related horizon: rH = —, and surface gravity
H0
kh = Hq.
In the cosmological dS patch, the fluctuation leads,
modolo a red-shift factor, to the Gibbons-Hawking H
temperature TH = ——. What about the temperature issue in this
static patch? One can answer to this question in general.
9 Static black hole as effective FRWL spacetime
Black holes are not black, and it is well known that they emit a quantum Hawking radiation in thermodynamical equilibrium at temperature TH. In order to (partially) investigate this fundamental issue, first let us show that for Kodama observers, one is
h
o
t
2 tT 2
oo
2
1
1
dealing with a special FRWL space-time. Start with a generic static BH solution
As a result
ds2 = -V (r)dt2 +
dr2 V (r)
+ r2dQ2
d72 = e-2KHr* V(r*)[-dT2 + dX2].
(^2(X))r =
48n2
2
(t)
t2 - 1
Using Kruskal coordinates definition and ts one gets
t = cosh kh
t
Wn / ' Wn
KH ■ u ( T smh kh
1 K2 / * 2 / w 1 K h
(X))R = 48^ V0 = 12 V
1 T 2
1 TH
(41)
with event horizon at V(rH) = 0 V— = 0. In order
to avoid the metric singularity at r = r— one can
make use of Kruskal coordinates. With tortoise radial dr
coordinate dr* = , the BH Kruskal metric is V (r)
ds2 = e-2KHr* V(r*)[—dT2 + dX2]+ r2(T,R)d^2 , (35)
V—
where now r* = r*(T, X), kh = being the Killing
surface gravity. The relevant part of BH Kruskal metric is its normal part, namely
(36)
Again a good thermometer: local temperature is TH
Tn =
(42)
kh
First key point : Kruskal normal space-time is conformally related to two-dimensional Minkowski space-time.
Second point: Kruskal space-time is effective flat FRWL space-time for Kodama observers r = r0
d72 = V0e-2KHr° (-dT2 + dX2), (37)
and we also have
d72 = -dt2 + a2dX2 = -di2V0 . (38)
Thus t = a/Vo e-KHr° T is a "cosmological" time, and a(r$) = V/Voe-KHr° is the related constant "expansion" factor, and dr2 = dt;; V0.
Since a2(r0) constant, i.e. H = 0, only acceleration term is present, and the general formula (27) gives
(39)
Wn'
.
with the Hawking temperature TH = 7—, while V0 =
—goo being the Tolman red-shift factor.
In the de Sitter space-time, V(r) = 1 — Hr2 and
TH = 1 = Ho, in full agreement with previous result.
10 Conclusion
Fluctuation related to massless conformally coupled scalar field has been used to probe the Quantum Vacuum in presence of gravity. Making use of this quantum probe, after a point-splitting regularization in the proper-time, the following renormalization prescription has been used: remove the Minkowski contribution related to inertial observers. In principle, other renormalization prescriptions are possible. However, one can show that our renormalization prescription is full agreement with the results obtained making use of Unruh-de Witt detector approach [9].
Acknowledgement
I thanks L. Vanzo, L. Bonetti and G. Acquaviva for useful discussions. This research has been supported by the INFN grant, project FLAG 2014.
References
[1] Elizalde E., Odintsov S. D., Romeo A., Bytsenko A. A. and Zerbini S., Zeta regularization techniques with applications, Singapore, Singapore: World Scientific (1994) 319 p.
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[3] Cognola G., Elizalde E. and Zerbini S., Phys. Rev. D 65, 085031 (2002) [hep-th/0201152],
[4] Cognola G. and Zerbini S., Int. J. Mod. Phys. A 18, 2067 (2003).
[5] Buchholz D. and Schlemmer J., Class. Quant. Grav. 24, F25 (2007) [gr-qc/0608133],
[6] Acquaviva G., Bonetti L., Vanzo L. and Zerbini S. (2014) Phys. Rev. D 89, 084031.
[7] Criscienzo R. Di, Hayward S. A., Nadalini M., Vanzo L. and Zerbini S., Class. Quant. Grav. 27, 015006 (2010).
1
[8] Hayward S. A., Di Criscienzo R„ Vanzo L., Nadalini M. and S. Zerbini S. (2009) Class. Quant. Grav. 26, 062001. [arXiv:0806.0014 [gr-qc]].
[9] Acquaviva G„ Di Criscienzo R„ Tolotti M., Vanzo L. and Zerbini S. (2012) Int. J. Theor. Phys. 51, 1555 [arXiv:1111.6389 [gr-qc]].
Received, 01.11.20Ц
С. Зербини
ВАКУУМНЫЕ ФЛУКТУАЦИИ КАК КВАНТОВЫЕ ЗОНДЫ В ПРОСТРАНСТВЕ-ВРЕМЕНИ ФРИДМАН А-РОБЕРТСОПА-УОКЕРА-ЛЕМЕТР А
Исследуются вакуумные флуктуации безмассового конформного скалярного поля в пространстве-времени Фридмана-Робертсона-Уокера-Леметра (ФРУЛ). Используется регуляризация раздвижки точек и обсуждается специальное ре-нормализационное предписание. Представлены приложения к черным дырам и ФРУЛ форме пространству-времени де Ситтера в форме пространства ФРУЛ.
Ключевые слова: флуктуации вакуума, черные дыры, температура Хокинга.
Зербини С., доктор, профессор. Университет г. Тренто.
Via Sommarivel460, 38123 Trento, Италия. E-mail: [email protected]
