Modanese - Large Dipolar Vacuum Fluctuations in Quantum Gravity (2000), Energy from the Vacuum
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Large \Dipolar" Vacuum Fluctuations
in Quantum Gravity
Giovanni Modanese
California Institute for Physics and Astrophysics
366 Cambridge Ave., Palo Alto, CA
and
University of Bolzano { Industrial Engineering
Via Sorrento 20, 39100 Bolzano, Italy
Abstract
sources satisfying (up to terms of order
G
2
) the simple condition
R
d
3
xT
00
(
x
) = 0. We give
two explicit examples of virtual sources: (i) a \mass dipole" consisting of two separated mass
distributions with dierent signs; (ii) two concentric \+/- shells". The eld uctuations can
be large even at macroscopic scale. There are some, for instance, which last
1
s
or more and
10
6
g
.This
appears paradoxical, for several reasons, both theoretical and phenomenological. We also give
an estimate of possible suppression eects following the addition to the pure Einstein action
of cosmological or
R
2
1
cm
and mass
terms.
04.20.-q Classical general relativity.
04.60.-m Quantum gravity.
1 Introduction
Vacuum uctuations are an essential ingredient of any quantum eld theory, and also in quantum
gravity they play an important role. The presence in the gravitational action of a dimensional
coupling of the order of 10
33
cm
{ the \Planck length" { indicates that the strongest uctuations
occurr at very small scale: this is the famous \spacetime foam", rst studied by Hawking and
Coleman through functional integral techniques
.
More recently, Ashtekar and others
analysed the possible occurrence of large uctuations
in 2+1 gravity coupled to matter. In this case the theory is classically solvable and admits a
standard Fock-space quantization. In 3+1 dimensions, however, Einstein quantum gravity is a
notoriously intractable theory. Exact implementation of the dieomorphism invariance, according
to Wheeler's geometrodynamical view of gravity, leads to an ample theory, called \loop quantum
1
e-mail address: giovanni.modanese@unibz.it
1
We study a novel set of gravitational eld congurations, called \dipolar zero modes",
which give an exactly null contribution to the Einstein action and are thus candidates to
become large uctuations in the quantized theory. They are generated by static unphysical
correspond to the eld generated by a virtual source with size
gravity", of dicult physical interpretation [
]. States, transition amplitudes, time... : everything
is highly non-trivial in quantum gravity.
The non-renormalizable UV divergences of the perturbative expansion may indicate that
quantum gravity is not a fundamental microscopic theory, but an eective low-energy limit [
], and
will be eventually replaced by a theory of strings or branes. On the other hand, it is known from
particle physics that the Einstein lagrangian can be obtained, without any geometrodynamical
assumption, as the only one which correctly accounts for a gravitational force mediated by helicity-
2 particles [
]. For this reason, it is important to investigate { besides the standard perturbative
expansion { all the basic properties of the Einstein lagrangian. In the past years we took an
interest into Wilson loops [
], vacuum correlations at geodesic distance [
], and the expression of
the static potential through correlations between particles worldlines
.
In this work we study a set of gravitational eld congurations, called \dipolar zero modes",
which were not considered earlier in the literature. They give an exactly null contribution to the
Einstein action, being thus candidates to become large uctuations in the quantized theory. We
give an explicit expression, to leading order in
G
, for some of the eld congurations of this
(actually quite large) set. We also give an estimate of possible suppression eects following the
addition to the pure Einstein action of cosmological or
R
2
terms.
Our zero modes have two peculiar features, which make them relatively easy to compute:
(i) they are solutions of the Einstein equations, though with unphysical sources; (ii) their typical
length scale is such that they can be treated in the weak eld approximation. We shall see that
these uctuations can be large even on a \macroscopic" scale. There are some, for instance, which
last
1
s
or more and correspond to the eld generated by a virtual source with size
1
cm
and
10
6
g
. This seems paradoxical, for several reasons, both theoretical and phenomenological.
We have therefore been looking for possible suppression processes. Our conclusion is that a
vacuum energy term (
=
8
G
)
R
d
4
x
p
g
(
x
) in the action could do the job, provided it was scale-
dependent and larger, at laboratory scale, than its observed cosmological value. This is at present
only a speculative hypothesis, however.
The dipolar uctuations owe their existence to the fact that the pure Einstein lagrangian
(1
=
8
G
)
p
g
(
x
)
R
(
x
) has indenite sign also for static elds. It is well known that the non-
positivity of the Einstein action makes an Euclidean formulation of quantum gravity dicult; in
that context, however, the \dangerous" eld congurations have small scale variations and could
be eliminated, for instance, by some UV cut-o. This is not the case of the dipolar zero modes.
They exist at any scale and do not make the Euclidean action unbounded from below, but have
instead null (or
h
)action.
A static virtual source will generate a zero mode provided it satises the condition
R
d
3
xT
00
(
x
) = 0 up to terms of order
G
2
. The cancellation of the terms of order
G
(Section
) is important from the practical point of view. In our earlier work on dipolar uctuations
we developed some general remarks based on the form of Einstein equations, and the result
was that in order to generate a zero mode the positive and negative masses of the source should
dier by a quantity of order
G
,namely
mr
Schw:
=r
; this is very small for weak elds,
but sucient to produce a \monopolar" component which complicates the situation. Explicit
calculations in Feynman gauge now have shown that the terms of order
G
cancel out exactly.
This opens the way to an amusing \virtual source engineering" work, to nd explicitly some zero
modes and give quantitative estimates in specic cases.
When analysing the Wilson loops, we had already pointed out some dierences in the
behavior of gravity and ordinary gauge theories, essentially due to the dierent signs of the
allowed physical sources. Here, again, these dierences are apparent. In gauge theories the real
sources can be both positive and negative; therefore one can close two Wilson lines at innity and
nd the static potential. The virtual sources cannot give rise to strong static dipolar uctuations,
Gm
2
=r
2
mass
since the lagrangian on-shell is indenite in sign and equal to
p
g
(
x
)Tr
T
(
x
), we can construct
static zero modes employing +/- virtual sources. Then, of course, we can Lorentz-boost these
modes in all possible ways.
The paper is composed of two main Sections. Section 2 is devoted to the analysis of the
dipolar elds and virtual sources. We start from some general features and then focus on two
examples. Section 3 contains an extensive discussion. For a summary of the main contents see
also the
Conclusions
Section.
1.1 Conventions. Sign of
vs. its classical eects
Let us dene here our conventions. We consider a gravitational eld in the standard metric
formalism; the action includes possibly a cosmological term:
S
=
S
Einstein
+
S
Z
d
4
x
q
g
(
x
)
R
(
x
)
(1)
1
16
G
S
Einstein
=
(2)
Z
d
4
x
q
g
(
x
)
8
G
S
=
(3)
with
g
(
x
)=
+
h
(
x
).
By varying this action with respect to
g
(
x
) and using the relation
p
g
g
=
2
p
gg
(4)
one nds the eld equations
R
(
x
)
1
2
g
(
x
)
R
(
x
)+
g
(
x
)=
8
GT
(
x
)
(5)
The energy-momentum tensor of a perfect uid has the form
[
T
] = diag(
;p;p;p
)
(6)
;
+
;
+
;
+), and the experimental estimates of are
mainly referred to this metric. It is important to x the sign of the cosmological term with
reference to the metric signature in a way which is clear both formally and intuitively.
If spacetime is nearly at, we can take the cosmological term in (
) to the r.h.s., set
g
(
x
)=
and regard it as a part of the source. We obtain, in matrix form
R
2
g
R
=
f
diag(
;
;
;
) + 8
G
diag(
;p;p;p
)
g
[metric (
;
+
;
+
;
+)] (7)
Which sign for allows to obtain a static solution? Even without nding explicitly this solution,
we see that for
>
0 the \pressure" due to the cosmological term is positive and can sustain the
system against gravitational collapse { especially in the case of a zero-pressure dust with
p
=0.
3
because the lagrangian is quadratic in the elds. On the contrary, in gravity there are no real
negative sources, the potential is always attractive and Wilson lines cannot be closed; however,
1
For a zero-pressure \dust" one has
p
=0.
Now let us introduce a signature for the metric. Articles in General Relativity or cosmology
usemostoftenthemetricwithsignature(
1
At the same time, the mass-energy density due to the cosmological term is negative and subtracts
from
, still opposing to the collapse.
In conclusion, with this convention on the metric signature a static solution of Einstein
equations with a cosmological term can be obtained for
>
0. If we are not interested into
a static solution, but into an expanding space a la Friedman-Walker, in that case the eect of
a cosmological term with
>
0 will be that of accelerating the expansion. The most recent
measurements of the Hubble constant from Type Ia supernovae [
] suggest indeed that there
is a cosmologically signicant positive in our universe.
In Quantum Field Theory, on the other hand, the signature (+
;
;
;
) is more popular,
2
. Since we shall introduce some coupling of
gravity to matter elds in the following, and make a correspondence to the Euclidean case, we
prefer to use this latter convention. We then have, instead of eq. (
)
R
=
t
2
j
x
j
2
g
R
=
f
diag(
;
;
;
) + 8
G
diag(
;p;p;p
)
g
[metric (+
;;;
)] (8)
In this case, a static solution { or an accelerated expansion { corresponds to
<
0.
2 The dipolar uctuations
We consider the functional integral of pure quantum gravity, which represents a sum over all
possible eld congurations weighed with the factor exp[
ihS
Einstein
] and possibly with a factor
due to the integration measure. The Minkowski space is a stationary point of the vacuum action
and has maximum probability. \O-shell" congurations, which are not solutions of the vacuum
Einstein equations, are admitted in the functional integration but are strongly suppressed by the
oscillations of the exponential factor.
Due to the appearance of the dimensional constant
G
in the Einstein action, the most
probable quantum uctuations of the gravitational eld \grow" at very short distances, of the
cannot exceed
G=d
4
. This means that the uctuations of
R
are stronger at short distances { down to
L
Planck
,
the minimum physical distance.
1 in absolute value, therefore
j
R
j
2.1 General features
There is another way, however, to obtain vacuum eld congurations with action smaller than 1
in natural units. This is due to the fact that the Einstein action has indenite sign. Consider the
pure Einstein equations (i.e. without the cosmological term; compare Section
)
R
(
x
)
1
2
g
(
x
)
R
(
x
)=
8
GT
(
x
)
(9)
4
such that the squared four-interval is
x
2
1
order of
L
Planck
=
p
Gh=c
3
10
33
cm
. This led Hawking, Coleman and others to depict
spacetime at the Planck scale as a \quantum foam"
, with high curvature and variable topology.
For a simple estimate (disregarding of course the possibility of topology changes, virtual black
holes nucleation etc.), suppose we start with a at conguration, and then a curvature uctuation
appears in a region of size
d
. How much can the uctuation grow before it is suppressed by the
oscillating factor exp[
iS
]? (We set
h
=1and
c
= 1 in the following.) The contribution of the
uctuation to the action is of order
Rd
4
; both for positive and for negative
R
, the uctuation
is suppressed when this contribution exceeds
and their covariant trace
R
(
x
)=8
G
Tr
T
(
x
)=8
Gg
(
x
)
T
(
x
)
(10)
Let us consider a solution
g
(
x
) of equation (
) with a source
T
(
x
) obeying the additional
integral condition
Z
d
4
x
q
g
(
x
)Tr
T
(
x
) = 0
(11)
Taking into account eq. (
) we see that the Einstein action (
) computed for this solution is
zero. Condition (
can be satised by energy-momentum tensors that are not identically zero,
provided they have a balance of negative and positive signs, such that their total integral is zero.
Of course, they do not represent any acceptable physical source, but the corresponding solutions
of (
) exist nonetheless, and are zero modes of the action.
We shall give two explicit examples of virtual sources: (i) a \mass dipole" consisting of two
separated mass distributions with dierent signs; (ii) two concentric \+/- shells". In both cases
there are some parameters of the source which can be varied: the total positive and negative
masses
m
, their distance, the spatial extension of the sources.
The procedure for the construction of the zero mode corresponding to the dipole is the
following. One rst considers Einstein equations with the virtual source without xing the pa-
rameters yet. Then one solves them with a suitable method, for instance in the weak eld
approximation when appropriate. Finally, knowing
g
(
x
) one adjusts the parameters in such a
way that condition (
) is satised.
2.2 Computation of
q
g
(
x
)
g
00
(
x
)
Now suppose we have a suitable virtual source, with some free parameters, and we want to adjust
them in such a way to generate a zero-mode
g
(
x
)forwhich
S
Einstein
[
g
] = 0. We shall always
consider static sources where only the component
T
00
is non vanishing. The action of their eld
is
Z
d
4
x
q
g
(
x
)
g
00
(
x
)
T
00
(
x
)
S
zeromode
=
(12)
To rst order in
G
, the eld
h
(
x
) generated by a given mass-energy distribution
T
(
x
)
is given by an integral of the eld propagator
P
(
x;y
) over the source:
h
(
x
)=
Z
d
4
yP
(
x;y
)
T
(
y
)
(13)
where in Feynman gauge
P
(
x;y
) is given, with our conventions on the metric signature, by
P
(
x;y
)=
2
G
+
(14)
(
x
y
)
2
+
i"
Computing the integral over time in eq. (
) we obtain for our source
h
(
x
)=
Z
+
1
1
dy
0
Z
d
3
yT
00
(
y
)
P
00
(
x;y
)
00
)
Z
+
1
1
dy
0
Z
d
3
y
=
2
G
(2
0
0
T
00
(
y
)
(
x
0
y
0
)
2
(
x
y
)
2
+
i"
00
)
Z
d
3
y
T
00
(
y
)
j
=2
G
(2
0
0
x
y
j
(15)
5
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