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The Paradox of Virtual Dipoles

in the Einstein Action

Giovanni Modanese 1

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

R d 4 x p g ( x )Tr T ( x ) = 0. These uctuations would exist also at macroscopic scales, with

paradoxical consequences. We set out their general features and give numerical estimates of

possible suppression processes.

04.20.-q Classical general relativity.

04.60.-m Quantum gravity.

There has been considerable interest, in the last years, for the subject of vacuum uctuations

in quantum gravity. Several authors studied the possible occurrence of large uctuations in

2+1 dimensions, in dierent contexts [ 1 ]. Other authors speculated whether the Planck scale

uctuations typical of quantum gravity, the so-called \spacetime foam", generate a noise which

could be observed in certain circumstances [ 2 ]. General phenomenological models for the eects

of the spacetime foam on high-energy scattering and dispersion relations were also proposed [ 3] ,

based on the idea that the amplitudes of these eects might be suppressed just by a M P factor

and not by M P .

In this work we describe 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. This letter is based upon the

paper [ 4] , which in turn settles and renes some partial previous work [ 5] .

These 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,

1 e-mail address: giovanni.modanese@unibz.it

The functional integral of pure Einstein 4D quantum gravity admits abnormally large

and long-lasting \dipolar uctuations", generated by virtual sources with the property

10 6 g . This seems paradoxical, for several reasons, both theoretical and

phenomenological. We have therefore been looking for possible suppression mechanisms. 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.

We shall 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 presence 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 order of

10 33 cm . This led Hawking, Coleman and others to depict spacetime at

the Planck scale as a \quantum foam" [6] , 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 =G ; both for positive and for negative R , the uctuation is suppressed

when this contribution exceeds

G=d 4 .This

means that the uctuations of R are stronger at short distances { down to L Planck , the minimum

physical distance.

There is another way, however, to obtain vacuum eld congurations with action smaller

than 1 in natural units. Consider the Einstein equations and their covariant trace

1 in absolute value, therefore

cannot exceed

R ( x )

2 g ( x ) R ( x )=

8 GT ( x );

(1)

R ( x )=8 G Tr T ( x )=8 Gg ( x ) T ( x ) :

(2)

Then consider a solution g ( x ) of equation ( 1) with a source T ( x ) obeying the additional

integral condition

Z d 4 x q g ( x )Tr T ( x )=0 :

(3)

Taking into account eq. ( 2) we see that the Einstein action computed for this solution is zero.

Condition ( 3) 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 ( 1 ) 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

which last 1 s or more and correspond to the eld generated by a virtual source with size

1 cm and mass

L Planck = p Gh=c 3

be varied: the total positive and negative masses m , their distance, the spatial extension of the

sources.

Suppose we have a suitable 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

S zeromode =

Z d 4 x q g ( x ) g 00 ( x ) T 00 ( x ) :

To rst order in G , one easily nds in Feynman gauge

h ( x )=2 G (2 0 0

00 ) Z d 3 y T 00 ( y )

R d 4 xT 00 ( x )+ o ( G 2 ). Therefore provided the integral of the mass-energy density vanishes,

the action of our eld conguration is of order G 2 , i.e., practically negligible, as we shall see now

with a numerical example. Let us choose the typical parameters of the source as follows:

1 cm ; m

10 k g

10 37+ k cm 1

(4)

10 29+ k ). We assume in general an adiabatic switch-on/o of the source,

thus the time integral contributes to the action a factor . We shall keep (in natural units)

very large, in order to preserve the static character of the eld. Here, for instance, let us take

1 s' 3 10 10 cm . With these parameters one nds

S order G 2

G 2 m 2

r 3

10 20+3 k :

R d 3 xT 00 ( x ) = 0, has negligible action even with k = 6 (corresponding to apparent matter

uctuations with a density of 10 6 g=cm 3 !) This should be compared to the huge action of

the eld of a single , unbalanced virtual mass m ; with the same values we have S single m =

10 47+ k .

This numerical estimate shows that the cancellation of the rst order term in S zeromode

allows to obtain a simple lower bound on the strength of the uctuations. In principle, however,

one could always nd all the terms in the classical weak eld expansion, proportional to G ,

G 2 , G 3 , etc., and adjust T 00 as to have S zeromode = 0 exactly. They can be represented by

those Feynman diagrams of perturbative quantum gravity which contain vertices with 3, 4 ...

gravitons but do not contain any loops. The ratio between each contribution to S and that of

lower order in G has typical magnitude r Schw: =r ,where r Schw: =2 Gm is the Schwarzschild

radius corresponding to one of the two masses and r is the typical size of the source. For a wide

range of parameters, this ratio is very small, so the expansion converges quickly.

As a rst example of unphysical source satisfying ( 3) , consider the static eld produced by

a mass dipole. This consists of a positive source with mass m + and radius r + and a negative

source with mass

m + o ( G 2 )

m and radius r , placed a distance 2 a apart. The radii of the two sources

r Schw: ,where r Schw: is the Schwarzschild radius corresponding to the

mass m + . The action is found to be S Dipole =

R d 4 xT 00 ( x )= ( m + m )+ o ( G 2 ). This

vanishes for m + = m , apart from terms of order G 2 (i.e., our dipoles have in reality a tiny

monopolar component). The values of the masses and the radii r (both of order r )canvary

in a continuous way { provided the condition above is satised. Therefore these (non singular)

It is straightforward to check that p g ( x ) g 00 ( x )=1+ o ( G 2 ) and thus S zeromode =

(implying r Schw: =r

zeromode

Thus the eld generated by a virtual source with typical size ( 4 ), satisfying the condition

R d 4 x p g ( x )Tr T ( x )

are such that a

\dipolar" elds constitute a subset with nonzero volume in the functional integration. In fact,

they are only a small subset of all solutions of the Einstein equations with sources satisfying eq.

( 3 ).

Another example is given by two concentric spherical shells, the internal one with radii

r 1 , r 2 , and the external one with radii r 2 , r 3 ( r 1 <r 2 <r 3 ). Let the internal shell have mass

density 1 and the external shell density 2 , with opposite sign. The condition for zero action

requires, up to terms of order G 2 , that the total positive mass equals the total negative mass,

i.e., 1 ( r 2

r 1 )+ 2 ( r 3

r 2 ) = 0. The spherical symmetry of this source oers some advantages

in the calculations.

One may think that large gravitational uctuations, if real, would not remain unnoticed.

Even though vacuum uctuations are homogeneous, isotropic and Lorentz-invariant, they could

manifest themselves as noise of some kind. Most authors are skeptic about the possibility of

detecting the noise due to spacetime foam [2, 3] , but the virtual dipole uctuations described in

this paper are much closer to the laboratory scale. Observable quantities, like for instance the

connection coecients could then exhibit strong uctuations.

The existence of these uctuations would be paradoxical, however, already at the purely

conceptual level. Common wisdom in particle physics states that the vacuum uctuations in

free space correspond to virtual particles or intermediate states which live very short, i.e., whose

lifetime is close to the minimum allowed by the Heisenberg indetermination relation.

Let us estimate the product E for the dipolar uctuations. The total energy of a static

gravitational eld conguration vanishing at innity is the ADM energy. Since the source of a

dipolar uctuation satises the condition R d 3 xT 00 ( x )=0uptotermsoforder G 2 , the dominant

contribution to the ADM energy is the Newtonian binding energy [ 7] .

The binding energy of the eld generated by a source of mass m and size r is of the order

Gm 2 =r , where the exact proportionality factor depends on the details of the mass

distribution. For a dipolar eld conguration characterized by masses m + and m and radii of

the sources r + and r , the total gravitational energy is of the order of E tot

Gm 2

( r 1

+ r 1

(disregarding the interaction energy between the two sources, proportional to 1 =a

1 =r ). With

10 12+ k cm 1 . Remembering that k can take values

up to k = 6, we nd for these dipolar uctuations

Gm 2

10 28 ! (For comparison, remember

the case of a \monopole" uctuation of virtual mass m and duration . The condition S< 1

implies m< 1. The dominant contribution to the ADM energy is just m ,sotherule E < 1is

respected.)

The Newtonian binding energy of the concentric +/- shells turns out to be of the same

magnitude order, more exactly E = Gm

E tot

(the repulsion between the two shells predominates) and negative if 1 < 2

(the attraction inside each shell predominates). From the physical point of view it is reasonable

to admit { remembering that we are in a weak eld regime and forgetting general covariance for

a minute { that the binding energy is localized within the surface of the outer shell (the eld

is o ( G 2 ) outside). The energy density is therefore of the order of jEj

1 j

2 j

r P ( ), where P ( )

P ( r 3 =r 2 ) is a polynomial which is

10 29+ k cm 4

(with the parameters ( 4 )), and can take both signs. This value looks quite large, even though

the Ford-Roman inequalities [8] or similar bounds do not apply to quantum gravity, where the

metric is not xed but free to uctuate, and there is in general no way to dene a local energy

density.

Concerning possible suppression processes of the dipolar uctuations, here we just quote

the results. The contribution to the cosmological term is S

r 3

r 4

m r 2

10 3+ k ,withthe

parameters above, and the contribution to an R 2 term is of the order of G 2 m 2

=r 3

10 48+2 k .

We see that only the cosmological term can act as a cut-o at macroscopic scales.

of E

+ )

the parameters ( 4) we have E tot

positive if

Acknowledgment - This work was supported in part by the California Institute for Physics

and Astrophysics via grant CIPA-MG7099.

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