**2.1 Classical equation of the zero-modes**

Now let us find at least some of these configurations. It is not obvious that eq. (1) has solutions with *R* not identically zero, because it is a difficult non-linear integro-differential equation.

In some previous work we used, to solve (1) in the weak field approximation, a method known as "virtual source method" or "reverse solution of the Einstein equations" (Modanese, 2007). According to this method, one solves the Einstein equations with nonphysical sources which satisfy some suitable condition, in our case 0 *<sup>v</sup> dx gg T* . Since

for solutions of the Einstein equations one has (trace of the equations) 4 <sup>8</sup> *<sup>G</sup> <sup>v</sup> R gT c* , it follows that such solutions will be zero-modes. The expression 0 *<sup>v</sup> dx gg T* is far simpler in the linear approximation. In that case the source must satisfy a condition like, for instance, 00 *dxT* <sup>0</sup> (supposing *T*ii is vanishing) and is therefore a "dipolar" virtual source.

A much more interesting class of zero-modes is obtained, however, in strong field regime, starting with a spherically-symmetric Ansatz. In other words, let us look for spherically symmetric solutions of (1). Consider the most general static spherically symmetric metric

Anomalous Gravitational Vacuum Fluctuations Which Act as Virtual Oscillating Dipoles 5

Fig. 2. Metric of an elementary static zero-mode of the Einstein action. Inside the radius *r*ext (region I) the *g*00 component is constant, and the *g*rr component goes to zero. On the outside (region II) both components have the form of a Schwarzschild solution with negative mass.

Now we can look for metrics close to (9), but with scalar curvature not identically zero. For large *M* and small *L*, the last term in eq. (7) is a small perturbation. Since never diverges and -1 does not appear in the equation, the perturbed solution is not very different from (7). For values of *M* of order 1 or smaller, the equation can be integrated numerically. If we choose a function *L*(*s*) with null integral on the interval (0,1), we obtain a metric which is a zero-mode of the action but not of the lagrangian density. One can take, for instance,

In conclusion, we have found a family of regular metrics with null scalar curvature, depending on a continuous parameter *M* . Furthermore, we have built a set of metrics close to the latter, by solving eq. (7) with *L* arbitrary but having null integral. These metrics do not have zero scalar curvature, but still have null action. They make up a full-dimensional

Our solutions of the zero-mode condition are, outside the radius *r*ext, Schwarzschild metrics with *M*<0. The quantity *Mc*2 coincides with the ADM energy of the metrics. At the origin of the coordinates the component *rr g* goes to zero, the integral of *gR* is finite and also the volume *dx <sup>g</sup>* is finite. The volume inside the radius *r*ext is smaller than the volume of a

According to our previous argument on the functional integral, these metrics give a significant contribution to the quantum averages, although they are neither classical solutions nor quantum fluctuations near the classical solutions. In the vacuum state, there exists a finite probability that the metric at any given point is not flat, but has the form of a zero-mode, i.e., seen from a distance, of a pseudo-particle of negative mass. In the language of Quantum Field Theory this could be called a vacuum fluctuation. Vacuum fluctuations

subset of the functional space (see proof in (Modanese, 2007)).

*L*(*s*)=*L*0sin(2*ns*), with *n* integer.

sphere with the same radius in flat space.

$$d\tau^2 = B(r)dt^2 - A(r)dr^2 - r^2(d\theta^2 + \sin^2\theta d\phi^2) \tag{4}$$

where *A*(*r*) and *B*(*r*) are arbitrary smooth functions. We add the requirement that outside a certain radius *r*ext, *A*(*r*) and *B*(*r*) take the Schwarzschild form, namely

$$B(r) = \left(1 - \frac{2GM}{c^2r}\right);\quad A(r) = \left(1 - \frac{2GM}{c^2r}\right)^{-1}\quad\text{for }r \ge r\_{ext} \tag{5}$$

This requirement serves two purposes: (1) It allows to give a physical meaning to these configurations, seen from the outside, as mass-energy fluctuations of strength *M*. For *r*>*r*ext their scalar curvature is zero. (2) More technically, the Gibbons-Hawking-York boundary term of the action is known to be constant in this case (Modanese, 2007).

Even with only the functions *A* and *B* to adjust, the condition (1) is very difficult to satisfy. We do find a set of solutions, however, if we make the drastic simplification *g*00=*B*(*r*)=*const*. The scalar curvature multiplied by the volume element becomes in this case

$$L = \sqrt{g}R = -8\pi\sqrt{|BA|} \left(\frac{rA'}{A^2} + 1 - \frac{1}{A}\right) \tag{6}$$

Apart from the constant *c*4/8*G*, *L* is the lagrangian density of the Einstein action, computed for this particular metric. Let us fix arbitrarily a reference radius *r*ext, and introduce reduced coordinates *s*=*r*/*r*ext. Define an auxiliary function =*A*-1. Regarding *L*(*s*) as known, eq. (6) becomes an explicit first-order differential equation for :

$$\alpha' = \frac{1}{s} - \frac{\alpha}{s} + \frac{L\sqrt{|\alpha|}}{8\pi s\sqrt{|B|}}\tag{7}$$

The boundary conditions (5) are written, in reduced coordinates

$$B(\mathbf{s} \ge 1) = \left(1 - \frac{\tilde{M}}{s}\right); \quad A(\mathbf{s} \ge 1) = \left(1 - \frac{\tilde{M}}{s}\right)^{-1} \tag{8}$$

where *M* is a free parameter, the total mass in reduced units: <sup>2</sup> *<sup>M</sup>* 2 / *GM c rext* . In the following we shall take *<sup>M</sup>* <sup>0</sup> , in order to avoid singularities. For *ext r r* , we have *BB M* (1) 1 .

It is interesting to note that putting *L*=0 in eq. (7) we can easily find an exact solution, ie a non-trivial static metric with *R*=0. Namely, if 1->0, then =1-econst/*s*, which does not satisfy the boundary condition; if 1-<0, then =1+econst/*s*, implying *cost e M* . The resulting *grr* component has the same form on the left and on the right of *s*=1, namely

$$\mathcal{g}\_{rr} = \left(1 + \frac{\lfloor \tilde{M} \rfloor}{s}\right)^{-1} \tag{9}$$

while *g*00 is constant and equal to (1 | |) *M* for *s*<1, and is equal to (1 | |/ ) *M s* for 1 *s* . Note that *g*rr goes to zero at the origin.

4 Quantum Gravity

2 2 222 2 2 *d B r dt A r dr r d d*

( ) ( ) ( sin )

where *A*(*r*) and *B*(*r*) are arbitrary smooth functions. We add the requirement that outside a

2 2 2 2 ( ) 1 ; ( ) 1 for *ext GM GM B r A r r r c r c r* 

This requirement serves two purposes: (1) It allows to give a physical meaning to these configurations, seen from the outside, as mass-energy fluctuations of strength *M*. For *r*>*r*ext their scalar curvature is zero. (2) More technically, the Gibbons-Hawking-York boundary

Even with only the functions *A* and *B* to adjust, the condition (1) is very difficult to satisfy. We do find a set of solutions, however, if we make the drastic simplification *g*00=*B*(*r*)=*const*.

> <sup>1</sup> 8| | 1 *rA L gR BA <sup>A</sup> <sup>A</sup>*

Apart from the constant *c*4/8*G*, *L* is the lagrangian density of the Einstein action, computed for this particular metric. Let us fix arbitrarily a reference radius *r*ext, and introduce reduced coordinates *s*=*r*/*r*ext. Define an auxiliary function =*A*-1. Regarding *L*(*s*) as known, eq. (6)

1 | |

( 1) 1 ; ( 1) 1 *M M B s A s*

where *M* is a free parameter, the total mass in reduced units: <sup>2</sup> *<sup>M</sup>* 2 / *GM c rext* . In the following we shall take *<sup>M</sup>* <sup>0</sup> , in order to avoid singularities. For *ext r r* , we have

It is interesting to note that putting *L*=0 in eq. (7) we can easily find an exact solution, ie a non-trivial static metric with *R*=0. Namely, if 1->0, then =1-econst/*s*, which does not satisfy

> | | <sup>1</sup> *rr M*

while *g*00 is constant and equal to (1 | |) *M* for *s*<1, and is equal to (1 | |/ ) *M s* for 1 *s* .

*s* 

8 || *L s s s B* 

*s s* 

1

2

 

1

(5)

(4)

(6)

(7)

1

(8)

*e M* . The resulting *grr*

(9)

certain radius *r*ext, *A*(*r*) and *B*(*r*) take the Schwarzschild form, namely

term of the action is known to be constant in this case (Modanese, 2007).

The scalar curvature multiplied by the volume element becomes in this case

The boundary conditions (5) are written, in reduced coordinates

the boundary condition; if 1-<0, then =1+econst/*s*, implying *cost*

component has the same form on the left and on the right of *s*=1, namely

*g*

becomes an explicit first-order differential equation for :

*BB M* (1) 1 .

Note that *g*rr goes to zero at the origin.

Fig. 2. Metric of an elementary static zero-mode of the Einstein action. Inside the radius *r*ext (region I) the *g*00 component is constant, and the *g*rr component goes to zero. On the outside (region II) both components have the form of a Schwarzschild solution with negative mass.

Now we can look for metrics close to (9), but with scalar curvature not identically zero. For large *M* and small *L*, the last term in eq. (7) is a small perturbation. Since never diverges and -1 does not appear in the equation, the perturbed solution is not very different from (7). For values of *M* of order 1 or smaller, the equation can be integrated numerically. If we choose a function *L*(*s*) with null integral on the interval (0,1), we obtain a metric which is a zero-mode of the action but not of the lagrangian density. One can take, for instance, *L*(*s*)=*L*0sin(2*ns*), with *n* integer.

In conclusion, we have found a family of regular metrics with null scalar curvature, depending on a continuous parameter *M* . Furthermore, we have built a set of metrics close to the latter, by solving eq. (7) with *L* arbitrary but having null integral. These metrics do not have zero scalar curvature, but still have null action. They make up a full-dimensional subset of the functional space (see proof in (Modanese, 2007)).

Our solutions of the zero-mode condition are, outside the radius *r*ext, Schwarzschild metrics with *M*<0. The quantity *Mc*2 coincides with the ADM energy of the metrics. At the origin of the coordinates the component *rr g* goes to zero, the integral of *gR* is finite and also the volume *dx <sup>g</sup>* is finite. The volume inside the radius *r*ext is smaller than the volume of a sphere with the same radius in flat space.

According to our previous argument on the functional integral, these metrics give a significant contribution to the quantum averages, although they are neither classical solutions nor quantum fluctuations near the classical solutions. In the vacuum state, there exists a finite probability that the metric at any given point is not flat, but has the form of a zero-mode, i.e., seen from a distance, of a pseudo-particle of negative mass. In the language of Quantum Field Theory this could be called a vacuum fluctuation. Vacuum fluctuations

Anomalous Gravitational Vacuum Fluctuations Which Act as Virtual Oscillating Dipoles 7

*<sup>c</sup> <sup>c</sup> rA S d x g x R x dt dr BA GG A <sup>A</sup>* 

where *A g rr* and *B g* 00 are functions of *r*. Define a radius *ext r* , the "external radius" of our configurations, on which we impose boundary conditions as in Sect. 2.1. This means that we integrate over configurations which outside the radius *ext r* appear like Schwarzschild metrics with mass *M*. In order to avoid singularities, we suppose 0 *M* . We can re-write the action as an integral on *r* with upper limit *r*ext, because the scalar curvature of the Schwarzschild metric is zero. We can also add the Gibbs-Hawking-York boundary term, which in this case takes the form *S M dt HGY* . For a fixed time interval, we can regard the

*g x*( ) 

Supposing *B* constant ( *B M* 1 ), the path integral over these field modes is written

1

4 ' 1 exp 1 exp

*<sup>i</sup> <sup>B</sup> sA <sup>i</sup> d A dt ds A M dt G A A*

The second exponential can be disregarded in the functional averages, because it cancels with the normalization factor in the denominator. In the first exponential, let us define a

0

0 0

*jA A d A e dA i A*

<sup>1</sup> <sup>1</sup>

*j j j j*

*<sup>N</sup> <sup>N</sup> j j*

 

*d ij*

2 1 exp 2

*j j*

   

 (14)

0 <sup>4</sup> ' 1 ()() <sup>1</sup>

> 

> > 2

and discretize the integral in *ds*. We divide the integration

2

*A A*

 

and replace the integral with a sum,

1 / *A* . This gives the

*j j j j*

 (13)

<sup>1</sup> <sup>1</sup>

<sup>1</sup> [ ] exp <sup>1</sup> *<sup>E</sup>*

The presence of the square root and of the fractions with *Aj* makes the integrals very complicated. Let us change variables. Suppose 0 *A* , which is physically a widely justified assumption (and remember we are looking for a sufficient condition, i.e. we want to show that there exist a set of gravitational configurations for which the functional average of a

*<sup>S</sup> <sup>N</sup> <sup>N</sup> j j i j*

(11)

2

(12)

For these metrics the Einstein action is written (Sect. 2.1)

*E HGY*

*dt*

where the derivative is written as a finite variation. We obtain

quadratic quantity is different from the classical value). Define

3 0 0 0

exp

*<sup>i</sup> dA S S*

*G* 

interval [0,1] in ( 1) *N* small intervals of length

*i*

8 *<sup>E</sup>*

integral *dt* as a constant.

constant factor <sup>1</sup> <sup>4</sup> *<sup>B</sup>*

new path integral

4

4 4

are created spontaneously and at zero energetic cost at any point of spacetime, in a homogeneous and isotropic way. Usually vacuum fluctuations have a very short life, as can be shown through the Schroedinger equation (time-energy uncertainty principle) or through a transformation to Euclidean time (when the action is positive-definite). These arguments on the lifetime of the fluctuations can not be applied here, because quantum gravity has neither a local Hamiltonian, nor a positive-definite action. Our fluctuations, if they were completely isolated, would be independent of time; in fact, their interaction causes a finite lifetime (Sect. 2.3). In Sect. 5 we shall give a comparison between this kind of vacuum fluctuations and other fluctuations present in quantum gravity, like the virtual gravitons which transmit the gravitational interactions.

In order to avoid a large global curvature, the *total* average effect of the virtual masses of the zero-modes must inevitably be renormalized to zero. This is, in our view, guaranteed by the "cosmological constant paradox": nature appears to be endowed with a dynamical mechanism which relaxes to zero any constant positive or negative contributions to the vacuum energy density, coming from particle physics or even from gravity itself. So, even though such contributions are formally infinite, in the end they do not affect the curvature of spacetime. The full explanation of this mechanism can only be achieved within a complete non-perturbative theory of Quantum Gravity. Some partial evidence of the dynamical emergence of flat spacetime has been obtained in the lattice theory, and in effective field theory approaches (Hamber, 2004, Dolgov, 1997).

Therefore we shall not be concerned with the global effect of our massive vacuum fluctuations on spacetime. We shall instead consider their interactions, which result in a novel pattern of purely gravitational excited states, above a ground state in which all fluctuations pairs with equal mass are in a symmetrical superposition. Freely speaking, it's a bit like studying the local effects of pressure variations, without worrying about how the total force due to atmospheric pressure affects the Earth.
