**1. Introduction**

In this work we would like to review some concepts developed over the last few years: that the gravitational vacuum has, even at scales much larger than the Planck length, a peculiar structure, with anomalously strong and long-lasting fluctuations called "zero-modes"; and that these vacuum fluctuations are virtual particles of negative mass and interact with each other, leading to the formation of weakly bound states. The bound states make up a continuum, allowing at each point of spacetime the local excitation of the gravitational vacuum through the coupling with matter in a coherent state. The spontaneous or stimulated decay of the excited states leads to the emission of virtual gravitons with spin 1 and large *p*/*E* ratio. The main results on the zero-modes and their properties have been given in (Modanese, 2011), but in this work we expand and discuss in physical terms several important details concerning the zero-mode interactions, the dynamics of virtual particles with negative mass and the properties of virtual gravitons.

Technically, our approach is based on the Lorenzian path integral of Einstein gravity in the usual metric formulation. We take the view that any fundamental theory of gravity has the Einstein action as its effective low-energy limit (Burgess, 2004). The technical problem of the non-renormalizability of the Einstein action is solved in effective quantum gravity through the asymptotic safety scheme (Niedermaier & Reuter, 2006; Percacci, 2009). According to this method, gravity can be nonperturbatively renormalizable and predictive if there exists a nontrivial renormalization group fixed point at which the infinite ultraviolet cutoff limit can be taken. All investigations carried out so far point in the direction that a fixed point with the desired properties indeed exists.

An important feature of the path integral approach is that it allows a clear visualization of the metric as a dynamical quantum variable, of which one can study averages and fluctuations also at the non-perturbative level. It is hard, however, to go much further than formal manipulations in the Lorenzian path integral; after proving the existence of the zeromodes we resort to semi-classical limits and standard perturbation theory. This method is clearly not always straightforward. At several points we proceed, by necessity, through physical induction and analogies with other interactions.

Anomalous Gravitational Vacuum Fluctuations Which Act as Virtual Oscillating Dipoles 3

ˆ ˆ *XE E* exp exp

Fig. 1. Subspace *X* of metrics with constant action. All the metrics (spacetime configurations)

The zero-modes can only give a significant contribution to the path integral if they are not isolated configurations (like a line in 2D, which has measure zero), but a whole fulldimensional subset of all the possible configurations. They are "classical" fields, not in the sense of being solutions of the Einstein equations in vacuum, but in the sense of being functions of spacetime coordinates which are weighed in the functional integral with non-

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

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 non-

physical sources which satisfy some suitable condition, in our case 0 *<sup>v</sup> dx gg T*

follows that such solutions will be zero-modes. The expression 0 *<sup>v</sup> dx gg T*

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

for solutions of the Einstein equations one has (trace of the equations) 4

*SE* . In particular, there exist a subspace whose metrics all have

(3)

*X* its measure. The

. Since

> <sup>8</sup> *<sup>G</sup> <sup>v</sup> R gT c*

is far

, it

*i i I S dg S X*

*X*

*SE* is the constant value of the action in the subspace and

where ˆ

case <sup>ˆ</sup> <sup>0</sup> *SE* is a special case of this.

in *X* have the same action ˆ

zero action.

equation.

vanishing measure.

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

The outline of the work is the following. In Section 2 we show the existence of the zeromodes and discuss their main features, using their classical equation and the path integral. This Section contains some definitely mathematical parts, but we have made an effort to translate all the concepts in physical terms along the way. Section 3 is about the pair interactions of zero-modes: symmetric and antisymmetric states, transitions between these states, virtual dipole emission and its *A* and *B* coefficients. Section 3.3 contains a digression on the elementary dynamics of virtual particles with negative mass. Section 4 is devoted to the interaction of the zero-modes with a time-variable -term. We discuss in detail the motivations behind the introduction of such a term and compare its effect to that of "regular" incoherent matter by evaluating their respective transition rates. Finally, in Section 5 we discuss in a simplified way the properties of virtual gravitons; the virtual gravitons exchanged in a quasi-static interaction are compared to virtual particles exchanged in a scattering process and to virtual gravitons emitted in the decay of an excited zero-mode.
