**3.3 Digression: Elementary dynamics of virtual particles with negative mass**

Real particles with negative mass cannot exist, because they would make the world terribly unstable, popping up spontaneously from the vacuum with production of energy. In this work, however, we hypothesize the existence of long-lived virtual particles with negative mass, whose creation from the vacuum does not require or generate any energy. We recognize that these virtual particles have negative mass by looking at their metric at infinity, which is Schwarzschild-like, but with negative *M* and negative ADM energy. We know that the dynamics of virtual particles, after their creation, is similar to that of real particles, and we have computed quantum amplitudes involving them.

We do not know any general principle about the "classical" dynamics of virtual particles with negative mass. Actually, virtual particles of this kind are an emergent phenomenon guessed from the path integral and can only be observed in a very indirect way. It is interesting, nonetheless, to make some reasonable hypothesis and check the consequences. Our basic assumption will be the following: for an isolated system comprising particles with positive and negative mass, the position of the center of mass, defined by

$$\mathbf{x}\_{\text{CM}} = \sum\_{i} \mathbf{M}\_{i} \mathbf{r}\_{i} \tag{25}$$

Anomalous Gravitational Vacuum Fluctuations Which Act as Virtual Oscillating Dipoles 15

which has a positive solution *E*g*E,* independently from . Furthermore, the recoil velocity *v*r turns out to be always non-relativistic. This means that the recoil of the zero-modes can always ensure conservation of momentum, independently from the value of the graviton

> 

*dx dx T m*

*d d* 

 

4

(29)

(28)

*c*

symmetric to an anti-symmetric state) can occur by absorption of a virtual graviton or by coupling to an external source. It is easy to show (Sect. 4.3) that the coupling of zero-modes

(Note that certain interactions between zero-modes and massive particles vanish exactly for symmetry reasons. For instance, a particle in uniform motion can never "lose energy in collisions with the zero-modes", because in its rest reference system the particle will see the vacuum, zero-modes included, as homogeneous and isotropic. There are possible exceptions

The coupling to a (*t*) term, or local time-dependent vacuum energy density, can lead to a significant transition probability. This is due to the presence of the non-linear *g* factor in the coupling, and corresponds physically to the fact that such a term does not describe

The Einstein equations with a cosmological constant, or vacuum energy term, are written

1 8

4 4

*c c S dx gR dx <sup>g</sup>*

*G G*

In this paper with use metric signature (+,-,-,-). With this convention, the cosmological

In perturbative quantum gravity on a flat background, this value of corresponds to a small real graviton mass (Datta *et al*., 2003, and ref.s). Actually, in the presence of a curved background the flat space quantization must be replaced by a suitable curved-space quantization (Novello & Neves, 2003). The limit *m*0 of a theory with massive gravitons is tricky, so this global value of still represents a challenge for quantum gravity (besides the

In our previous work we introduced the idea that at the *local* level, the coupling of gravity with certain coherent condensed-matter systems could give an effective local positive

 

4 4

*<sup>G</sup> R gR g T*

to this argument: accelerated particles, or particles in states with large *p* uncertainty.)

(transition of a zero-modes pair from a

is exceedingly weak.

with emission of

**4. Interaction of the zero-modes with a variable -term** 

In Sect. 3 we have computed the probability of the decay process

energy-momentum ratio .

a virtual graviton. The excitation process

to "ordinary" matter with energy-momentum

isolated particles, but coherent, delocalized matter.

**4.1 Summary of conventions and of some previous results** 

The corresponding action (without the boundary term) is

need to explain its origin; compare Sect. 2.1).

2

 

8 8 *<sup>E</sup>*

(repulsive) background experimentally observed is of the order of *c*4/*G*=-10-9 J/m3.

is invariant in time. From this assumption one can prove in a straightforward way several strange properties of particles with negative mass. These properties can be summarized by saying that in the usual dynamical rules their mass really behaves like a negative number, namely: (a) The acceleration of the virtual particle is opposite to the applied force. (b) The momentum is opposite to the velocity. (c) The kinetic energy is negative. The kinetic energy is defined as usual through the work of the applied force, in such a way that the sum *E*kin+*E*pot is conserved.

Applying these rules one obtains a bizarre behaviour in the scattering processes and in the dynamics. For instance, although the gravitational potential energy of two virtual particles with negative mass is negative, *E*pot =-*GM*1*M*2/*r* (compare Sect. 3.1), the two particles experience a repulsion, due to Property (a). They tend to run away from each other; while their distance increases, their *E*pot decreases in absolute value, and their (negative) *E*kin increases in absolute value. If the particles were initially at rest at some distance *r*0 (Fig. 5), when their distance goes to infinity they gain a *E*kin equal to their initial *E*pot.

Fig. 5. "Classical" motion of two virtual particles with negative mass initially at rest at distance *r*0. Although their potential energy is negative, they feel a repulsion and their (negative) kinetic energy increases in absolute value as their distance goes to infinity.

In the decay (Sect. 3.2) the momentum of the emitted graviton is balanced by the recoil of the zero-modes (in the same direction of the emission). The conservation equations give

$$\begin{cases} \Delta \boldsymbol{v}\_r^2 + \boldsymbol{E}\_\mathcal{g} = \Delta \boldsymbol{E} \\ 2\Delta \boldsymbol{w}\_r - \boldsymbol{p}\_\mathcal{g} = 0 \end{cases} \tag{26}$$

where *E* is the energy gap, *E*g and *p*g are the graviton energy and momentum, *v*r is the recoil velocity of the zero-mode and 2*M*-10-13 kg is the zero-mode mass. After replacing *p*g=*E*g, the system (26) leads to the equation

$$\frac{1}{M\alpha^2} E\_g^2 + E\_g - \Delta E = 0 \tag{27}$$

which has a positive solution *E*g*E,* independently from . Furthermore, the recoil velocity *v*r turns out to be always non-relativistic. This means that the recoil of the zero-modes can always ensure conservation of momentum, independently from the value of the graviton energy-momentum ratio .
