###### Abstract

In theories with a hidden ghost sector that couples to visible matter through gravity only, empty space can decay into ghosts and ordinary matter by graviton exchange. Perturbatively, such processes can be very slow provided that the gravity sector violates Lorentz invariance above some cut-off scale. Here, we investigate non-perturbative decay processes involving ghosts, such as the spontaneous creation of self-gravitating lumps of ghost matter, as well as pairs of Bondi dipoles (i.e., lumps of ghost matter chasing after positive energy objects). We find the corresponding instantons and calculate their Euclidean action. In some cases, the instantons induce topology change or have negative Euclidean action. To shed some light on the meaning of such peculiarities, we also consider the nucleation of concentrical domain walls of ordinary and ghost matter, where the Euclidean calculation can be compared with the canonical (Lorentzian) description of tunneling. We conclude that non-perturbative ghost nucleation processes can be safely suppressed in phenomenological scenarios.

hep-th/0512274

Non-perturbative materialization of ghosts

Roberto Emparan and Jaume Garriga

Departament de Física Fonamental

Universitat de Barcelona, Diagonal 647, E-08028, Barcelona, Spain

Institució Catalana de Recerca i Estudis Avançats (ICREA)

,

## 1 Introduction

Recently it has been suggested that the smallness of the observed cosmological constant can be attributed to an approximate “energy symmetry” [1]. The idea is that Nature is endowed with an exact copy of the matter sector, but with an overall minus sign in the action [2, 1],

(1) |

Here, is the metric, are ordinary matter fields (including those of the Standard Model), and are the ghost fields. Energy parity is defined by Ignoring gravity, the Hamiltonian transforms as

(2) |

The vacuum state is defined as parity invariant, and from (2) the corresponding vacuum energy vanishes to all orders in perturbation theory. However, gravity breaks the energy symmetry and a cosmological constant is induced. It was argued in [1] that the magnitude of this vacuum energy can be comparable to the one suggested by observations provided that the gravitational cut-off scale is sufficiently low (lower than the inverse of 30 microns, about an order of magnitude beyond the reaches of ongoing short distance probes of gravity).

“Phantom” matter has also been invoked in phenomenological studies of dark energy [3, 4, 5], as a way of obtaining an effective equation of state parameter . This violates all the standard energy conditions, but is not at all disfavoured by observations. A ghost sector is also present in the recently proposed B-inflation, based on effective theories with only second derivatives of a scalar field [6].

In all of these cases, the Hamiltonian is unbounded below, and disaster would follow unless one postulates that the Lorentz symmetry is broken at a certain energy scale [5]. The reason is simple. In the model (1), empty space can decay into a pair of ordinary particles and a pair of ghost particles

(3) |

(if particles are charged, one in each pair should be understood as the antiparticle). Let the momenta of the ordinary particles be and , and the momenta of the ghost particles and . From translation invariance, the decay amplitude takes the form , which after integration over external momenta leads to the vacuum decay rate per unit volume

(4) |

where . In a Lorentz invariant theory, is just a function of , and Defining , we have

(5) |

Physically, the last integral corresponds to the fact that there
is no preferred reference frame, and the total momentum
of the pair of particles (or the pair of ghosts) is equally likely
to fall anywhere on the mass-shell of radius . Particles
only interact with ghosts gravitationally, and so the momentum
is transferred by gravitons. The decay rate is in
principle infinite (due to the mass-shell integral) but it can be
rendered finite if we postulate that Lorentz invariance is broken
in the gravitational sector at some scale [5]. The
remaining integral over can be finite in a theory where
gravity becomes soft at a certain cut-off scale , as it is in
fact assumed. The process becomes completely negligible if is comparable to the cut-off scale discussed above [1]. Similarly, empty
space can decay to ghosts and gravitons
,^{1}^{1}1Ref. [1] actually considered the decay , where is an “excited”
(soft-scale) graviton, which is a more dominant process than
(6). We shall not consider the non-perturbative analogue
of this process, since it cannot be described in terms of the low
energy effective action (1).

(6) |

In this case, the integrals over the momenta of the external gravitons must be cut-off at the Lorentz violating energy scale . In this way, the vacuum can be made sufficiently stable to perturbative decay processes, in spite of the ghosts [1, 4].

Although perturbative processes may be suppressed by the Lorentz-violating physics, it is conceivable that non-perturbative processes may quickly destabilize the present vacuum, through the production of lumps of non-relativistic ghost matter. The purpose of the present paper is to investigate the non-perturbative analogues of (3) and (6). Decays that proceed via non-perturbative tunneling are typically slower than their perturbative counterparts, but when ghosts are involved there are several reasons why this is not so obvious.

In accordance with the equivalence principle, a lump of ghost matter tends to fall towards the potential well created by a positive energy object. On the other hand, the repulsive gravitational field it produces tends to push the positive energy object away. It has been known for some time that this leads to a runaway behaviour, where the positive energy object is chased after by the ghost [7, 8, 9, 10], with a constant acceleration. Such configuration is known as a Bondi dipole [7]. As we shall see, such self-accelerating solutions can be continued to the Euclidean section, leading to a semiclassical description of the spontaneous nucleation of pairs of Bondi dipoles. This would be the non-perturbative analogue of (3). A simple estimate (which we will confirm by rigorous calculation) gives the Euclidean action of this process as , where is the size of the dipole and is the mass of its ordinary positive-mass component. Even if we impose , we see that the action can dangerously approach a value of order one if the Compton wavelength of the particles produced is also close to the gravitational cutoff.

The analogue of (6) is the pair creation of self-gravitating lumps of ghost matter, which repel each other. The possibility of this process is suggested by the following weak field argument. The interaction energy of two ghost particles at rest, with identical mass , is given by . Here is Newton’s constant and is the distance between the masses. For the positive gravitational energy is equal to minus the rest mass energy of the pair , so this configuration can in principle pop out of the vacuum without violating energy conservation. Also, the initial acceleration of each particle is given by , suggesting that there is a Euclidean solution where the ghost matter runs around a circle of radius . Note, however, that for the gravitational field is of order one, and non-linear gravity must be taken into account. The corresponding instantons still exist, and have interesting peculiarities which make their interpretation non-trivial. First of all, they can produce a topology change, and second, the corresponding Euclidean action (defined as the bounce action minus the background action) can be negative.

To shed some light into the meaning of such peculiarities, we shall first consider the simpler example of vacuum decay in a theory where the matter and the ghost sectors support domain wall solutions. In this case, the process of spontaneous nucleation of concentrical spherical domain walls of ordinary and ghost matter chasing after each other is the analogue of (3). The analogue of (6) is the spontaneous creation of spherical domain walls of ghost matter. If the cosmological constant is exactly vanishing, the instanton for the latter process changes the topology of space and a new boundary appears inside of the domain wall. This makes the calculation of the corresponding Euclidean action somewhat ill-defined. On the other hand, in the presence of a small cosmological constant (such as the one present in our universe), the topology does not change at all and the Euclidean action can be calculated unambiguously. Interestingly, it turns out to be negative. These features are quite similar to what happens in the case of pair creation of lumps of ghost matter, but the advantage here is that the geometry is much simpler, and the Euclidean calculation can be compared with a canonical (Lorentzian) description of tunneling.

The plan of the paper is the following. Since the subject of tunneling in theories with ghosts is fraught with many subtleties, we have developed it in quite some detail in Sections 2 through 4. The specific application of our results to the energy-symmetric scenario of [1] is then discussed in the concluding Section 5, to which phenomenologically-minded readers might want to jump directly if not interested in the theory of ghost tunneling.

The more technical sections 2–4 consist of a discussion of: the spontaneous nucleation of domain walls (Section 2); the pair creation of Bondi dipoles, i.e., the non-perturbative analogue of (3) (Section 3, where also we briefly review several technical aspects of the axisymmetric class of solutions for the benefit of readers unfamiliar with them); the spontaneous creation of self-gravitating lumps of ghost matter, i.e., the non-perturbative analogue of (6) (Section 4). Some details of the canonical WKB construction of tunneling paths are deferred to an Appendix.

## 2 Ghosts through the tunnel

In field theory, semiclassical tunneling rates are usually estimated through the expression

(7) |

Here, is the action of the Euclidean instanton describing the decay, minus the action of the background, and we have omitted the prefactor arising from integration of fluctuations around these solutions. For ordinary fields, with the rotation , the Euclidean action is positive definite, , and the above formula gives an exponential suppression in the limit when the semiclassical approximation is valid, . With the same rotation , the Euclidean action for ghost matter is negative definite, , and from a naive application of (7) one may be tempted to conclude that ghosts lead to catastrophic decay rates. However, this conclusion would be premature. Rather, the problem is that the Euclidean path integral is ill-defined: in order to make it convergent, ordinary matter and ghosts would require opposite Wick rotations. Hence, the standard Euclidean methods are not directly applicable in the present context.

For instance, in the limit when gravity is neglected , the theory (1) is symmetric under energy parity. In this limit, ordinary and ghost matter are decoupled and have exactly the same dynamics. Note that, as a consequence, in e.g., Schwinger pair production, it is just as hard to screen a ghost electric field by nucleation of charged ghosts, as it is to screen an ordinary electric field by nucleation of ordinary charged particles. So in both cases, instanton processes must be exponentially suppressed, as

(8) |

Thus, when gravity is switched off, nonperturbative processes in the ghost sector will not bring any disaster, even if with the standard rotation the Euclidean action is negative.

When gravity is switched on, both sectors are coupled and, as mentioned above, the standard Euclidean methods do not apply. Hence, we should try to develop some understanding of the problem from the canonical approach to tunneling. In the WKB approximation, the wave function is of the form , where for a simple quantum mechanical system , and the integral is taken along a semiclassical trajectory. Under the barrier, the momentum is imaginary and becomes a superposition of growing and decaying exponentials. We are interested in the situation where the wave function is outgoing after tunneling, so generically we have a comparable contribution of the growing and decaying modes at the turning point after the barrier. This means that the amplitude after the barrier is exponentially smaller than the amplitude before the barrier

(9) |

Here, is the difference in evaluated at the two turning points. From this perspective, we should expect that a tunneling process is suppressed, whether it involves ghosts or not.

In order to gain some intuition on this problem, we shall consider in the following subsections the spontaneous nucleation of spherical domain walls. The dynamics of domain walls is sufficiently simple to be discussed in the canonical formalism. First, in Subsections 2.2, 2.3 and 2.4 we shall describe some instanton solutions which in the standard interpretation would correspond to the nucleation of walls in flat and in deSitter space. In Subsection 2.4 we consider the same processes in the canonical approach, without reference to Euclidean methods. Finally, in Subsection 2.5, and in the light of the examples considered, we ellaborate on the possible relation between the Euclidean action and the nucleation rates.

### 2.1 Nucleation of diwalls from flat space

The gravitational field of an ordinary domain wall is repulsive [11]. On the other hand, a ghost wall will be attractive, and we may expect to find solutions where a wall of ghost matter is chased after by a wall of usual matter. By analogy with the Bondi dipoles discussed in the introduction, we may call such a configuration a diwall.

Domain walls are rather easy to treat as distributional sources in General Relativity. Their effect is a discontinuity in the extrinsic curvature accross the worldsheet [12]

(10) |

where is the tension of the wall and is the induced metric. Consider a Euclidean spherically symmetric solution with a wall of positive tension and a wall of negative tension . The metric takes the form

(11) |

where is the line element on the 3-sphere and is a radial coordinate. Outside the sources, the metric is flat, and the warp factor is piecewise linear with slope . At the location of the sources, the slope is discontinuous, to account for the jump in the extrinsic curvature. Hence, starting from the center of symmetry at the solution is given by (see fig. 1)

(12) |

Eq. (10) demands that and . The radius of the 3-spheres increases up to , backtracks to , and then increases to infinity. Note that , which requires For our illustrative purposes, we shall simply assume that the theory supports domain walls satisfying this inequality.

The solution Eq. (12) is perfectly regular and asymptotically flat. It can be thought of as a semiclassical trajectory which interpolates between flat empty space (at infinity), and the equatorial slice of the metric (11). This “turning point” slice contains two concentrical domain walls of radii and .

After nucleation, the evolution is given by the analytic continuation of (11) to Lorentzian time, which converts the 3-spheres into time-like hyperboloids. In particular, the positive and negative tension walls expand with constant proper acceleration and respectively. Due to the peculiar “backtracking” form of the metric between and , it is the wall of larger radius (and positive tension) which chases after the one of smaller radius (and negative tension) as they both expand. This is as it should be, since the wall of positive tension is repulsive and the other one is attractive.

The Euclidean action can be easily calculated from

(13) |

where is the sign of the wall tension (and is as usual its absolute value). On shell, the Ricci scalar is related to the source [13]

(14) |

and integrating over the volume of the 3-spheres, we easily find

(15) |

Note that the Euclidean action is negative. In the limit , we have . This is likely to be very large, unless the wall tensions are very nearly degenerate, or unless they are very close to the Planck scale .

### 2.2 Ghost walls from flat space?

An instanton for a single spherical domain wall of negative tension in an asymptotically flat space can be constructed along the same lines as in the previous subsection. The warp factor is now given by (see fig. 2)

(16) |

From large radii (), the worldsheet is seen as a spherical object of radius embedded in an otherwise flat Euclidean space. However, if we cross the worldsheet towards negative , we discover that the radius does not shrink to zero. Rather, it grows as we go in. This “throat” is of course an identical copy of the geometry for . The topology of the instanton is not , but , and the solution has two disconnected boundaries, one at and another one “inside” the domain wall, at .

Calculating the Euclidean action as we did in the previous subsection would yield

(17) |

But the fact that the instanton has different topology than the background should give us pause. The Gibbons-Hawking boundary term at the new inner boundary () is given by , where is some large cut-off radius. It is unclear what subtraction (if any) should be performed on this divergent contribution, since the original flat space background does not include such inner boundary. Also, from a canonical point of view, it doesn’t seem possible to go from the original to the final turning point geometry (with two asymptotically flat regions) by continuous slicing of the instanton. Because of that, the interpretation of the solution (16) as a semiclassical path describing the nucleation of a ghost wall remains unclear.

Fortunately, the situation is clarified by considering nucleation from a vacuum with a small cosmological constant.

### 2.3 Walls from deSitter space

As mentioned in the introduction, gravity breaks the energy symmetry (2). This may give rise the small observed cosmological constant . In this case, the Euclidean background is the 4-sphere of radius , with warp factor

(18) |

Nucleation of positive tension domain walls in deSitter has been thoroughly studied in Ref. [13]. The relevant instanton is constructed from two caps of the 4-sphere, joined at the worldsheet of the domain wall (which is an ),

(19) |

The angular span of the spherical cap is determined through the junction condition (10)

(20) |

where is the tension of the domain wall. Thus, the radius of the worldsheet is given by

(21) |

The Euclidean action can be calculated from (13) with an additional vacuum energy term 14) as well as the bulk equations of motion . . On shell, we can use (

Properly speaking, this instanton would describe the creation from nothing of a deSitter space containing a wall. If we want to describe instead the transition from empty deSitter to deSitter with a wall, then we must face the well-known problem that no non-singular instanton mediates between both spacetimes. For the time being, we will follow the common procedure of calculating the action for the whole process by subtracting the action of the background deSitter,

(22) |

This will be justified in more detail in the next subsection. Some straightforward algebra leads to [13]

(23) |

Let us now consider the nucleation of negative tension walls in de Sitter. In fact, Eqs. (20-23) still hold, with the replacement . The angle , determined by

(24) |

will be larger than , so instead of having two spherical caps glued to the worldsheet, it will now be two capped spheres which are glued. The corresponding euclidean action is negative

(25) |

Unlike the case of ghost walls from flat space discussed in the previous subsection, the topology of the instanton is here the same as for the deSitter background, and no additional boundaries appear. Note that in the limit of a small cosmological constant , the action (25) tends to negative infinity, unlike the naive result (17).

In the limit of small tension, where the walls do not deform the geometry of deSitter, the equations of motion for positive and negative tension are exactly the same, and therefore it is expected on general grounds that it is just as hard to create a positive tension wall as it is to create a negative tension one. Here, we find that has the same form in both cases, which supports the use of Eq. (8) [rather than Eq. (7)] for the calculation of the nucleation rates.

It is interesting to observe also that has the same form for both signs of the tension even when this tension is large and the background geometry is considerably deformed (note that the shape of the instantons for positive and negative tension walls is quite different in this case).

### 2.4 Canonical approach

In this subsection, we shall consider the processes discussed in the previous ones in the canonical WKB approach, without reference to Euclidean methods.

Let us consider the system of a spherically symmetric domain wall in the presence of a cosmological constant . Inside the wall, the metric is just empty deSitter space, whereas outside the wall, and by Birkhoff’s theorem, the metric is Schwarzschid-deSitter, characterized by a mass parameter . This system has a single degree of freedom, which is the radius of the wall. A domain wall of very small radius, and correspondingly very small mass , can tunnel to a big domain wall of size comparable to the cosmological horizon (while the mass parameter remains of course a small constant). In the limit when , this process corresponds to the spontaneous nucleation of a large domain wall (from an infinitessimally small seed) in an otherwise empty deSitter space. This is schematically represented in Figs. 3 and 4 for the case of ordinary walls and ghost walls respectively.

In order to describe this process, we shall closely follow the procedure developed in Ref. [14]. The action is given by

(26) |

The spherically symmetric metric takes the form

(27) |

where and are functions of and , and is the metric on the two-sphere. The action can be written as

(28) |

where is the radial coordinate of the wall at time , the momenta

(29) |

are conjugate to and respectively, and the Hamiltonian densities are given in terms of the canonical variables as

(31) |

Here , and the primes indicate derivatives with respect to . Time derivatives of lapse and shift do not appear in the action, which leads to the constraints . When imposed on the wave function, this leads to

Derivative of the action with respect to lapse and shift gives the constraints .

With the ansatz

the WKB approximation is obtained from a solution of the Hamilton-Jacobi equations

We are interested in a solution which interpolates between suitable turning points.

Away from the source at , we may define

(32) |

and we have , where we have used the Hamiltonian constraints. This implies that is constant for . At the turning point geometries the momenta vanish, , and one has

(33) |

where . This is of course the relation between the intervals of proper length and proper radius of the 2-spheres in the Schwarzschild-(A)dS geometry (in static coordinates). We are mostly interested here in the case with , and with , where turning point geometries and before and after the tunneling will be the equatorial slices of the 4-sphere and of the domain-wall deSitter solution. The functions and are given by Eqs. (18) and (19) respectively.

From (32), we have

(34) |

where we have introduced

At the wall , the constraints are consistent with and being continuous, while they imply the following discontinuities for and :

(35) |

[note that at the turning point, the first equation is trivial since the momenta vanish, and the second reproduces Eq. (10)]. Moreover, from Eq. (31) we have

(36) |

which by virtue of (35) does not have delta-function contributions on the wall. Note that from (34) and (36) we can write and in terms of and . In particular

(37) |

where and is given by (34). Also, the momentum is determined by (35) in terms of and in the neighborhood of .

The solution of the Hamilton-Jacobi equation satisfies

(38) |

for arbitrary variations and . Under the barrier, where , we may define [14]

(39) |

where we have introduced the function

(40) |

Here, the inverse cosine is defined in the range (so that the sine is positive). It is straightforward to check that under arbitrary variations of the arguments of , we have

(41) |

The second term in the right hand side arises from partial integration in the variation with respect to , and it is non-vanishing because is discontinuous across the wall. The square brackets in the last term indicate the discontinuity (that is, the quantity evaluated at minus the quantity evaluated at ). This term arises from variation with respect to of the second term in (39) (variation of the first term in (39) produces the term , which has been included in ). The last two terms in the right hand side of (41) only depend on quantities evaluated near the wall. Using , where is the proper radius of the wall, they can be combined into . Hence, we have

(42) |

Since is a function which depends on and , it may appear that the last integral cannot be done unless a semiclassical path is completely specified. However, the system is integrable, and can in fact be found from the discontinuities (35) in terms of . From Eqs. (34) and (40), we have

(43) |

It follows from (35) that

(44) |

Squaring and adding both equations in (44), we immediately find

(45) |

where the subindices and refer to the limiting values on both sides of the wall. Note that and can be different if the cosmological constants are different on both sides of the wall, or if the bubble configuration has non-vanishing mass . Eq. (44) determines in terms of , but only up to a sign. Since , we have

(46) |

In particular, we shall see in the Appendix that in the applications we are interested in, one of these two conditions are met: either we can construct a semiclassical path for which , or we can construct a semiclassical path for which . In both cases, it follows that

(47) |

We are interested in the change between the turning point geometries and ,

(48) |

where it is now understood that the inverse cosine lies between and . The value of at the turning point geometries is easy to evaluate. The first term in the integrand of (39) vanishes, and in the second term , where is the step function. As noted in [14], the integral only receives contributions where the geometry is backtracking , and the integral yields

(49) |

where is the absolute value of the change in in the backtracking part of the geometry.

Let us now particularize the general formula (48) to the problem of nucleation of walls in deSitter space, where the turning point geometries are given by (18) and (19). For deSitter space

(50) |

For the nucleation of positive tension walls, the geometry backtracks from , given by (21), to , and so

(51) |

On the other hand, for the nucleation of negative tension walls, we have two backtracking pieces. First from to , and then from to . Both contribute and give rise to

(52) |

In both cases, we have

(53) |

The integral in (48) is taken between , corresponding to a wall of infinitessimal size before tunneling, and . Also, since we are discussing the case where the cosmological constant outside and inside the wall are the same, and where the initial bubble has infinitessimally small mass, we have . The third term in (48) becomes

(54) |

where . Adding (53) and (54), we have

(55) |

where is given by Eq. (23) [or by Eq. (25)] for a positive [or negative] tension wall.

Eq. (48) applies also to the case of diwalls. In this case, and the calculation simplifies somewhat. The turning point before tunneling is given by a flat three dimensional space , whereas the turning point after tunneling is given by Eq. (12). The first and second terms in the right hand side of (48) can still be calculated from (49). The flat space has no backtracking part, as is monotonic, and doesn’t contribute, whereas for the turning point after tunneling we have

The third term in the right hand side of (48) includes a separate contribution from each one of the walls, which is readily calculated and contributes minus one half of the previous terms. Thus, we find

(56) |

where now is given by (15).

Thus, both for diwalls nucleating in flat space and for walls (ghost or ordinary) nucleating in deSitter, the WKB ”suppression factor” is given by

(57) |

In the next Subsection we shall further elaborate on the relation between the Euclidean action and the tunneling rates in general.

Note that, since the system is integrable, the precise form of the semiclassical path has not been used in deriving (48) (except in determining the sign of , see the discussion around Eq. (47)). In the Appendix, we construct a WKB tunneling path for the cases discussed in this subsection. The trajectories considered are such that the radii of the walls grow monotonically from an infinitessimally small size to the turning point size . It is interesting to note, in particular, that for the case of creation of negative tension walls in deSitter, the deformation of the geometry is smooth and the three volume is never zero (This is in contrast with the instanton picture where the deSitter geometry first disappears and then reappears with a large bubble in it). Note also that, due to the change in topology, we do not have the possibility of constructing a semiclassical path for the case of nucleation of negative tension walls in exactly flat space. Physically, this seems to be in agreement with the fact that the tunneling suppression for nucleation of such objects in deSitter becomes insurmountable in the limit .

### 2.5 The Euclidean action and the tunneling rates

Eq. (57) suggests that given a semiclassical path with Euclidean action , the corresponding tunneling rate is given by

(58) |

For systems with a single degree of freedom, the tunneling suppression is indeed given by . However, for loosely bound systems of ordinary and ghost matter, the situation is not so clear, and the above formula need not be of general validity.

To illustrate this point, let us consider the Euclidean solution corresponding to the nucleation of diwalls in flat space, considered in Subsection 2.1. We may in fact study the analogous solution in the presence of a cosmological constant. It is straightforward to show that the corresponding Euclidean action is given by

(59) |

where are the actions for nucleation of positive and negative domain walls in deSitter, given by Eqs. (23) and (25). Note that, since and have opposite sign, we have

(60) |

This means that if Eq. (58) is valid, it should be easier to form a diwall, than it is to create a wall of negative tension and then, subsequently, another wall of positive tension. Is this a reasonable expectation?

A simple system wich bears a useful analogy with the system of diwalls is that of an electron and a proton crossing an electric barrier. Consider a one-dimensional barrier (fig. 5) where from left to right, the electric field is negative in a segment of width , vanishing in a large segment of length , and positive with strength in a third segment of width . A proton of very low kinetic energy impinging from the left bounces off the first segment, repelled by the electric field. The probability for it to go through the barrier is exponentially small, and practically vanishing in the limit of large . Likewise, an electron of very low kinetic energy has no trouble going through the first two segments of the barrier, but will never make it to the asymptotic region on the right since it simply doesn’t have enough energy. On the other hand, if the proton and electron are bound together, the resulting hydrogen atom has no trouble going through the barrier. The electron pulls the proton through the negative electric field, and the proton pulls the electron through the third segment with positive electric field.

Going back to the system of diwalls, let us first consider the limit , when the Hubble radius is much larger than the scale characterizing the gravitational field of the walls. In this limit, the walls can be thought as a tightly bound system. In flat space () both and diverge. Physically, it is impossible to nucleate either a positive energy wall or a negative energy wall in Minkowski. Separately, both processes would lead to a breakdown of energy conservation (barring the possibility of topology change). Yet, the diwall process seems to be possible and it should occur at a finite rate. The two walls push and pull each other through a barrier which none of them would be able to penetrate separately. In this sense, it is not surprising that Eq. (58) gives a higher probability to nucleation of diwalls than to the separate nucleation of walls of either tension.

On the other hand, in the limit the gravitational field of the walls (and their interaction) becomes negligible. In this limit, the walls can be treated as non-gravitating objects in the background external field of a fixed deSitter space. The instanton for nucleation of walls of positive or negative tension is just a maximal worldsheet 3-sphere of radius embedded in . The corresponding action is . The action for the diwall is , and in the limit when the positive and negative tension walls have approximately the same modulus, , we have . Eq. (58) would suggest that if , diwall production is unsuppressed, even if . However, this cannot be true, since in the limit we are considering the interaction between walls is negligible. Production of walls (and diwalls) should be suppressed, with an exponent which is parametrically of order . In the example of the proton and electron crossing an electric barrier, this limit of weakly bound components would correspond to the case when there is a thermal bath of temperature , where is the ionization energy, or when the electric field is intense enough to ionize the atom , where is the Bohr radius. In both cases, the proton cannot get hold of the electron, and both have to go through the barrier on their own, so tunneling is highly suppressed.

The above examples suggest that Eq. (58) is a good estimate of the nucleation rate only in the case when we have a tightly bound system of ghosts and ordinary matter. However, it overestimates the rate when the gravitational binding energy between ghost and matter components is weak compared to any other force involved in the tunneling (e.g., the expansion of the universe caused by in the case of wall production in deSitter, the difference in pressure on both sides of the wall if we consider simultaneous false vacuum decay in ghost and matter sectors, or the electric force if we consider the Schwinger process occurring simultaneously in both sectors). In the general case, Eq. (58) is at best a conservative upper bound to the nucleation rate.

In the following sections, we shall be interested in nucleation of lumps of ghost and ordinary matter in flat space. In this case, the system can be considered to be tightly bound, since the gravitational energy-momentum transfer between matter and ghosts is what allows the system to nucleate (there is no other driving force). In the light of the above discussion, we shall adopt Eq. (58) as our estimate for the nucleation rate. Investigation of the general case of weakly bound systems is left for further research.

## 3 Nucleation of Bondi dipoles

We now turn to discussing the spontaneous nucleation of particle-like ghosts, accompanied with ordinary particles.

### 3.1 Preliminaries

A spherically symmetric lump of neutral ghost matter of mass coupled to non-ghost gravity creates the gravitational field

(61) |

This metric exhibits a naked singularity at . This singularity is usually regarded as a ‘valuable’ one [15], i.e., one not to be regularized since it signals a pathological negative-energy spectrum unbounded below. However, here we intend to explore how deadly this pathology is. So we assume that (61) is a valid description of the gravitational field, though only down to distances of the order of the gravitational cutoff distance scale , or, for a light ghost lump , down to the Compton wavelength .

We are interested in configurations with several collinear ghost and normal particles. The class of metrics that describes these are the axisymmetric Weyl solutions

(62) |

where and are functions of and . This is a well-studied system (see e.g., [16]) so we shall only sketch its solution. The vacuum Einstein equations can be completely integrated in an explicit form: the function satisfies a Laplace equation in the auxiliary space , and then can be obtained from by quadratures. Hence the solutions are fully determined once we specify the sources for the axisymmetric potential . For example, the Schwarzschild solution with positive mass corresponds to taking as a source an infinitely thin rod of length and linear mass density along the axis (fig. 6), so

(63) |

and

(64) |

where we introduce notation that is useful for this class of solutions [17],

(65) | |||||

(66) | |||||

(67) |

and, for this particular solution, .

It may seem odd that the spherically symmetric Schwarzschild solution is obtained from a non-spherical source, but this is just an artifact of the Weyl representation. The rod at , , in fact corresponds to the Schwarzschild horizon, and it is possible to see in general that a regular horizon is present iff the rod density is , otherwise a naked singularity will appear. In this paper, however, we will only consider situations where the geometry is cutoff before the horizon (or the naked singularity) is reached.

The field of a ghost particle (61) is given by a configuration similar to the one above, but now the density of the rod is instead . A simple way to obtain this from the solution (63)-(64) is to take , with . For definiteness and simplicity we shall only consider normal and ghost particles with rod linear densities —other densities do not introduce any significant qualitative changes.

It is now easy to construct axisymmetric configurations with arbitrarily many collinear particles. One simply superposes rods corresponding to each of the particles. is then the linear superposition of the fields of each rod, and then the function can be explicitly integrated [8, 17, 18]. Of course, in general the particles will attract each other (or possibly repel if ghosts are present) so there will be uncancelled forces among the particles. This reflects itself in the geometry through the presence of conical singularities on the portions of the -axis away from the rods. For Weyl metrics (62), the conical deficit angle at a given point on the axis away from any rod, is given by

(68) |

A conical deficit angle on a certain segment of the axis can be interpreted as a string stretched along the segment and pulling together the objects at its endpoints, while a conical excess angle is a strut pushing the objects apart. An integration constant in (corresponding to constant rescalings of ) can be used to eliminate possible conical singularities on a given segment, but in general, once this is fixed, there will remain conical singularities at other segments. In some cases it will be possible to tune parameters in the solution to cancel all conical singularities: the system is then balanced. If it is impossible to cancel all of them, then external forces are needed to keep the configuration in place.

Finally, we note that a semi-infinite rod of density corresponds to an acceleration horizon. The Weyl coordinates cover in this case only a Rindler wedge of the whole spacetime, but they can be extended across the Rindler horizon to provide a maximal analytic extension of the geometry, which contains a second copy of the Rindler wedge [8, 9]. For instance, the configuration corresponding to a finite, positive mass rod, and a semi-infinite rod, is known as the C-metric (see fig. 7) and describes two black holes accelerating away with uniform acceleration, each in a separate Rindler wedge, and being pulled apart by strings [19, 20].

### 3.2 Accelerating Bondi dipoles

A Bondi dipole consists of a positive and a negative mass particle
[7]. If we construct the gravitational field of this
configuration as the Weyl metric for a positive density and a negative
density rod, and nothing else, then it is straightforward to see that no
choice of the rod parameters is able to cancel all conical singularities
on all segments of the axis away from the rods. In particular, if we
eliminate the conical defects at infinity, then a strut remains
inbetween the two particles. This is of course expected, since the two
particles can not remain in static equilibrium. The strut pushes the
normal particle away from the ghost, while the ghost is also pushed
away, but having negative mass, it will tend to accelerate in a
direction opposite to the one it is pushed in^{2}^{2}2Hence Gamow’s
term “donkey particle” for such ghosts [21].. This indicates
that the Bondi dipole will naturally accelerate together, the ghost
chasing after the normal particle in a runaway fashion.

From our previous discussion, it should be obvious how to construct
the Weyl metric for such a configuration^{3}^{3}3A slightly different
version of the Bondi dipole was constructed in [9], with
point-like sources for the normal and ghost particle. These are then
“Chazy-Curzon particles”. For our purposes, the qualitative properties
are basically the same as in our solution. In particular, the result for
the action of the corresponding instanton takes the same form as in
eq. (87). Other instances of self-accelerating
positive+negative mass bound states are discussed in [22]..
From left to right, take the sources for the potential to be (figure
9): a semi-infinite rod at (and of density )
for the acceleration horizon; a negative density rod at
for the ghost; and a positive density rod at for the
normal particle. Note that again we have chosen to label the rod
endpoints in such a way that the conventional correlative order is
inverted for the negative density rod, i.e., we take . This is
done in order to facilitate a direct
comparison to the solution in [17], which described two normal
particles (black holes) accelerating together. Then we can directly
read off our solution from [17] as

(69) |

with

(70) |

and a constant with dimension of inverse length whose value can be changed by coordinate rescalings. For later convenience we choose it as

(71) |

The inversion in the ordering between and in (70) compared to [17] changes a positive mass into a negative one. Note also that we can always perform a shift in to fix one of the , so the solution contains only four physical parameters. To identify the physical quantities, we take the approximation where the rods are well separated from each other, and the dipole has a size smaller than the acceleration length scale, i.e.,

(72) |

so the gravitational forces between the particles involved are weak. A
careful examination of the solution in different limits then allows to
identify^{4}^{4}4These identifications are not uniquely
determined: they can be modified by terms that are small in the limit
(74). Our choices simplify some expressions below.

(73) | |||||

All these quantities are invariant under shifts along the axis. The approximation (74) is now equivalent to

(74) |

and in this limit we can approximately interpret , and , , respectively, as the masses and accelerations of the positive and negative particles, and as the distance between them. In particular, in the approximation (74) the harmonic difference of accelerations satisfies (see figure 8)

(75) |

Note that with our definition, is positive, so the ghost mass is . We can take the four physical parameters of the solution to be , and , while is fixed in terms of them.

In (3.2) we have already fixed the integration constant in so as to have

(76) |

and therefore, according to (68), there is no conical defect at infinity as long as . On the other hand, at the segment inbetween the two rods we have

(77) |

and between the ghost rod and the acceleration horizon,

(78) |

To cancel conical singularities on all segments we must require equality of (76), (77), (78), i.e.,