Casimir densities from coexisting vacua
Wightman function, the vacuum expectation values (VEVs) of the field squared and the energy-momentum tensor are investigated for a massive scalar field with general curvature coupling in a spherically symmetric static background geometry described by two distinct metric tensors inside and outside a spherical boundary. The exterior and interior geometries can correspond to different vacuum states of the same theory. In the region outside the sphere, the contributions in the VEVs, induced by the interior geometry, are explicitly separated. For the special case of the Minkowskian exterior geometry, the asymptotics of the VEVs near the boundary and at large distances are discussed in detail. In particular, it has been shown that the divergences on the boundary are weaker than in the problem of a spherical boundary in Minkowski spacetime with Dirichlet or Neumann boundary conditions. As an application of general results, dS and AdS spaces are considered as examples of the interior geometry. For AdS interior there are no bound states. In the case of dS geometry and for nonminimally coupled fields, bound states appear for a radius of the separating boundary sufficiently close to the dS horizon. Starting from a critical value of the radius the Minkowskian vacuum in the exterior region becomes unstable. For small values of the AdS curvature radius, to the leading order, the VEVs in the exterior region coincide with those for a spherical boundary in Minkowski spacetime with Dirichlet boundary condition. The exceptions are the cases of minimal and conformal couplings: for a minimal coupling the VEVs are reduced to the case with Neumann boundary condition, whereas for a conformally coupled field there is no reduction to Dirichlet or Neumann results.
PACS numbers: 03.70.+k, 04.62.+v, 11.10.Kk
In many physical problems, the model is formulated in backgrounds having boundaries on which the dynamical variables obey prescribed boundary conditions. The boundaries can have different physical origins, like interfaces between two media with different electromagnetic properties in condensed matter physics, horizons in gravitational physics, domain walls of various physical nature in the theory of phase transitions and critical phenomena, branes in string theory and in higher-dimensional cosmologies. In quantum field theory, the imposition of boundary conditions on a field operator gives rise to modifications of the spectrum for the vacuum fluctuations of a quantum field and, as a result, to the change of physical characteristics of the vacuum state, such as the energy density and vacuum stresses. As a consequence of this, vacuum forces arise acting on constraining boundaries. This is the familiar Casimir effect, first predicted for the electromagnetic field by Casimir in 1948 . This effect can have important implications on all scales, from subnuclear to cosmological, and it has been investigated for various types of bulk and boundary geometries (for reviews see -). The features of the Casimir forces depend on the nature of a quantum field, on the type of the spacetime manifold, on the geometry of boundaries, and on the specific boundary conditions imposed on the field. The explicit dependence can be found for highly symmetric geometries only.
In consideration of the Casimir effect, usually, the boundaries separate the regions with different electromagnetic properties (for example, media with different dielectric permittivities). Another type of effect related to the Casimir physics arises in a class of models with boundaries separating the spatial regions with different gravitational backgrounds. It can be referred to as gravitationally induced Casimir effect. The different gravitational backgrounds on both sides of the separating boundary can correspond to different vacuum states of the same theory. For example, one can consider a bubble of a false vacuum embedded in true vacuum or vice versa. Simple examples of vacuum bubbles are de Sitter (dS) and anti-de Sitter (AdS) spacetimes embedded in the Minkowski spacetime. In these examples, a physical boundary separates two regions with different values of the cosmological constant. It serves as a thin-wall approximation of a domain wall interpolating between two coexisting vacua (for a discussion see ).
In a configuration with coexisting gravitational backgrounds, the geometry of one region affects the properties of the quantum vacuum in the other region. Previously, we have considered several examples of this type of vacuum polarization. In , the Casimir densities are investigated for a scalar field in the geometry of a cosmic string for a core with finite support. In the corresponding model, the cylindrical boundary separates two different background geometries: the spacetime outside the boundary is described by the idealized cosmic string geometry with a planar angle deficit and for the interior geometry a general cylindrically symmetric static model is employed. Two specific models of the core have been considered: the ’ballpoint pen’ model [9, 10], with a constant curvature interior metric, and the ’flower pot’ model  with an interior Minkowskian spacetime. Similar problems for the exterior geometry of a global monopole are discussed in  and  for scalar and fermionic fields, respectively. In the corresponding models the boundary separating different spatial geometries is a sphere. The model with a sphere as a boundary and with an exterior dS metric, described in planar inflationary coordinates, has been considered in . The vacuum expectation values of the field squared and the energy-momentum tensor induced by a -symmetric brane with finite thickness located on AdS background are evaluated in [15, 16] for a massive scalar field. The general case of a static plane symmetric interior structure for the brane is considered, and the exterior AdS geometry is described in Poincaré coordinates. In the corresponding problem the separating boundaries are plane symmetric.
In the present paper, we consider the vacuum densities for a massive scalar field with a general curvature coupling parameter in a spherically symmetric static geometry described by two distinct metric tensors inside and outside a spherical boundary. In addition, the presence of a surface energy-momentum tensor located on the separating boundary is assumed. Among the most important characteristics of the quantum vacuum are the expectation values of the field squared and the energy-momentum tensor. Although the corresponding operators are local, due to the global nature of the vacuum state, they carry an important information about the global properties of the bulk. Moreover, in addition to describing the physical structure of the quantum field at a given point, the vacuum expectation value (VEV) of the energy-momentum tensor acts as a source of gravity in the quasiclassical Einstein equations. Consequently, it plays a crucial role in modelling a self-consistent dynamics of the background spacetime. For the evaluation of the VEVs we first construct the positive frequency Wightman function by the direct summation over a complete set of scalar modes. This function also determines the excitation probability of a Unruh-DeWitt detector (see, for instance, ). The quantum effects induced by distinct geometries in the exterior and interior regions should be taken into account, in particular, in discussions of the dynamics of vacuum bubbles during the phase transitions in the early Universe.
The organization of the paper is as follows. In the next section we describe the background spacetime under consideration and the matching conditions on a spherical boundary separating the interior and exterior geometries. A complete set of normalized mode functions for a scalar field with a general curvature coupling parameter is constructed in Section 3. By using the mode functions, in Section 4 we evaluate the positive frequency Wightman function for the general case of static spherically symmetric interior and exterior geometries. This function is presented in the form where the contribution induced by the interior geometry is explicitly separated. A special case of the exterior Minkowskian background is considered in Section 5. Explicit expressions for the VEVs of the field squared and of the energy-momentum tensor are provided and their behavior in asymptotic regions of the parameters is investigated. As an application of general results, in Section 6, two special cases of the interior geometry are discussed corresponding to maximally symmetric spaces with positive and negative cosmological constants (dS and AdS spaces). Section 7 summarizes the main results of the paper. In Appendix A, the coefficient in the asymptotic expansion of the logarithmic derivative of the hypergeometric function is determined, which is used for the evaluation of the leading terms in the asymptotic expansions of the VEVs near the boundary for the cases of the interior dS and AdS spaces.
2 Background geometry
Consider a -dimensional spherically symmetric static spacetime described by two distinct metric tensors inside and outside of a spherical boundary of coordinate radius . In the interior region, , the spacetime geometry is regular with the line element
where is the line element on a -dimensional sphere with a unit radius. The corresponding hyperspherical angular coordinates will be denoted by , where , , , and . The value of the radial coordinate corresponding to the center of the configuration will be denoted by . Of course, we could rescale the radial coordinate in order to have for the center, but for the further discussion it is convenient to keep general. Introducing a new coordinate
with the center at , the angular components of the metric tensor coincide with the corresponding components in the Minkowski spacetime described in the standard hyperspherical coordinates.
In the exterior region, , the geometry has a similar structure with different radial functions:
The metric tensor is continuous at the separating boundary :
Although the scheme described below can be generalized for metric tensors with horizons, for the sake of simplicity we will assume that if the line elements (2.1) and (2.3) have horizons at and , respectively, then . This means that the combined geometry contains no horizons.
The Ricci tensors for the interior and exterior geometries are diagonal with the mixed components (no summation over )
where and for the interior and exterior regions respectively and the prime means the derivative with respect to the radial coordinate (we adopt the convention of Ref.  for the curvature tensor). For the corresponding Ricci scalars we get the expression
In general, we assume the presence of an infinitely thin spherical shell at , having a surface energy-momentum tensor with nonzero components and . Let , , be the normal to the shell which points into the bulk on both sides. For the interior () and exterior () regions one has with and . We denote by the induced metric on the shell, , and is the extrinsic curvature. In the geometry under consideration, for the non-zero components of the latter we obtain
From the Israel matching conditions on the sphere one has
where is the gravitational constant and is the trace of the extrinsic curvature tensor. From these conditions, by taking into account (2.7), we find (no summation over ):
where is understood as the limit . Note that from (2.9) the relation
is obtained for the trace of the surface energy-momentum tensor. For given interior and exterior geometries, the relations (2.9) determine the surface energy-momentum tensor needed for the matching of these geometries.
3 Mode functions for a scalar field
3.1 Modes of continuous spectrum
Having described the background geometry, now we turn to the field content. We will consider a scalar field with curvature coupling parameter on background described by (2.1) and (2.3). The corresponding field equation reads
where is the covariant derivative operator. The most important special cases of the curvature coupling parameter and correspond to minimally and to conformally coupled fields, respectively.
In addition to the field equation in the regions and , the matching conditions for the field should be specified at . The field is continuous on the separating surface: . In order to find the matching condition for the radial derivative of the field, we note that the discontinuity of the functions and at leads to the delta function term
in the Ricci scalar and, hence, in the field equation (3.1), if we require its validity everywhere in the space. The expression (3.2) is given in terms of the trace of the surface energy-momentum tensor by using the formula (2.10). As a result of the presence of the delta function term in the field equation, the radial derivative of the field has a discontinuity at . The jump condition is obtained by integrating the field equation through the point . This gives
For a minimally coupled field the radial derivative is continuous.
In what follows, we are interested in the VEVs of the field squared and of the energy-momentum tensor induced in the region by the geometry in . In the model under consideration all the information about the properties of the vacuum is encoded in two-point functions. As such we will use the positive frequency Wightman function defined as the VEV , where stands for the vacuum state. In addition to describing the local properties of the vacuum, this function also determines the response of the Unruh-DeWitt type particle detectors . For the evaluation of the Wightman function we will use the direct summation over a complete set of positive- and negative-energy mode functions , obeying the field equation (3.1) and the matching conditions described above. Here, the set of quantum numbers specifies the solutions. Expanding the field operator over the complete set and using the standard commutation relations for the annihilation and creation operators, the following mode-sum formula is readily obtained:
where we assume summation over discrete quantum numbers and integration over continuous ones.
In the problem under consideration, the mode functions can be presented in the factorized form
where , is the hyperspherical harmonic of degree , , with being integers such that
Presenting the radial function as
the equations for the exterior and interior functions are obtained from (3.1)
where the Ricci scalar is given by the expression (2.6). From the matching conditions on the separating boundary, given above, for the radial functions in the interior and exterior regions, we find and
Note that, introducing a new radial coordinate, the equation (3.8) can be written in the Schrödinger-like form
and for the potential function we have
Let , with , be the solution of the equation (3.8) in the interior region which is regular at the origin. It can be taken as a real function. In addition, by taking into account that enters in the equation in the form , without loss of generality we can assume that . From the regularity of the geometry at the center and from (3.8) it follows that near the center the interior regular solution behaves as .
Now, the radial parts of the mode functions are presented as
where and are the two linearly independent solutions of the radial equation in the exterior region (equation (3.8) with ). We assume that the functions , , are taken to be real. The coefficients in (3.14) are determined by the continuity condition for the radial functions and by the jump condition (3.9) for their radial derivatives. From these conditions we get
with the notations
In (3.16) we have defined the functions
where is the Wronskian. The Wronskian can be found from the equation (3.8) with :
where the constant is determined by the choice of the functions and . Here we will assume that the exterior metric is asymptotically flat at large distances from the boundary, . With this assumption, we can see that for large the solution for the exterior equation is given by , where is a cylinder function of the order
If we take the functions and such that , , for , with and being the Bessel and the Neumann functions, then for the constant in (3.18) we find . In what follows we will assume this choice of the normalization for the exterior mode functions. In this way, as a complete set of quantum numbers specifying the mode functions we can take the set . Here we assume that is real. In addition, bound states can be present with purely imaginary . These states are discussed below.
The remaining coefficient is determined by the normalization condition for the mode functions given by
The integral over is finite and the divergence for comes from the upper limit of the integration over . As a consequence of this, we can replace the functions and by their asymptotics for . In this way, for the normalization coefficient one finds
with given by (3.16). Hence, for the radial mode-functions we get
where the notation
An equivalent form of the exterior mode functions is given by
with the notation
and with the normalization coefficient
Here and in what follows, for a given function , we use the notation
where . Note that one has the relation
for the coefficients in the exterior and interior regions.
3.2 Bound states
In the previous subsection we have considered the modes with real . In addition to them, the modes with imaginary can be present which correspond to possible bound states. For these states the exterior radial mode functions in the region behave as , where and is the Macdonald function. In order to have a stable vacuum state we will assume that . For the radial functions corresponding to the bound states one has
where for . The continuity of the mode functions at leads to the relation
From the jump condition for the radial derivative we see that the allowed values of for bound states are solutions of the equation
where for a function we define
The possible solutions of the equation (3.31) will be denoted by , .
The remaining coefficient in the mode functions (3.29) is determined from the normalization condition for the bound states
with . In order to evaluate the integrals in this formula we note that for a solution to the radial equation (3.8) the following formula can be proved:
In particular, in the limit , from here one can obtain
The coefficient is found from (3.30).
An equivalent expression for the normalization coefficient is obtained by using the Wronskian relation
for two linearly independent solutions of the radial equation in the exterior region. Here, the function is normalized by the relation for , with being the modified Bessel function. From (3.37) we get
By taking into account that for the bound states one has the equation (3.31), this gives
Hence, the normalization constant for the exterior modes is written in the form
4 Wightman function
Having a complete set of modes we can proceed to the evaluation of the Wightman function by using the mode sum formula (3.4). First we consider the case with no bound states. Substituting the functions (3.5) in (3.4), the summation over is done by using the addition formula for the hyperspherical harmonics 
where is the angle between the directions determined by the angles and . In (4.1), is the surface area of the unit sphere in -dimensional space and is the Gegenbauer polynomial of degree and order . With the modes (3.24) and the normalization coefficient (3.26), the expression for the Wightman function in the exterior region reads:
where and the function is defined by (3.25).
In order to separate from the Wightman function the contribution induced by the interior geometry, firstly we introduce the functions
Note that, as the functions and are real, one has . For these new functions, at large distances, , one has the asymptotics
with being the Hankel functions. Now, it can be seen that the following identity takes place:
with the functions
The function is the Wightman function in the case of the background when the geometry is described by the line element (2.3) for all values of the radial coordinate . As a radial function in the corresponding modes the function is taken. Recall that we have for and, hence, for these modes the vacuum state at asymptotic infinity coincides with the Minkowskian vacuum. Thus, the function can be interpreted as the contribution to the Wightman function induced by the geometry in the region with the line element (2.1).