# Quasi-local stress-tensor formalism and the Casimir effect

Borzoo Nazari<EMAIL_ADDRESS>University of Tehran, Tehran, Iran,
P.O.Box: 11155-4563

###### Abstract

We apply the quasi-local stress-energy tensor formalism to the Casimir effect
of a scalar field confined between conducting planes located in a static
spacetime. We show that the surface energy vanishes for both Neumann and
Dirichlet boundary conditions and consequently the volume Casimir energy
reduces to the famous zero point energy of the quantum field, i.e.
$E^{vol.}=\sum\frac{\hbar\omega}{2}$. This enables us to reinforce previous
results in the literature and extend the calculations to the case of massive
and arbitrarily coupled scalar field. We found that there exists a first order
perturbation correction to the Casimir energy contrary to previous claims
which state that it vanishes. This shows many orders of magnitude greater than
previous estimations for the energy corrections and makes it detectable by
near future experiments.

##  I Introduction

The Casimir effect has been the subject of numerous studies since its
discovery in 1948 by H.B.G. Casimir Casimir . The effect appears as an
attractive/repulsive force on boundaries in the presence of some quantum
field. The periodic or anti-periodic boundary condition may arise due to the
presence of a material boundary or when the spacetime admits some compact
dimensions without necessarily having a material boundary DeWitt ; Ford .
Thus, in the context of extra dimensions, many studies concerning the role of
Casimir force in the vacuum structure of the underlying theories have been
accomplished. See Teo2  and references therein. Some other studies refer to
the long standing question of the effect of the gravitational field on the
quantum vacuum energy, i.e. the zero point energy of quantum field theory. The
Casimir effect in curved spacetime has been analysed by many authors Bezerra ;
SorgeNew ; Muniz1 ; Muniz2 ; Geyer ; Bimonte ; Bimonte1 ; Sorge ; Milton ;
Esposito ; Napolitano ; NouriNazari ; Dowker  and unfortunately most of the
calculations have been ended up with implicit and complicated formulas for the
stress energy tensor including the energy and pressure respectively. A brief
literature review on explicit results would be instructive.
Sorge Sorge  calculated corrections to the Casimir energy of a massless
minimally coupled scalar field between two parallel planes in Schwarzschild
spacetime using an expanded isotropic metric of the form
$g_{\mu\nu}=diag(1+2\gamma+2\lambda z,-1-2\gamma-2\lambda
z,-1-2\gamma-2\lambda z,-1-2\gamma-2\lambda z)$ and found the Casimir energy
per unit volume as

$\displaystyle E=-\frac{\pi^{2}}{1440l_{p}^{3}}(1-\lambda
l_{p}),\;\;\;\hbar=c=1,$ (1)

in which $l_{P}$ is the proper distance between the planes and
$\gamma=O(\epsilon),\;\lambda=O(\epsilon^{2})$ are small parameters. In a
recent paper Sorge2019 , the same author confirmed this result through
different approach of the Schwinger’s action principle. Note that the energy
does not depend on $\gamma$.
Bimonte et al Bimonte , see specially the three published errata, calculated
the electromagnetic energy-momentum tensor for parallel planes in Fermi
coordinates which is described by $g_{\mu\nu}=diag(1+2gz,-1,-1,-1)$ for some
small parameter $g$ and concluded that the total Casimir energy decreases
according to

$\displaystyle E=-\frac{\pi^{2}}{720l^{3}}(1+\frac{l}{2}g),\;\;\;\hbar=c=1.$
(2)

Fermi coordinate, which describes the spacetime of a hovering observer in a
static spacetime, is equivalent to the well-known accelerated observer of flat
spacetime via weak principle of equivalence. In Ref.Bimonte1  they confirmed
the above result and extend the calculations to next order of perturbations
using such equivalency. Using the same method, Esposito et al Esposito  and
Napolitano et al Napolitano  repeated the calculations for the case of a
massless scalar field under Dirichlet and Neumann boundary conditions (D.N.
B.C.s) on the planes. The obtained correction, as expected, was the same as
equation (2) times a factor of $\frac{1}{2}$ due to the fact that the
electromagnetic field has two degrees of freedom relative to that of the
scalar field. Nazari BorzooEPJC  extended the calculations for scalar and
electromagnetic fields to the more general metric of the form BorzooEPJC

$\displaystyle
ds^{2}=(1+2\gamma_{0}+2\lambda_{0}z)dt^{2}-(1+2\gamma_{1}+2\lambda_{1}z)\left(dx^{2}+dy^{2}+dz^{2}\right),$
(3)

in which $\lambda_{0}z,\lambda_{1}z,\gamma_{0}$ and $\gamma_{1}$ were small.
It was shown in BorzooEPJC  that the metric (3) describes any static spacetime
(after expanding in the space between the planes) and the total energy for
D.N. B.C.s was found.

Another trend of computations in the literature concerns the force exerted to
the apparatus by the gravitational field in which the planes are located in
Calloni . In a series of papers Fulling et al Fulling , Milton et al Milton
and Shajesh et al Shajesh  considered the force to parallel planes seeking
confirmation of the weak principle of equivalence in Fermi coordinates. They
showed the Casimir energy, and probably the quantum vacuum, gravitates just as
required by the principle of equivalence and confirmed the energy (2).
However, they failed to derive the correct formula for the force. Bimonte
Bimonte2  found a way according to which one can find the correct formula for
the force exerted to any matter configuration by a constant gravitational
field.

Calculations related to the Casimir energy in curved spacetime are complex and
lengthy. As has been pointed out in a recent work BorzooCQG , the usual
procedure consists of calculating a typical summation

$\displaystyle
E=\frac{\hbar}{2}\int\sum_{\omega_{n},k}<T_{00}(\omega_{n},k,\textbf{x})>dV$
(4)

Many authors Bezerra ; SorgeNew ; Muniz1 ; Muniz2 ; Sorge ; Bimonte ; Bimonte1
; Esposito ; Napolitano ; NouriNazari ; Sorge2019 ; Lima ; Blasone ;
Buoninfante ; Lambiase  performed such a procedure to find the Casimir energy
for D.N. BCs in various spacetimes. The point is that none of the above
studies distinguished between the volume Casimir energy and the total one
which is composed of a surface part resides on boundaries and a volume part in
the bulk. Employing the quasi-local approach to the gravitational action,
Saharian Saharian2  had previously shown such a point. Accordingly, we know
that the volume energy always satisfies

$\displaystyle
E^{vol.}=\frac{\hbar}{2}\int\sum_{\omega_{n},k}\omega_{n}d^{2}k-E^{surf.},$
(5)

in any static spacetime. In addition, this is true regardless of the shape of
the boundary. Thus, most of the complexities encountered in calculation of
equation (4) is due to the surface term $E^{surf.}$ because we know that the
first term in the right side of (5) is nothing but the zero point energy of
the corresponding quantum field which can be computed much easier.

Our purpose is to apply the quasi-local formalism to the Casimir effect for
parallel plates. This approach has many advantages. First of all, the Casimir
energy is a phenomenon under the presence of boundaries. Thus, it is natural
to investigate it using the formalism of gravitational action on manifolds
with boundaries. Second, it is simple and unambiguous and notably reduces the
calculations in a way that enables us to extend previous results found by
other studies to the case of massive and arbitrarily coupled scalar field.
Moreover, we find that the correction to the Casimir energy do not vanish
within first order perturbation calculations, contrary to previous studies in
the literature. Therefore, it could be measured by precise experiments in the
near future.

The structure of the paper is as follows. Most of section II is a review of
the Brown-York notation and formalism for stress tensor on manifolds with
boundaries. In section III we show that the surface energy vanishes for N.D.
BCs. An important discussion and calculation in justification of the
appearance of the first order corrections into the Casimir energy is
presented. Conclusion is the final section.

##  II The stress tensor for manifolds with boundaries

Suppose a compact $(D+1)-$dimensional spacetime manifold $M$ along with a
metric $g_{\mu\nu}$ and boundary $\partial M$. The spacetime has been foliated
by typical spacelike hypersurfaces $\Sigma$ each of them has a boundary
$\partial\Sigma$. As Fig.1 shows, $\partial M=\partial
M_{s}\cup\Sigma_{1}\cup\Sigma_{2}$ where $\Sigma_{1}$ and $\Sigma_{2}$ are
$D-$dimensional initial and final spacelike hypersurfaces. Evidently,
$\partial M_{s}$ is nothing but the evolution of $\partial\Sigma$, i.e.
$\partial\Sigma=\Sigma\cap\partial M_{s}$. Normal vector to $\partial M$ is
denoted by $n^{\mu}$ and $n^{\mu}n_{\mu}=\epsilon$ in which $\epsilon=1$ for
spacelike hypersurfaces $\Sigma_{1},\;\Sigma_{2}$ and $\epsilon=-1$ for
$\partial M_{s}$. Unit timelike vector $u^{\mu}$ is normal to the
hypersurfaces $\Sigma$. Thus, $u^{\mu}=n^{\mu}$ on $\Sigma_{1}$ and
$u^{\mu}=-n^{\mu}$ for $\Sigma_{2}$. The unit vector $\overline{n}^{\mu}$ is
defined as the vector field normal to $\partial\Sigma$ and lies on $\Sigma$.
$\overline{u}^{\mu}$ is also normal to $\partial\Sigma$ and tangent to
$\partial M_{s}$. For now we assume $M$ is not orthogonally foliated.

Figure 1: Schematics of spacetime compact manifold foliated with hypersurfaces
$\Sigma$. The boundary is defined by $\partial M=\partial
M_{s}\cup\Sigma_{1}\cup\Sigma_{2}$. Unit vectors $n^{\mu}$ and $u^{\mu}$ are
normal to $\partial M$ and $\Sigma$. Two vectors $\overline{n}^{\mu}$ and
$\overline{u}^{\mu}$ are both normals to $\partial\Sigma$, the former tangent
to $\Sigma$ and the later tangent to $\partial M_{s}$.

The action describing a massive arbitrarily coupled scalar field consists of
bulk and surface parts which are given by Saharian2

$\displaystyle S=S_{b}+S_{s},$ (6a) $\displaystyle
S_{b}=\frac{1}{2}\int_{M}d^{D+1}x\sqrt{|g|}\left(\nabla_{\mu}\phi\nabla^{\mu}\phi-m^{2}\phi^{2}-\zeta
R\phi^{2}\right),$ (6b) $\displaystyle S_{s}=-\epsilon\int_{\partial
M}d^{D}x\sqrt{|h|}\left(\zeta\phi^{2}K+m_{s}\phi^{2}\right).$ (6c)

In the surface action, $h$ is the determinant of the projection tensor
$h_{ik}=g_{ik}-\epsilon n_{i}n_{k}$ and $K$ the scalar obtained by contraction
of extrinsic curvature $K_{ik}=h^{l}_{i}h^{m}_{k}\nabla_{l}n_{m}$. The
parameter $m_{s}$ is free to keep the consistency of boundary conditions. The
first term in the surface action has a counterpart in general relativity known
as the Gibbons-Hawking term and its presence is necessary to obtain correct
equations of motion when the induced metric held fix on the boundaries.
Another point which is worth to mention is related to the case when the unit
normal $n^{\mu}$ abruptly changes direction on the surface, i.e. when the
surface is non-smooth on some edges. In this case, there must be taken into
account some extra terms in the action to get rid of such potentially
divergent terms. In Fig.1 sharp edges are the boundaries of $\Sigma_{1}$ and
$\Sigma_{2}$. Therefore, we assume there is no such non-smoothness as we are
interested here to take variation when the metric and field are held fixed on
$\Sigma_{1}$ and $\Sigma_{2}$.

###  II.1 Variation with respect to metric

Taking variation of (6a) with respect to metric results in Saharian2

$\displaystyle\delta_{g}S$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{M}d^{D+1}x\sqrt{|g|}\,\delta
g^{\mu\nu}T_{\mu\nu}^{\mathrm{(vol)}}+\frac{1}{2}\int_{\partial
M_{s}}d^{D}x\sqrt{|h|}\,\delta g^{\mu\nu}\tau_{\mu\nu}$ (7)
$\displaystyle+\frac{1}{2}\int_{\partial M_{s}}d^{D}x\sqrt{|h|}\zeta
D_{\lambda}(\phi^{2}n^{\mu}h^{\lambda\theta}\delta g_{\mu\theta})$

in which the volume and surface energy momentum tensors
$T_{\mu\nu}^{(\mathrm{vol})}$ and
$T_{\mu\nu}^{\mathrm{(surf)}}=\tau_{\mu\nu}\delta(x;\partial M_{s})$ are
defined by

$\displaystyle
T_{\mu\nu}^{\mathrm{(vol)}}=\nabla_{\mu}\phi\nabla_{\nu}\phi-\frac{1}{2}g_{\mu\nu}g^{\lambda\theta}\nabla_{\lambda}\phi\nabla_{\theta}\phi+\frac{m^{2}}{2}\phi^{2}g_{\mu\nu}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-\zeta\phi^{2}G_{\mu\nu}+\zeta
g_{\mu\nu}g^{\lambda\theta}\nabla_{\lambda}\nabla_{\theta}\phi^{2}-\zeta\nabla_{\mu}\nabla_{\nu}\phi^{2}$
(8a) $\displaystyle
T_{\mu\nu}^{\mathrm{(surf)}}=\left\\{\zeta\phi^{2}K_{\mu\nu}-h_{\mu\nu}\left(\zeta\phi^{2}K+\zeta
n^{\theta}\nabla_{\theta}\phi^{2}+m_{s}\phi^{2}\right)\right\\}\delta(x;\partial
M_{s}).$ (8b)

One can easily check that $T_{\mu\nu}^{(1)}$ is divergence-free onshell, i.e.
when the Klein-Gordon equation of motion holds. An expectable feature of
surface energy momentum tensor is that it satisfies
$T_{\mu\nu}^{\mathrm{(surf)}}n^{\mu}=0$.

On ground of discussion after (6c), the last term in (7) is nothing but the
divergent term representing possible non-smoothness due to edges on
boundaries. We drop it by now.

###  II.2 Variation with respect to $\phi(x)$

After taking variation with respect to $\phi(x)$, the equations of motion are
found to be

$\displaystyle g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi+m^{2}\phi+\zeta
R\phi=0,$ (9a) $\displaystyle 2(\zeta K+m_{s})\phi+n^{\mu}\nabla_{\mu}\phi=0.$
(9b)

Equation (9b) is a Robin boundary condition and valid only on $\partial M$.
Note that this boundary condition has sufficient flexility by letting the
parameter $m_{s}$ to be free. In fact, the term $(\zeta K+m_{s})$ is a free
parameter for the case of minimal coupling or $K$ being a constant on
boundaries. This is the case for well-known boundary geometries like sphere,
cylinder or planes. Putting $m_{s}\rightarrow\infty$ reinforces the Dirichlet
B.C.. As we will see in the next section, for the case of parallel planes, and
in fact for any configuration with constant $K$, it is possible to obtain the
Neumann B.C. from (9b) as well.

###  II.3 The surface and volume energies

Using (8a) and Birrell

$\langle
0|T_{\mu\nu}(x)|0\rangle=\sum_{\alpha}T_{\mu\nu}\\{\phi{{}_{\alpha}(x),\phi_{\alpha}^{\ast}(x)\\}},$
(10)

the vacuum expectation value $\langle 0|T_{0}^{{\mathrm{(vol)}}0}|0\rangle$
has been obtained as follows Saharian2

$\displaystyle E^{{\mathrm{(vol)}}}$ $\displaystyle=$
$\displaystyle\int_{{\Sigma}}d^{D}x\sqrt{|g|}\langle
0|T_{0}^{{\mathrm{(vol)}}0}|0\rangle$ (11) $\displaystyle=$
$\displaystyle\sum_{\alpha}\left\\{\frac{\omega_{\alpha}}{2}+\int_{\partial{\Sigma}}d^{D-1}x\sqrt{|g|}n^{i}\left[(\zeta-1/4)\partial_{i}-\zeta(\partial_{i}\ln\sqrt{g_{00}})\right]\phi_{\alpha}(x)\phi_{\alpha}^{\ast}(x)\right\\},$

in which $n^{i}$ is the spatial part of $n^{\mu}=(0,n^{i})$, the unit normal
to $\partial M_{s}$.

Same calculation for the surface energy will result in Saharian2

$\displaystyle E^{{\mathrm{(surf)}}}$ $\displaystyle=$
$\displaystyle\int_{{\Sigma}}d^{D}x\sqrt{|g|}\langle
0|T_{0}^{{\mathrm{(surf)}}0}|0\rangle$ (12) $\displaystyle=$
$\displaystyle-\sum_{\alpha}\int_{\partial{\Sigma}}d^{D-1}x\sqrt{|g|}n^{i}\left[(\zeta-1/4)\partial_{i}-\zeta(\partial_{i}\ln\sqrt{g_{00}})\right]\phi_{\alpha}(x)\phi_{\alpha}^{\ast}(x).$

Equation (11) is quite important since the second term in (11) is nothing but
the surface energy $E^{{\mathrm{(surf)}}}$. In fact, we see that
$E=E^{{\mathrm{(vol)}}}+E^{{\mathrm{(surf)}}}=\frac{1}{2}\sum_{\alpha}\omega_{\alpha}.$
As is shown in Appendix A (see after (49)), since $n^{\mu}\eta_{\mu}=0$ for
any static spacetime, the total energy $E$ is a conserved quantity.

Calculation of $E^{{\mathrm{(surf)}}}$ or $E^{{\mathrm{(vol)}}}$ is not as
easy task as one might suspect at first. For such a complicated and lengthy
process see for example Ref.BorzooCQG  and references therein.

In the next subsection, we see that the surface energy vanishes under
Dirichlet B.C.s in the spacetime described by (3). We show the same statement
for the case of parallel planes under Neumann B.C.s. This is an interesting
results according to which it suffices simply to sum over mode frequencies
$\omega_{\alpha}$ instead of complicated calculation of (11) which is usually
done in the literature.

##  III the vacuum energy for Neumann B.C.

In this section we find the total energy $E$ for two conducting parallel
planes subject to a class of gravitational fields described by metric (3)
which first used in BorzooEPJC . It can be shown that any static, and even
stationary, weak gravitational field can be transformed into the form (3) by
using perturbation expansion of the metric in the space between the
planesSorge . The coordinate system is described by $(x,y,z,t)$ with the
origin on one of the planes, i.e. the one nearest to the source of the
gravitational field. The planes are separated by a distance $l$ in such a way
that each plane is perpendicular to $z-$axis. Therefore, the planes are
characterized by $z=0$ and $z=l$ respectively.

###  III.1 The mode functions

To find $\phi$, we notice the translational symmetry in $x$ and $y$ directions
between the plates and use the ansatz

$\displaystyle\phi(x)=C(\omega,k_{\perp})e^{-i\omega
t}e^{ik_{x}x}e^{ik_{y}y}Z(z),$ (13)

for the wave function. Thus, the Klein-Gordon equation simplifies to

$\displaystyle
Z^{\prime\prime}(z)+\partial_{z}\ln{|\sqrt{-g}g^{11}|}Z^{\prime}(z)+(-g^{00}g_{11}\omega^{2}-k_{\perp}^{2}+m^{2}g_{11})Z(z)=0,$
(14)

for a typical diagonal static spacetime. Here $k_{\perp}^{2}\equiv
k_{x}^{2}+k_{y}^{2}$. Note that the coupling to curvature has been ignored in
the Klein-Gordon equation as it can be easily shown that the Riemann’s scalar
contributes only within second order perturbations. We use the solution to
(14) which has been found by Nazari BorzooCQG  as follows

$\displaystyle
Z(z)=C(\omega,k_{\perp})\left(1-(\frac{\lambda}{2}+\frac{a}{4b})z\right)\sin\left(\sqrt{b}z(1+\frac{a}{4b}z)+\Theta_{0}\right),$
(15)

in which

$\displaystyle a$
$\displaystyle=-2(\lambda_{0}-\lambda_{1})\omega^{2}-2\lambda_{1}m^{2},$ (16a)
$\displaystyle b$
$\displaystyle=(1-2(\gamma_{0}-\gamma_{1}))\omega^{2}-k_{\perp}^{2}-(1+2\gamma_{1})m^{2},$
(16b) $\displaystyle\lambda$ $\displaystyle=\lambda_{1}+\lambda_{0}.$ (16c)

Evidently, parameters $\omega$ and $\Theta_{0}$ should be determined by
imposition of boundary conditions. After using the Neumann B.C. and doing some
algebra it founds that

$\displaystyle\cot(\Theta_{0})=\frac{1}{\sqrt{b}}(\frac{\lambda}{2}+\frac{a}{4b})=O(\lambda)=\epsilon,$
(17a)
$\displaystyle\cot(\sqrt{b}l+\frac{a}{4\sqrt{b}}l^{2}+\Theta_{0})=\frac{1}{\sqrt{b}}\left(\frac{\lambda}{2}+\frac{a}{4b}\right)=\epsilon,$
(17b)

which in turn gives

$\displaystyle\sqrt{b}l+\frac{a}{4\sqrt{b}}l^{2}=n\pi,\;\Theta_{0}=\frac{\pi}{2}-\epsilon.$
(18)

Further algebra on this equations results in

$\displaystyle\omega$
$\displaystyle=(1+\gamma_{0}-\gamma_{1}+\frac{\lambda_{0}-\lambda_{1}}{2}l)\sqrt{\omega_{0}^{2}+M^{2}},$
(19a) $\displaystyle\Theta_{0}$ $\displaystyle=\frac{\pi}{2}-\epsilon,$ (19b)

in which $\omega_{0}^{2}=k_{\perp}^{2}+(\frac{n\pi}{l})^{2}$ and
$M^{2}=(1+2\gamma_{1}+\lambda_{1}l)m^{2}$. The same calculation shows that
(19a) holds also for Dirichlet condition.

###  III.2 Energy for Dirichlet B.C.

By substituting (13) into (12) we find

$\displaystyle
E^{{\mathrm{(surf)}}}=-\sum_{\omega,k_{\perp}}C^{2}\int_{\partial{\Sigma}}d^{D-1}x\sqrt{|g|}n^{i}\left[(2\zeta-1/2)\partial_{i}Z(z)-\zeta(\partial_{i}\ln\sqrt{g_{00}})Z(z)\right]Z(z).$
(20)

Since for Dirichlet B.C. we have $Z(z)|_{\partial M}=0$, quite generally the
surface energy vanishes regardless of which material configuration and
background (static) spacetime is used.

###  III.3 Energy for Neumann B.C.

The Neumann B.C., i.e. $n^{\mu}\partial_{\mu}\phi|_{\partial M}=0$, is
compatible with (9b) by setting either $\zeta=m_{s}=0$ or more generally
$m_{s}=-\zeta K$. We take a short analysis on both cases.

####  III.3.1 $\zeta=m_{s}=0$

In this case, the surface energy momentum tensor (8b) vanishes upon imposing
the Neumann B.C.. Thus, for massless minimally coupled scalar field the total
energy equals the zero point energy of the field, i.e. $E=\sum\hbar\omega/2$.
This result is also a general statement regardless of which material
configuration and background (static) spacetime is used.

####  III.3.2 $m_{s}=-\zeta K$

As $m_{s}$ and $\zeta$ are constants, this case is impossible unless $K$ be a
constant which means that $\partial M_{s}$ should have constant extrinsic
curvature. This is the case for some well-known geometries like parallel
planes, sphere, cylinder and so on. In this case, the first term containing
$\partial_{i}Z(z)$ vanishes due to Neumann B.C. and we see from (20) that

$\displaystyle\begin{split}E^{{\mathrm{(surf)}}}&=\zeta\sum_{\omega,k_{\perp}}C^{2}\int_{\partial\Sigma}d^{D-1}x\sqrt{|g|}(n^{i}\partial_{i}\ln\sqrt{g_{00}})Z^{2}(z)\\\
&=\zeta\sum_{\omega,k_{\perp}}C^{2}\int_{upper\;plate}+\zeta\sum_{\omega,k_{\perp}}C^{2}\int_{lower\;plate}+\zeta\sum_{\omega,k_{\perp}}C^{2}\int_{side\;surf.},\end{split}$
(21)

where side surfaces are perpendicular to the $z-direction$. The normal vector
$n^{i}$ is given by

$\displaystyle
n^{\mu}|_{up.}=+\frac{\delta^{\mu}_{3}}{\sqrt{|g_{33}|}},\;\;n^{\mu}|_{low.}=-\frac{\delta^{\mu}_{3}}{\sqrt{|g_{33}|}},\;\
;n^{\mu}|_{x-dir.}=+\frac{\delta^{\mu}_{1}}{\sqrt{|g_{11}|}},\;\;n^{\mu}|_{y-dir.}=+\frac{\delta^{\mu}_{2}}{\sqrt{|g_{22}|}}.$
(22)

By this, we mean that the vacuum state is that of an observer which is
hovering in a spacetime point and static relative to the plates. Such an
observer is inevitably accelerated relative to the source of the gravity. The
energy on surfaces perpendicular to the $x$ and $y$ directions vanishes. In
fact, in the $x$-direction, $n^{i}=(0,1,0,0)$ and we saw in (12) that
$n^{i}\partial_{i}\phi\phi^{*}=\partial_{x}\phi\phi^{*}=0$ because
$\phi\phi^{*}$ is independent of $x$ according to (13). The same argument is
true for $y$-direction. Therefore, the third term on right side of (21)
vanishes and the rest is simplified as follows:

$\displaystyle E^{{\mathrm{(surf)}}}$
$\displaystyle=A\zeta\lambda_{1}\sum_{\omega,k_{\perp}}C^{2}\left\\{g_{33}\sqrt{g_{00}}Z^{2}(z)\right\\}|_{z=l}-A\zeta\lambda_{1}\sum_{\omega,k_{\perp}}C^{2}\left\\{g_{33}\sqrt{g_{00}}Z^{2}(z)\right\\}|_{z=0}.$
(23)

By using (15) we find for the first term that

$\displaystyle\begin{split}&A\zeta\lambda_{1}\sum_{\omega,k_{\perp}}C^{2}\left\\{g_{33}\sqrt{g_{00}}Z^{2}(z)\right\\}|_{z=l}\\\
&=A\zeta\lambda_{1}\sum_{\omega,k_{\perp}}C^{2}\left(1-(\lambda+\frac{a}{2b})l\right)\sin^{2}\left(\sqrt{b}l+\frac{a}{4\sqrt{b}}l+\Theta_{0}\right)\\\
&=A\zeta\lambda_{1}\sum_{\omega,k_{\perp}}C^{2},\end{split}$ (24)

in which we have used (18) and
$\lambda_{1}\left(1-(\lambda+\frac{a}{2b})l\right)=\lambda_{1}+O(\lambda^{2})$.
Note that $\sin^{2}(\Theta_{0})=1+O(\epsilon^{2})$. Some algebra shows that
the second term in (23) is also equal to (24) hence $E^{{\mathrm{(surf)}}}=0$.

####  III.3.3 Volume energy for Neumann and Dirichlet B.C.s

So far we have shown that the total surface energy vanishes for both Neumann
and Dirichlet boundary conditions and the conserved volume (total) energy
reads

$E=\frac{1}{2}\sum_{\alpha}\omega_{\alpha}=(1+\gamma_{0}-\gamma_{1}+\frac{\lambda_{0}-\lambda_{1}}{2}l)\;\frac{1}{2}\int\frac{k_{\perp}dk_{\perp}}{(2\pi)^{2}}\sum_{n}\sqrt{\omega_{0}^{2}+M^{2}},$
(25)

in which we have used (19a). The massive problem will be calculated in section
(V). For now, we assume $M=0$. So, the summation on the right side of (25) is
nothing but the famous case of the Casimir effect in flat spacetime which can
be regularized using standard procedures MostepanenkoBook  and the result is
as follows

$E=-(1+\gamma_{0}-\gamma_{1}+\frac{\lambda_{0}-\lambda_{1}}{2}l)\;\frac{\pi^{2}\hbar
c}{1440l^{3}},$ (26)

which in terms of proper distance

$l_{p}=\int_{0}^{l}\sqrt{-g_{33}}dz=l(1+\gamma_{1}+\frac{1}{2}\lambda_{1}l),$
(27)

between the planes can be written as

$E=-(1+\gamma_{0}+2\gamma_{1}+\frac{\lambda_{0}+2\lambda_{1}}{2}l_{p})\frac{\pi^{2}\hbar
c}{1440l_{p}^{3}}.$ (28)

This result is in agreement with the previous studies which have used a
different approach. See Eq.(82) in BorzooCQG  and the discussions after. Note
that equation (28) is valid also for $\zeta\neq 0,\;m\neq 0$ while it was
obtained in BorzooCQG  for $\zeta=0,\;m=0$. Another consistency check is Eq.
(5.4) in Esposito  which has been found using a special case of the metric
(3), i.e. with $\gamma_{0}=\gamma_{1}=\lambda_{1}=0,\;\lambda_{0}=g,$ and
$\;\zeta=0,\;m=0$.

Note that the first order constants $\gamma_{0}$ and $\gamma_{1}$ are present
in (28). We return to this point in the next section. Before that, we need to
carry out some estimations.

####  III.3.4 Some estimations

The weak field limit of the the Schwarzschild metric in isotropic form is
given by

$\displaystyle
ds^{2}=(1-\frac{R_{s}}{r})dt^{2}-(1+\frac{R_{s}}{r})d\Omega^{2},$ (29)

where $R_{s}=2\frac{GM}{c^{2}}$ is the Schwarzschild radius. Suppose the
planes are located in a distance $R$ from the center of the source. If $0\leq
z\leq l$ and the planes be small enough, we may expand Sorge

$\displaystyle\frac{1}{r}=\frac{1}{R+z}=\frac{1}{R}-\frac{1}{R^{2}}z+O(R^{-2}).$
(30)

Thus, the metric (29) recasts into

$\displaystyle
ds^{2}=(1+\frac{R_{s}}{R}-\frac{R_{s}}{R^{2}}z)dt^{2}-(1-\frac{R_{s}}{R}+\frac{R_{s}}{R^{2}}z)d\Omega^{2},$
(31)

which in comparison with (3) gives
$\gamma_{0}=-\gamma_{1}=\frac{R_{s}}{2R},\;\lambda_{0}=-\lambda_{1}=-\frac{R_{s}}{2R^{2}}$
and the energy reads

$E=-(1-\frac{R_{s}}{R}+\frac{R_{s}}{2R^{2}}l_{p})\;\frac{\pi^{2}\hbar
c}{1440l_{p}^{3}}.$ (32)

This shows that the energy increases and the leading order correction is of
order $\frac{R_{s}}{R}\approx 7\times 10^{-10}$ for the Earth while previous
studies Sorge2019 ; Calloni ; Sorge ; Milton  predict the leading order
correction to be $\frac{R_{s}l_{p}}{R^{2}}\approx 1.1\times 10^{-22}$.

The force by which the planes attract each other is given by

$\displaystyle F=-\frac{\partial E}{\partial
l_{p}}=-(1+\gamma_{0}+2\gamma_{1}-\frac{\lambda_{0}+2\lambda_{1}}{3}l_{p})\;\frac{\pi^{2}\hbar
c}{480l_{p}^{4}},$ (33)

which for the Schwarzschild spacetime gives

$\displaystyle F=-\frac{\partial E}{\partial
l_{p}}=-(1-\frac{R_{s}}{R}-\frac{R_{s}}{3R^{2}}l_{p})\;\frac{\pi^{2}\hbar
c}{480l_{p}^{4}}.$ (34)

This shows that the mutual force between the planes decreases for
Schwarzschild spacetime.

##  IV more analysis on the presence of first order corrections $\gamma_{0}$
and $\gamma_{1}$ in the energy

Although the energy in (28) has been found in a quite unambiguous manner, one
may still think that the constants $\gamma_{0}$ and $\gamma_{1}$ should not be
present in the energy (28) because we can initially absorb them into the time
and space parts of the metric (3) by using a simple rescaling
$t\rightarrow(1+\gamma_{0})t,\;\overrightarrow{x}\rightarrow(1+\gamma_{1})\overrightarrow{x}$.
This idea has been used in some previous studies in the subject and we show
that it is not correct. For instance see equation (8) in BorzooEPJC , the
discussion before equation (17) in Fulling  , equation (2.5) in Sorge2019  and
equation (2.15) in Lima . In the following, we give more elementary and
satisfactory analysis to justify the presence of the constants $\gamma_{0}$
and $\gamma_{1}$ in (28).

First we should observe that the above-mentioned rescaling is acceptable only
when there is no boundary in the problem under consideration or the boundary
be at spatial infinity. When the spacetime manifold has boundary, any
transformation should be applied to the boundary as well, otherwise the
physics will change. In fact, to achieve consistent results, any
transformation must be applied to the whole problem (which consists of the
metric, the boundaries and the Klein-Gordon equation) rather than only to the
metric. Second, to be more clear, we now apply the rescalings to both the
spacetime and boundaries and calculate mode frequencies again.

To proceed, let’s apply the transformations to the Klein-Gordon equation
first. By
$t^{\prime}=(1+\gamma_{0})t,\;\overrightarrow{x}^{\prime}=(1+\gamma_{1})\overrightarrow{x}$
we find

$\displaystyle\partial_{0}$
$\displaystyle=(1+\gamma_{0})\partial_{0^{\prime}}\;,\;\partial^{2}_{0}=(1+2\gamma_{0})\partial^{2}_{0^{\prime}}\;,$
(35a) $\displaystyle\partial_{i}$
$\displaystyle=(1+\gamma_{1})\partial_{i^{\prime}}\;,\;\partial^{2}_{i}=(1+2\gamma_{1})\partial^{2}_{i^{\prime}}\;,$
(35b) $\displaystyle g_{00}$ $\displaystyle=(1+2\gamma_{0})g_{0\prime
0^{\prime}}\;,\;g^{00}=(1-2\gamma_{0})g^{\prime 00}\;,$ (35c) $\displaystyle
g_{11}$
$\displaystyle=(1+2\gamma_{1})g^{\prime}_{33}\;,\;g^{33}=(1-2\gamma_{1})g^{\prime
11}\;.$ (35d)

Now the metric recasts into

$\displaystyle ds^{2}=(1+2\lambda_{0}z^{\prime})dt^{\prime
2}-(1+2\lambda_{1}z^{\prime})\left(dx^{\prime 2}+dy^{\prime 2}+dz^{\prime
2}\right),$ (36)

It can be easily checked out that the Klein-Gordon equation is invariant under
the rescalings. Thus we use

$\displaystyle g^{\prime 00}\partial_{t^{\prime}}^{2}\Phi+g^{\prime
33}(\partial_{x^{\prime}}^{2}\Phi+\partial_{y^{\prime}}^{2}\Phi)+\frac{1}{\sqrt{-g^{\prime}}}\partial_{z^{\prime}}(\sqrt{-g^{\prime}}g^{\prime
33}\partial_{z^{\prime}}\Phi)+m^{2}\Phi=0.$ (37)

The transformation changes the boundaries as well

$\displaystyle\begin{split}&z=0\;\rightarrow z^{\prime}=0,\\\
&z=l\;\rightarrow z^{\prime}=l^{\prime}=l(1+\gamma_{1}).\end{split}$ (38)

For the wave function we have

$\displaystyle\begin{split}\Phi(x)&=Ce^{-i\omega
t}e^{i\textbf{k}_{\perp}.\textbf{x}_{\perp}}Z(z)=Ce^{-i\omega(1-\gamma_{0})(1+\gamma_{0})t}e^{i\textbf{k}_{\perp}(1-\gamma_{1}).(1+\gamma_{1})\textbf{x}_{\perp}}Z(z)\\\
&=Ce^{-i\omega^{\prime}t^{\prime}}e^{i\textbf{k}_{\perp}^{\prime}.\textbf{x}^{\prime}_{\perp}}Z(z^{\prime}),\end{split}$
(39)

where

$\displaystyle\omega^{\prime}$ $\displaystyle=\omega(1-\gamma_{0}),$ (40a)
$\displaystyle k_{\perp}^{\prime}$ $\displaystyle=k_{\perp}(1-\gamma_{1}).$
(40b)

Now, the new problem consists of the metric (36), the Klein-Gordon equation
(37) and the boundary condition (38). To find mode frequencies we use the
ansatz in (39) and do the same process as already done through (13) to (19a).
The result is given by

$\displaystyle\omega^{\prime}$
$\displaystyle=(1+\frac{\lambda_{0}-\lambda_{1}}{2}l^{\prime})\sqrt{\omega_{0}^{\prime
2}+M^{\prime 2}},$ (41)

where $\omega_{0}^{\prime}=(k_{\perp}^{\prime
2}+(\frac{n\pi}{l^{\prime}})^{2})^{\frac{1}{2}}$ and $M^{\prime
2}=(1+\lambda_{1}l)m^{2}$. In (40a) we saw
$\omega=(1+\gamma_{0})\omega^{\prime}$. Equations (38) and (40b) also give
$\omega_{0}^{\prime 2}=(1-2\gamma_{1})(k_{\perp}^{2}+(\frac{n\pi}{l})^{2})$.
Moreover, note that $M^{\prime 2}=(1-2\gamma_{1})M^{2}$. Substituting all
these back into (41) recovers (19a) again.

The non-vanishing corrections in first order perturbation is not related to
the vacuum state. As we see in various spacetimes, for example, in
Schwrazschild with $\gamma_{0}=-\gamma_{1}\neq 0$ or in Kerr spacetime with
$\gamma_{0}\neq-\gamma_{1}\neq 0$, the corrections does not vanish. Since the
vacuum state is different for this examples, we conclude that the non-
vanishing contribution of $\gamma_{0}$ and $\gamma_{1}$ to the energy is not a
vacuum effect. It is really a boundary effect and the main concern of this
section is to justify such a point. In fact in the presence of boundary we do
not eligible to perform gauge transformations to throw away and ignore the
constants $\gamma_{0}$ and $\gamma_{1}$.

Another point is that the total energy (28) does not depend on the coupling
constant $\zeta$. Although we ignored $\zeta$ previously through the term
$\zeta R$ , we expect to see it in the final result for the energy because
$\zeta$ presents in the energy-momentum tensor. However, contrary to this
expectation, we saw that $\zeta$ presents only in the surface sector of the
energy-momentum tensor which was shown that vanishes for D.N. B.C.s.

##  V Energy for massive and finite temperature scalar field

Regularizing and renormalizing the summation in equation (25) follows standard
procedures. However, it differs from that of usually done by
$M=(1+\gamma_{1}+\frac{l}{2}\lambda_{1})m$ which contains not only $m$ but
also the spacetime parameters $\gamma_{1},\;\lambda_{1}$. Therefore, we need
to carefully redo the procedure to find the explicit contribution of
$\gamma_{1}$ and $\lambda_{1}$. In fact, the energy appears in (28) has the
form $(1+\gamma_{0}+2\gamma_{1}+...)E^{flat}_{m=0}$ in which
$E^{flat}_{m=0}=-\frac{\pi^{2}}{1440l_{p}^{3}}$ is the massless flat space
Casimir energy. For the massive case, the Casimir energy has the same form
with $E^{flat}_{m=0}$ being replaced by $E^{flat}_{m\neq 0}$.

The summation in (25) can be written as MostepanenkoBook

$\displaystyle\begin{split}\mathbb{E}&=\frac{\hbar}{2}\int\frac{k_{\perp}dk_{\perp}}{(2\pi)^{2}}\sum_{n}\sqrt{\omega_{0}^{2}+M^{2}}\\\
&=\frac{\hbar}{2}\int_{0}^{\infty}\frac{k_{\perp}dk_{\perp}}{2\pi}\left[\sum_{n=0}^{\infty}\sqrt{k^{2}_{\perp}+\left(\frac{n\pi}{l}\right)^{2}+M^{2}}-\frac{l}{\pi}\int_{0}^{\infty}dk_{z}\sqrt{k^{2}_{\perp}+k_{z}^{2}+M^{2}}\right]\end{split}$
$\displaystyle=\frac{\pi^{2}}{4l^{3}}\int_{a}^{\infty}ydy\left[\sum_{n=0}^{\infty}\sqrt{y^{2}+n^{2}}-\int_{0}^{\infty}dt\sqrt{y^{2}+t^{2}}\right],$
(42)

in which $a=\frac{l}{\pi}M,\;\hbar=c=1$. Using the Abel-Plana formulae

$\displaystyle\sum_{0}^{\infty}F(n)-\int_{0}^{\infty}F(t)dt=\frac{1}{2}F(0)+i\int_{0}^{\infty}\frac{F(it)-F(-it)}{e^{2\pi
t}-1}dt,$ (43)

it can be shown that

$\displaystyle\begin{split}\mathbb{E}=-\frac{\pi^{2}}{6(2\pi)^{4}l^{3}}\int_{p_{0}}^{\infty}\frac{\left(u^{2}-p_{0}^{2}\right)^{\frac{3}{2}}}{e^{u}-1}du,\end{split}$
(44)

in which $p_{0}=2lM$. Now, the important observation is that

$\displaystyle
p_{0}=2lM=2(1-\gamma_{1}-\frac{\lambda_{1}}{2}l_{p})l_{p}(1+\gamma_{1}+\frac{\lambda_{1}}{2}l_{p})m=2l_{p}m,$
(45)

where we have used $l=(1-\gamma_{1}-\frac{\lambda_{1}}{2}l_{p})l_{p}$ again.
Equation (45) proves that $\mathbb{E}$ is free of spacetime parameters and
$\mathbb{E}=E^{flat}_{m\neq 0}$ which is the massive Casimir energy in flat
spacetime MostepanenkoBook . Thus, we have shown that

$\displaystyle E_{m\neq
0}=(1+\gamma_{0}+2\gamma_{1}+\frac{\lambda_{0}+2\lambda_{1}}{2}l_{p})E^{flat}_{m\neq
0}.$ (46)

The above formalism can be extended to the case of the finite temperature
field using the Matsubara formalism Matsubara . Some studies has been done
previously in the literature borzoothermal1 ; borzoothermal2 ; SERNELIUS . The
point is that it suffices to find modified frequencies and simply use the
Matsubara summation formula for the field in its excited states. See equations
(23)-(28) in borzoothermal1 .

##  VI A glance on corner (joint) terms

If we have a discontinuity or abrupt change in the unit normal to $\partial
M_{s}$ then the last term in (7) should be kept. Since our setting consists of
two parallel plates, there is a discontinuity in the normal vector on
transition from one plate to another one at the edges of the plates.

The role of corner terms in the quasi-local approach to gravitational action
has been explored in many studies. Joint terms may be on the intersection of
$\Sigma_{1}$(or $\Sigma_{2}$) and $\partial M_{s}$ or completely on $\partial
M_{s}$ itself. The later fits our problem as we know that the discontinuity in
$n^{\mu}$ breaks the $2D$ surface $\partial\Sigma$ into parts. To find the
probable contribution of the corresponding joint terms in the surface stress-
tensor we use the following typical form of the joint terms in the
gravitational action BrownLawYork

$\displaystyle\frac{1}{2}\int_{\partial\Sigma}d^{D-1}x\sqrt{|\sigma|}\phi^{2}(u^{\mu}\delta
n^{\mu}+\overline{n}_{\mu}\delta u^{\mu})$ (47)

This form equals the third term in (7) in the case of timelike joint. Now
assume that $\partial\Sigma$ be broken into four parts corresponding to four
edges of the apparatus. Each of this edges give $\delta
n^{\mu}=n^{\mu}|_{z=l}-n^{\mu}|_{z=0}=n^{\mu}-(-n^{\mu})=2n^{\mu}$. On the
other side, we know that $n_{\mu}u^{\mu}=0$ and consequently $u^{\mu}\delta
n^{\mu}=0$. For the second term in (47) we use $\delta u^{\mu}=0$. Therefore,
the joint term contribution to the Casimir energy vanishes. The same manner
can be employed for timelike joint terms in our setting Hayward ; Lehner .

##  VII conclusion

Since the Casimir effect is a phenomena which occur under the presence of
boundaries, it is acceptable and natural to study it in curved spacetime
through the quasi-local approach to the gravitational action and energy. In
this paper, we applied the quasi-local approach to the Casimir energy and
showed that the calculations become much easier and reliable. We found
previous results in the literature (see after equation (28)) and extended the
calculations to the case of massive and arbitrarily coupled scalar field. We
found that for the case of massive field the energy equals the flat space
Casimir energy times a factor composed of spacetime parameters (see section
V). The coupling constant was absent in the final result for the energy due to
the fact that it contributed only in the surface part of the energy. However,
we showed that the surface energy vanishes under Dirichlet and Neuman boundary
conditions. As an aside, the curvature coupling did not affect the energy up
to second order perturbation under the influence of weak gravitational field.
We showed that the above results are also valid for the case of Neumann
boundary conditions.

The common thought was that the impact of the gravitational field on the
Casimir energy appears only in second order perturbation expansion hence it
can not be measured within the current precision of the experiments. We showed
that the correction to the Casimir energy appears also within the first order
perturbation calculations. As we shown in section IV, the source of error in
previous studies was the fact that the lowest order perturbations had been
omitted in the course of an unappropriate transformation of the spacetime
metric. Therefore, the corrections are twelve orders of magnitudes larger than
what previously found and hopefully be measured by the current precision of
the experiments.

There was another issue related to the use of quasi-local approach for the
Casimir effect. The joint terms should be incorporated in the gravitational
action due to the presence of discontinuities in the normal vector of the
boundaries. This terms may have contributions in the energy-momentum tensor
and we shown that they vanish for the case of parallel plates.

##  Appendix A conserved quantities

If a spacetime admits some killing symmetries, one finds conserved quantities
according to the Noether theorem. An important probable conserved quantity has
been shown to be Saharian2

$P_{\Sigma_{2}}-P_{\Sigma_{1}}=\int_{\partial
M_{s}}d^{D}x\sqrt{|h|}\,n_{\mu}n_{\nu}n^{\lambda}\eta_{\lambda}T^{{\rm(vol)}\mu\nu},$
(48)

where $P_{\Sigma}$ is defined by:

$P_{\Sigma}=\int_{{\Sigma}}d^{D}x\sqrt{|\gamma|}\,u_{\mu}\eta_{\nu}T^{{\rm(vol)}\mu\nu}+\int_{{\partial\Sigma}}d^{D-1}x\sqrt{|\sigma|}\,\eta_{\nu}\overline{u}_{\mu}\tau^{\mu\nu}.$
(49)

The first term in the right side is the volume part $P^{{\rm(vol)}}_{\Sigma}$
and the second one is $P^{{\rm(surf)}}_{\Sigma}$. The possibility to have
$P_{\Sigma}$ conserved is available by letting $n^{\lambda}\eta_{\lambda}=0$
on the right side of (48). This, in turn, needs to have some killing vectors
tangent to boundaries of the spacetime manifold $M$.

Any static spacetime has a timelike hypersurface-ortghogonal killing vector
field. Suppose $\eta^{\mu}=\delta^{\mu}_{0}$ be such a killing vector. The
hypersurface $\Sigma$ is described by $t=const.$ hence the unit normal vector
is given by

$u^{\mu}=\frac{\delta^{\mu}_{0}}{\sqrt{|g_{00}|}},\;u_{\mu}=\sqrt{|g_{00}|}\,\delta_{\mu
0},\;\eta_{\mu}=g_{\mu 0},$ (50)

in which the hypersurface-orthogonal condition has been supposed to be
$u^{\mu}=a\eta^{\mu}$ for some constant $a$. Putting (50) into (49) we find
for the volume part of (49)

$\begin{split}P^{{\rm(vol)}}_{\Sigma}&=\int_{{\Sigma}}d^{D}x\sqrt{|\gamma|}\,u_{\mu}\eta_{\nu}T^{{\rm(vol)}\mu\nu}\\\
&=\int_{{\Sigma}}d^{D}x\sqrt{|g|}\,T^{{\rm(vol)}0}_{0},\end{split}$ (51)

where use is made of $\sqrt{|g|}=\sqrt{|\gamma|}\sqrt{|g_{00}|}$ and
$g^{00}=g_{00}^{-1}$ for a static spacetime. A similar result can be found for
$P^{{\rm(surf)}}_{\Sigma}$.

## Acknowledgments

The author would like to thank University of Tehran for supporting this
research under the grant No. 30102/1/01.

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