Nonminimal coupling, quantum scalar field stress-energy tensor and energy conditions on global anti-de Sitter space-time

The vacuum Green functions $G_{0}^{{{D}}/{{N}}}(s)$ with Dirichlet ($D$) and Neumann ($N$) boundary conditions applied are Namasivayam:2022bky

where ${}_{2}F_{1}(a,b,c;z)$ are hypergeometric functions and $s(x,x^{\prime})$ is the proper distance between the points $x,x^{\prime}$, given by Avis:1977yn

with $\Delta t=t^{\prime}-t$ and $\Delta\theta=\theta^{\prime}-\theta$. The only difference between the Dirichlet and Neumann vacuum Green functions is the sign of the first term.

To find the RSET, we first subtract from the Green functions (2.3) the singular part $G_{\text{sing}}(x,x^{\prime})$ of the Hadamard parametrix, leaving the state-dependent biscalar $W(x,x^{\prime})$:

In three space-time dimensions, the RSET is then given, in terms of the coincidence limits of $W(x,x^{\prime})$ and its derivatives, by Decanini:2005eg

where

The resulting v.e.v.s of the RSET when Dirichlet or Neumann boundary conditions are applied are then Kent:2014nya

The v.e.v.s (2.8) differ only by an overall sign. Both are proportional to the metric, indicating that the corresponding vacuum states respect the maximal symmetry of the underlying space-time, and hence the v.e.v. of the RSET is determined entirely by its trace. For a massless scalar field, we find that $\langle{\hat{T}}_{\alpha\gamma}\rangle_{0}^{{{D}}/{{N}}}=0$, as expected since there is no trace anomaly in three space-time dimensions.

Since $\langle{\hat{T}}_{\alpha\gamma}\rangle_{0}^{{{D}}/{{N}}}$ is proportional to the metric, the NEC is satisfied trivially for both Dirichlet and Neumann boundary conditions, and all scalar field masses and couplings. Furthermore, $\langle{\hat{T}}_{\alpha\gamma}\rangle_{0}^{{{D}}/{{N}}}$ has the perfect fluid (1.2) form with

where the $-$ sign is for Dirichlet boundary conditions and the $+$ sign for Neumann boundary conditions. Therefore, when $0<\nu<1$, the WEC is satisfied by the v.e.v. of the RSET for Neumann but not Dirichlet boundary conditions. However, if we consider the semiclassical Einstein equations (1.6), then the RSETs (2.8) correspond to a renormalization of the cosmological constant Kent:2014nya ; Thompson:2024vuj , so there are no deep consequences of any violation of the WEC.

The thermal Green functions, $G_{\beta}^{D/N}(x,x^{\prime})$, for inverse temperature $\beta$, can be expressed as an infinite sum involving the vacuum Green function $G_{0}^{D/N}(x,x^{\prime})$ Birrell:1982ix :

giving Namasivayam:2022bky

where $s_{\beta}$ is given by

To determine the t.e.v. of the RSET with Dirichlet and Neumann boundary conditions, we apply the methodology used in Namasivayam:2022bky and consider the difference between the expectation values of the RSET in the thermal and vacuum states. As the vacuum Green functions in (2.3) are given by the $j=0$ contributions to the sums in (2.11), we define a regular biscalar $W_{\beta}^{D/N}(x,x^{\prime})$ as follows:

We then use this biscalar in (2.6) to give the difference between the t.e.v. and the v.e.v. of the RSET. We can then simply add the relevant v.e.v. to the final answer, to give the t.e.v.. Despite the simplicity of the expression (2.13), the resulting components of the RSET are algebraically lengthy, these can be found in appendix A. We compute the t.e.v.s numerically in Mathematica, and find that the sum over $j$ converges very rapidly.

In Fig. 2 we show the energy density component of the RSET, $-\langle\hat{T}_{t}^{t}\rangle_{\beta}^{D/N}$, for Dirichlet (left) and Neumann (right) boundary conditions when $\nu=3/4$, $\xi=1/8$ and a selection of values of the inverse temperature $\beta$. The red dotted lines denote the v.e.v., which corresponds to $\beta\rightarrow\infty$. For both boundary conditions, we see that the energy density as a function of $\rho$ has a maximum at the space-time origin and is monotonically decreasing as $\rho$ increases. This is to be expected, as the local inverse temperature ${\widetilde{\beta}}$ is related to the (fixed) inverse temperature $\beta$ via Tolman:1930zza ; Tolman:1930ona ; Ambrus:2018olh

and hence the local temperature ${\widetilde{\beta}}^{-1}$ has a maximum at $\rho=0$ and tends to zero as $\rho\rightarrow\pi/2$ and the boundary is approached. The value of the maximum in the energy density at $\rho=0$, unsurprisingly, increases with increasing temperature $\beta^{-1}$.

Of more interest is the difference between Dirichlet (left) and Neumann (right) boundary conditions. In particular, we see that the energy density is significantly greater when Neumann boundary conditions are applied, compared to Dirichlet boundary conditions. Similar behaviour is seen for the vacuum polarization on adS in three dimensions Namasivayam:2022bky . We also see that the energy density decreases more rapidly as the space-time boundary is approached for Dirichlet boundary conditions than for Neumann boundary conditions.

For both Dirichlet and Neumann boundary conditions, for the values of $\nu$ and $\xi$ considered in Fig. 2 (which correspond to a massive but conformally coupled scalar field), we see that the energy density in the thermal states is always greater than that in the corresponding vacuum state. For Dirichlet and Neumann boundary conditions, it is the difference in expectation values between the thermal and vacuum states which affects the backreaction of the quantum field on the space-time background, due to the maximal symmetry of the vacuum states with these boundary conditions applied Thompson:2024vuj . To see this, following Thompson:2024vuj , we write (1.6) in the alternative form

where

is the difference between the thermal and vacuum expectation values. Using the v.e.v.s (2.8), we can write (2.15) as

where ${\widetilde{\Lambda}}=\Lambda\pm m\nu^{2}/(12\pi L)$ is the renormalized cosmological constant. From Fig. 2, we see that the effective RSET $\Delta{\hat{T}}_{\alpha}^{\gamma}$ for the conformally coupled field has a positive energy density.

To explore whether this remains the case for nonconformally-coupled fields, in Fig. 3 we show the energy density $-\langle\hat{T}_{t}^{t}\rangle_{\beta}^{D/N}$ as a function of both the radial coordinate $\rho$ and coupling constant $\xi$, for fixed $\nu=1/4$, and two values of the inverse temperature $\beta$, considering Dirichlet boundary conditions on the left, and Neumann boundary conditions on the right. For all values of $\xi$, the t.e.v. energy density approaches the v.e.v. energy density (2.9) as $\rho\rightarrow\pi/2$ and the space-time boundary is approached. However, the profiles of the energy density as a function of $\rho$ depend strongly on the value of the coupling constant $\xi$ (bearing in mind that varying $\xi$ for fixed $\nu$ (2.2) corresponds to varying $m^{2}$, the squared mass of the field). For sufficiently large $\xi$, the energy density at the origin is a maximum and decreases as $\rho$ increases towards the boundary, as seen in Fig. 2 for a conformally coupled field (albeit with a different value of $\nu$). However, as $\xi$ decreases and becomes negative, the energy density takes its minimum value at the origin and is monotonically increasing as $\rho$ increases. In this case the effective RSET (2.16) no longer satisfies the WEC. The violations of the WEC are maximal at the origin, and tend to zero on the space-time boundary.

To examine whether the NEC is satisfied even though the WEC is not, we need to consider other components of the RSET. Fig. 4 shows the component $\langle T_{\rho}^{\rho}\rangle_{\beta}^{D/N}$ of the RSET as a function of $\rho$ and $\xi$, with fixed inverse temperature $\beta=1$ and $\nu=3/4$. Once again we see a strong dependence on the coupling constant $\xi$, but rather different behaviour from that observed for the $-\langle T_{t}^{t}\rangle_{\beta}^{D/N}$ component in Fig. 3. For sufficiently large and positive $\xi$, the value of $\langle T_{\rho}^{\rho}\rangle_{\beta}^{D/N}$ has a minimum at the origin and increases towards the boundary, while for sufficiently large and negative $\xi$ the value at the origin is a maximum and $\langle T_{\rho}^{\rho}\rangle_{\beta}^{D/N}$ decreases as the boundary is approached.

The surface plots of the remaining RSET component, $\langle T_{\theta}^{\theta}\rangle_{\beta}^{D/N}$, are indistinguishable by eye from those for $\langle T_{\rho}^{\rho}\rangle_{\beta}^{D/N}$. However, these two components of the RSET are not, in fact, equal. We therefore employ the Landau decomposition of the RSET in an analogous way to that in four space-time dimensions Ambrus:2018olh

where $E_{\beta}^{D/N}$ is the energy density, $P_{\beta}^{D/N}$ the pressure and $\Pi_{\beta}^{D/N}$ is the pressure deviator Ambrus:2018olh . For a classical thermal gas of particles, the pressure deviator vanishes identically. In Fig. 5 we show $-\Pi_{\beta}^{D/N}$ as a function of the radial coordinate $\rho$ . For both Dirichlet and Neumann boundary conditions and all values of $\xi$ considered, the pressure deviators $\Pi_{\beta}^{D/N}$ attain their maximum magnitude between the space-time origin and boundary and are zero at the space-time origin. At the space-time boundary however, the difference between the two boundary conditions is noticeable. For Dirichlet boundary conditions, $\Pi_{\beta}^{D}$ vanishes on the boundary whereas for Neumann boundary conditions, $\Pi_{\beta}^{N}$ is nonzero for all $\rho$ considered, with its magnitude increasing with increasing $|\xi|$. However, for both Dirichlet and Neumann boundary conditions, $\Pi_{\beta}^{D/N}$ have their minimum magnitudes at fixed $\rho$ for conformal coupling ($\xi=1/8$), suggesting that it is for conformal coupling that the quantum scalar field most closely resembles its classical counterpart.

Although the RSET no longer has the perfect fluid form (1.2), the NEC will be satisfied if $\langle{\hat{T}}_{\rho}^{\rho}\rangle_{\beta}^{D/N}-\langle{\hat{T}}_{t}^{t}\rangle_{\beta}^{D/N}\geq 0$ and $\langle{\hat{T}}_{\theta}^{\theta}\rangle_{\beta}^{D/N}-\langle{\hat{T}}_{t}^{t}\rangle_{\beta}^{D/N}\geq 0$. However, since the pressure deviator $\Pi_{\beta}^{D/N}$ is roughly an order of magnitude smaller than the corresponding RSET components, in order to avoid a proliferation of plots displaying qualitatively similar results, in this work we focus on the combination of RSET components $-\langle{\hat{T}}_{t}^{t}\rangle_{\beta}^{D/N}+\langle{\hat{T}}_{\rho}^{\rho}\rangle_{\beta}^{D/N}$ (further plots and discussion of the RSET components can be found in NamasivayamPhD ). The positivity of this combination is a necessary (but not sufficient) condition for the NEC to be satisfied. Since for the v.e.v., $\langle{\hat{T}}_{\rho}^{\rho}\rangle_{0}^{D/N}=\langle{\hat{T}}_{t}^{t}\rangle_{\beta}^{D/N}$ (2.8), when studying the NEC in this section we do not need to consider the effective RSET (2.16).

In Figs. 6 and 7 respectively, we show the combination of RSET components $\langle{\hat{T}}_{\rho}^{\rho}\rangle_{\beta}^{D/N}-\langle{\hat{T}}_{t}^{t}\rangle_{\beta}^{D/N}$ for Dirichlet and Neumann boundary conditions, and selected values of the coupling constant $\xi$, parameter $\nu$ (2.2) and inverse temperature $\beta$. The NEC is violated when this combination of RSET components is less than zero.

Considering first Dirichlet boundary conditions (Fig. 6), at the higher temperature ($\beta=1$, left-hand plots) we see that, for the values of the coupling constant $\xi$ considered, there are no violations of the NEC in a neighbourhood of the origin. However, for sufficiently large positive $\xi$, there are small regions close to the boundary where the NEC is violated. At the lower temperature ($\beta=3$, right-hand plots), the NEC is again violated in a neighbourhood of the boundary for sufficiently large and positive $\xi$, but there are also violations of the NEC close to the origin for sufficiently large and negative $\xi$, at least for some values of the parameter $\nu$ (2.2).

The situation for Neumann boundary conditions (Fig. 7) is similar at the lower temperature ($\beta=3$, right-hand plots), with violations of the NEC in a region including the origin for sufficiently large and negative $\xi$, and violations in a neighbourhood of the boundary for sufficiently large and positive $\xi$. At the higher temperature ($\beta=1$, left-hand plots), we again find that the NEC is violated close to the boundary for sufficiently large and positive $\xi$, but, unlike Dirichlet boundary conditions, there are also small violations of the NEC close to the origin for sufficiently large and negative $\xi$ when $\nu=3/4$. We conjecture that similar violations of the NEC would occur for Neumann boundary conditions and other values of $\nu$, and for Dirichlet boundary conditions, if we considered coupling constants $\xi$ having larger magnitudes.

For both Dirichlet and Neumann boundary conditions, and for both inverse temperatures considered, we do not find any significant violations of the NEC for either minimally ($\xi=0$) or conformally coupled ($\xi=1/8$) fields, and for values of $\xi$ lying in a sufficiently small interval containing these two particular values (there may be very small violations of the NEC very close to the space-time boundary).

All of the plots in Figs. 6 and 7 have an additional feature in common, namely that the curves for different $\xi$ and fixed $\nu$, $\beta$ all intersect at a particular value of $\rho\approx\pi/6$. This phenomenon can be understood as follows. The quantities $w$ (and its derivatives) and $w_{\alpha\gamma}$ (2.7) which appear in the RSET (2.6) are constructed from the biscalar $W(x,x^{\prime})$ (2.5) which depends on the scalar field mass $m$ and coupling constant $\xi$ only through the combination $\nu$ (2.2). Therefore the terms in the RSET (2.6) which are proportional to $\xi$ depend on $\nu$ and not on $\xi$, $m^{2}$ separately. In all our figures, we consider fixed $\nu$, even when varying the coupling constant $\xi$. The curves for varying values of $\xi$ intersect when the terms in (2.6) proportional to $\xi$ vanish for a particular value of the radial coordinate $\rho$.

We perform a Wick rotation, $t\to i\tau$, to give the Euclidean adS₃ metric

Suitable ansatze for the vacuum, $G^{E}_{0}(x,x^{\prime})$, and thermal, $G^{E}_{\beta}(x,x^{\prime})$, Euclidean Green functions are Namasivayam:2022bky

where $\Delta\tau=\tau-\tau^{\prime}$ and $\Delta\theta=\theta-\theta^{\prime}$ are the separation of the time and angular coordinates respectively, $\kappa=2\pi/\beta$ (with $\beta$ the inverse temperature) and $g_{\omega\ell}(\rho,\rho^{\prime})$, $g_{nl}(\rho,\rho^{\prime})$ are, respectively, the vacuum and thermal radial Green functions.

Applying Robin boundary conditions to the scalar field, the vacuum radial Green function $g_{\omega\ell}^{\zeta}(\rho,\rho^{\prime})$ takes the form Namasivayam:2022bky

where $\rho_{<}=\min\{\rho,\rho^{\prime}\}$ and $\rho_{>}=\max\{\rho,\rho^{\prime}\}$. The normalization constant $\mathcal{N}_{\omega l}^{\zeta}$ is given by

The corresponding thermal radial Green function, $g_{n\ell}^{\zeta}(\rho,\rho^{\prime})$, is given by substituting $\omega=n\kappa$ into the vacuum radial Green function (3.3). In (3.3, 3.4), the quantity $\zeta$ parameterizes the Robin boundary conditions, with $\zeta=0$ corresponding to Dirichlet boundary conditions, and $\zeta=\pi/2$ corresponding to Neumann boundary conditions. The denominator of (3.4) becomes zero if $\zeta$ satisfies the equation

in which case the vacuum and thermal Green functions become divergent. Denoting the smallest positive root of (3.5) by $\zeta_{\text{crit}}$ (where $\zeta_{\text{crit}}>\pi/2$), we restrict attention to values of the Robin parameter below this critical value, that is, $0\leq\zeta<\zeta_{\text{crit}}$ Namasivayam:2022bky . In our numerical results in the following subsections, we will focus particularly on the values $\nu=1/4$, $1/2$ and $3/4$ for the parameter $\nu$ (2.2), for which $\zeta_{\text{crit}}$ is approximately given by

Since the divergences in the Green functions are independent of the quantum state under consideration, we follow Namasivayam:2022bky and employ a state-subtraction technique to find the RSET when Robin boundary conditions are applied to the quantum scalar field. Previous studies of the expectation value of the square of the scalar field (the “vacuum polarization”) in both three Namasivayam:2022bky and four space-time dimensions Morley:2020ayr and of the RSET for a massless, conformally-coupled scalar field on four-dimensional global adS Morley:2023exv have shown that it is Neumann boundary conditions which give the generic behaviour on the space-time boundary. We therefore follow the methodology of Namasivayam:2022bky and consider the difference in RSET between vacuum and thermal states with Robin and Neumann boundary conditions applied. To implement this approach, we subtract the vacuum/thermal Euclidean Green function with Neumann boundary conditions applied from the corresponding Green function with Robin boundary conditions applied, to give the regular biscalars

where $G^{\zeta}_{0/\beta}(x,x^{\prime})$ are the vacuum/thermal Green functions (3.2) with Robin boundary conditions applied, and $G^{N}_{0/\beta}(x,x^{\prime})=G^{\frac{\pi}{2}}_{0/\beta}(x,x^{\prime})$ are the vacuum/thermal Green functions for Neumann boundary conditions. Similarly, we denote the radial Green functions for Neumann boundary conditions by $g_{\omega\ell}^{N}(\rho,\rho^{\prime})$ and $g_{n\ell}^{N}(\rho,\rho^{\prime})$ for vacuum and thermal states, respectively.

The biscalars (3.7) are substituted into (2.6) to give the difference in RSET expectation values between the relevant states with Robin and Neumann boundary conditions. The final v.e.v.s and t.e.v.s of the RSET for Robin boundary conditions are then found by adding the corresponding RSET expectation values for Neumann boundary conditions, computed in section 2.

Substituting the biscalar (3.7a) into (2.6) gives an expression for the difference in v.e.v.s of the RSET with Robin and Neumann boundary conditions, involving an infinite sum over the quantum number $\ell$ and an integral over the frequency $\omega$. The lengthy expressions for the quantities to be summed/integrated over can be found in appendix B. The sums and integrals are performed numerically using Mathematica, following the methodology of Namasivayam:2022bky . The $\omega$ integral is computed first, followed by the sum over $\ell$. The integrals converge rapidly for large $|\omega|$, and, for fixed values of the radial coordinate $\rho$, the sum over $\ell$ is also rapidly converging. We find that truncating the integral at $|\omega|=100$ and the sum at $|\ell|=100$ gives final results which are sufficiently accurate for our purposes, away from the space-time boundary. For example, for a representative value of the Robin parameter $\zeta=\pi/10$ with $\nu=1/2$ and $\xi=1/8$, the relative error in the $\langle{\hat{T}}_{\tau}^{\tau}\rangle$ component of the RSET at $\rho=94\pi/200$ is $\sim 4\times 10^{-6}$ comparing to performing the sum and integral over $|\omega|,|\ell|\leq 200$. As observed in Namasivayam:2022bky , the sum over $\ell$ is not uniformly convergent as the radial coordinate $\rho$ varies. Accordingly, the relative error is much smaller for smaller values of the radius $\rho$.

We begin the discussion of our numerical results by examining, in Figs. 8, 9 and 10 respectively, the v.e.v.s $-\langle\hat{T}_{\tau}^{\tau}\rangle_{0}^{\zeta}$, $\langle\hat{T}_{\rho}^{\rho}\rangle_{0}^{\zeta}$ and minus the vacuum pressure deviator $-\Pi_{0}^{\zeta}$ with Robin boundary conditions applied to the quantum scalar field (additional plots and discussion can be found in NamasivayamPhD ). We take the field to be conformally coupled, $\xi=1/8$, and consider three values of the parameter $\nu$ (2.2), namely $\nu=1/4$, $\nu=1/2$ and $\nu=3/4$. When $\nu=1/4$, the scalar field has negative squared mass $m^{2}<0$; for $\nu=1/2$ the scalar field is massless and conformally coupled, while for $\nu=3/4$ we have $m^{2}>0$. In adS, a negative squared mass does not necessarily indicate an instability; we only consider values of $\nu$ for which the Breitenlohner-Freedman bound is satisfied Breitenlohner:1982bm ; Breitenlohner:1982jf , and Robin boundary conditions for which the scalar field is classically stable.

First consider the energy density component, shown in Fig. 8. For all values of $\nu$, when either Dirichlet ($\zeta=0$) or Neumann ($\zeta=\pi/2$) boundary conditions are applied, this does not depend on the radial coordinate $\rho$ (2.8) (note that for a massless, conformally coupled field with $\nu=1/2$ the v.e.v. vanishes for Dirichlet/Neumann boundary conditions). We can see that this is no longer the case when Robin boundary conditions are applied; the maximal symmetry of the underlying space-time is broken Pitelli:2019svx . When the field is massless ($\nu=1/2$, middle row), the profiles of the energy density have similar qualitative features to those seen for a massless, conformally coupled scalar field in four space-time dimensions Morley:2023exv . In particular, the energy density is positive throughout the space-time. The energy density is a maximum at the origin and monotonically decreasing towards the space-time boundary. The value of the energy density at the origin is zero for Dirichlet boundary conditions, then increases as the Robin parameter $\zeta$ increases, reaches a maximum, then decreases with increasing $\zeta$ until it is zero again for Neumann boundary conditions. When $\zeta>\pi/2$, the magnitude of the v.e.v. of the energy density at the space-time origin increases rapidly as $\zeta$ increases, up to the critical value $\zeta_{\mathrm{crit}}$ (3.6), beyond which the field is classically unstable.

The energy density profiles for massive fields ($\nu=1/4$ and $\nu=3/4$) are qualitatively very different. When $m^{2}\neq 0$, the v.e.v.s for Dirichlet and Neumann boundary conditions (2.8) are no longer the same. As a function of $\rho$, the v.e.v. of the energy density is either a monotonically increasing or monotonically decreasing function. As $\rho$ increases towards the space-time boundary, it seems to be the case that the v.e.v. is approaching the v.e.v. when Neumann boundary conditions are applied. This suggests that it is Neumann rather than Dirichlet boundary conditions which give the generic behaviour on the boundary, as observed previously for the renormalized vacuum polarization Namasivayam:2022bky . When $\nu=1/4$ and the scalar field has negative squared mass, the v.e.v. of the energy density is monotonically decreasing when $0<\zeta<\pi/2$ and monotonically increasing when $\zeta>\pi/2$. The opposite is the case when $\nu=3/4$ and the scalar field has positive squared mass; in this case $-\langle\hat{T}_{\tau}^{\tau}\rangle_{0}^{\zeta}$ is monotonically increasing for $0<\zeta<\pi/2$ and monotonically decreasing for $\zeta>\pi/2$.

When $-\langle\hat{T}_{\tau}^{\tau}\rangle_{0}^{\zeta}<0$, the WEC is violated. For $\nu=1/4$, this is the case close to the boundary for nearly all $\zeta$ studied, and for sufficiently large $\zeta$ it is the case throughout the space-time. When $\nu=3/4$ the energy density is positive (a necessary but not sufficient condition for the WEC to be satisfied) throughout the space-time for sufficiently large $\zeta$, but for $\zeta$ close to zero, the WEC is violated in a neighbourhood of the origin. Since the energy density depends on $\rho$ for $\zeta\neq 0$, $\pi/2$, when Robin boundary conditions are applied it is not possible to absorb the v.e.v. of the RSET into a renormalization of the cosmological constant, as in section 2.2. However, we could decide to take the vacuum state with either Dirichlet or Neumann boundary conditions applied as a reference state, and, since the RSET for these boundary conditions is a constant multiple of the a metric (2.8), absorb this into a renormalization of the cosmological constant, so that we are interested in an effective stress-energy tensor $\Delta{\hat{T}}_{\alpha}^{\gamma}$ (as in (2.17)), where now

Since it is Neumann boundary conditions which give the generic behaviour of the energy density at the space-time boundary, it would be natural to use these boundary conditions in defining the reference state. With this choice, the effective RSET will violate the WEC everywhere in the space-time for $\zeta>\pi/2$ for $\nu=1/4$ and for $0<\zeta<\pi/2$ for $\nu=3/4$. Alternatively, if we choose Dirichlet boundary conditions to give the reference state, the effective RSET violates the WEC for all $0<\zeta<\zeta_{\mathrm{crit}}$ when $\nu=1/4$ but the WEC can potentially be satisfied for all $\zeta$ and $\nu=3/4$ (whether or not the WEC is satisfied depends also on the sign of $\langle\hat{T}_{\rho}^{\rho}\rangle_{0}^{\zeta}-\langle\hat{T}_{\tau}^{\tau}\rangle_{0}^{\zeta}$, which we consider later in this subsection).

The v.e.v.s of the component $\langle\hat{T}_{\rho}^{\rho}\rangle_{0}^{\zeta}$ of the RSET, shown in Fig. 9, have similar properties to those of the energy density. In particular, as $\rho$ increases towards the space-time boundary, they seem to approach the v.e.v. when Neumann boundary conditions are applied, and the profiles as functions of $\rho$ are always either monotonically increasing or decreasing. The (negative) vacuum pressure deviators $-\Pi_{0}^{\zeta}$ (Fig. 10) also show significant differences between the different values of $\nu$ that we consider. For all $\nu$ studied, the pressure deviator is zero at the space-time origin at $\rho=0$, and has a maximum magnitude at a point between the origin and the space-time boundary at $\rho=\pi/2$. For $\nu=1/2$, the pressure deviator $-\Pi_{0}^{\zeta}$ also becomes zero at the space-time boundary (similar behaviour is found for a massless, conformally coupled scalar field in four space-time dimensions Morley:2023exv ). However, for the other values of $\nu$ studied ($\nu=1/4$, $3/4$), while the pressure deviators are decreasing in magnitude as the space-time boundary is approached, our numerical results suggest that they are no longer zero at the boundary.

In Figs. 11 and 12 we explore the consequences for the WEC and NEC on varying the coupling constant $\xi$ and Robin parameter $\zeta$. Given that the pressure deviator (see Fig. 10) is roughly an order of magnitude smaller than the RSET components, we consider only the combination $\langle{\hat{T}}_{\rho}^{\rho}\rangle_{0}^{\zeta}-\langle{\hat{T}}_{\tau}^{\tau}\rangle_{0}^{\zeta}$ in our study of the NEC. Recall, as discussed in section 2.2, whether the NEC is satisfied or violated is independent of the choice of reference state to use in renormalizing the cosmological constant. In Fig. 11, we fix the Robin parameter $\zeta=3\pi/10$, consider three values of the parameter $\nu$ (2.2), namely $\nu=1/4$, $\nu=1/2$ and $\nu=3/4$ and, in each plot, a selection of values of the coupling constant $\xi$. The WEC is violated if the energy density $-\langle\hat{T}_{\tau}^{\tau}\rangle_{0}^{\zeta}<0$ (left-hand column), while the NEC (and hence also the WEC) is violated if $\langle{\hat{T}}_{\rho}^{\rho}\rangle_{0}^{\zeta}-\langle{\hat{T}}_{\tau}^{\tau}\rangle_{0}^{\zeta}<0$ (right-hand column). The WEC can be satisfied only if both quantities in the left- and right-hand columns are positive.

The plots in Fig. 11 have qualitative features in common. As seen previously in Figs. 6 and 7, in each plot the curves for different values of the coupling constant $\xi$ intersect at one value of the radial coordinate $\rho$ (where the terms proportional to $\xi$ in (2.6) vanish). The values of $-\langle\hat{T}_{\tau}^{\tau}\rangle_{0}^{\zeta}$ and $\langle{\hat{T}}_{\rho}^{\rho}\rangle_{0}^{\zeta}-\langle{\hat{T}}_{\tau}^{\tau}\rangle_{0}^{\zeta}$ at the intersection point are not exactly zero; by definition they are equal to their values for a conformally coupled scalar field which are small but nonzero (as can be seen in Figs. 8 and 9). For nearly all values of $\xi$ studied, both the WEC and the NEC are violated in some region of the space-time, either in a neighbourhood of the origin (for sufficiently large and positive $\xi$) or in a neighbourhood of the space-time boundary (for sufficiently large and negative $\xi$). For a minimally coupled scalar field with $\xi=0$, the NEC (and hence also the WEC) is violated in a neighbourhood of the space-time boundary. The degree to which the NEC is violated increases with increasing $|\xi|$.

We explore the effect of changing the Robin parameter $\zeta$ (as well as the coupling constant $\xi$) in Fig. 12, where we have fixed $\nu=1/4$. Examining first the energy density $-\langle\hat{T}_{\tau}^{\tau}\rangle_{0}^{\zeta}$ (left-hand column), the qualitative features of the profiles of this quantity as a function of the radial coordinate $\rho$ depend strongly on the value of the Robin parameter $\zeta$. When $\zeta=\pi/10$, the energy density has the same sign for all $\rho$, and is positive for sufficiently large and negative $\xi$, but negative for sufficiently large and positive $\xi$. When $\zeta=3\pi/10$, the curves for different values of $\xi$ intersect at a particular value of the radial coordinate $\rho$, as seen in Fig. 11. For $\zeta=4\pi/10$ and $\zeta=6\pi/10$, the sign of the energy density is again the same for all values of $\rho$, but is now positive for sufficiently large and positive $\xi$ but negative for sufficiently large and negative $\xi$. For both $\zeta=\pi/10$ and $\zeta=6\pi/10$, the magnitude of the energy density is monotonically decreasing as the radial coordinate $\rho$ increases; in contrast, for $\zeta=4\pi/10$, the magnitude of the energy density is monotonically increasing as $\rho$ increases. From the plots in the left-hand column of Fig. 12, we see that the WEC is violated (due to the energy density being negative) throughout the space-time for some combinations of the parameters $\zeta$ and $\xi$, in particular for $\xi$ sufficiently large and positive when $\zeta=\pi/10$, and for $\xi$ sufficiently large and negative for $\zeta=4\pi/10$, $6\pi/10$.

Turning now to the plots in the right-hand column of Fig. 12, the profiles of the quantity $\langle{\hat{T}}_{\rho}^{\rho}\rangle_{0}^{\zeta}-\langle{\hat{T}}_{\tau}^{\tau}\rangle_{0}^{\zeta}$ share qualitative features with those in Fig. 11. For all fixed values of $\zeta$ studied, the curves for the different values of the coupling constant $\xi$ intersect at a particular value of the radial coordinate $\rho$ (the precise value depending on $\zeta$). We find violations of the NEC (corresponding to $\langle{\hat{T}}_{\rho}^{\rho}\rangle_{0}^{\zeta}-\langle{\hat{T}}_{\tau}^{\tau}\rangle_{0}^{\zeta}<0$) in some region of the space-time for almost all values of $\zeta$ and $\xi$ shown in Fig. 12. When $0<\zeta<\pi/2$ (top three rows in Fig. 12), the NEC is violated in a neighbourhood of the origin for $\xi$ sufficiently large and positive, and in a neighbourhood of the space-time boundary for $\xi$ sufficiently large and negative. When $\zeta>\pi/2$ (bottom row in Fig. 12), the NEC is violated in a neighbourhood of the origin for $\xi$ sufficiently large and negative, and in a neighbourhood of the space-time boundary for $\xi$ sufficiently large and positive.

Substituting the biscalar (3.7b) into (2.6) gives an expression for the difference in t.e.v.s of the RSET with Robin and Neumann boundary conditions, involving infinite sums over the quantum numbers $n$ and $\ell$. The lengthy expressions for the summands can be found in appendix C. Again we follow Namasivayam:2022bky and our numerical computations are performed in Mathematica. We sum over $n$ first, and then $\ell$. In order to have a reasonable computation time, we sum over $|n|\leq 20$ and $|\ell|\leq 50$. Taking a representative value of the Robin parameter, $\zeta=\pi/10$, with $\nu=1/2$ and $\beta=1$, for $\rho=94\pi/200$ we find that the relative error in the RSET components is of order between $10^{-5}$ and $10^{-7}$ compared to summing over $|n|,|\ell|\leq 150$. As in the vacuum case, the relative error is substantially smaller further away from the space-time boundary.