High-Temperature Modifications of Charged Casimir Wormholes

In this work, we extend the investigation of the consequences of thermal fluctuations on the Casimir effect within the context of a traversable wormhole, recently proposed by Garattini \&Faizal, arXiv:2403.15174 [gr-qc], subject to charge contributions. Specifically, we focus on scenarios where the plates exhibit both constant and radial variations. In our analysis, we initially concentrate on the high temperature approximation, considering solely the influence of charge on the thermal Casimir wormholes. Additionally, upon incorporating Generalized Uncertainty Principle (GUP) corrections to the Casimir energy, we obtain a new class of wormhole solutions. Notably, we establish that the flare-out condition remains consistently satisfied. Intriguingly, our findings reveal that both the charge and GUP contributions serve to further enlarge the throat's size in the radial variation.


I. INTRODUCTION
The Casimir effect is based on the assumption of perfectly conducting surfaces.However, real-world plates are never perfectly conducting; they exhibit a complex permittivity, varying with frequency depending on the material.This consideration alters the original Casimir force at zero temperature, transforming it into a finite-temperature thermal effect.This thermal effect incorporates both thermal and quantum fluctuations, as described by the Lifshitz theory, which accounts for correlated fluctuating charges and currents in the plates [1].This adaptation of the conventional Casimir effect to finite temperature is termed the thermal Casimir effect [2,3].It has been shown that at separations beyond a critical value, the force generated by thermal fluctuations surpasses that of zero-temperature quantum fluctuation [4,5].This suggests the potential significance of thermal effects in scaling up the size of traversable wormholes (TWs).Very recently, the consequences of thermal fluctuations to the Casimir effect on a traversable wormhole have been investigated [6].The authors illustrated how thermal fluctuations alter the throat of the wormhole in both high and low temperature regimes.Furthermore, they elucidated the impact of finite temperature on the size of a wormhole.In this work, we investigate new exact and analytic solutions of the Einstein-Maxwell field equations describing Casimir wormholes including the hightemperature contribution to the Casimir energy source for a traversable wormhole with the plates radially varying and discuss the effect of the Generalized Uncertainty Principle (GUP).Notice that the electromagnetic field combined with the Casimir source has been used to examine the impacts of such an electrovacuum on a traversable wormhole [7,8].
The main purpose of this work is to extend the analysis present in Ref. [6] for the thermal Casimir wormholes by incorporating the electric charge underlying the GUP.This work is organized as follows: In Section (II), we formulate the Einstein-Maxwell field equations describing Casimir wormholes with charge.We next consider the thermal Casimir wormholes employing energy densities featuring the thermal modifications to the chared Casimir wormholes and solve to obtian the wormholes solutions in Section (III).The results also include the GUP effects.We conclude of findings in the last section.

II. SETUP
Our aim is to model a wormhole solution and typically we consider an anisotropic fluid stress-energy tensor to describe the matter distribution, which is described by the following form where u µ is the fluid 4-velocity, X µ is the unit space like vector in the direction of radial vector and g µν is the metric tensor.Furthermore, ρ = ρ(r) is the energy density, p r = p r (r) and p t = p t (r) are the radial and tangential pressures, respectively.The energy-momentum tensor of electromagnetic field defined by We define an appropriate frame of the fluid velocity vectors [9] so that u a u a = −1 and X a X a = 1 as required.Now, in its covariant form, we construct the geometrical Einstein tensor on the left-hand side with an effective energy-momentum tensor on the right-hand side, as where We start by considering a static, spherically symmetric and asymptotically flat space-time with the metric where Φ and b are arbitrary functions of the variable 'r,' where Φ is referred to as the 'redshift function', and 'b' is known as the 'shape function.Taking into account the Eqs.
(1) and ( 5), the nonzero components of the field equations ( 4) take the form: where ρ represents the effective energy density, p r is the effective radial pressure and p t is the effective tangential pressure respectively.

III. WORMHOLE SOLUTIONS
Following Ref. [11], we can consider the plates positioned at a distance either parametrically fixed and radially varying.Thus, we consider a more realistic scenario, incorporating the effects of thermal fluctuations as a source of a traversable wormhole.To examine the dependence on the thermal Casimir effect, we consider plates positioned either at a fixed parametric distance or with radial variation.

A. Constant Plates separation
In this case, the form of the energy density has been proposed in Ref. [6].It can be written as a shape function can be straightforwardly computed using Eq.( 7) to obtain where c 1 is a integration constant.Taking b(r 0 ) = r 0 , it can be determined to yield Therefore, the first EFE leads to the following shape function with ζ(x) being the Riemann zeta function and T eff = 1/(2dk B ) in our convention.Setting q = 0, the above result become that of Ref. [6].From the shape function (13), we can verify that the flare-out condition is always satisfied since This implies that To obtain a TW, we need to compute the redshift function.We have to impose the EoS p r (r) = ωρ(r).From Eq.( 8) and Eq.( 13), one finds Close to the throat, the r.h.s. of Eq.( 38) can be approximated by Following Ref. [11] we can choose ω such that Φ ′ (r) = 0 to avoid the appearance of a horizon.
This can be seen if Note that when q = 0, the above result reduces to that of Ref. [6], i.e., ω = 2d 4 T eff ζ(3)r 2 0 T .With a flare-out condition Eq.( 24), we find that and can be negative when r 0 < q.We can generalize the above results by incorporating the GUP corrected energy density.A new class of Casimir wormholes can be basically constructed.Following [7], it is straightforward to calculate the shape function of the traverasable wormholes with the electric charge underlying a so-called Generalized Uncertainty Principle (GUP).As proposed in Ref. [10], we consider the GUP corrected Casimir energy density in a constant plate scenario.We finally find b where c 2 is a integration constant, D i s are the constants, see Ref. [10].Taking b(r 0 ) = r 0 , it can be determined to yield Therefore, the following shape function is this case becomes that b(r)/r → 0 when r → ∞.From the shape function ( 22), one finds that the flare-out condition is always satisfied since This implies that To obtain a TW, we have to determine the redshift function.We have to impose the EoS p r (r) = ωρ(r).From Eq.( 8), one finds Close to the throat, the r.h.s. of Eq.( 25) can be approximated by Following Ref. [11], we can choose ω such that Φ ′ (r) = 0 to avoid the appearance of a horizon.
This can be done if The result is more general compared to the previous case.Note that when q = 0, β = 0, the above result reduces to that of Ref. [6], i.e., ω

B. Variable Plates separation
In the same manner of previous work on zero temperature Casimir wormholes [11], the plates separation distance d can be promoted to a radial variable r.We can compute the form of the energy density.For the high temperature case, the thermal energy density is given by [6] where for in this case, we also have that the EoS leads to ω T C = 2. Following Ref. [6] for high temperature regimes, we come up with the following shape function: where c 1 is a integration constant.Taking b(r 0 ) = r 0 , it can be determined to yield Therefore, the first EFE leads to the following shape function with ζ(x) being the Riemann zeta function.Having considered the charge contribution, the above result is a generalization of what is found in Ref. [6].We also find that b(r)/r → 0 when r → ∞.From the shape function (31), one finds that the flare-out condition is always implying that Moreover from the condition 1 − b(r)/r > 0, it is possible to clarify that there exists r = r * such that b(r * ) = 0, where with and W (χ) is a Lambert W function or product logarithm and we have used the identity e W (x) = x W (x) .From the above result, Let us focus on a small charge approximation, ε = q 2 /r 2 0 ≪ 1.In this regime, we find The above result shows that the point r * where b(r * ) = 0 is positive and greater than r 0 .
Notice that at the zeroth order, it was already found in Ref. [6].The second term emerges thank to the charge contribution.To obtain a TW we need to compute the redshift function.
We have to impose the EoS p r (r) = ωρ(r).From Eq.( 7), one finds Substituting b(r) from Eq.( 31), we come up with Close to the throat, the r.h.s. of Eq.( 38) can be approximated by Following Ref. [11] we can choose ω such that Φ ′ (r) = 0 to avoid the appearance of a horizon.
This can be seen if Note that when q = 0, the above result reduces to that of Ref. [6], i.e., ω = r 0 /(ζ(3)r 2 0 T k B ).However, we can separately consider for the two cases: Interestingly, this allow us to compute the throat of the wormhole: r 0 .Here using Eq.( 40) and incorporating with ω V P = 2, we simply find a positive solution: Having compared with Ref. [6], in our convention, we take l P = 1.Notice from the above result that the size of the throat is enlarged further by the effect of the charge.Moreover, it can be more generalized by following the work present in Ref. [10] for elaborating in more details about the GUP corrected energy density.This allow to construct a new class of Casimir wormholes.Following [7], it is straightforward to calculate the shape function