Evolution of Wormholes under f(R, T) Theory, the Karmarkar Condition and the Casimir Energy

Murat Metehan Turkoglu

REVIEW 4 major objections 5 minor 85 references

Evolution of Wormholes under f(R, T) Theory, the Karmarkar Condition and the Casimir Energy

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read In f(R,T) gravity, one model supports traversable wormholes without exotic matter, while another requires it and the Karmarkar condition does not change that.

desk verdict A competent f(R,T) wormhole scan whose stability test uses GR TOV despite the theory's non-conservation, and whose Karmarkar shape function has a pole at the throat. read the letter →

arxiv 2506.02074 v1 pith:2C4EFWFV submitted 2025-06-02 gr-qc astro-ph.CO

Murat Metehan Turkoglu This is my paper · ORCID

classification gr-qcastro-ph.CO MSC 83D0583C15 PACS 04.50.Kd

keywords f(RT)gravitytraversablewormholesenergyconditionsKarmarkarconditionCasimirshapefunctionexoticmatterTolman-Oppenheimer-Volkoffequation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks whether traversable wormholes can exist without exotic matter within f(R,T) gravity, and answers conditionally: the Lagrangian matters. For the model $R - a_1^2/R + a_2 T$, the computed density and pressures satisfy the null, weak, and strong energy conditions, with the dominant condition holding in the tangential direction. For the model $R + a_1^2 R^2 + a_2 T$, energy conditions are only partially met, so exotic matter remains necessary. The author derives a generalized shape function from the Karmarkar condition, reports it as new, and shows it does not remove the exotic-matter requirement. Finally, for $\beta = 3$ the equation-of-state parameter matches the Casimir wormhole, which the paper takes as evidence that Casimir energy can stabilize the throat.

What carries the argument

The argument is carried by three named objects. The f(R,T) Lagrangians $R - a_1^2/R + a_2 T$ and $R + a_1^2R^2 + a_2T$, drawn from cosmological models of accelerated expansion, determine the field equations whose solutions give $\rho$, $p_r$, and $p_t$. The generalized Tolman-Oppenheimer-Volkoff equation, $dp_r/dr + \delta'(\rho+p_r)/2 + 2(p_r-p_t)/r = 0$, is the stability test that balances gravitational, hydrostatic, and anisotropic forces. The Karmarkar condition, $R_{1414} = (R_{1212}R_{3434}+R_{1224}R_{1334})/R_{2323}$, ties the metric potentials together and yields the generalized shape function. Finally, the Casimir wormhole equation-of-state parameter $w_{\rm casimir} = \frac13(3 - 2(9r+r_0)/(3r+r_0))$ is the comparison target that picks out $\beta=3$.

What would settle it

Evaluate the covariant divergence $\nabla_\mu T^{\mu\nu}$ using Eq. (13) for the anisotropic fluid (14) with $L_m = (p_r + 2p_t)/3$: if the result is nonzero, the TOV equation (36) omits source terms and the derived stability condition is not the equilibrium condition of this theory.

Watch

Extended reading notes

Core claim

The central claim is that the gravitational Lagrangian's form decides whether a static, spherically symmetric wormhole needs exotic matter. Starting from the Morris-Thorne metric with redshift $\delta(r)=\delta_0(r_0/r)^\alpha$ and shape $b(r)=b_0(r_0/r)^\beta$, the paper solves the f(R,T) field equations for the two models. In the first model the weak, strong, and null energy conditions all hold (with DEC partially), so no exotic matter appears; in the second one, increasing $\beta$ drives the energy density negative and the conditions fail, so exotic matter is required. Applying the Karmarkar condition to the second model produces the shape function $b(r) = r - r^{2\alpha+3}/(r^{2\alpha+2} + r_0^{2\alpha+2}(r_0 - C))$, but the energy-condition violations persist. The paper then identifies $\beta = 3$ as the case where $w = (p_r + 2p_t)/(3\rho)$ agrees with the Casimir wormhole parameter, concluding that Casimir energy can support the throat while the TOV equation indicates equilibrium.

Load-bearing premise

The stability analysis assumes the standard TOV equilibrium equation applies unchanged in f(R,T) gravity, even though the theory's own equations imply the stress-energy tensor is not conserved; if f(R,T) correction terms belong in the force balance, the equilibrium and Casimir-support conclusions would need revision.

Editorial extensions

If this is right

For the model $R - a_1^2/R + a_2T$, traversable wormholes can satisfy the null, weak, and strong energy conditions, so this branch of f(R,T) gravity may host wormholes without exotic matter.
For the model $R + a_1^2R^2 + a_2T$, energy-condition violations persist, so exotic matter remains necessary even after the Karmarkar condition is imposed.
At $\beta=3$, the wormhole equation of state matches the Casimir wormhole, implying quantum vacuum (Casimir) energy can stabilize the throat in this configuration.
The TOV force balance holds around the throat for both models in the studied parameter sets, which the paper interprets as static stability.
Increasing $\beta$ changes the wormhole geometry and, in the second model, turns the energy density negative, linking the shape-parameter evolution to the sign of the matter energy.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the paper itself writes $\nabla_\mu T^{\mu\nu} \neq 0$ in f(R,T) gravity, a natural next step is to re-derive the TOV equilibrium with the non-conservation terms; until that is done, the stability conclusion rests on an assumption rather than on the theory's own equations.
The $\beta=3$ Casimir match suggests a concrete test: compute the renormalized vacuum stress-energy tensor of a quantum field in the $\beta=3$ wormhole geometry and check whether it equals the Casimir form at the throat.
The generalized Karmarkar shape function could serve as a scan tool across modified-gravity theories: since it isolates the role of the redshift function, inserting it into other gravitational Lagrangians would reveal which theories can avoid exotic matter.
If the proposed positive-to-negative energy transition near the throat is real, wormhole geometry itself could act as a vacuum-energy source; a search direction would be to look for anomalous lensing or tidal signatures in compact objects whose internal geometry evolves toward $\beta=3$.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Briefly: the paper runs a standard Morris-Thorne wormhole analysis on two f(R,T) Lagrangians and wraps it in two extra claims—a 'first' Karmarkar-derived shape function and a Casimir-energy stabilization result. The scan is competent; the extra claims are not. The TOV stability test is the load-bearing problem.

What is good: the field equations for both models are derived carefully, the density and pressure expressions are explicit, and the energy-condition plots give a consistent picture: R - a1^2/R + a2T satisfies WEC/SEC for the chosen parameters, and R + a1^2R^2 + a2T requires exotic matter. That's a modest but concrete result.

Where it falls down. First, Eq. 16 states the matter stress tensor is not conserved, and for a2 = -9π the prefactor is nonzero. The paper then uses the ordinary GR TOV equation (Eq. 36) as the equilibrium condition. That is not the correct force balance in f(R,T); the non-conservation contributes an extra source term. The stability verdicts and the Casimir-support claim both rest on this equation, so they are not established. Second, the Karmarkar shape function in Eq. 44 does not satisfy the throat condition for the stated parameters. For r0=1, C=2 the denominator r^(2α+2)+1*(1-2) vanishes at r=1, so b(r) has a pole at the throat. The paper notes the unmodified condition yields r0=0 and 'we can add a constant C' — that is an ad hoc patch, not a derivation. The Karmarkar energy-condition plots in Section 5 are therefore undefined at the throat. Third, the Casimir match at β=3 is parameter selection: the paper computes w for β=1,2,3 and picks the one that tracks w_casimir. That is curve fitting, not a prediction. The couplings a1=9π, a2=-9π are also asserted without motivation.

These are central flaws, not polish issues. But the paper is not incoherent; it's a routine scan with two interpretation-layer failures. The 'first time' claim for the Karmarkar shape function is unsubstantiated because no comparison with existing Karmarkar wormhole work is given.

Recommendation: I would not accept this in its current form. If the journal wants to engage, send it to a referee who knows f(R,T) and wormhole stability; the main job is to identify the TOV issue and the Karmarkar pole. That is a legitimate use of referee time, so I would not desk reject it out of hand. But the paper needs major revision or, more likely, a different set of claims.

Referee Report

4 major / 5 minor

Summary. The paper investigates static, spherically symmetric traversable wormholes in f(R,T) gravity for two Lagrangians, R - a1^2/R + a2 T and R + a1^2 R^2 + a2 T. For each model and for beta = 1, 2, 3, the authors derive explicit algebraic expressions for the energy density and pressures, plot the energy conditions, and use the standard Tolman-Oppenheimer-Volkoff (TOV) equation to claim hydrostatic stability. They also construct a shape function from the Karmarkar condition, examine whether that condition removes the need for exotic matter, and identify beta = 3 as matching the equation-of-state parameter of a Casimir wormhole, concluding that Casimir energy can support the throat. The main advertised results are that the first model satisfies the WEC, SEC, and NEC without exotic matter, that the second model requires exotic matter, and that the Karmarkar-based analysis does not eliminate that requirement.

Significance. If the claims were correct, the paper would be of moderate interest to the modified-gravity wormhole community: it would provide explicit f(R,T) wormhole solutions with no exotic matter in one model and a possible Casimir-energy stabilization mechanism in another. The manuscript has some positive features: it presents the field equations in detail, gives long analytic expressions for the fluid variables, and explicitly acknowledges in Eq. (16) that the matter stress-energy tensor is not conserved in f(R,T) theory. However, the central claims are currently supported by two invalid or incomplete pieces of analysis: the use of the GR TOV equation despite the non-conservation of T, and a Karmarkar shape function that fails the throat condition. The Casimir conclusion is also a retrospective curve match rather than an independent derivation. As a result, the significance of the paper is not established by the present calculation.

major comments (4)

[Section 4, Eq. (36)] The stability analysis uses the standard GR TOV equation dp_r/dr + delta'(rho + p_r)/2 + (2/r)(p_r - p_t) = 0 and interprets F_g + F_a + F_h = 0 as hydrostatic equilibrium. In f(R,T) theory, however, Eqs. (13) and (16) give nabla_mu T^{mu nu} = [a_2/(8 pi + a_2)] nabla^nu(L_m - T/2), which is nonzero for the chosen value a_2 = -9 pi. The correct equilibrium condition must contain an extra source term proportional to nabla^r(L_m - T/2). Since the stability verdicts for both models (Figs. 6, 8, 10, 13, 15, 17) and the final claim that beta = 3 is stabilized by Casimir energy all rest on Eq. (36), those conclusions are not supported by the calculation as presented.
[Section 5, Eq. (44)] The Karmarkar shape function in Eq. (44) does not satisfy the throat condition b(r_0) = r_0. Direct substitution gives b(r_0) = r_0 - r_0/(1 + r_0 - C), which equals r_0 only if r_0/(1 + r_0 - C) = 0, an equation that has no solution for finite C. For the values used in Section 5, r_0 = 1 and C = 2, the denominator r^{2 alpha + 2} + r_0^{2 alpha + 2}(r_0 - C) reduces to r^{2 alpha + 2} - 1 and vanishes at r = r_0, making b(r) singular at the throat. Consequently the energy-condition plots in Figs. 19-21 and the conclusion that the Karmarkar condition does not remove the need for exotic matter are based on an invalid wormhole geometry.
[Section 6 and Fig. 22] The claimed agreement between beta = 3 and the Casimir equation-of-state parameter is obtained by computing w for beta = 1, 2, 3 and selecting the value that lies closest to w_casimir. The paper does not derive beta = 3 from the condition w = w_casimir, does not provide a quantitative measure of the agreement, and does not specify the radial interval over which the match holds. Moreover, the statement that Casimir energy can stabilize the throat is presented as a stability conclusion, but the only stability analysis in the paper is the invalid TOV analysis of Section 4. The beta = 3 matching therefore does not constitute a demonstration that Casimir energy supports the throat.
[Section 4, Figs. 5-17] The energy-condition plots for both models and all beta values begin at r/M = 2, whereas the wormhole throat is at r_0 = 1 and the traversability conditions are evaluated at r_0. The claims that the WEC, SEC, and NEC are satisfied for the R - a_1^2/R + a_2 T model are therefore demonstrated only on the interval [2, 3], not on the full wormhole domain r >= r_0. Since the no-exotic-matter conclusion depends on the energy conditions being satisfied throughout the wormhole, the paper should verify the conditions for all r >= r_0 or explicitly restrict the claim to the plotted range.

minor comments (5)

[Eq. (12) and Eq. (26)] The notation (g_mu nu nabla_mu nabla_nu - nabla_mu nabla_nu) f_R should be written with a box operator, e.g., (g_mu nu Box - nabla_mu nabla_nu) f_R; the current notation is confusing and appears in both field equations.
[Section 3] The abstract and Section 3 use inconsistent notation for the matter coupling: the abstract writes a_2 g(T) while the field equations are derived for a_2 T. This should be reconciled, since L_m is later chosen as (p_r + 2p_t)/3.
[Section 5, after Eq. (44)] The expression for Gamma, Gamma = r_0^2 (r_0 - C)/(e^delta alpha^2 b_0^2), introduces b_0 and delta_0 that do not appear in Eq. (44), and the definition of b_0 in this context is unclear.
[Figure captions, Figs. 19-21] The captions for Figs. 19 and 20 state 'alpha = 1' while the titles and text refer to alpha = 1, 2, and 3; the captions should report the alpha value actually used in each figure.
[Fig. 22 and surrounding text] The text says 'upper part of figure 22' and 'lower part of figure 22' when the figure has left and right panels; this should be corrected. The right panel's y-axis range makes the w_Casimir curve difficult to read, and the claimed overlap with beta = 3 should be shown in a zoomed panel.

Circularity Check

1 steps flagged · score 6.0 of 10

Casimir-support conclusion is obtained by choosing β=3 to match w_Casimir, turning the target equation of state into a selected model parameter.

fitted input called prediction [Section 6 (Conclusion), Casimir analysis following Eq. (45) and Fig. 22]
"As can be seen, the w parameter for R + a2 1R2 + a2T is in strong agreement with the Casimir wormhole parameter wcasimir for a specific value of the β parameter (β = 3). ... Therefore, it can be concluded that specifically for β = 3 the wormhole is supported by Casimir energy around the throat radius."

The shape function in Eq. (4) contains β as a free parameter: b(r) = b0(r0/r)^β. The paper computes the equation-of-state parameter w = (pr + 2pt)/(3ρ) for β = 1, 2, 3 and compares it with w_casimir imported from Garattini’s Eq. (45). The value β = 3 is selected only because this comparison shows agreement; it is not fixed by the field equations or by any independent physical condition. The concluding statement that the β = 3 throat is supported by Casimir energy is therefore a restatement of the selection rule “choose β so that w matches w_casimir” rather than an independent prediction. Had a different target equation of state been chosen, a different β could have been selected; the agreement is an input to the model-selection argument, not an output of the derivation.

full rationale

The energy-condition computations for the two f(R,T) models and the Karmarkar-derived shape function are self-contained: the field equations (15) and (26) are solved with the stated ansatz and plotted, and the Karmarkar condition is applied to obtain a new shape function without reducing to its own output. The load-bearing circularity is confined to the Casimir claim. There, β is a free parameter in the shape function, and the agreement between w(β) and w_Casimir is achieved by choosing β = 3. The subsequent conclusion that the wormhole is supported by Casimir energy is thus equivalent to the selection procedure, not a prediction derived from the theory. I do not score the use of the GR TOV equation as circularity because the issue is a consistency/correctness problem with the non-conservation equation (16), not a reduction of an output to an input. There are no self-citations, so the self-citation load-bearing patterns do not apply.

Assumptions & free parameters 8 free parameters · 6 assumptions · 0 invented entities

The central derivation depends on the f(R,T) field equations, an anisotropic fluid ansatz, a specific wormhole metric, the Karmarkar condition, and the unmodified GR TOV equation, plus several hand-set parameters (a1=9*pi, a2=-9*pi, delta0=1, r0=1, M=1, C=2). No new particles or fields are introduced.

free parameters (8)

a1 = 9*pi
Coupling constant of the R-a1^2/R and R+a1^2R^2 terms, fixed to 9*pi in Sections 4.1, 4.2 and the Karmarkar analysis without derivation.
a2 = -9*pi
Coupling constant of the a2T term, fixed to -9*pi in all numerical and plotted analyses.
beta = 1, 2, 3
Exponent in the shape function b(r)=b0(r0/r)^beta; varied discretely. The value beta=3 is later selected to match the Casimir equation of state.
alpha = 1 (and 2,3 in Section 5)
Exponent in the redshift function delta=delta0(r0/r)^alpha; set to 1 for the main analysis and varied in the Karmarkar section.
delta0 = 1
Amplitude of the redshift function, set to unity.
r0 = 1
Throat radius, normalized to 1.
M = 1
Mass used to normalize r/M, set to 1.
C = 2
Constant added ad hoc to the Karmarkar shape function in Eq. 44, fixed to 2.

assumptions (6)

domain assumption f(R,T) field equations (Eq. 12) correctly describe wormhole geometry.
The paper relies on the f(R,T) action and field equations from Harko et al. without modification.
domain assumption Anisotropic perfect fluid energy-momentum tensor (Eq. 14) with Lm=(pr+2pt)/3 represents wormhole matter.
This choice is imposed, not derived.
domain assumption Morris-Thorne metric ansatz with delta=delta0(r0/r)^alpha and b=b0(r0/r)^beta is a valid wormhole geometry.
Used throughout Sections 2-4.
domain assumption Karmarkar condition (Eq. 40) is applicable and the derived shape function satisfies wormhole conditions.
Applied in Section 5; the derivation is questionable.
ad hoc to paper Standard GR TOV equation (Eq. 36) is valid in f(R,T) gravity despite non-conservation of T.
No derivation or citation is provided for this modification; Eq. 13 shows the matter tensor is not conserved.
domain assumption Casimir equation of state (Eq. 45) from Garattini can be compared to the model's w=P/rho.
Used in Section 6 to identify beta=3 as the Casimir-supported case.

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Cite this review

Pith. "Pith review of Evolution of Wormholes under f(R, T) Theory, the Karmarkar Condition and the Casimir Energy." pith.science (2026). https://pith.science/paper/2C4EFWFV

@misc{pith2026250602074,
  author       = {Pith},
  title        = {Pith review of: Evolution of Wormholes under f(R, T) Theory, the Karmarkar Condition and the Casimir Energy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2C4EFWFV}},
  note         = {Machine review of arXiv:2506.02074}
}

abstract

In this study, both the evolution of wormholes (by examining both the energy conditions and using the TOV equations) and the effects of the Karmarkar condition on the solutions obtained under certain specific cases were examined in the light of the $f(R,T)$ gravity theory, using two $f(R,T)$ functions predicted to describe the accelerated expansion of the universe. In this context, for the first time in the literature, a generalized shape function was obtained using the Karmarkar condition. It was observed that solutions of the type $R-a_{1}^2/R+a_{2}g(T)$ satisfy the energy conditions (with the dominant energy condition being partially satisfied), whereas solutions of the type $R+a_{1}^2R^2+a_{2}g(T)$ require the presence of exotic matter. In both cases, stable, static, and traversable wormhole solutions were obtained. By applying the Karmarkar condition to the $R+a_{1}^2R^2+a_{2}g(T)$ type solutions, which violate the energy conditions, the relationship between wormhole geometry and energy conditions was investigated. The study examined whether the Karmarkar condition eliminates the need for exotic matter, and it was found that the solutions do not remove the necessity of exotic matter. Additionally, it was demonstrated that a specific value of the parameter, ${\beta}$, which determines the radial variation of the shape function, could ensure the stability of the wormhole throat with the aid of Casimir energy. In other words, it is considered possible that the geometric evolution of the wormhole throat could trigger the transition from positive energy (baryonic matter) to negative energy (dark matter, dark energy, or other exotic matter) by inducing Casimir forces.

Figures

Figures reproduced from arXiv: 2506.02074 by the authors.

**Figure 1.** Embedding diagram for different β values according to eq. 10. The throat radius has been set to r0 = 1. According to Morris and Thorne [3], for wormholes to be traversable, the shape function must satisfy the following conditions: 5 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** The behavior of the function z(r) for different values of β with b0 = 1. b(r) b(r)/r db(r)/dr b(r) - r b(r)-rb'(r) 2 4 6 8 10 -3 -2 -1 0 1 2 3 r r [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Evaluation of wormhole shape function b(r) for r0 = 1. 3. f (R, T) field equations The f (R, T) action is given as [49] S = Z d 4x √ −g 1 16π f(R, T) + Lm . (11) 6 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (19 more)

**Figure 4.** Figure 4: Signs of the anisotropy parameter for the case [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]

**Figure 5.** Figure 5: Behaviour of energy conditions of the R − a 2 1 /R + a2T case for β = 1 and the values r0 = 1, δ0 = 1, α = 1, M = 1 are chosen. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]

**Figure 6.** Figure 6: Stability of the R − a 2 1 /R + a2T case for β = 1 and the values r0 = 1, δ0 = 1, α = 1, M = 1 are chosen. Fg , Fa and Fh represent gravitational force, anisotropic force and hydrostatic force, respectively. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]

**Figure 7.** Figure 7: Behaviour of energy conditions of the R − a 2 1 /R + a2T case for β = 2 and the values r0 = 1, δ0 = 1, α = 1, M = 1 are chosen. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]

**Figure 8.** Figure 8: Stability of the R − a 2 1 /R + a2T case for β = 2 and the values r0 = 1, δ0 = 1, α = 1, M = 1 are chosen. Fg , Fa and Fh represent gravitational force, anisotropic force and hydrostatic force, respectively. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_8.png]

**Figure 9.** Figure 9: Behaviour of energy conditions of the R − a 2 1 /R + a2T case for β = 3 and the values r0 = 1, δ0 = 1, α = 1, M = 1 are chosen. 23 [PITH_FULL_IMAGE:figures/full_fig_p023_9.png]

**Figure 10.** Figure 10: Stability of the R − a 2 1 /R + a2T case for β = 3 and the values r0 = 1, δ0 = 1, α = 1, M = 1 are chosen. Fg , Fa and Fh represent gravitational force, anisotropic force and hydrostatic force, respectively. 4.2. Energy and Stability Conditions for the case R + a 2 1R…

**Figure 11.** Figure 11: Signs of the anisotropy parameter for the case [PITH_FULL_IMAGE:figures/full_fig_p025_11.png]

**Figure 12.** Figure 12: Behaviour of energy conditions of the R+a 2 1R2+a2T case for β = 1 and the values r0 = 1, δ0 = 1, α = 1, M = 1 are chosen. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_12.png]

**Figure 13.** Figure 13: Stability of the R + a 2 1R2 + a2T case for β = 1 and the values r0 = 1, δ0 = 1, α = 1, M = 1 are chosen. Fg , Fa and Fh represent gravitational force, anisotropic force and hydrostatic force, respectively. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_13.png]

**Figure 14.** Figure 14: Behaviour of energy conditions of the R+a 2 1R2+a2T case for β = 2 and the values r0 = 1, δ0 = 1, α = 1, M = 1 are chosen. 28 [PITH_FULL_IMAGE:figures/full_fig_p028_14.png]

**Figure 15.** Figure 15: Stability of the R + a 2 1R2 + a2T case for β = 2 and the values r0 = 1, δ0 = 1, α = 1, M = 1 are chosen. Fg , Fa and Fh represent gravitational force, anisotropic force and hydrostatic force, respectively. 29 [PITH_FULL_IMAGE:figures/full_fig_p029_15.png]

**Figure 16.** Figure 16: Behaviour of energy conditions of the R+a 2 1R2+a2T case for β = 3 and the values r0 = 1, δ0 = 1, α = 1, M = 1 are chosen. 30 [PITH_FULL_IMAGE:figures/full_fig_p030_16.png]

**Figure 17.** Figure 17: Stability of the R + a 2 1R2 + a2T case for β = 3 and the values r0 = 1, δ0 = 1, α = 1, M = 1 are chosen. Fg , Fa and Fh represent gravitational force, anisotropic force and hydrostatic force, respectively. 5. Karmarkar Condition In this study, to understand the effec…

**Figure 18.** Figure 18: Embedding diagram obtained by applying the Karmakar condition for different [PITH_FULL_IMAGE:figures/full_fig_p033_18.png]

**Figure 19.** Figure 19: Karmarkar analysis of energy conditions for [PITH_FULL_IMAGE:figures/full_fig_p033_19.png]

**Figure 20.** Figure 20: Karmarkar analysis of energy conditions for [PITH_FULL_IMAGE:figures/full_fig_p034_20.png]

**Figure 21.** Figure 21: Karmarkar analysis of energy conditions for [PITH_FULL_IMAGE:figures/full_fig_p035_21.png]

**Figure 22.** Figure 22: The relationship between the parameter w, obtained for different values of β , and the Casimir wormhole parameter wcasimir is shown by R − a 2 1 /R + a2T (left graph) and R + a 2 1R2 + a2T (right graph). As can be seen, the w parameter for R + a 2 1R2 + a2T is in stro…

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