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Traversable wormhole solutions admitting Karmarkar condition in f(R, T) theory

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Abstract

In this paper, we evaluate traversable wormhole solutions through Karmarkar condition in f(R, T) theory, where T is the trace of the energy-momentum tensor and R represents the Ricci scalar. We develop a wormhole shape function for the static traversable wormhole geometry by using the embedding class-I technique. The resulting shape function is used to construct wormhole geometry that fulfills all the necessary conditions and joins the two asymptotically flat regions of the spacetime. We investigate the existence of viable traversable wormhole solutions for anisotropic matter configuration and examine the stable state of these solutions for different f(R, T) gravity models. We analyze the graphical behavior of null energy bound to examine the presence of physically viable wormhole geometry. It is found that viable and stable traversable wormhole solutions exist in this modified theory of gravity.

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Acknowledgements

We are thankful to Dr. Zeeshan Gul for fruitful discussions during the write up of this paper.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, The University of Lahore, 1-KM Defence Road, Lahore, Pakistan

    M. Sharif & Arooj Fatima

Authors
  1. M. Sharif
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  2. Arooj Fatima
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Corresponding author

Correspondence to M. Sharif.

Appendix A

Appendix A

The field equations corresponding to the model (17) are

$$\begin{aligned} {\rho}= & {} \frac{1}{{e^{\nu }}2(2\lambda +1)(\lambda +1)}\bigg [(5\lambda +2) \bigg \{-\frac{f}{2}e^{\nu }+\bigg (\frac{\mu '}{r}-\frac{\mu '\nu '}{4} +\frac{\mu ''}{2} \nonumber \\{} & {} +\frac{\mu ^{\prime 2}}{4}\bigg )f_{R}+\bigg (\frac{\nu '}{2}-\frac{2}{r}\bigg ) f'_{R}-f''_{R}\bigg \}+\lambda \bigg \{\frac{f}{2}e^{\nu }+\bigg (\frac{\mu ''}{2}-\frac{\nu '}{r}-\frac{\mu '\nu '}{4} \nonumber \\{} & {} +\frac{\mu ^{\prime 2}}{4}\bigg )f_{R} +\bigg (\frac{\mu '}{2}+\frac{2}{r}\bigg )f'_{R}\bigg \}+2\lambda \bigg \{\frac{f}{2}e^{\nu }-\bigg (\frac{(\mu '-\nu ')r}{2}-e^{\nu }+1\bigg ) \nonumber \\{} & {} \times \frac{f_{R}}{r^{2}}+\bigg (\frac{\mu '-\nu '}{2}+\frac{1}{r}\bigg ) f'_{R}+f''_{R}\bigg \}\bigg ], \end{aligned}$$
(A1)
$$\begin{aligned} P_{r}= & {} \frac{1}{{e^{\nu }}2(2\lambda +1)(\lambda +1)}\bigg [-\lambda \bigg \{-\frac{f}{2}e^{\nu }+\bigg (\frac{\mu '}{r}-\frac{\mu '\nu '}{4} +\frac{\mu ''}{2} \nonumber \\{} & {} +\frac{\mu ^{\prime 2}}{4}\bigg )f_{R}+\bigg (\frac{\nu '}{2}-\frac{2}{r}\bigg ) f'_{R}-f''_{R}\bigg \}+(3\lambda +2)\bigg \{\frac{f}{2}e^{\nu }+\bigg (\frac{\mu ''}{2}-\frac{\nu '}{r} \nonumber \\{} & {} -\frac{\mu '\nu '}{4}+\frac{\mu ^{\prime 2}}{4}\bigg )f_{R} +\bigg (\frac{\mu '}{2}+\frac{2}{r}\bigg )f'_{R}\bigg \}-2\lambda \bigg \{\frac{f}{2}e^{\nu }+\bigg (\frac{(\mu '-\nu ')r}{2}\nonumber \\{} & {} -e^{\nu }+1\bigg ) \frac{-f_{R}}{r^{2}}+\bigg (\frac{\mu '-\nu '}{2}+\frac{1}{r}\bigg )f'_{R} +f''_{R}\bigg \}\bigg ], \end{aligned}$$
(A2)
$$\begin{aligned} P_{t}= & {} \frac{1}{{e^{\nu }}2(2\lambda +1)(\lambda +1)}\bigg [-\lambda \bigg \{-\frac{f}{2}e^{\nu }+\bigg (\frac{\mu '}{r}-\frac{\mu '\nu '}{4}+\frac{\mu ''}{2} \nonumber \\{} & {} +\frac{\mu ^{\prime 2}}{4}\bigg )f_{R} +\bigg (\frac{\nu '}{2}-\frac{2}{r}\bigg )f'_{R}-f''_{R}\bigg \}+\lambda \bigg \{\frac{f}{2}e^{\nu }+\bigg (\frac{\mu ''}{2}-\frac{\nu '}{r}-\frac{\mu '\nu '}{4} \nonumber \\{} & {} +\frac{\mu ^{\prime 2}}{4}\bigg )f_{R} +\bigg (\frac{\mu '}{2}+\frac{2}{r}\bigg )f'_{R}\bigg \}-2(\lambda +1) \bigg \{\frac{f}{2}e^{\nu }-\bigg (\frac{(\mu '-\nu ')r}{2}-e^{\nu }+1\bigg ) \nonumber \\{} & {} \times \frac{f_{R}}{r^{2}}+\bigg (\frac{\mu '-\nu '}{2}+\frac{1}{r}\bigg ) f'_{R}+f''_{R}\bigg \}\bigg ]. \end{aligned}$$
(A3)

The resulting field equations corresponding to the model 1 turn out to be

$$\begin{aligned} {\rho}= & {} \frac{1}{{e^{\nu }}2(2\lambda +1)(\lambda +1)}\bigg [(5\lambda +2)\bigg \{-\frac{f}{2}e^{\nu }+\bigg (\frac{\mu '}{r}-\frac{\mu '\nu '}{4}+\frac{\mu ''}{2}+\frac{\mu ^{\prime 2}}{4}\bigg ) \nonumber \\{} & {} \times \bigg (1-K e^{\frac{-R}{B}}\bigg )+\bigg (\frac{\nu '}{2}-\frac{2}{r}\bigg )\bigg (\frac{1}{B}K e^{\frac{-R}{B}}\bigg )R' -\bigg \{\bigg (\frac{1}{B}K e^{\frac{-R}{B}}\bigg )R''-\bigg (\frac{1}{B^{2}}K e^{\frac{-R}{B}}\bigg ) \nonumber \\{} & {} \times R^{\prime 2}\bigg \}\bigg \}+\lambda \bigg \{\frac{f}{2}e^{\nu }+\bigg (\frac{\mu ''}{2}-\frac{\nu '}{r}-\frac{\mu '\nu '}{4}+\frac{\mu ^{\prime 2}}{4}\bigg )(1-Ke^{\frac{-R}{B}})+\bigg (\frac{\mu '}{2}+\frac{2}{r}\bigg ) \nonumber \\{} & {} \times \bigg (\frac{1}{B}Ke^{\frac{-R}{B}}\bigg )R'\bigg \}+2\lambda \bigg \{\frac{f}{2}e^{\nu }+\bigg (\frac{(\mu '-\nu ')r}{2}-e^{\nu }+1\bigg ) \frac{(-1+K e^{\frac{-R}{B}})}{r^{2}} \nonumber \\{} & {} +\bigg (\frac{\mu '-\nu '}{2}+\frac{1}{r}\bigg )\bigg (\frac{1}{B}K e^{\frac{-R}{B}}\bigg )R'+\bigg \{\bigg (\frac{1}{B}K e^{\frac{-R}{B}}\bigg )R'' \nonumber \\{} & {} +\bigg (\frac{-1}{B^{2}}K e^{\frac{-R}{B}}\bigg )R^{\prime 2}\bigg \}\bigg \}\bigg ], \end{aligned}$$
(A4)
$$\begin{aligned} P_{r}= & {} \frac{1}{{e^{\nu }}2(2\lambda +1)(\lambda +1)}\bigg [-\lambda \bigg \{-\frac{f}{2}e^{\nu }+\bigg (\frac{\mu '}{r}-\frac{\mu '\nu '}{4}+\frac{\mu ''}{2}+\frac{\mu ^{\prime 2}}{4}\bigg ) \nonumber \\{} & {} \times \bigg (1-K e^{\frac{-R}{B}}\bigg )+\bigg (\frac{\nu '}{2}-\frac{2}{r}\bigg )\bigg (\frac{1}{B}K e^{\frac{-R}{B}}\bigg )R'-\bigg \{\bigg (\frac{1}{B}K e^{\frac{-R}{B}}\bigg )R'' \nonumber \\{} & {} +\bigg (\frac{-1}{B^{2}}K e^{\frac{-R}{B}}\bigg )R^{\prime 2}\bigg \}\bigg \}+(3\lambda +2) \bigg \{\frac{f}{2}e^{\nu }+\bigg (\frac{\mu ''}{2}-\frac{\nu '}{r}-\frac{\mu '\nu '}{4}+\frac{\mu ^{\prime 2}}{4}\bigg ) \nonumber \\{} & {} \times \bigg (1-K e^{\frac{-R}{B}}\bigg )+\bigg (\frac{\mu '}{2}+\frac{2}{r}\bigg )\bigg (\frac{1}{B}K e^{\frac{-R}{B}}\bigg )R'\bigg \}-2\lambda \bigg \{\frac{f}{2}e^{\nu }-\bigg (\frac{(\mu '-\nu ')r}{2} \nonumber \\{} & {} -e^{\nu }+1\bigg ) \frac{\bigg (1-K e^{\frac{-R}{B}}\bigg )}{r^{2}}+\bigg (\frac{\mu '-\nu '}{2}+\frac{1}{r}\bigg )\bigg (\frac{1}{B}K e^{\frac{-R}{B}}\bigg )R'+\bigg \{\bigg (\frac{1}{B}K e^{\frac{-R}{B}}\bigg ) \nonumber \\{} & {} \times R''+\bigg (\frac{-1}{B^{2}}K e^{\frac{-R}{B}}\bigg )R^{\prime 2}\bigg \}\bigg \}\bigg ], \end{aligned}$$
(A5)
$$\begin{aligned} P_{t}= & {} \frac{1}{{e^{\nu }}2(2\lambda +1)(\lambda +1)}\bigg [-\lambda \bigg \{-\frac{f}{2}e^{\nu }+\bigg (\frac{\mu '}{r}-\frac{\mu '\nu '}{4}+\frac{\mu ''}{2}+\frac{\mu ^{\prime 2}}{4}\bigg ) \nonumber \\{} & {} \times \bigg (1-K e^{\frac{-R}{B}}\bigg )+\bigg (\frac{\nu '}{2}-\frac{2}{r}\bigg )\bigg (\frac{1}{B}K e^{\frac{-R}{B}}\bigg )R'-\bigg \{\bigg (\frac{1}{B}K e^{\frac{-R}{B}}\bigg )R'' \nonumber \\{} & {} +\bigg (\frac{-1}{B^{2}}K e^{\frac{-R}{B}}\bigg )R^{\prime 2}\bigg \}\bigg \}+\lambda \bigg \{\frac{f}{2}e^{\nu }+\bigg (\frac{\mu ''}{2}-\frac{\nu '}{r}-\frac{\mu '\nu '}{4}+\frac{\mu ^{\prime 2}}{4}\bigg ) \nonumber \\{} & {} \times \bigg (1-K e^{\frac{-R}{B}}\bigg ) +\bigg (\frac{\mu '}{2}+\frac{2}{r}\bigg )\bigg (\frac{1}{B}K e^{\frac{-R}{B}}\bigg )R'\bigg \}-2(\lambda +1)\bigg \{\frac{f}{2}e^{\nu } \nonumber \\{} & {} -\bigg (\frac{(\mu '-\nu ')r}{2}-e^{\nu }+1\bigg )\frac{\bigg (1-K e^{\frac{-R}{B}}\bigg )}{r^{2}}+\bigg (\frac{\mu '-\nu '}{2}+\frac{1}{r}\bigg )\bigg (\frac{1}{B}K e^{\frac{-R}{B}}\bigg )R' \nonumber \\{} & {} +\bigg \{\bigg (\frac{1}{B}K e^{\frac{-R}{B}}\bigg )R''+\bigg (\frac{-1}{B^{2}}K e^{\frac{-R}{B}}\bigg )R^{\prime 2}\bigg \}\bigg \}\bigg ]. \end{aligned}$$
(A6)

The corresponding field equations corresponding to the model 2 are

$$\begin{aligned} {\rho}= & {} \frac{1}{{e^{\nu }}2(2\lambda +1)(\lambda +1)}\bigg [(5\lambda +2) \frac{1}{e^{\nu }}\bigg \{-\frac{f}{2}e^{\nu }+\bigg (\frac{\mu '}{r}- \frac{\mu '\nu '}{4}+\frac{\mu ''}{2} \nonumber \\{} & {} +\frac{\mu ^{\prime 2}}{4}\bigg )\bigg (1-\frac{2nR\gamma (1+\frac{R^{2}}{B^{2}})^{-1-n}}{B}\bigg ) +\bigg (\frac{\nu '}{2}-\frac{2}{r}\bigg )\bigg \{\frac{2n\gamma }{B(1+\frac{R^{2}}{B^{2}})^{1+n}} \nonumber \\{} & {} \times \bigg (\frac{(-2-2n)R^{2}}{B^{2}(1+\frac{R^{2}}{B^{2}})}-1\bigg ) R'\bigg \}-\bigg \{\frac{2n\gamma }{B(1+\frac{R^{2}}{B^{2}})^{1+n}}\bigg (\frac{(-2-2n) R^{2}}{B^{2}(1+\frac{R^{2}}{B^{2}})}-1\bigg ) \nonumber \\{} & {} \times R''+\frac{4nR\gamma }{B^{3}(1+\frac{R^{2}}{B^{2}})^{2+n}}\bigg ((3+ 3n)-\frac{(8+6n+2n^{2})R^{2}}{B^{2}(1+\frac{R^{2}}{B^{2}})}\bigg )R^{\prime 2} \bigg \}\bigg \} \nonumber \\{} & {} +\lambda \bigg \{\frac{f}{2}e^{\nu }+\bigg (\frac{\mu ''}{2}-\frac{\nu '}{r}-\frac{\mu '\nu '}{4}+\frac{\mu ^{\prime 2}}{4}\bigg )\bigg (1-\frac{2nR\gamma (1+\frac{R^{2}}{B^{2}})^{-1-n}}{B}\bigg ) \nonumber \\{} & {} +\bigg (\frac{\mu '}{2}+\frac{2}{r}\bigg )\bigg \{\frac{2n\gamma }{B(1 +\frac{R^{2}}{B^{2}})^{1+n}}\bigg (\frac{(-2-2n)R^{2}}{B^{2}(1+\frac{R^{2}}{B^{2}})}-1\bigg )R'\bigg \}\bigg \}+2\lambda \bigg \{\frac{f}{2 }e^{\nu } \nonumber \\{} & {} +\bigg (\frac{(\mu '-\nu ')r}{2}-e^{\nu }+1\bigg )\frac{\bigg (-1+\frac{2nR\gamma (1+\frac{R^{2}}{B^{2}})^{-1-n}}{B}\bigg )}{r^{2}}+\bigg (\frac{\mu '-\nu '}{2}+\frac{1}{r}\bigg ) \nonumber \\{} & {} \times \bigg \{\frac{2n\gamma }{B(1+\frac{R^{2}}{B^{2}})^{1+n}}\bigg (\frac{(-2-2n)R^{2}}{B^{2}(1+\frac{R^{2}}{B^{2}})}-1\bigg )R'\bigg \} +\bigg \{\frac{2n\gamma }{B(1+\frac{R^{2}}{B^{2}})^{1+n}} \nonumber \\{} & {} \times \bigg (\frac{(-2-2n)R^{2}}{B^{2}(1+\frac{R^{2}}{B^{2}})}-1\bigg ) R''+\frac{4nR\gamma }{B^{3}(1+\frac{R^{2}}{B^{2}})^{2+n}} \bigg ((3+3n) \nonumber \\{} & {} -\frac{(8+6n+2n^{2})R^{2}}{B^{2}(1+\frac{R^{2}}{B^{2}})} \bigg )R^{\prime 2}\bigg \}\bigg \}\bigg ], \end{aligned}$$
(A7)
$$\begin{aligned} P_{r}= & {} \frac{1}{{e^{\nu }}2(2\lambda +1)(\lambda +1)}\bigg [-\lambda \bigg \{-\frac{f}{2}e^{\nu }+\bigg (\frac{\mu '}{r}-\frac{\mu '\nu '}{4}+\frac{\mu ''}{2}+\frac{\mu ^{\prime 2}}{4}\bigg )\bigg (1 \nonumber \\{} & {} -\frac{2nR\gamma (1+\frac{R^{2}}{B^{2}})^{-1-n}}{B}\bigg )+\bigg (\frac{\nu '}{2}-\frac{2}{r} \bigg )\bigg \{\frac{2n\gamma }{B(1+\frac{R^{2}}{B^{2}})^{1+n}}\bigg (\frac{ (-2-2n)R^{2}}{B^{2}(1+\frac{R^{2}}{B^{2}})} \nonumber \\{} & {} -1\bigg )R'\bigg \}-\bigg \{\frac{2n\gamma }{B(1+\frac{R^{2}}{B^{2}})^{1+n}} \bigg (\frac{(-2-2n)R^{2}}{B^{2}(1+\frac{R^{2}}{B^{2}})}-1\bigg )R''+\frac{4nR\gamma }{B^{3}(1+\frac{R^{2}}{B^{2}})^{2+n}} \nonumber \\{} & {} \times \bigg ((3+3n)-\frac{(8+6n+2n^{2})R^{2}}{B^{2}(1+\frac{R^{2}}{B^{2} })}\bigg )R^{\prime 2}\bigg \}\bigg \}+(3\lambda +2)\bigg \{\frac{f}{2}e^{\nu }+\bigg (\frac{\mu ''}{2}-\frac{\nu '}{r} \nonumber \\{} & {} -\frac{\mu '\nu '}{4}+\frac{\mu ^{\prime 2}}{4}\bigg )\bigg (1-\frac{2nR\gamma (1+\frac{R^{2}}{B^{2}}) ^{-1-n}}{B}\bigg )+\bigg (\frac{\mu '}{2}+\frac{2}{r}\bigg )\bigg \{\frac{2n\gamma }{B(1+\frac{R^{2}}{B^{2}})^{1+n}} \nonumber \\{} & {} \times \bigg (\frac{(-2-2n)R^{2}}{B^{2}(1+\frac{R^{2}}{B^{2}})}-1\bigg ) R'\bigg \}\bigg \}-2\lambda \bigg \{\frac{f}{2}e^{\nu }+\bigg (\frac{(\mu '-\nu ') r}{2}-e^{\nu }+1\bigg ) \nonumber \\{} & {} \times \frac{\bigg (-1+\frac{2nR\gamma (1+\frac{R^{2}}{B^{2}})^{-1-n}}{B} \bigg )}{r^{2}}+\bigg (\frac{\mu '-\nu '}{2}+\frac{1}{r}\bigg )\bigg \{\frac{2n\gamma }{B(1+\frac{R^{2}}{B^{2}})^{1+n}} \nonumber \\{} & {} \times \bigg (\frac{(-2-2n)R^{2}}{B^{2}(1+\frac{R^{2}}{B^{2}})}-1\bigg )R' \bigg \}+\bigg \{\frac{2n\gamma }{B(1+\frac{R^{2}}{B^{2}})^{1+n}}\bigg (\frac{ (-2-2n)R^{2}}{B^{2}(1+\frac{R^{2}}{B^{2}})}-1\bigg )R'' \nonumber \\{} & {} +\frac{4nR\gamma }{B^{3}(1+\frac{R^{2}}{B^{2}})^{2+n}}\bigg ((3+3n)- \frac{(8+6n+2n^{2})R^{2}}{B^{2}(1+\frac{R^{2}}{B^{2}})}\bigg )R^{\prime 2} \bigg \}\bigg \}\bigg ], \end{aligned}$$
(A8)
$$\begin{aligned} P_{t}= & {} \frac{1}{{e^{\nu }}2(2\lambda +1)(\lambda +1)}\bigg [-\lambda \bigg \{-\frac{f}{2}e^{\nu }+\bigg (\frac{\mu '}{r}-\frac{\mu '\nu '}{4}+\frac{\mu ''}{2}+\frac{\mu ^{\prime 2}}{4}\bigg )\bigg (1 \nonumber \\{} & {} -\frac{2nR\gamma (1+\frac{R^{2}}{B^{2}})^{-1-n}}{B}\bigg )+\bigg (\frac{\nu '}{2}-\frac{2}{r}\bigg )\bigg \{\frac{2n\gamma }{B(1+\frac{R^{2}}{B^{2}})^{1+n}}\bigg (\frac{(-2-2n)R^{2}}{B^{2}(1+\frac{R^{2}}{B^{2}})} \nonumber \\{} & {} -1\bigg )R'\bigg \}-\bigg \{\frac{2n\gamma }{B(1+\frac{R^{2}}{B^{2}} )^{1+n}}\bigg (\frac{(-2-2n)R^{2}}{B^{2}(1+\frac{R^{2}}{B^{2}})}-1 \bigg )R''+\frac{4nR\gamma }{B^{3}(1+\frac{R^{2}}{B^{2}})^{2+n}} \nonumber \\{} & {} \times \bigg ((3+3n)-\frac{(8+6n+2n^{2})R^{2}}{B^{2}(1+\frac{R^{2}}{B^{2}})}\bigg )R^{\prime 2}\bigg \}\bigg \}+\lambda \bigg \{\frac{f}{2}e^{\nu } +\bigg (\frac{\mu ''}{2}-\frac{\nu '}{r} \nonumber \\{} & {} -\frac{\mu '\nu '}{4}+\frac{\mu ^{\prime 2}}{4}\bigg )\bigg (1-\frac{2nR \gamma (1+\frac{R^{2}}{B^{2}})^{-1-n}}{B}\bigg )+\bigg (\frac{\mu '}{2} +\frac{2}{r}\bigg )\bigg \{\frac{2n\gamma }{B(1+\frac{R^{2}}{B^{2}})^{1+n}} \nonumber \\{} & {} \times \bigg (\frac{(-2-2n)R^{2}}{B^{2}(1+\frac{R^{2}}{B^{2}})}-1\bigg ) R'\bigg \}\bigg \}-2(\lambda +1)\bigg \{\frac{f}{2}e^{\nu }+\bigg (\frac{ (\mu '-\nu ')r}{2}-e^{\nu }+1\bigg ) \nonumber \\{} & {} \times \frac{\bigg (-1+\frac{2nR\gamma (1+\frac{R^{2}}{B^{2}})^{-1-n}}{B}\bigg )}{r^{2}}+\bigg (\frac{\mu '-\nu '}{2}+\frac{1}{r}\bigg )\bigg \{\frac{2n\gamma }{B(1+\frac{R^{2}}{B^{2}})^{1+n}} \nonumber \\{} & {} \times \bigg (\frac{(-2-2n)R^{2}}{B^{2}(1+\frac{R^{2}}{B^{2}})} -1\bigg )R'\bigg \}+\bigg \{\frac{2n\gamma }{B(1+\frac{R^{2}}{B^{2}}) ^{1+n}}\bigg (\frac{(-2-2n)R^{2}}{B^{2}(1+\frac{R^{2}}{B^{2}})} -1\bigg )R'' \nonumber \\{} & {} +\frac{4nR\gamma }{B^{3}(1+\frac{R^{2}}{B^{2}})^{2+n}}\bigg ( (3+3n)-\frac{(8+6n+2n^{2})R^{2}}{B^{2}(1+\frac{R^{2}}{B^{2}})} \bigg )R^{\prime 2}\bigg \}\bigg \}\bigg ]. \end{aligned}$$
(A9)

The field equations for the model 3 take the following form

$$\begin{aligned} {\rho}= & {} \frac{1}{{e^{\nu }}2(2\lambda +1)(\lambda +1)}\bigg [(5\lambda +2) \bigg \{-\frac{f}{2}e^{\nu }+\bigg (\frac{\mu '}{r}-\frac{\mu '\nu '}{4}+\frac{\mu ''}{2}+\frac{\mu ^{\prime 2}}{4}\bigg ) \nonumber \\{} & {} \times \bigg (1-\omega \sec h^{2}\bigg (\frac{R}{B}\bigg )\bigg )+\bigg (\frac{\nu '}{2}-\frac{2}{r}\bigg )\frac{2\omega \sec h^{2}(\frac{R}{B})\tanh (\frac{R}{B})}{B}R' \nonumber \\{} & {} -\bigg \{\frac{2\omega \sec h^{2}(\frac{R}{B})\tanh (\frac{R}{B})}{B}R''+\frac{2\omega \sec h^{4}(\frac{R}{B})}{B^{2}}\bigg (1-2\sinh ^ {2}\bigg (\frac{R}{B}\bigg )\bigg )R^{\prime 2}\bigg \}\bigg \} \nonumber \\{} & {} +\lambda \bigg \{\frac{f}{2}e^{\nu }+\bigg (\frac{\mu ''}{2}-\frac{\nu '}{r}-\frac{\mu '\nu '}{4}+\frac{\mu ^{\prime 2}}{4}\bigg )\bigg (1- \omega \sec h^{2}\bigg (\frac{R}{B}\bigg )\bigg )+\bigg (\frac{\mu '}{2} +\frac{2}{r}\bigg ) \nonumber \\{} & {} \times \frac{2\omega \sec h^{2}(\frac{R}{B})\tanh (\frac{R}{B})}{B}R'\bigg \}+2\lambda \bigg \{\frac{f}{2}e^{\nu }+\bigg (\frac{ (\mu '-\nu ')r}{2}-e^{\nu }+1\bigg ) \nonumber \\{} & {} \times \frac{\bigg (-1+\omega \sec h^{2}\bigg (\frac{R}{B}\bigg )\bigg )}{r^{2}}+\bigg (\frac{\mu '-\nu '}{2}+\frac{1}{r}\bigg )\bigg (\frac{2 \omega \sec h^{2}(\frac{R}{B})\tanh (\frac{R}{B})}{B}\bigg ) \nonumber \\{} & {} \times R'+\bigg \{\frac{2\omega \sec h^{2}(\frac{R}{B})\tanh (\frac{R}{B})}{B}R''+\frac{2\omega \sec h^{4}(\frac{R}{B})}{B^{2}} \nonumber \\{} & {} \times \bigg (1-2\sinh ^{2}\bigg (\frac{R}{B}\bigg )\bigg )R^{\prime 2}\bigg \} \bigg \}\bigg ], \end{aligned}$$
(A10)
$$\begin{aligned} P_{r}= & {} \frac{1}{{e^{\nu }}2(2\lambda +1)(\lambda +1)}\bigg \{-\frac{f}{2}e^{\nu }+\bigg (\frac{\mu '}{r}-\frac{\mu '\nu '}{4}+\frac{\mu ''}{2}+\frac{\mu ^{\prime 2}}{4}\bigg )\bigg (1 \nonumber \\{} & {} -\omega \sec h^{2}\bigg (\frac{R}{B}\bigg )\bigg )+\bigg (\frac{\nu '}{2} -\frac{2}{r}\bigg )\frac{2\omega \sec h^{2}(\frac{R}{B})\tanh (\frac{R}{B})}{B}R' \nonumber \\{} & {} -\bigg \{\frac{2\omega \sec h^{2}(\frac{R}{B})\tanh (\frac{R}{B})}{B}R''+\frac{2\omega \sec h^{4}(\frac{R}{B})}{B^{2}}\bigg (1-2\sinh ^ {2}\bigg (\frac{R}{B}\bigg )\bigg )R^{\prime 2}\bigg \}\bigg \} \nonumber \\{} & {} +(3\lambda +2)\bigg \{\frac{f}{2}e^{\nu }+\bigg (\frac{\mu ''}{2} -\frac{\nu '}{r}-\frac{\mu '\nu '}{4}+\frac{\mu ^{\prime 2}}{4}\bigg ) \bigg (1-\omega \sec h^{2}\bigg (\frac{R}{B}\bigg )\bigg ) \nonumber \\{} & {} +\bigg (\frac{\mu '}{2}+\frac{2}{r}\bigg )\frac{2\omega \sec h^{2}(\frac{R}{B})\tanh (\frac{R}{B})}{B}R'\bigg \}-2\lambda \bigg \{\frac{f}{2}e^{\nu }+\bigg (\frac{(\mu '-\nu ')r}{2} \nonumber \\{} & {} -e^{\nu }+1\bigg )\frac{\bigg (-1+\omega \sec h^{2}\bigg (\frac{R}{B}\bigg )\bigg )}{r^{2}}+\bigg (\frac{\mu '-\nu '}{2} +\frac{1}{r}\bigg )\bigg (\frac{2\omega \sec h^{2}(\frac{R}{B}) \tanh (\frac{R}{B})}{B}\bigg )R' \nonumber \\{} & {} +\bigg \{\frac{2\omega \sec h^{2}(\frac{R}{B})\tanh (\frac{R}{B})}{B}R'' \nonumber \\{} & {} +\frac{2\omega \sec h^{4}(\frac{R}{B})}{B^{2}}\bigg (1-2\sinh ^ {2}\bigg (\frac{R}{B}\bigg )\bigg )R^{\prime 2}\bigg \}\bigg \}\bigg ], \end{aligned}$$
(A11)
$$\begin{aligned} P_{t}= & {} \frac{1}{{e^{\nu }}2(2\lambda +1)(\lambda +1)}\bigg [-\lambda \bigg \{-\frac{f}{2}e^{\nu }+\bigg (\frac{\mu '}{r}-\frac{\mu '\nu '}{4}+\frac{\mu ''}{2}+\frac{\mu ^{\prime 2}}{4}\bigg ) \nonumber \\{} & {} \times \bigg (1-\omega \sec h^{2}\bigg (\frac{R}{B}\bigg )\bigg )+\bigg (\frac{\nu '}{2}-\frac{2}{r}\bigg )\frac{2\omega \sec h^{2}(\frac{R}{B})\tanh (\frac{R}{B})}{B}R' \nonumber \\{} & {} -\bigg \{\frac{2\omega \sec h^{2}(\frac{R}{B})\tanh (\frac{R}{B})}{B}R''+\frac{2\omega \sec h^{4}(\frac{R}{B})}{B^{2}}\bigg (1-2\sinh ^{2} \bigg (\frac{R}{B}\bigg )\bigg )R^{\prime 2}\bigg \}\bigg \} \nonumber \\{} & {} +\lambda \bigg \{\frac{f}{2}e^{\nu }+\bigg (\frac{\mu ''}{2}-\frac{\nu '}{r}-\frac{\mu '\nu '}{4}+\frac{\mu ^{\prime 2}}{4}\bigg )\bigg (1-\omega \sec h^{2}\bigg (\frac{R}{B}\bigg )\bigg )+\bigg (\frac{\mu '}{2}+\frac{2}{r}\bigg ) \nonumber \\{} & {} \times \frac{2\omega \sec h^{2}(\frac{R}{B})\tanh (\frac{R}{B})}{B}R' \bigg \}-2(\lambda +1)\bigg \{\frac{f}{2}e^{\nu }+\bigg (\frac{(\mu '-\nu ') r}{2}-e^{\nu }+1\bigg ) \nonumber \\{} & {} \times \frac{\bigg (-1+\omega \sec h^{2}\bigg (\frac{R}{B}\bigg )\bigg )}{r^{2}}+\bigg (\frac{\mu '-\nu '}{2}+\frac{1}{r}\bigg )\bigg (\frac{2\omega \sec h^{2}(\frac{R}{B})\tanh (\frac{R}{B})}{B}\bigg )R' \nonumber \\{} & {} +\bigg \{\frac{2R''\omega \sec h^{2}(\frac{R}{B})\tanh (\frac{R}{B})}{B} \nonumber \\{} & {} +\frac{2\omega \sec h^{4}(\frac{R}{B})\bigg (1-2\sinh ^{2} \bigg (\frac{R}{B}\bigg )\bigg )R^{\prime 2}}{B^{2}}\bigg \}\bigg \}\bigg ]. \end{aligned}$$
(A12)

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Sharif, M., Fatima, A. Traversable wormhole solutions admitting Karmarkar condition in f(R, T) theory. Eur. Phys. J. Plus 138, 196 (2023). https://doi.org/10.1140/epjp/s13360-023-03825-5

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