A study of important solutions in Chern–Simons modified gravity


In this manuscript, we have presented the compatible solutions under the framework of dynamical and non-dynamical Chern–Simons modified gravity. By taking an open choice of external field as a function of angular parameter \(\theta \), we obtained Gödel-type solutions of non-dynamical Chern–Simons modified gravity. It is mentioned that a vacuum Gödel-type solution exists in this theory. Non-static spherical symmetric solutions are also described for dynamical Chern–Simons modified gravity by considering external field as a function of radial parameter r. Later, results are similar to those of well-known Tolman–Bondi solutions found in the context of general relativity. It is observed that in case of \(\varLambda \rightarrow \kappa p_{0}\) both parameters A(tr) and B(tr) turned to be undefined.

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  1. [1]

    A G Riess, A V Filippenko, P Challis, A Clocchiatti, A Diercks, P M Garnavich, R L Gilliland, C J Hogan, S Jha, R P Kirshner, B Leibundgut, M M Phillips, D Reiss, B P Schmidt, R A Schommer, R C Smith, J Spyromilio, C Stubbs, B S Nicholas and J Tonry Astron. J. 116 1009 (1998)

    ADS  Article  Google Scholar 

  2. [2]

    R Jackiw and S Y Pi Phys. Rev. Lett. 83 1506 (1999). 104012 (2003)

  3. [3]

    S Alexander and N Yunes Phys. Rev. D75 124022 (2007)

    ADS  MathSciNet  Article  Google Scholar 

  4. [4]

    A Lue, L Wang and M Kamionkowski Phys. Rev. Lett. 83 1506 (1999)

    ADS  Article  Google Scholar 

  5. [5]

    S Alexander Phys. Lett. B660 444 (2008)

    ADS  Article  Google Scholar 

  6. [6]

    C Furtado, J R Nascimento, A Yu Petrov and A F Santos Phys. Lett. B693 494 (2010)

    ADS  Article  Google Scholar 

  7. [7]

    C Furtado, T Mariz, J R Nascimento, A Y Petrov, and A F Santos Int. J. Mod. Phys. Conf. Ser. 18 145 (2012)

    Article  Google Scholar 

  8. [8]

    D Grumiller and N Yunes Phys. Rev. D77 044015 (2008)

    ADS  MathSciNet  Article  Google Scholar 

  9. [9]

    C Furtado, T Mariz, J R Nascimento, A Y Petrov and A F Santos Phys. Rev. D79 124039 (2009)

    ADS  Google Scholar 

  10. [10]

    H Ahmedov and A N Aliev Phys. Rev. D82 024043 (2010)

    ADS  MathSciNet  Article  Google Scholar 

  11. [11]

    N Yunes and F Pretorius Phys. Rev. D79 084043 (2009)

    ADS  Article  Google Scholar 

  12. [12]

    D Guarrera and A J Hariton Phys. Rev. D76 044011 (2007)

    ADS  MathSciNet  Article  Google Scholar 

  13. [13]

    Y S Myung and T Moon JCAP 10 043 (2014)

    ADS  Article  Google Scholar 

  14. [14]

    L E Qianga and P Xu Eur. Phys. J. C 75 390 (2015)

    ADS  Article  Google Scholar 

  15. [15]

    M Sharif Int. J. Mod. Phys. D13 1019 (2004)

    ADS  Article  Google Scholar 

  16. [16]

    S Hawking Phys. Rev. D46 603 (1992)

    ADS  MathSciNet  Article  Google Scholar 

  17. [17]

    K A Bronnikov and V N Melnikov Gen. Relat. Grav. 27 465 (1995)

    ADS  Article  Google Scholar 

  18. [18]

    J T Jebsen Gen. Relat. Grav. 37 2253 (2005)

    ADS  MathSciNet  Article  Google Scholar 

  19. [19]

    R M Wald General Relativity (Chicago: The University of Chicago Press) (1984)

  20. [20]

    D M Eardley and Smarr L Phys. Rev. D19 2239 (1979)

    ADS  Google Scholar 

  21. [21]

    K Lake Phys. Rev. D62 027301 (2000)

  22. [22]

    M Sharif and Z Ahmad Mod. Phys. Lett. A22 1493 (2007)

    ADS  Article  Google Scholar 

  23. [23]

    M Sharif and Z Ahmad Mod. Phys. Lett. A22 2947 (2007)

    ADS  Article  Google Scholar 

  24. [24]

    M Sharif and Z Ahmad J. Korean Phys. Soc. 52 980 (2008)

    ADS  Article  Google Scholar 

  25. [25]

    M Sharif and Z Ahmad Acta Phys. Pol. B39 1337 (2008)

    ADS  Google Scholar 

  26. [26]

    M Sharif and K Iqbal Mod. Phys. Lett. A24 1533 (2009)

    ADS  Article  Google Scholar 

  27. [27]

    K Kohkichi, M Toyoki and T Satoshi Prog. Theor. Phys. 2 122 (2009)

    Google Scholar 

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Appendix 1

According to the product law of differentiation, the covariant differentiation is very similar to ordinary differentiations. But there is an important property of ordinary differentiation, i.e., if we perform two differentiations in succession, their order does not matter but it does not hold for covariant differentiation in general.

$$\begin{aligned} A_{\mu ,\nu }=A_{\mu :\nu }-\varGamma ^{\alpha } _{\mu \nu }A_{\alpha }. \end{aligned}$$

This is symmetric w.r.t. \(\mu \) and \(\nu \). So in this case, the order of covariant differentiations does not matter. Now, we take a vector \(A_{\nu }\) and apply two covariant differentiations on it.

$$\begin{aligned} A_{\nu ;\rho ;\sigma }&= A_{\nu ;\rho ,\sigma }- \varGamma ^{\alpha } _{\nu \sigma }A_{\alpha ;\rho } - \varGamma ^{\alpha } _{\rho \sigma }A_{\nu ;\alpha },\\&= \left( A_{\nu ,\rho }-{\varGamma ^{\alpha }}_{\nu \rho } A_{\alpha }\right) ,\sigma -{\varGamma ^{\alpha }} _{\nu \sigma } \left( A_{\alpha ,\rho } - {\varGamma ^{\beta }} _{\alpha \rho }A_{\beta }\right) \\&-{\varGamma ^{\alpha }}_{\rho \sigma }\left( A_{\nu ,\alpha } - {\varGamma ^{\beta }}_{\nu \alpha } A_{\beta }\right) ,\\&= A_{\nu ,\rho ,\sigma }-\varGamma ^{\alpha }_{\nu \rho }A_{\alpha ,\sigma }- \varGamma ^{\alpha }_{\nu \sigma }A_{\alpha ,\rho } -\varGamma ^{\alpha }_{\rho \sigma }A_{\nu ,\alpha }\\&- A_{\beta }\left( \varGamma ^{\beta }_{\nu \rho ,\sigma }-\varGamma ^{\alpha } _{\nu \rho }\varGamma ^{\beta }_{\alpha \rho }-\varGamma ^{\alpha } _{\rho \sigma }\varGamma ^{\beta }_{\nu \alpha }\right) . \end{aligned}$$

Interchanging \({\rho }\) and \({\sigma }\) here and subtract from the previous expression. The result is

$$\begin{aligned} A_{\nu ;\rho ;\sigma }-A_{\nu ;\sigma ;\rho }=A_{\beta } {R^{\beta }}_{\nu \rho \sigma }, \end{aligned}$$


$$\begin{aligned} {R^{\beta }}_{\nu \rho \sigma }={\varGamma ^{\beta }}_{\nu \sigma ,\rho } -{\varGamma ^{\beta }}_{\nu \rho ,\sigma }+{\varGamma ^{\alpha }}_{\nu \sigma } {\varGamma ^{\beta }}_{\alpha \rho }-{\varGamma ^{\alpha }}_{\nu \rho } {\varGamma ^{\beta }}_{\alpha \sigma }. \end{aligned}$$

The left-hand side of Eq. (67) is a tensor. It follows that the right-hand side must be a tensor. Thus, by quotient theorem \({{R}}^{\beta }_{\nu \rho \sigma }\) is tensor. It is called curvature tensor. It is antisymmetric in its lower index, i.e.,

$$\begin{aligned} {R^{\beta }}_{\nu \rho \sigma }=-{R^{\beta }}_{\nu \sigma \rho }. \end{aligned}$$

also from Eq. (68), we have

$$\begin{aligned} {R^{\beta }}_{\nu \rho \sigma }+{R^{\beta }}_{\rho \sigma \nu }+{R^{\beta }}_{\sigma \nu \rho }=0. \end{aligned}$$

Let us lower the suffix \({{\beta }}\) and put it as the first suffix. We get,

$$\begin{aligned} R_{\mu \nu \rho \sigma }=g_{\mu \beta }{R^{\beta }}_{\nu \rho \sigma }=g_{\mu \beta } {\varGamma ^{\beta }}_{\nu \sigma ,\rho }+{\varGamma ^{\alpha }}_{\nu \sigma } \varGamma _{\mu \alpha \rho }-\langle \rho \sigma \rangle . \end{aligned}$$

where the symbol \(\langle \rho \sigma \rangle \) is used to denote the proceeding terms with \(\rho \) and \(\sigma \) interchanged.

$$\begin{aligned} R_{\mu \nu \rho \sigma }&= \varGamma _{\mu \nu \sigma ,\rho } - g_{\mu \beta ,\rho }{\varGamma ^{\beta }}_{\nu \sigma }+ \varGamma _{\mu \beta \rho }{\varGamma ^{\beta }}_{\nu \sigma }- \langle \rho \sigma \rangle ,\\&= \varGamma _{\mu \nu \sigma ,\rho }-\varGamma _{\beta \mu \rho } {\varGamma ^{\beta }}_{\nu \sigma }-\langle \rho \sigma \rangle . \end{aligned}$$


$$\begin{aligned} R_{\mu \nu \rho \sigma }=\frac{1}{2}(g_{\mu \sigma ,\nu \rho } -g_{\nu \sigma ,\mu \rho }-g_{\mu \rho ,\nu \sigma }+g_{\nu \rho ,\mu \sigma }) +\varGamma _{\beta \mu \sigma } \varGamma ^{\beta } _{\nu \rho }-\varGamma _{\mu \beta \rho }{\varGamma ^{\beta }}_{\nu \sigma }. \end{aligned}$$

So further symmetries now show up, namely

$$\begin{aligned} R_{\mu \nu \rho \sigma }=-R_{\nu \mu \rho \sigma }, \end{aligned}$$


$$\begin{aligned} R_{\mu \nu \rho \sigma }=R_{\rho \sigma \mu \nu }=R_{\sigma \rho \mu \nu }. \end{aligned}$$

The result of all these symmetries is that of the 256 components of \(R_{\mu \nu \rho \sigma }\), and only 20 are independent.

Let us contract two of the suffixes in \(R_{\mu \nu \rho \sigma }\). If we take two suffixes with respect to which it is antisymmetrical, we get zero, of course. If we take any other two, we get the same result, because of these symmetries,

Let us take the first and last suffixes

$$\begin{aligned} {R^{\mu }}_{\nu \rho \mu } = R_{\nu \rho }. \end{aligned}$$

It is called Ricci tensor. Contracting Eq. (67) by \(g^{\mu \sigma }\), we get

$$\begin{aligned} R_{\nu \rho }=R_{\rho \nu }. \end{aligned}$$

It can be contracted again and formed

$$\begin{aligned} g^{\nu \rho }R_{\nu \rho }={R^{\nu }}_{\nu }=R. \end{aligned}$$

This is a scalar and is called scaler curvature or total curvature. It is defined in such a way that it is positive for the surface of a sphere in three dimensions.

Appendix 2

The corresponding non-vanishing components of Riemann tensor for the Gödel-type metrics are given as:

$$\begin{aligned} R_{0002}&= R_{2020} = \frac{H(r)H^{\prime ^{2}}(r)}{4B^{2}(r)},\\ R_{1010}&= \frac{H^{\prime ^{2}}(r)}{4B^{2}(r)},\\ R_{1012}&= \frac{-2B(r)B^{\prime }(r)H^{\prime }(r) + H(r)H^{\prime ^{2}}(r) + 2B^{2}(r)H^{\prime \prime }(r)}{4B^{2}(r)},\\ R_{0101}&= \frac{B(r)H^{\prime ^{2}}(r) + H(r)(-2B^{\prime }(r)H^{\prime }(r)+2B(r)H^{\prime \prime }(r))}{4B^{3}(r)},\\ R_{0121}&= \frac{1}{2B^{3}(r)}\left[ H^{2}(r)B^{\prime }(r)H^{\prime }(r) \right. \\&\left.\quad + B^{2}(r)(B^{\prime }(r)H^{\prime }(r) + 2H(r)B^{\prime \prime }(r))\right. \\&\left. \quad - B^{3}(r)H^{\prime \prime }(r) - B(r)H(r)\left( 2H^{\prime ^{2}}(r) + H(r)H^{\prime \prime }(r)\right) \right] ,\\ R_{2110}&= \frac{B^{\prime }(r)H^{\prime }(r) - B(r)H^{\prime \prime }(r)}{2B^{3}(r)},\\ R_{2121}&= \frac{B(r)\left( 3H^{\prime ^{2}}(r) - 4B(r)B^{\prime \prime }(r)\right) + H(r)(- 2B^{\prime }(r)H^{\prime }(r) + 2B(r)H^{\prime \prime }(r))}{4B^{3}(r)},\\ R_{0220}&= \frac{\left( B^{2}(r) - H^{2}(r)\right) H^{\prime ^{2}}(r)}{4B^{2}(r)},\\ R_{1210}&= \frac{1}{4}\left( \frac{H^{\prime }(r)(- 2B(r)B^{\prime }(r) + H(r)H^{\prime }(r))}{B^{2}(r)} \right. \\&\left. \quad + 2H^{\prime \prime }(r)\right) ,\\ R_{1221}&= \frac{H(r)B^{\prime }(r)H^{\prime }(r)}{B(r)} \\&\quad- \frac{3}{4}H^{\prime ^{2}}(r) - \frac{H^{2}(r)H^{\prime ^{2}}(r)}{4B^{2}(r)} + B(r)B^{\prime \prime }(r) - H(r)H^{\prime \prime }(r),\\ R_{2020}&= -\frac{H^{\prime ^{2}}(r)}{4B^{2}(r)},\\ R_{2202}&= -\frac{H(r)H^{\prime ^{2}}(r)}{4B^{2}(r)}. \end{aligned}$$

The surviving components of C-tensor are given as:

$$\begin{aligned} C^{00}&= \frac{1}{2B^{5}(r)}\left( H^{2}(r)\partial _3\varTheta H^{\prime }(r)\left( 2B^{\prime ^{2}}(r) + H^{\prime ^{2}}(r)\right) \right. \nonumber \\&\left.\quad - B(r)H(r)\left( 6\partial _3\varTheta B^{\prime }(r)H^{\prime ^{2}}(r)\right. \right. \nonumber \\&\left. \left. \quad+ H(r)(2\partial _3\varTheta H^{\prime }(r)B^{\prime \prime }(r)\right. \right. \nonumber \\&\left. \left.\quad +B^{\prime }(r)(\partial _3\partial _1\varTheta H^{\prime }(r)+ 2\partial _3\varTheta H^{\prime \prime }(r)))\right) \right. \nonumber \\&\left. \quad - B^{3}(r)(3\partial _3\varTheta H^{\prime }(r)B^{\prime \prime }(r)\right. \nonumber \\&\left. \quad+ B^{\prime }(r)(\partial _3\partial _1\varTheta H^{\prime }(r)+ \partial _3\varTheta H^{\prime \prime }(r))\right. \nonumber \\&\left. \quad+ 2H(r)(\partial _3\partial _1\varTheta B^{\prime \prime }(r)\right. \nonumber \\&\left. \quad+ \partial _3\varTheta B^{3}(r)))+ B^{4}(r)(\partial _3\partial _1\varTheta H^{\prime \prime }(r) + \partial _3\varTheta H^{3}(r))\right. \nonumber \\&\left. \quad + B^{2}(r)\left( \partial _3\varTheta B^{\prime ^{2}}(r)H^{\prime }(r)+ 2\partial _3\varTheta H^{\prime ^{3}}(r) \right. \right. \nonumber \\&\left. \left.\quad + 2H(r)\partial _3\varTheta B^{\prime }(r)B^{\prime \prime }(r)\nonumber \right. \right. \\&\left. \left.\quad + 2H(r)H^{\prime }(r)(\partial _3\partial _1\varTheta H^{\prime }(r)+3\partial _3\varTheta H^{\prime \prime }(r))\right. \right. \nonumber \\&\left. \left. \quad+H^{2}(r)(\partial _3\partial _1\varTheta H^{\prime \prime }(r)\right. \right. \nonumber \\&\left. \left. \quad+ \partial _3\varTheta H^{3}(r))\right) \right) ,\nonumber \\ C^{01}&= \frac{1}{4B^{4}(r)}(H(r)(H(r)\partial _3\partial _0\varTheta - \partial _3\partial _2\varTheta )B^{\prime }(r)H^{\prime }(r)\nonumber \\&\quad+ B^{2}(r)(\partial _3\partial _0\varTheta B^{\prime }(r)H^{\prime }(r)\nonumber \\&\quad+2(H(r)\partial _3\partial _0\varTheta - \partial _3\partial _2\varTheta )B^{\prime \prime }(r))\nonumber \\&\quad - B^{3}(r)\partial _3\partial _0\varTheta H^{\prime \prime }(r)-B(r)(H(r)\partial _3\partial _0\varTheta \nonumber \\&\quad - \partial _3\partial _2\varTheta )(2H^{\prime }(r)\nonumber \\&\quad + H(r)H^{\prime \prime }(r))),\nonumber \\ C^{02}&= \frac{1}{2B^{5}(r)}\left( -H(r)\partial _3\varTheta H^{\prime }(r)\left( 2B^{\prime ^{2}}(r) + H^{\prime ^{2}}(r)\right) \right. \nonumber \\&\left. \quad + B(r)\left( 3\partial _3\varTheta B^{\prime }(r)H^{\prime ^{2}}(r)\right. \right. \nonumber \\&\left. \left. \quad + H(r)(2\partial _3\varTheta H^{\prime }(r)B^{\prime \prime }(r)\right. \right. \nonumber \\&\left. \left.\quad +B^{\prime }(r)(\partial _3\partial _1\varTheta H^{\prime }(r) + 2\partial _3\varTheta H^{\prime \prime }(r)))\right) + B^{3}(r)(\partial _3\partial _1\varTheta B^{\prime \prime }(r)\right. \nonumber \\&\left.\quad +\partial _3\varTheta B^{(3)}(r))- B^{2}(r)\left( \partial _3\partial _1\varTheta H^{\prime ^{2}}(r)+ \partial _3\varTheta B^{\prime }(r)B^{\prime \prime }(r)\right. \right. \nonumber \\&\left. \left. \quad + H(r)\partial _3\partial _1\varTheta H^{\prime \prime }(r)+ 3\partial _3\varTheta H^{\prime }(r)H^{\prime \prime }(r)+ H(r)\partial _3\varTheta H^{3}(r)\right) \right) ,\nonumber \\ C^{03}&= \frac{1}{4B^{5}(r)}\left( 3H(r)(- H(r)\partial _0\varTheta +\partial _2\varTheta )B^{\prime ^{2}}(r)H^{\prime }(r)\right. \nonumber \\&\left. \quad+ B(r)\left( - 2\partial _2\varTheta B^{\prime }(r)H^{\prime ^{2}}(r)+ H^{2}(r)(\partial _0\varTheta H^{\prime }(r)B^{\prime \prime }(r)\right. \right. \nonumber \\&\left. \left.\quad + B^{\prime }(r)(\partial _1\partial _0\varTheta H^{\prime }(r)+ 3\partial _0\varTheta H^{\prime \prime }(r)))- H(r)(\partial _2\varTheta H^{\prime }(r)B^{\prime \prime }(r) \right. \right. \nonumber \\&\left. \left. \quad+ B^{\prime }(r)(\partial _2\partial _1\varTheta H^{\prime }(r)- 6\partial _0\varTheta H^{\prime ^{2}}(r)+ 3\partial _2\varTheta H^{\prime \prime }(r)))\right) \right. \nonumber \\&\left. \quad+ B^{3}(r)(3\partial _0\varTheta H^{\prime }(r)B^{\prime \prime }(r)+ B^{\prime }(r)(\partial _1\partial _0\varTheta H^{\prime }(r) +\partial _0\varTheta H^{\prime \prime }(r))\right. \nonumber \\&\left.\quad + 2H(r)(\partial _1\partial _0\varTheta B^{\prime \prime }(r)+ \partial _0\varTheta B^{3}(r))) - B^{4}(r)(\partial _1\partial _0\varTheta H^{\prime \prime }(r)\right. \nonumber \\&\left. \quad+ \partial _0\varTheta H^{(3)}(r))- B^{2}(r)\left( \partial _0\varTheta B^{\prime ^{2}}(r)H^{\prime }(r) + 2\partial _0\varTheta H^{\prime ^{3}}(r)\right. \right. \nonumber \\&\left. \left.\quad + 2H(r)\partial _0\varTheta B^{\prime }(r)B^{\prime \prime }(r)- 2\partial _2\varTheta H^{\prime }(r)H^{\prime \prime }(r)+ H^{2}(r)(\partial _1\partial _0\varTheta H^{\prime \prime }(r)\right. \right. \nonumber \\&\left. \left. \quad+ \partial _0\varTheta H^{(3)}(r))+ H(r)\left( 2\partial _1\partial _0\varTheta H^{\prime ^{2}}(r)-\partial _2\partial _1\varTheta H^{\prime \prime }(r)\right. \right. \right. \nonumber \\&\left. \left. \left. \quad+ 6\partial _0\varTheta H^{\prime }(r)H^{\prime \prime }(r)- \partial _2\varTheta H^{(3)}(r)\right) \right) \right) ,\nonumber \\ C^{11}&= \frac{1}{2B^{3}(r)}\left( \partial _{3}\varTheta \left( - B^{\prime ^2}(r)H^{\prime }(r)+ H^{\prime ^{3}}(r) - B(r)H^{\prime }(r)B^{\prime \prime }(r)\right. \right. \nonumber \\&\left. \left. \quad+ B(r)B^{\prime }(r) H^{\prime \prime }(r)\right) \right) ,\nonumber \\ C^{12}&= \frac{1}{4B^{4}(r)}(H(r)\partial _3 \partial _0 \varTheta - \partial _3 \partial _2 \varTheta )( - B^{\prime }(r)H^{\prime }(r) + B(r)H^{\prime \prime }(r)),\nonumber \\ C^{13}&= \frac{1}{4B^{4}(r)}\left( -(H^{2}(r)\partial _0 \partial _0\varTheta -2H(r)\partial _2\partial _0\varTheta +\partial _2\partial _2 \varTheta )B^{\prime }(r) H^{\prime }(r)\right. \nonumber \\&\left. \quad- B^{2}(r)(\partial _0\partial _0\varTheta B^{\prime }(r)H^{\prime }(r)+2(H(r)\partial _0\partial _0\varTheta - \partial _2\partial _0\varTheta )B^{\prime \prime }(r))\right. \nonumber \\&\left. \quad+ B^{3}(r)\partial _0\partial _0\varTheta H^{\prime \prime }(r)+ B(r)\left( -2\partial _2\partial _0\varTheta H^{\prime ^{2}}(r) + H^{2}(r)\partial _0\partial _0\varTheta H^{\prime \prime }(r)\right. \right. \nonumber \\&\left. \left. \quad+\partial _2\partial _2\varTheta H^{\prime \prime }(r)+ 2H(r)\left( \partial _0\partial _0\varTheta H^{\prime ^{2}}(r) - \partial _2\partial _0\varTheta H^{\prime \prime }(r)\right) \right) \right) ,\nonumber \\ C^{22}&= \frac{1}{2B^{5}(r)}\left( {\partial _3}\varTheta H^{\prime }(r)\left( 2B^{\prime ^{2}}(r) + H^{\prime ^{2}}(r)\right) -B(r)(2{\partial _3}\varTheta H^{\prime }(r)B^{\prime \prime }(r)\right. \nonumber \\&\left. \quad+ B^{\prime }(r)(\partial _3 \partial _1 \varTheta H^{\prime }(r)+ 2\partial _3 \varTheta H^{\prime \prime }(r))) + B^{2}(r)(\partial _3 \partial _1 \varTheta H^{\prime \prime }(r)\right. \nonumber \\&\left.\quad +\partial _3 \varTheta H^{(3)}(r))\right) ,\nonumber \\ C^{23}&= \frac{1}{4B^{5}(r)}\left( 3(H(r)\partial _0 \varTheta -\partial _2 \varTheta ) B^{\prime ^{2}}(r)H^{\prime }(r) \right. \nonumber \\&\left. \quad+ B(r)({\partial _2}\varTheta H^{\prime }(r)B^{\prime \prime }(r)\right. \nonumber \\&\left.\quad + B^{\prime }(r)\left( \partial _2\partial _1\varTheta H^{\prime }(r) - 4 \partial _0\varTheta H^{\prime ^{2}}(r) + 3 \partial _2\varTheta H^{\prime \prime }(r)\right) \right. \nonumber \\&\left. \quad - H(r)(\partial _0 \varTheta H^{\prime }(r)B^{\prime \prime }(r)+ B^{\prime }(r)(\partial _1\partial _0\varTheta H^{\prime }(r) + 3 \partial _0 \varTheta H^{\prime \prime }(r))))\right. \nonumber \\&\left. \quad -2B^{3}(r)(\partial _1 \partial _0 \varTheta B^{\prime \prime }(r)+ \partial _0 \varTheta B^{(3)}(r))+ B^{2}(r)(2 \partial _1 \partial _0 \varTheta H^{\prime ^{2}}(r)\right. \nonumber \\&\left. \quad+2 \partial _0 \varTheta B^{\prime }(r)B^{\prime \prime }(r)+ H(r)\partial _1 \partial _0 \varTheta H^{\prime \prime }(r)- \partial _2\partial _1\varTheta H^{\prime \prime }(r)\right. \nonumber \\&\left. \quad+ 4\partial _0\varTheta H^{\prime }(r)H^{\prime \prime }(r)+ H(r)\partial _0\varTheta H^{(3)}(r)-\partial _2\varTheta H^{(3)}(r))\right) . \end{aligned}$$

The nonzero components of C-tensor for the spherically symmetric metrics are given by

$$\begin{aligned}&C_{02}=C_{20}=-C_{03}=-C_{30}\nonumber \\&\quad = -\frac{l}{2A^5 B^4} \csc \theta \left( -3\partial _{3}\varTheta B A^{\prime 2}B^{\prime } + A(-\partial _{3} \varTheta A^{\prime }B^{\prime 2}\right. \nonumber \\&\left. \qquad + B(\partial _{3} \varTheta B^{\prime }A^{\prime \prime }+A^{\prime }( \partial _{3}\partial _{1}\varTheta B^{\prime }+ 3\partial _{3} \varTheta B^{\prime \prime })))+ A^2(\partial _{3}\partial _{1}\varTheta B^{\prime 2}\right. \nonumber \\&\left. \qquad + \partial _{3} \varTheta B^{\prime }A^{\prime \prime }+B(- \partial _{3}\partial _{1}\varTheta B^{\prime \prime } - \partial _{3} \varTheta (B^{\prime \prime \prime }+ A^{\prime }(\dot{A}\dot{B} - B\ddot{A}))))\right. \nonumber \\&\left. \qquad + A^{3}(\partial _{3} \varTheta B^{\prime }\dot{A}\dot{B}+B(\dot{A}(\partial _{3}\partial _{1}\varTheta \dot{B} + \partial _{3} \varTheta \dot{B^{\prime }}) + \partial _{3} \varTheta (\dot{B}\dot{A}^{\prime } - 2B^{\prime }\ddot{A}))\right. \nonumber \\&\left. \qquad - B^{2}(\partial _{3}\partial _{1}\varTheta \ddot{A} + \partial _{3} \varTheta \ddot{A^{\prime }}))+ A^{4}(- \partial _{3}\partial _{1}\varTheta - \partial _{3}\partial _{1}\varTheta \dot{B}^{2} - 2\partial _{3} \varTheta \dot{B}\dot{B}^{\prime }\right. \nonumber \\&\left. \qquad +\partial _{3} \varTheta B^{\prime }\ddot{B}+ B(\partial _{3}\partial _{1}\varTheta \ddot{B} + \partial _{3} \varTheta \ddot{B}^{\prime }))\right) , \end{aligned}$$
$$\begin{aligned}&C_{12}=C_{21}=-C_{13}=-C_{31}\nonumber \\&\quad =\frac{l}{2A^5B^4} \csc \theta \Bigg (- 3\partial _{2} \varTheta BA^{\prime 2}B^{\prime } + A(-\partial _{2} \varTheta A^{\prime }B^{\prime 2}\nonumber \\&\qquad +B(\partial _{2} \varTheta B^{\prime }A^{\prime \prime }+A^{\prime }( \partial _{2}\partial _{0}\varTheta B^{\prime }+3\partial _{2} \varTheta B^{\prime \prime })))+ A^2(\partial _{2}\partial _{0}\varTheta B^{\prime 2} + \partial _{2} \varTheta B^{\prime }A^{\prime \prime }\nonumber \\&\qquad +B(- \partial _{2}\partial _{0}\varTheta B^{\prime \prime } - \partial _{2} \varTheta (B^{\prime \prime \prime }+ A^{\prime }(\dot{A}\dot{B} - B\ddot{A}))))+ A^{3}(\partial _{2} \varTheta B^{\prime }\dot{A}\dot{B}\nonumber \\&\qquad +B(\dot{A}(\partial _{2}\partial _{0}\varTheta \dot{B}+\partial _{2} \varTheta \dot{B^{\prime }}) + \partial _{2} \varTheta (\dot{B}\dot{A}^{\prime }-2B^{\prime }\ddot{A}))- B^{2}(\partial _{2}\partial _{0}\varTheta \ddot{A} + \partial _{2} \varTheta \ddot{A^{\prime }}))\nonumber \\&\qquad +A^{4}\Bigg (- \partial _{2}\partial _{0}\varTheta -\partial _{2}\partial _{0}\varTheta \dot{B}^{2} - 2\partial _{2} \varTheta \dot{B}\dot{B}^{\prime }+ \partial _{2} \varTheta B^{\prime }\ddot{B}\nonumber \\&\qquad + B(\partial _{2}\partial _{0}\varTheta \ddot{B} + \partial _{2} \varTheta \ddot{B}^{\prime })\Bigg )\Bigg ). \end{aligned}$$

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Ali, S., Jamil Amir, M. A study of important solutions in Chern–Simons modified gravity. Indian J Phys 94, 1837–1845 (2020). https://doi.org/10.1007/s12648-019-01616-2

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  • Chern–Simons modified gravity
  • Gödel-type metric
  • Spherical symmetric metric


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