A study of important solutions in Chern–Simons modified gravity

Abstract

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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Appendices

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.

$$\begin{aligned} A_{\mu ,\nu }=A_{\mu :\nu }-\varGamma ^{\alpha } _{\mu \nu }A_{\alpha }. \end{aligned}$$
(65)

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

$$\begin{aligned} A_{\nu ;\rho ;\sigma }-A_{\nu ;\sigma ;\rho }=A_{\beta } {R^{\beta }}_{\nu \rho \sigma }, \end{aligned}$$
(66)

where

$$\begin{aligned} {R^{\beta }}_{\nu \rho \sigma }={\varGamma ^{\beta }}_{\nu \sigma ,\rho } -{\varGamma ^{\beta }}_{\nu \rho ,\sigma }+{\varGamma ^{\alpha }}_{\nu \sigma } {\varGamma ^{\beta }}_{\alpha \rho }-{\varGamma ^{\alpha }}_{\nu \rho } {\varGamma ^{\beta }}_{\alpha \sigma }. \end{aligned}$$
(67)

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.,

$$\begin{aligned} {R^{\beta }}_{\nu \rho \sigma }=-{R^{\beta }}_{\nu \sigma \rho }. \end{aligned}$$
(68)

also from Eq. (68), we have

$$\begin{aligned} {R^{\beta }}_{\nu \rho \sigma }+{R^{\beta }}_{\rho \sigma \nu }+{R^{\beta }}_{\sigma \nu \rho }=0. \end{aligned}$$
(69)

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.

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

Hence,

$$\begin{aligned} R_{\mu \nu \rho \sigma }=\frac{1}{2}(g_{\mu \sigma ,\nu \rho } -g_{\nu \sigma ,\mu \rho }-g_{\mu \rho ,\nu \sigma }+g_{\nu \rho ,\mu \sigma }) +\varGamma _{\beta \mu \sigma } \varGamma ^{\beta } _{\nu \rho }-\varGamma _{\mu \beta \rho }{\varGamma ^{\beta }}_{\nu \sigma }. \end{aligned}$$
(70)

So further symmetries now show up, namely

$$\begin{aligned} R_{\mu \nu \rho \sigma }=-R_{\nu \mu \rho \sigma }, \end{aligned}$$
(71)

and

$$\begin{aligned} R_{\mu \nu \rho \sigma }=R_{\rho \sigma \mu \nu }=R_{\sigma \rho \mu \nu }. \end{aligned}$$
(72)

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

$$\begin{aligned} {R^{\mu }}_{\nu \rho \mu } = R_{\nu \rho }. \end{aligned}$$
(73)

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

$$\begin{aligned} R_{\nu \rho }=R_{\rho \nu }. \end{aligned}$$
(74)

It can be contracted again and formed

$$\begin{aligned} g^{\nu \rho }R_{\nu \rho }={R^{\nu }}_{\nu }=R. \end{aligned}$$
(75)

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}$$
(76)

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

$$\begin{aligned}&C_{02}=C_{20}=-C_{03}=-C_{30}\nonumber \\&\quad = -\frac{l}{2A^5 B^4} \csc \theta \left( -3\partial _{3}\varTheta B A^{\prime 2}B^{\prime } + A(-\partial _{3} \varTheta A^{\prime }B^{\prime 2}\right. \nonumber \\&\left. \qquad + B(\partial _{3} \varTheta B^{\prime }A^{\prime \prime }+A^{\prime }( \partial _{3}\partial _{1}\varTheta B^{\prime }+ 3\partial _{3} \varTheta B^{\prime \prime })))+ A^2(\partial _{3}\partial _{1}\varTheta B^{\prime 2}\right. \nonumber \\&\left. \qquad + \partial _{3} \varTheta B^{\prime }A^{\prime \prime }+B(- \partial _{3}\partial _{1}\varTheta B^{\prime \prime } - \partial _{3} \varTheta (B^{\prime \prime \prime }+ A^{\prime }(\dot{A}\dot{B} - B\ddot{A}))))\right. \nonumber \\&\left. \qquad + A^{3}(\partial _{3} \varTheta B^{\prime }\dot{A}\dot{B}+B(\dot{A}(\partial _{3}\partial _{1}\varTheta \dot{B} + \partial _{3} \varTheta \dot{B^{\prime }}) + \partial _{3} \varTheta (\dot{B}\dot{A}^{\prime } - 2B^{\prime }\ddot{A}))\right. \nonumber \\&\left. \qquad - B^{2}(\partial _{3}\partial _{1}\varTheta \ddot{A} + \partial _{3} \varTheta \ddot{A^{\prime }}))+ A^{4}(- \partial _{3}\partial _{1}\varTheta - \partial _{3}\partial _{1}\varTheta \dot{B}^{2} - 2\partial _{3} \varTheta \dot{B}\dot{B}^{\prime }\right. \nonumber \\&\left. \qquad +\partial _{3} \varTheta B^{\prime }\ddot{B}+ B(\partial _{3}\partial _{1}\varTheta \ddot{B} + \partial _{3} \varTheta \ddot{B}^{\prime }))\right) , \end{aligned}$$
(77)
$$\begin{aligned}&C_{12}=C_{21}=-C_{13}=-C_{31}\nonumber \\&\quad =\frac{l}{2A^5B^4} \csc \theta \Bigg (- 3\partial _{2} \varTheta BA^{\prime 2}B^{\prime } + A(-\partial _{2} \varTheta A^{\prime }B^{\prime 2}\nonumber \\&\qquad +B(\partial _{2} \varTheta B^{\prime }A^{\prime \prime }+A^{\prime }( \partial _{2}\partial _{0}\varTheta B^{\prime }+3\partial _{2} \varTheta B^{\prime \prime })))+ A^2(\partial _{2}\partial _{0}\varTheta B^{\prime 2} + \partial _{2} \varTheta B^{\prime }A^{\prime \prime }\nonumber \\&\qquad +B(- \partial _{2}\partial _{0}\varTheta B^{\prime \prime } - \partial _{2} \varTheta (B^{\prime \prime \prime }+ A^{\prime }(\dot{A}\dot{B} - B\ddot{A}))))+ A^{3}(\partial _{2} \varTheta B^{\prime }\dot{A}\dot{B}\nonumber \\&\qquad +B(\dot{A}(\partial _{2}\partial _{0}\varTheta \dot{B}+\partial _{2} \varTheta \dot{B^{\prime }}) + \partial _{2} \varTheta (\dot{B}\dot{A}^{\prime }-2B^{\prime }\ddot{A}))- B^{2}(\partial _{2}\partial _{0}\varTheta \ddot{A} + \partial _{2} \varTheta \ddot{A^{\prime }}))\nonumber \\&\qquad +A^{4}\Bigg (- \partial _{2}\partial _{0}\varTheta -\partial _{2}\partial _{0}\varTheta \dot{B}^{2} - 2\partial _{2} \varTheta \dot{B}\dot{B}^{\prime }+ \partial _{2} \varTheta B^{\prime }\ddot{B}\nonumber \\&\qquad + B(\partial _{2}\partial _{0}\varTheta \ddot{B} + \partial _{2} \varTheta \ddot{B}^{\prime })\Bigg )\Bigg ). \end{aligned}$$
(78)

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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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Keywords

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

PACS Nos.

  • 95.30.Sf
  • 97.60.Lf
  • 04.20.q