Cosmic string in gravity’s rainbow

Abstract

In this paper, we study the various cylindrical solutions (cosmic strings) in gravity’s rainbow scenario. In particular, we calculate the gravitational field equations corresponding to energy-dependent background. Further, we discuss the possible Kasner, quasi-Kasner and non-Kasner exact solutions of the field equations. In this framework, we find that quasi-Kasner solutions cannot be realized in gravity’s rainbow. Assuming only time-dependent metric functions, we also analyse the time-dependent vacuum cosmic strings in gravity’s rainbow, which are completely different than the other GR solutions.

Appendix: Mathematical details

In this appendix, we present explicit forms of different geometrical quantities which are used to derive field equations.

1.1 A.1 Case $A=A(r)$, $B=B(r)$ and $C=C(r)$

The nonzero components of the metric tensor are as following:

$$\begin{aligned} {}&g_{00} = \frac{A(r)}{f^{2} (E/E_{p} )} , \qquad g_{11} = - \frac{1}{g^{2} (E/E_{p} )}, \\ & g_{22} = - \frac{B(r)}{g^{2} (E/E_{p} )}, \qquad g_{33} = - \frac{C(r)}{g^{2} (E/E_{p} )}. \end{aligned} $$

The inverse of these metric components are given by

$$\begin{aligned} {}& g^{00} = \frac{f^{2} (E/E_{p} )}{A(r)}, \qquad g^{11}= - g^{2} (E/E_{p} ),\\ & g^{22} = - \frac{g^{2} (E/E_{p} )}{B(r)}, \qquad g^{33} = - {\frac{g^{2} (E/E_{p} )}{C(r)}}. \end{aligned} $$

Utilizing definition (30), the nonzero values of the Christoffel symbols are as follows,

$$\begin{aligned} \begin{aligned} {}&\varGamma^{0}_{10}=\frac{1}{2} \frac{A^{\prime}}{A},\qquad \varGamma^{1}_{00}=\frac{1}{2} \frac{A^{\prime}}{f^{2}}g^{2},\qquad \varGamma^{1}_{22}=-\frac{1}{2} B^{\prime},\\ &\varGamma^{1}_{33}=-\frac{1}{2} C^{\prime},\qquad \varGamma^{2}_{21}=\frac{1}{2} \frac{B^{\prime}}{B},\qquad \varGamma^{3}_{31}=\frac{1}{2} \frac{C^{\prime}}{C}. \end{aligned} \end{aligned}$$

(81)

Exploiting the definition (32), the covariant components of Ricci tensor are calculated as

$$\begin{aligned} &{R_{00}=-\frac{1}{4}\frac{g^{2}}{f^{2}}\frac{A^{\prime 2}}{A}+ \frac{1}{4}\frac{g^{2}}{f^{2}}\frac{A^{\prime}B^{\prime}}{B}+\frac{1}{4} \frac{g^{2}}{f^{2}}\frac{A^{\prime}C^{\prime}}{C}+\frac{1}{2}\frac{g^{2}}{f^{2}}A^{\prime\prime},} \\ \end{aligned}$$

(82)

$$\begin{aligned} &{R_{11}=\frac{1}{4}\frac{A^{\prime 2}}{A^{2}}+\frac{1}{4} \frac{B^{\prime 2}}{B^{2}}+\frac{1}{4}\frac{C^{\prime 2}}{C^{2}}-\frac{1}{2} \frac{A^{\prime\prime}}{A}-\frac{1}{2}\frac{B^{\prime\prime}}{B}-\frac{1}{2} \frac{C^{\prime\prime}}{C},} \\ \end{aligned}$$

(83)

$$\begin{aligned} &{R_{22}=-\frac{1}{4}\frac{A^{\prime}B^{\prime}}{A}+\frac{1}{4} \frac{B^{\prime 2}}{B}-\frac{1}{4}\frac{B^{\prime}C^{\prime}}{C}-\frac{1}{2}B^{\prime\prime},} \end{aligned}$$

(84)

$$\begin{aligned} &{R_{33}=-\frac{1}{4}\frac{A^{\prime}C^{\prime}}{A}-\frac{1}{4} \frac{B^{\prime}C^{\prime}}{C}+\frac{1}{4}\frac{C^{\prime 2}}{C}-\frac{1}{2}C^{\prime\prime},} \end{aligned}$$

(85)

and similarly the mixed components are computed as

$$\begin{aligned} &{R_{0}^{0}=-\frac{g^{2}}{4} \frac{A^{\prime 2}}{A^{2}}+ \frac{g^{2}}{4} \biggl(\frac{A^{\prime}B^{\prime}}{AB} \biggr)^{2}+ \frac{g^{2}}{4} \biggl(\frac{A^{\prime}C^{\prime}}{AC} \biggr)^{2}+ \frac{g^{2}}{2} \frac{A^{\prime\prime}}{A},} \\ \end{aligned}$$

(86)

$$\begin{aligned} &{R_{1}^{1}=-\frac{g^{2}}{4}\frac{A^{\prime 2}}{A^{2}}- \frac{g^{2}}{4}\frac{B^{\prime 2}}{B^{2}}-\frac{g^{2}}{4}\frac{C^{\prime 2}}{C^{2}}+ \frac{g^{2}}{2} \frac{A^{\prime\prime}}{A}+\frac{g^{2}}{2} \frac{B^{\prime\prime}}{B}} \\ &{\phantom{R_{1}^{1}=}{}+ \frac{g^{2}}{2} \frac{C^{\prime\prime}}{C},} \end{aligned}$$

(87)

$$\begin{aligned} &{R_{2}^{2}=\frac{g^{2}}{4}\frac{A^{\prime}B^{\prime}}{AB}- \frac{g^{2}}{4}\frac{B^{\prime 2}}{B^{2}}+\frac{g^{2}}{4}\frac{B^{\prime}C^{\prime}}{BC}+ \frac{g^{2}}{2} \frac{B^{\prime\prime}}{B},} \end{aligned}$$

(88)

$$\begin{aligned} &{R_{3}^{3}=\frac{g^{2}}{4}\frac{A^{\prime}C^{\prime}}{AC}- \frac{g^{2}}{4}\frac{C^{\prime 2}}{C^{2}}+\frac{g^{2}}{4}\frac{B^{\prime}C^{\prime}}{BC}+ \frac{g^{2}}{2} \frac{C^{\prime\prime}}{C},} \end{aligned}$$

(89)

Finally, using definition $R = g^{\mu\nu}R_{\mu\nu}$, the expression for Ricci scalar is given by,

$$\begin{aligned} R =&-\frac{g^{2}}{2} \frac{A^{\prime 2}}{A^{2}}+\frac{g^{2}}{2} \frac{A^{\prime}B^{\prime}}{AB}-\frac{g^{2}}{2}\frac{B^{\prime 2}}{B^{2}}+\frac{g^{2}}{2} \frac{A^{\prime}C^{\prime}}{AC}+\frac{g^{2}}{2}\frac{B^{\prime}C^{\prime}}{BC} \\ &{}-\frac{g^{2}}{2} \frac{C^{\prime 2}}{C^{2}}+g^{2}\frac{A^{\prime\prime}}{A}+g^{2} \frac{B^{\prime\prime}}{B}+g^{2}\frac{C^{\prime\prime}}{C}. \end{aligned}$$

(90)

1.2 A.2 Case $A=A(t,r)$, $B=B(t,r)$ and $C=C(t,r)$

The nonzero components of time-dependent metric tensor are

$$\begin{aligned} {}&g_{00} = \frac{A(t,r)}{f^{2} (E/E_{p} )} , \qquad g_{11} = - \frac{1}{g^{2} (E/E_{p} )}, \\ &g_{22} = - \frac{B(t,r)}{g^{2} (E/E_{p} )}, \qquad g_{33} = - \frac{C(t,r)}{g^{2} (E/E_{p} )}. \end{aligned} $$

The inverse of these metric components are

$$\begin{aligned} {}& G^{00} = \frac{f^{2} (E/E_{p} )}{A(t,r)}, \qquad g^{11}= - g^{2} (E/E_{p} ),\\ & g^{22} = - \frac{g^{2} (E/E_{p} )}{B(t,r)}, \qquad g^{33} = - {\frac {g^{2} (E/E_{p} )}{C(t,r)}}. \end{aligned} $$

The Christoffel symbols of the second kind are,

$$\begin{aligned} \begin{aligned} {}&\varGamma^{0}_{00}=\frac{1}{2} \frac{\dot{A}}{A},\qquad \varGamma^{0}_{10}=\frac{1}{2} \frac{A^{\prime}}{A},\qquad \varGamma^{0}_{22}=\frac{1}{2}\frac{f^{2}}{g^{2}} \frac{\dot{B}}{A},\\ &\varGamma^{0}_{33}=\frac{1}{2} \frac{f^{2}}{g^{2}}\frac{\dot{C}}{A},\qquad \varGamma^{1}_{00}= \frac{1}{2}\frac{g^{2}}{f^{2}}A^{\prime},\qquad\varGamma^{1}_{22}=-\frac{1}{2} B^{\prime},\\ &\varGamma^{1}_{33}=-\frac{1}{2} C^{\prime},\qquad \varGamma^{2}_{20}=\frac{1}{2} \frac{\dot{B}}{B},\qquad \varGamma^{2}_{21}=\frac{1}{2} \frac{B^{\prime}}{B},\\ &\varGamma^{3}_{30}=\frac{1}{2} \frac{\dot{C}}{C},\qquad \varGamma^{3}_{31}=\frac{1}{2} \frac{C^{\prime}}{C}. \end{aligned} \end{aligned}$$

(91)

These induce the following forms of Ricci tensor:

$$\begin{aligned} &{R_{00}=\frac{1}{4}\frac{g^{2}}{f^{2}}\frac{A^{\prime}B^{\prime}}{B}+ \frac{1}{4}\frac{\dot{A}\dot{B}}{AB}+\frac{1}{4}\frac{g^{2}}{f^{2}} \frac{A^{\prime}C^{\prime}}{C}+\frac{1}{4}\frac{\dot{A}\dot{C}}{AC}} \\ &{\phantom{R_{00}=}{}+\frac{1}{2} \frac{g^{2}}{f^{2}}A^{\prime\prime}-\frac{1}{4}\frac{g^{2}}{f^{2}}\frac{A^{\prime 2}}{A}-\frac{1}{2} \frac{\ddot{B}}{B}+\frac{1}{4}\frac{\dot{B^{2}}}{B^{2}}} \\ &{\phantom{R_{00}=}{}-\frac{1}{2} \frac{\ddot{C}}{C}+\frac{1}{4}\frac{\dot{C^{2}}}{C^{2}},} \end{aligned}$$

(92)

$$\begin{aligned} &{R_{10}=\frac{1}{4}\frac{A^{\prime}B^{\prime}}{AB}+\frac{1}{4} \frac{A^{\prime}\dot{C}}{AC}-\frac{1}{2}\frac{\dot{B}^{\prime}}{B}+\frac{1}{4} \frac{B^{\prime}\dot{B}}{B^{2}}-\frac{1}{2}\frac{\dot{C}^{\prime}}{C}} \\ &{\phantom{R_{00}=}{}+\frac{1}{4} \frac{C^{\prime}\dot{C}}{C^{2}},} \end{aligned}$$

(93)

$$\begin{aligned} &{R_{11}=-\frac{1}{2}\frac{A^{\prime\prime}}{A}+\frac{1}{4} \frac{A^{\prime 2}}{A^{2}}-\frac{1}{2}\frac{B^{\prime\prime}}{B}+\frac{1}{4} \frac{B^{\prime 2}}{B^{2}}-\frac{1}{2}\frac{C^{\prime\prime}}{C}} \\ &{\phantom{R_{00}=}{}+\frac{1}{4} \frac{C^{\prime 2}}{C^{2}},} \end{aligned}$$

(94)

$$\begin{aligned} &{R_{22}=-\frac{1}{4}\frac{A^{\prime}B^{\prime}}{A}-\frac{1}{4} \frac{f^{2}}{g^{2}}\frac{\dot{A}\dot{B}}{A^{2}}+\frac{1}{4}\frac{f^{2}}{g^{2}} \frac{\dot{B}\dot{C}}{AC}+\frac{1}{2}\frac{f^{2}}{g^{2}}\frac{\ddot{B}}{A}} \\ &{\phantom{R_{00}=}{}- \frac{1}{4}\frac{f^{2}}{g^{2}}\frac{\dot{B}^{2}}{AB}-\frac{1}{4} \frac{B^{\prime}C^{\prime}}{C}+\frac{1}{4}\frac{B^{\prime 2}}{B}-\frac{1}{2}B^{\prime\prime},} \end{aligned}$$

(95)

$$\begin{aligned} &{R_{33}=-\frac{A^{\prime} C^{\prime}}{4 A}-\frac{f^{2} \dot{A}\dot{C}}{4 A^{2} g^{2}}+\frac{f^{2} \dot{B}\dot{C}}{4 A B g^{2}}+ \frac{f^{2} \ddot{C}}{2 A g^{2}}-\frac{f^{2} \dot{C}^{2}}{4 A C g^{2}}} \\ &{\phantom{R_{00}=}{}-\frac {B^{\prime} C^{\prime}}{4 B}+\frac{C^{\prime 2}}{4 C}- \frac{1}{2} C^{\prime\prime}.} \end{aligned}$$

(96)

The Ricci scalar is this case is given by

$$\begin{aligned} R =&\frac{g^{2}}{2}\frac{A^{\prime}B^{\prime}}{AB}+\frac{g^{2}}{2} \frac{A^{\prime}C^{\prime}}{AC}+g^{2}\frac{A^{\prime\prime}}{A}+\frac{f^{2}}{2} \frac{\dot{A}\dot{B}}{A^{2}B}+\frac{f^{2}}{2}\frac{\dot{A}\dot {C}}{A^{2}C} \\ &{}-\frac{g^{2}}{2} \frac{A^{\prime 2}}{A^{2}}-\frac{f^{2}}{2}\frac{\dot{B}\dot{C}}{ABC}-f^{2} \frac{\ddot{B}}{AB}+\frac{f^{2}}{2}\frac{\dot{B}^{2}}{AB^{2}}-f^{2} \frac{\ddot{C}}{AC} \\ &{}+\frac{f^{2}}{2}\frac{\dot{C}^{2}}{AC^{2}}+\frac{g^{2}}{2} \frac{B^{\prime}C^{\prime}}{BC}+g^{2}\frac{B^{\prime\prime}}{B}-\frac{g^{2}}{2} \frac{B^{\prime 2}}{B^{2}} \\ &{}+g^{2}\frac{C^{\prime\prime}}{C}-\frac{g^{2}}{2} \frac{C^{\prime 2}}{C^{2}}. \end{aligned}$$

(97)