Geometry of Quark and Strange Quark Matter in Higher Dimensional General Relativity

Abstract

In this paper we study quark matter and strange quark matter in higher-dimensional spherical symmetric space-times. We analyze strange quark matter for the different equations of state and obtain the space-time geometry of quark and strange quark matter. We also discuss the features of the obtained solutions in the context of higher-dimensional general theory of relativity.

Appendix

For the line element (5) the non zero component of $R_{jij}^{i}$ is

$$ R_{jij}^{i} = (1 + e^{-2\nu}\dot{R^{2}} - e^{-2\varphi}R'^{2}),$$

(42)

where i=3,4,5,…,n j=2,3,4,…,n−1 and i>j.

Examination of this expression reveals that if R=r our line element (5) reduces to

$$ ds^{2} = e^{-2\nu}dt^{2} - \frac{dr^{2}}{1 - R_{jij}^{i}}- r^{2}d\Omega^{2}.$$

(43)

If curvature coordinates used in empty region then spherically symmetric five dimensions (5D) Schwarzschild’s like metric is given by

$$ ds^{2} = \biggl(1 - \frac{A}{r^{2}}\biggr)dt^{2} - \frac{dr^{2}}{(1 -\frac{A}{r^{2}})} - r^{2}d\Omega^{2},$$

(44)

where A is a constant and from Newtonian approximation A=M. Comparison of (43) and (44) gives

$$ (R_{jij}^{i})_{b} = \frac{M}{r_{b}^{2}},$$

(45)

where the subscript b means evaluation at the boundary. We can, at this stage, follow the arguments of Cahill and McVittie [21] in defining intuitively a mass function in 5D as

$$ m(r,t) = R^{2}(1 + e^{-2\nu}\dot{R^{2}} - e^{-2\varphi}R'^{2}).$$

(46)