부산대학교 도서관

This paper is part of a series of works [see M.-N. Célérier, Phys. Rev. D {\bf 104} (2021), no.~6, Paper No. 064040; MR4334571; J. Math. Phys. {\bf 64} (2023), no.~3, Paper No. 032501; MR4557615; J. Math. Phys. {\bf 64} (2023), no.~4, Paper No. 042501; MR4574847; J. Math. Phys. {\bf 64} (2023), no.~5, Paper No. 052502; MR4585816; ``Study of stationary rigidly rotating anisotropic cylindrical fluids with new exact interior solutions of GR. 5. Dust limit and discussion'', preprint, \arx{2209.05060}] dealing with the subject of exact interior solutions of Einstein's equations of General Relativity in the presence of a rigidly rotating, cylindrically symmetric, perfect fluid, and their properties. In this work, a class of exact solutions of this form is derived for a fluid exhibiting one nonzero pressure component. \par The basic setting is the following. The spacetime is assumed to be stationary and cylindrically symmetric, described by the line element $$ ds^2 = g_{\alpha\beta} dx^\alpha dx^\beta= - f(r) t^2 +2 k(r) dt d\phi +e^{\mu(r)} (dr^2 +dz^2) + l(r) d\phi^2 , $$ where the domain of each of the variables is $$ t,z \in \Bbb{R}, \quad r\geq 0 \quad \text{and} \quad \phi \in [0,2\pi]. $$ The energy momentum tensor for the perfect fluid is given by $$ T_{\alpha \beta} = (\rho(r)+P(r)) V_\alpha V_\beta + P(r) g_{\alpha\beta} , $$ with the four-velocity $V^\alpha$ being of course a time-like vector, $V^\alpha V_{\alpha}=-1$. \par The integration of Einstein's field equations is achieved by introducing two new functions defined as $D(r)^2=f(r) l(r)+k(r)^2$ and $h(r)=\frac{P(r)}{\rho(r)}$. By manipulating the equations of motion, one obtains a relation which can be satisfied by setting to zero one of two multiplicative factors. By setting to zero one of them, the exact solution studied in this paper emerges. \par In the final solution, all the involved functions are expressed as functions of $h(r)$, which by itself satisfies a particular ordinary differential equation. This means that the equation of state for this solution is fixed by the field equations. The resulting equation of state has a term corresponding to a radiation fluid plus another, more complicated term, which signals the departure from the radiation matter content. \par A further analysis follows, where specific relations among the integration constants of the solution are imposed. This is achieved by considering physical arguments and the symmetries of the metric. For example, this includes restrictions so that the metric is real and of Lorentzian signature. The constraints implied by the junction condition as well as the weak and strong energy conditions are also presented. In addition to the above there is also a discussion on the regularity conditions and their implications on constraining the integration constants. These are imposed in order to avoid singularities on the axis of the cylindrical symmetry, although as also discussed these conditions do not necessarily ensure the smoothness of the manifold on the axis. \par A comparison with previous results in the literature is also provided.