Interior solution for the Kerr metric

A recently presented general procedure to find static and axially symmetric, interior solutions to the Einstein equations is extended to the stationary case, and applied to find an interior solution for the Kerr metric. The solution, which is generated by an anisotropic fluid, verifies the energy conditions for a wide range of values of the parameters, and matches smoothly to the Kerr solution, thereby representing a globally regular model describing a nonspherical and rotating source of gravitational field. In the spherically symmetric limit, our model converges to the well-known incompressible perfect fluid solution. The key stone of our approach is based on an ansatz allowing to define the interior metric in terms of the exterior metric functions evaluated at the boundary source. The physical variables of the energy-momentum tensor are calculated explicitly, as well as the geometry of the source in terms of the relativistic multipole moments.

Figure 1

Functions $\exp (\widehat{g}(s,y=0)-\widehat{a}(s,y=0))$ and $\exp (-\widehat{a}(s,y=1))$ for values of the rotation parameter $j=\pm 0.1$ and $\tau =2.7$.

Figure 2

Relation between the ellipticity of the source $e$ and its rotation parameter $j$ for different values of $\tau$.

Figure 3

Four profiles of $r_{\mathrm{\Sigma }}^2{P}_{rr}\equiv r_{\mathrm{\Sigma }}^2{g}_{rr}{T}_1^1$, as a function of $s$, for $y=0.3$ (a), and $y=1$ (b) with $\tau =2.7$, and different values of $j$.

Figure 4

$-r_{\mathrm{\Sigma }}^2{T}_0^0$ (a), and $r_{\mathrm{\Sigma }}^2{T}_1^1$ (b), as functions of $y=\cos \theta$ and $s$, with $j=0.1$ and $\tau =2.7$.

Figure 5

(a) $r_{\mathrm{\Sigma }}^2{T}_3^3$ and (b) $r_{\mathrm{\Sigma }}^3{T}_1^2$ as functions of $y$ and $s$.

Figure 6

$r_{\mathrm{\Sigma }}^2[(-T_0^0)-T_1^1]$ as a function of $y$ and $s$, with $j=0.1$ and $\tau =2.7$.

Figure 7

$r_{\mathrm{\Sigma }}^3{T}_0^3$ as a function of $y$ and $s$, with $j=0.1$ and $\tau =2.7$.