Kerr black hole in the formalism of isolated horizons

We revise the work of Scholtz, Flandera, and Gürlebeck [Phys. Rev. D 96, 064024 (2017)]. We cast the Kerr metric explicitly in the form suitable for the framework of isolated horizons. We proceed in a geometrical fashion and are capable to provide the results in a compact closed manner, without any unevaluated integrals. We also discuss the uniqueness and drawbacks of this construction. We suggest a new vector field to generate the null geodesic foliation.

Figure 1

Red lines represent projection of integral lines of $\mathit{n}$ (i.e. lines of constant $\mathrm{\Theta }$) in BL coordinates meanwhile the light blue lines are equipotentials of constant affine parameter $s(r,\theta )$. The thick blue half-circle is black hole horizon $r=r_p$, axis of rotation is vertical. The black line denotes vanishing expansion of the congruence $\mathit{n}$. Calculations are done in BL coordinates, which are later identified with polar coordinates $(R=r\sin \theta ,z=r\cos \theta$).

Figure 2

The integral lines of $\mathit{n}$ and equipotentials of affine parameter in BL coordinates for ${\epsilon }_r=1$ emanating from the horizon with initial conditions given by Eq. (124). The geodesics, i.e. lines of constant $\mathrm{\Theta }$, are slightly bent which can be difficult to observe by naked eye, therefore we included lines of constant $\theta$ as green dashed lines (with half density of lines). The explanation of structures is the same as in Fig. 1.

Figure 3

The value of $\mathrm{\Delta }K\equiv K(r(z,\mathrm{\Theta }),\theta (z,\theta ))-a^2{\sin }^2\theta$ above the horizon (small half-circle in the middle) in BL coordinates. The red line is a rotational axis ($z$), the black line depicts the equatorial plane, in the green direction the function value is plotted. The exact values have no particular meaning. The yellow line is one particular geodesic with $\mathrm{\Theta }=\pi /4$ for which the function is in greater detail in Fig. 4. The blue line is a black hole horizon.

Figure 4

The vertical axis represents the value of $\mathrm{\Delta }K\equiv K(r(z,\mathrm{\Theta }),\theta (z,\theta ))-a^2{\sin }^2\theta$ for one particular geodesic from Fig. 3. The values of the black hole are $r_p=2,r_m=1$ and $\mathrm{\Theta }=\pi /4$. On the horizontal axis is the BL coordinate $r$.

Figure 5

The integral lines of $\mathit{n}$ and equipotentials of affine parameter in BL coordinates for ${\epsilon }_r={\epsilon }_{\theta }=1$ emanating from the horizon. The detailed explanation as in Fig. 1.