Kerr–Newman–de–Sitter metric

Null coordinates

To get ${\displaystyle g_{\mathrm{r}\mathrm{r}}=0}$ coordinates we apply the transformation

${\displaystyle \mathrm{d}\mathrm{t}=\mathrm{d}\mathrm{u}-\frac{\mathrm{d}\mathrm{r}\left({\mathrm{a}}^2\mathrm{\Lambda }/3+1\right)\left({\mathrm{a}}^2+{\mathrm{r}}^2\right)}{\left({\mathrm{a}}^2+{\mathrm{r}}^2\right)\left(1-\mathrm{\Lambda }{\mathrm{r}}^2/3\right)-2\mathrm{r}+{\mho }^2}}$

${\displaystyle \mathrm{d}\varphi =\mathrm{d}\phi -\frac{\mathrm{a}\mathrm{d}\mathrm{r}\left({\mathrm{a}}^2\mathrm{\Lambda }/3+1\right)}{\left({\mathrm{a}}^2+{\mathrm{r}}^2\right)\left(1-\mathrm{\Lambda }{\mathrm{r}}^2/3\right)-2\mathrm{r}+{\mho }^2}}$

and get the metric coefficients

${\displaystyle g_{\mathrm{u}\mathrm{r}}=-\frac{3}{{\mathrm{a}}^2\mathrm{\Lambda }+3}}$

${\displaystyle g_{\mathrm{r}\phi }=\frac{3\mathrm{a}{\sin }^2\theta }{{\mathrm{a}}^2\mathrm{\Lambda }+3}}$

${\displaystyle g_{\mathrm{u}\mathrm{u}}=g_{\mathrm{t}\mathrm{t}},g_{\theta \theta }=g_{\theta \theta },g_{\phi \phi }=g_{\varphi \varphi },g_{\mathrm{u}\phi }=g_{\mathrm{t}\varphi }}$

and all the other ${\displaystyle g_{\mu \nu }=0}$, with the electromagnetic vector potential

${\displaystyle {\mathrm{A}}_{\mu }=\left\{\frac{3\mathrm{r}\mho }{\left({\mathrm{a}}^2\mathrm{\Lambda }+3\right)\left({\mathrm{a}}^2{\cos }^2\theta +{\mathrm{r}}^2\right)},\frac{3\mathrm{r}\mho }{{\mathrm{a}}^2\left(\mathrm{\Lambda }{\mathrm{r}}^2-3\right)+6\mathrm{r}+\mathrm{\Lambda }{\mathrm{r}}^4-3\left({\mathrm{r}}^2+{\mho }^2\right)},0,-\frac{3\mathrm{a}\mathrm{r}\mho {\sin }^2\theta }{\left({\mathrm{a}}^2\mathrm{\Lambda }+3\right)\left({\mathrm{a}}^2{\cos }^2\theta +{\mathrm{r}}^2\right)}\right\}}$

Defining ${\displaystyle \overline{\mathrm{t}}=\mathrm{u}-\mathrm{r}}$ ingoing lightlike worldlines give a ${\displaystyle {45}^{\circ }}$ light cone on a ${\displaystyle \{\overline{\mathrm{t}},\mathrm{r}\}}$ spacetime diagram.

Horizons and ergospheres

The horizons are at ${\displaystyle g^{\mathrm{r}\mathrm{r}}=0}$ and the ergospheres at ${\displaystyle g_{\mathrm{t}\mathrm{t}}||g_{\mathrm{u}\mathrm{u}}=0}$. This can be solved numerically or analytically. Like in the Kerr and Kerr–Newman metrics, the horizons have constant Boyer–Lindquist ${\displaystyle \mathrm{r}}$, while the ergospheres' radii also depend on the polar angle ${\displaystyle \theta }$.

This gives 3 positive solutions each (including the black hole's inner and outer horizons and ergospheres as well as the cosmic ones) and a negative solution for the space at ${\displaystyle \mathrm{r}<0}$ in the antiverse behind the ring singularity, which is part of the probably unphysical extended solution of the metric.

With a negative ${\displaystyle \mathrm{\Lambda }}$ (the anti–de–Sitter variant with an attractive cosmological constant), there are no cosmic horizon and ergosphere, only the black hole-related ones.

In the Nariai limit the black hole's outer horizon and ergosphere coincide with the cosmic ones (in the Schwarzschild–de–Sitter metric to which the KNdS reduces with ${\displaystyle \mathrm{a}=\mho =0}$ that would be the case when ${\displaystyle \mathrm{\Lambda }=1/9}$).

Invariants

The Ricci scalar for the KNdS metric is ${\displaystyle \mathrm{R}=-4\mathrm{\Lambda }}$, and the Kretschmann scalar is

${\displaystyle \mathrm{K}=\{220{\mathrm{a}}^{12}{\mathrm{\Lambda }}^2\cos (6\theta )+66{\mathrm{a}}^{12}{\mathrm{\Lambda }}^2\cos (8\theta )+12{\mathrm{a}}^{12}{\mathrm{\Lambda }}^2\cos (10\theta )+{\mathrm{a}}^{12}{\mathrm{\Lambda }}^2\cos (12\theta )+}$

${\displaystyle 462{\mathrm{a}}^{12}{\mathrm{\Lambda }}^2+1080{\mathrm{a}}^{10}{\mathrm{\Lambda }}^2{\mathrm{r}}^2\cos (6\theta )+240{\mathrm{a}}^{10}{\mathrm{\Lambda }}^2{\mathrm{r}}^2\cos (8\theta )+24{\mathrm{a}}^{10}{\mathrm{\Lambda }}^2{\mathrm{r}}^2\cos (10\theta )+}$

${\displaystyle 3024{\mathrm{a}}^{10}{\mathrm{\Lambda }}^2{\mathrm{r}}^2+1920{\mathrm{a}}^8{\mathrm{\Lambda }}^2{\mathrm{r}}^4\cos (6\theta )+240{\mathrm{a}}^8{\mathrm{\Lambda }}^2{\mathrm{r}}^4\cos (8\theta )+8400{\mathrm{a}}^8{\mathrm{\Lambda }}^2{\mathrm{r}}^4-}$

${\displaystyle 1152{\mathrm{a}}^6\cos (6\theta )-11520{\mathrm{a}}^6+1280{\mathrm{a}}^6{\mathrm{\Lambda }}^2{\mathrm{r}}^6\cos (6\theta )+12800{\mathrm{a}}^6{\mathrm{\Lambda }}^2{\mathrm{r}}^6+207360{\mathrm{a}}^4{\mathrm{r}}^2-}$

${\displaystyle 138240{\mathrm{a}}^4\mathrm{r}{\mho }^2+11520{\mathrm{a}}^4{\mathrm{\Lambda }}^2{\mathrm{r}}^8+16128{\mathrm{a}}^4{\mho }^4-276480{\mathrm{a}}^2{\mathrm{r}}^4+368640{\mathrm{a}}^2{\mathrm{r}}^3{\mho }^2+}$

${\displaystyle 6144{\mathrm{a}}^2{\mathrm{\Lambda }}^2{\mathrm{r}}^{10}-104448{\mathrm{a}}^2{\mathrm{r}}^2{\mho }^4+3{\mathrm{a}}^4\cos (4\theta )[165{\mathrm{a}}^8{\mathrm{\Lambda }}^2+960{\mathrm{a}}^6{\mathrm{\Lambda }}^2{\mathrm{r}}^2+2240{\mathrm{a}}^4{\mathrm{\Lambda }}^2{\mathrm{r}}^4-}$

${\displaystyle 256{\mathrm{a}}^2(9-10{\mathrm{\Lambda }}^2{\mathrm{r}}^6)+256(90{\mathrm{r}}^2-60\mathrm{r}{\mho }^2+5{\mathrm{\Lambda }}^2{\mathrm{r}}^8+7{\mho }^4)]+24{\mathrm{a}}^2\cos (2\theta )[33{\mathrm{a}}^{10}{\mathrm{\Lambda }}^2+}$

${\displaystyle 210{\mathrm{a}}^8{\mathrm{\Lambda }}^2{\mathrm{r}}^2+560{\mathrm{a}}^6{\mathrm{\Lambda }}^2{\mathrm{r}}^4-80{\mathrm{a}}^4(9-10{\mathrm{\Lambda }}^2{\mathrm{r}}^6)+128{\mathrm{a}}^2(90{\mathrm{r}}^2-60\mathrm{r}{\mho }^2+5{\mathrm{\Lambda }}^2{\mathrm{r}}^8+}$

${\displaystyle 7{\mho }^4)+256{\mathrm{r}}^2(-45{\mathrm{r}}^2+60\mathrm{r}{\mho }^2+{\mathrm{\Lambda }}^2{\mathrm{r}}^8-17{\mho }^4)]+36864{\mathrm{r}}^6-73728{\mathrm{r}}^5{\mho }^2+}$

${\displaystyle 2048{\mathrm{\Lambda }}^2{\mathrm{r}}^{12}+43008{\mathrm{r}}^4{\mho }^4\}÷\{12[{\mathrm{a}}^2\cos (2\theta )+{\mathrm{a}}^2+2{\mathrm{r}}^2{]}^6\}\text{.}}$