# Reissner–Nordström -de Siiter black hole in Kretschmann Gravity

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We consider here a  Kretschmann gravity with a Lagrangian density with the form

$L = K - a R_{u v} R^ {u v}$

where $K$ is the  Kretschmann invariant and $a$ is constant coupling..

We look for  a metric with the form

${{\it ds}}^{2}=f \left( r \right) d{t}^{2}-{\frac {d{r}^{2}}{f \left( r \right) }}-{r}^{2}d{\theta}^{2}-{r}^{2} \sin ^2 \left( \theta \right) d{\phi}^{2}$

For this metric we have

$K={\frac {4-8\,f \left( r \right) +4\, \left( f \left( r \right) \right) ^{2}+4\, \left( {\frac {d}{dr}}f \left( r \right) \right) ^{ 2}{r}^{2}+ \left( {\frac {d^{2}}{d{r}^{2}}}f \left( r \right) \right) ^{2}{r}^{4}}{{r}^{4}}}$

and

$R_{{{\it uv}}}{R}^{{\it uv}}=\,{\frac { \left( {\frac {d^{2}}{d{r}^ {2}}}f \left( r \right) \right) ^{2}{r}^{4}+4\,{r}^{3} \left( {\frac {d^{2}}{d{r}^{2}}}f \left( r \right) \right) {\frac {d}{dr}}f \left( r \right) +8\, \left( {\frac {d}{dr}}f \left( r \right) \right) ^{2}{ r}^{2}-8\,r{\frac {d}{dr}}f \left( r \right) +8\,f \left( r \right) r{ \frac {d}{dr}}f \left( r \right) +4-8\,f \left( r \right) +4\, \left( f \left( r \right) \right) ^{2}}{2{r}^{4}}}$

The corresponding Euler-Lagrange equation is given by

$-4\,{r}^{2} \left( -1+\alpha \right) {\frac {d^{2}}{d{r}^{2}}}f \left( r \right) +4\,{r}^{3} \left( -2+\alpha \right) {\frac {d^{3}}{ d{r}^{3}}}f \left( r \right) +{r}^{4} \left( -2+\alpha \right) {\frac {d^{4}}{d{r}^{4}}}f \left( r \right) \\+8\, \left( f \left( r \right) -1 \right) \left( -1+\alpha \right) =0$

and the solution has the form

$f \left( r \right) =1+{\frac {A}{r}}+B{r}^{2}+C{r}^{n}+E{r}^{m}$

where

$n={\frac {-2+\alpha+\sqrt {36-52\,\alpha+17\,{\alpha}^{2}}}{2(-2+ \alpha)}}$

$m=-\,{\frac {2-\alpha+\sqrt {36-52\,\alpha+17\,{\alpha}^{2}}}{2(-2+ \alpha(}}$

In the particular case when $a = 4$ we obtain

$f \left( r \right) =1+{\frac {A}{r}}+B{r}^{2}+C{r}^{3}+{\frac {E}{{r}^ {2}}}$

it is to say we obtain a  Reissner–Nordström -de Siiter black hole when

$A=-2\,M$

$B={\frac {\Lambda}{3}}$

$E={Q}^{2}$

$C = 0$

Then my questions are:

1.  There is any reference on which this solution can be founded?

2.  It is possible to have a physical solution with $C \neq 0$.?

3.  It is possible to have a physical solution with $a \neq 4$.?

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