We consider here a Kretschmann gravity with a Lagrangian density with the form
L=K−aRuvRuv
where K is the Kretschmann invariant and a is constant coupling..
We look for a metric with the form
ds2=f(r)dt2−dr2f(r)−r2dθ2−r2sin2(θ)dϕ2
For this metric we have
K=4−8f(r)+4(f(r))2+4(ddrf(r))2r2+(d2dr2f(r))2r4r4
and
RuvRuv=(d2dr2f(r))2r4+4r3(d2dr2f(r))ddrf(r)+8(ddrf(r))2r2−8rddrf(r)+8f(r)rddrf(r)+4−8f(r)+4(f(r))22r4
The corresponding Euler-Lagrange equation is given by
−4r2(−1+α)d2dr2f(r)+4r3(−2+α)d3dr3f(r)+r4(−2+α)d4dr4f(r)+8(f(r)−1)(−1+α)=0
and the solution has the form
f(r)=1+Ar+Br2+Crn+Erm
where
n=−2+α+√36−52α+17α22(−2+α)
m=−2−α+√36−52α+17α22(−2+α(
In the particular case when a=4 we obtain
f(r)=1+Ar+Br2+Cr3+Er2
it is to say we obtain a Reissner–Nordström -de Siiter black hole when
A=−2M
B=Λ3
E=Q2
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≠0.?
3. It is possible to have a physical solution with a≠4.?