twist-free, see [@Stephanietal:2003; @GriffithsPodolsky:2009]. In fact, it is a subclass of Kundt spacetimes which is the *direct-product of two 2-spaces*, and is of Weyl algebraic type D and Ricci type II [@GriffithsPodolsky:2009; @PravdaPravdovaPodolskySvarc:2017]. The first part of [[[d]{}]{}]{}s\^2\_=[[[d]{}]{}]{}\^2+\^2[[[d]{}]{}]{}\^2 -2[[[d]{}]{}]{}u[[[d]{}]{}]{}r+[H]{}(r)[[[d]{}]{}]{}u\^2 \[Kundt seed\] spanned by ${\theta, \phi}$ is a round 2-sphere of Gaussian curvature ${K=1}$, while the second part spanned by ${u, r}$ is a 2-dim Lorentzian spacetime. With the usual stereographic representation of a 2-sphere given by ${x+\hbox{i}\, y = 2\tan(\theta/2)\exp(\hbox{i}\phi)}$, this *Kundt seed* metric can be rewritten as [[[d]{}]{}]{}s\^2\_= -2[[[d]{}]{}]{}u[[[d]{}]{}]{}r+[H]{}(r)[[[d]{}]{}]{}u\^2 . \[Kundt seed xy\]

The black hole horizon {#BH horizon}
----------------------

In the usual metric form (\[Einstein-WeylBH\]), the Schwarzschild horizon is defined by the zeros of the same two metric functions $h{({\bar r})=f({\bar r})}$. Due to (\[SchwarzschildBH\]), it is located at ${{\bar r}_h=2m}$, where $m$ denotes the total mass of the black hole.

In a general case, such a horizon can be defined as the *Killing horizon* associated with the vector field ${\partial_t}$. Its norm is determined by the metric function $-h({\bar r})$. In the regions where ${h({\bar r})>0}$, the spacetime is static and $t$ is the corresponding temporal coordinate. The Killing horizon is generated by the *null vector field* 