Black holes in p-adic gravity/ultra-metric metric field? As a radically different to beyond standard general relativity, at least from the type of geometry it deals with:


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*Consider p-adic gravity and/or general relativity defined on certain ultra-metric spacetime where we have "a metric". 

*Consider the concept of black hole solution (I have no reference of any ansatz of this typer for these theories), if any!
Question: do black hole solutions exist, in the classical set-up (i.e.vacuum solutions) for p-adic gravity and/or general relativity formulated on p-adic numbers or an ultrametric (likely adelic) field on which we define "a metric"? 
(Note: I am well aware maybe I can NOT define a standard metric on a p-adic or adelic sense).
 A: One of the simpler (from a mathematical perspective) black holes is the BTZ black hole which is the solution of a 3D gravity with negative cosmological constant. Einstein equations ensure that spacetime is locally an $\text{AdS}_3$  space and this solution could be seen as a factor of $\text{AdS}_3$  space by a discrete group. So, the p-adic generalization of the BTZ black hole would be a p-adic version of $\text{AdS}_3$  factored by some discrete group. Gravity as such does not  enter this reasoning. Such realization of p-adic BTZ black hole has been constructed by  Matthew Heydeman, Matilde Marcolli and others in [1,2].
Here is a picture of a 3-adic BTZ black hole from [ 1]:
 
Some explanation. p-adic version of $\text{AdS}_3$ space is represented by the  Bruhat–Tits tree for $\mathrm{PGL}(2,\mathbb{Q}_p)$, the infinite tree of uniform valence $p+1$, such as this one (here $p=2$):

Factoring this tree by a discrete Schottky group means that there would be different points of the tree that we must identify, thus forming cycles. And such a cycle means that we now have the black hole. 
While such a construct may seem very abstract, it is straightforward to think about realization of Bruhat–Tits tree on a hyperbolic space $H_2$ as a spatial slice of the $\text{AdS}_3$  space, and the trees growing from a cycle (as in the first image) would be a spatial slice of a BTZ black hole spacetime.
Original papers:


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*Matthew Heydeman, Matilde Marcolli, Ingmar Saberi, Bogdan Stoica, Tensor networks, p-adic fields, and algebraic curves: arithmetic and the $\text{AdS}_3/\text{CFT}_2$  correspondence, arXiv:1605.07639.

*Heydeman, M., Marcolli, M., Parikh, S., & Saberi, I. Nonarchimedean Holographic Entropy from Networks of Perfect Tensors, arXiv:1812.04057
Talks, lectures by the authors:


*Talk “Non-Archimedean Holography” by M. Marcolli: Slides, Video here.

*Talk “AdS/CFT and p-Adic Numbers: A Model of Discrete Holography” by M. Heydeman: Slides, Video.
