Topology-changing horizons at large D as Ricci flows

Abstract

The topology-changing transition between black strings and black holes localized in a Kaluza-Klein circle is investigated in an expansion in the inverse of the number of dimensions D. Performing a new kind of large-D scaling reduces the problem to a Ricci flow of the near-horizon geometry as it varies along the circle direction. The flows of interest here simplify to a non-linear logarithmic diffusion equation, with solutions known in the literature which are interpreted as the smoothed conifold geometries involved in the transition, namely, split and fused cones, which connect to black holes and non-uniform black strings away from the conical region. Our study demonstrates the adaptability of the 1/D expansion to deal with all the regimes and aspects of the static black hole/black string system, and provides another instance of the manner in which the large D limit reduces the task of solving Einstein's equations to a simpler but compelling mathematical problem. arXiv:1905.01062v3 [hep-th] 22 Jul 20191 See [1, 2] for early reviews, and the book monograph [3] for a more up to date view. 2 The evolution of black strings to black holes in solution space seems to be unrelated to the dynamical evolution of black strings towards (and across) a naked singularity. Although the change in topology is expected to be the same in both cases, the dynamical evolution appears to be at all moments strongly time-dependent, and occurs far from the static configurations around the transition in solution space. The singularities also appear to be very different in the two cases.3 Again, these black hole mergers in solution space are rather different than the dynamical mergers of event horizons. The latter admit simple local models studied in [6] and do not involve naked singularities. The evolution in solution space is adiabatic, while the dynamical merger is irreversible.

Black hole collisions, instabilities, and cosmic censorship violation at large D

“…Their interpretation as bars that rotate rigidly with angular velocity Ω becomes clearer using co-rotating Cartesian coordinates 24) in which (2.21) becomes the oblong Gaussian profile…”

Black hole interactions at large D: brane blobology

“…[22], by numerically solving 4NLO effective equation, the zero tension condition admitted D 1/4 dependence on the maximum period for the nonuniform black string. This scaling is a little smaller than L = O( √ D) in which the topology-changing transition was observed in the large D conifold ansatz [10]. The blob approximation with 1/D correction should solve this puzzle.…”

“…Lastly, we consider the limit L ∼ √ D, which gives a neck amplitude of R ∼ −D and the 1/D expansion breaks down there. In this regime, the physical compactification length reaches O (1), in which one can observe the smooth topology-changing transition by taking the proper scaling on the neck [10]. On the other hand, eq.…”

“…For L O( √ D), this scaling hierarchy would smoothly continue to the near-neck structure in the fused conifold[10].…”

“…Though, recent studies[9,10] sheded some light on this sector 2. Blobs are almost solitons except that the collision does not preserve the number of blobs due to the thermalization by the viscosity.…”