Matched Asymptotic Expansion for Caged Black Holes - Regularization of the Post-Newtonian Order

TL;DR

The paper develops a Hadamard-regularized, two-zone matched asymptotic framework to extend black-hole in a compact dimension analyses to Post-Newtonian order in arbitrary dimensions. It identifies and regularizes divergences at PN order, derives the PN corrections to mass and tension, and demonstrates a Newtonian basis for leading tension. The results align with previous leading-order work and numerical phase-diagram data, while providing evidence for a second-order transition at high dimensions and hints toward Sorkin's critical dimension. The work deepens understanding of small black holes in compactified spaces and sharpens the connection between regularization, self-interaction subtraction, and phase-structure predictions.

Abstract

The "dialogue of multipoles" matched asymptotic expansion for small black holes in the presence of compact dimensions is extended to the Post-Newtonian order for arbitrary dimensions. Divergences are identified and are regularized through the matching constants, a method valid to all orders and known as Hadamard's partie finie. It is closely related to "subtraction of self-interaction" and shows similarities with the regularization of quantum field theories. The black hole's mass and tension (and the "black hole Archimedes effect") are obtained explicitly at this order, and a Newtonian derivation for the leading term in the tension is demonstrated. Implications for the phase diagram are analyzed, finding agreement with numerical results and extrapolation shows hints for Sorkin's critical dimension - a dimension where the transition turns second order.

Figures (4)

• Figure 1: Illustration of the $(r,z)$ "cylindrical" coordinates and the $(\rho,\chi)$ "spherical" near horizon coordinates. The period of the compact dimension (in the $z$ direction) is denoted by $L$ and the radius of the black hole is denoted by $\rho_{0}$.
• Figure 3: The first steps in the matching procedure. Each box in the top row denotes a specific order in the asymptotic zone, and the lower row depicts the near zone. Arrows denote the flow of matching information between the zones: Dark (red) arrows denote monopole matching ($l=0$), light arrows (light-blue) denote the quadrupole ($l=2$), and higher $l$ arrows are not shown.
• Figure 4: The analytic linear approximation line (red) compared to the numerical data points (green) of KudohWiseman2 for the complete phase diagram in $d=6$ consisting of two branches: the black hole and the non-uniform black string. The vertical axis is proportional to the mass, and the horizontal axis to the tension (see text for exact definitions). For small black holes (lower right) we see excellent agreement. By extrapolation, the thick (blue) point defines $\mu_X$, the intersection point of the extrapolated linear approximation with the line of uniform black strings, $n=n_{st}$.
• Figure 5: The ratio $\mu_X/\mu_{crit}$ of extrapolated vs. actual Gregory-Laflamme points for various dimensions. For large $d$ the ratio tends to zero strongly. We interpret that as an indication for a higher (second) order phase transition, and thus as an indication for Sorkin's critical dimension $D^*="13.5"$. Interestingly, our graph shows a maximum around $12 \le d \le 14$ which coincides with $D^*$.