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Probing Penrose-type singularities in sonic black holes

Satadal Datta, Uwe R. Fischer

TL;DR

This work uses a $2+1$D analogue black hole (ABH) realized in a polytropic, inviscid, irrotational axisymmetric flow to probe Penrose-type singularities in a non-Einsteinian setting, addressing whether such singularities require Einstein gravity. It develops a rigorous framework to locate the analogue event horizon via $g^{rr}=0$ and horizon conditions, analyzes null geodesics and their affine-parameter behavior inside the trapped region, and demonstrates a physically realized mechanism to avoid the singularity through a finite-radius drain in the laboratory. The authors prove key results (Theorems 1–4) about the maximal physical flow domain, the necessity of a diverging external potential for steady nonzero flux, a relation between horizon radii, and a non-Einsteinian singularity theorem based on the Raychaudhuri equation, illustrating that Penrose-type singularities can occur without Einstein equations. Collectively, the work provides a concrete, testable platform to study foundational questions about spacetime singularities, their avoidance, and the possible influence of quantum corrections in gravity.

Abstract

Addressing the general question whether Penrose singularities physically exist inside black holes, we investigate the problem in the context of an analogue system, a flowing laboratory liquid, for which the governing equations are at least in principle known to all relevant scales, and in all regions of the effective spacetime. We suggest to probe the physical phenomena taking place close to the singularity in the interior of a $2+1$D analogue black hole arising from a polytropic, inviscid, irrotational, and axisymmetric steady flow, and propose to this end an experimental setup in a Bose-Einstein condensate. Our study provides concrete evidence, for a well understood dynamical system, that the Einstein equations are not necessary for a singularity to form, demonstrating that Penrose-type spacetime singularities can potentially also exist in non-Einsteinian theories of gravity. Finally, we demonstrate how the singularity is physically avoided in our proposed laboratory setup.

Probing Penrose-type singularities in sonic black holes

TL;DR

This work uses a D analogue black hole (ABH) realized in a polytropic, inviscid, irrotational axisymmetric flow to probe Penrose-type singularities in a non-Einsteinian setting, addressing whether such singularities require Einstein gravity. It develops a rigorous framework to locate the analogue event horizon via and horizon conditions, analyzes null geodesics and their affine-parameter behavior inside the trapped region, and demonstrates a physically realized mechanism to avoid the singularity through a finite-radius drain in the laboratory. The authors prove key results (Theorems 1–4) about the maximal physical flow domain, the necessity of a diverging external potential for steady nonzero flux, a relation between horizon radii, and a non-Einsteinian singularity theorem based on the Raychaudhuri equation, illustrating that Penrose-type singularities can occur without Einstein equations. Collectively, the work provides a concrete, testable platform to study foundational questions about spacetime singularities, their avoidance, and the possible influence of quantum corrections in gravity.

Abstract

Addressing the general question whether Penrose singularities physically exist inside black holes, we investigate the problem in the context of an analogue system, a flowing laboratory liquid, for which the governing equations are at least in principle known to all relevant scales, and in all regions of the effective spacetime. We suggest to probe the physical phenomena taking place close to the singularity in the interior of a D analogue black hole arising from a polytropic, inviscid, irrotational, and axisymmetric steady flow, and propose to this end an experimental setup in a Bose-Einstein condensate. Our study provides concrete evidence, for a well understood dynamical system, that the Einstein equations are not necessary for a singularity to form, demonstrating that Penrose-type spacetime singularities can potentially also exist in non-Einsteinian theories of gravity. Finally, we demonstrate how the singularity is physically avoided in our proposed laboratory setup.
Paper Structure (7 sections, 4 theorems, 43 equations, 4 figures)

This paper contains 7 sections, 4 theorems, 43 equations, 4 figures.

Table of Contents

  1. Supplemental Material
  2. Proof of main text theorems
  3. Hypersurfaces in a $2+1$D ABH geometry
  4. Calculating the location of the analogue black hole event horizon and Mach number profile
  5. Null geodesics traveling away from the trapped surface
  6. Null expansion for radial null geodesics inside trapped surface
  7. Singularity theorem in non-Einsteinian gravity

Key Result

Theorem 1

Given an IVP at $r=r_0$ with finite $\rho_{0}(r_0)>0$, and finite $v^r_{0}(r_0)$ and $v^\phi_{0}(r_0)$, the maximal domain in $r$ of an axisymmetric steady physical flow $M_P$, with nonzero mass flux rate, is the region where the functions $f(c_{s0}, v^r_{0}, r)$ and $g(c_{s0}, v^r_{0},r)$ exist and

Figures (4)

  • Figure 1: Proposed setup for axisymmetric purely radial BEC flow creating an analogue black hole, BEC atoms are strongly confined in the $z$ direction (quasi-2D geometry). A steady current along $z$ creates an azimuthal magnetic field $\vec{B}$ according to Ampère's law, leading to an external potential for atoms with magnetic moment flowing radially inward. A coherent source at the boundary and a drain at the center (outcoupling the atoms BlochcwlaserO) maintains a steady flow.
  • Figure 2: Singularity in our ABH from a transonic flow with $C_1=1$, $C_2=0.5$. (Left) The null expansion $\Theta$ starts from a negative value from the trapped surface at $r=0.6$ and it reaches minus infinity (see suppl) when $r$ goes to zero. (Right) Affine parameter interval (see for further discussion in main text) between two given radii for radially in- and outward pointing null geodesics is the sea green area under the $c_{s0}^3 (r)$ curve (solid green curve). The affine interval is bounded for such geodesics traveling from the trapped surface to arbitrarily close to the origin, because $c_{s0}$ decays even less fast than $r^{-1/4}$ in the supersonic flow region, cf. Eq. \ref{['affine']}.
  • Figure 3: Causal structure of the ABH akin to the Penrose construction Penrose65PRL. Parameters as in Fig. \ref{['fig1']}. Outward and inward pointing null geodesics in red and blue, respectively. The horizon, denoted by $H-H'$, forms a cylindrical hypersurface (in our units located at $r=1$), and the singularity ($S$) is at $r=0$. At $t=0$, a trapped surface $T^1$ is a one-dimensional circle $S^1$, here of radius 0.6, and the three dimensional $F^3$ is the future region occupied by smooth timelike curves from $T^1$, with the two-dimensional null boundary $B^2$, a closed hypersurface. Atoms escape from the $x-y$ plane through the drain around $r=0$, cf. Fig. \ref{['fsetup']}.
  • Figure S1: Steady radial flow transonic solution. Solid green lines correspond to an ABH for radial inflow. Mach number vs $r$; with a fixed $C_1$ and $C_2$, we get two transonic solutions. For a BEC with $V_{\rm ext}(r)=-\frac{1}{r}$ and $C_2=0.5$, we have $r_H=1$, $c_{sH}=1$

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • proof
  • proof
  • proof
  • Corollary 2
  • proof
  • proof