N-spheres: regular black holes without apparent horizons, static wormholes with event horizons and gravastars with a tube-like core O. B. Zaslavskii, Department of Mechanics and Mathematics, Kharkov V. N. Karazin National University

Alternative: under BH horizon - singular or regular centre? Third option: tube like geometry, everywhere regular Regular BH classical model of elementary particle: Lorentz electron Outside: black hole. Inside: tube-like core Poincare-Lorentz electron Pure electromagnetic forces Interior: Bertotti-Robinson (BR) metric Exterior – Reissner-Nordström (RN).

Matching 2 regions to replace “bad” siingular inner region by “good” one Appearance of surface stresses: shell at is Lanczos tensor. BR metric a)extremal limit of non-extremal RN BH (O..Z., PRL 1996, PRD 1997) b)no horizon c) extremal Smooth gluing: m(outside)=m(inside) q (outside)=q(inside)

For smooth gluing is impossible but becomes possible in horizon liomit in case c) only, RN is extremal: BR as limiting form of RN throat In horizon limit no bare sources. Wheeler’s idea: mass without mass, charge without charge Mass defect: for external observer m=e (extremal RN) Proper mass Classical analogue of electron in GR: extremal charged black hole with infinitely long tube inside (O. Z. PRD 2004)

Geometry:sphere of constant radius R-region T-region N-region Generalization of RN – BR gluing inside For an external observer: object of finite areal radius and ADM mass Inside: tube of constant radius and infinite proper mass N – sphere (cf. T – sphere, Ruban, 1969)

Behavior of surface stresses. It follows from 00 and 11 Einstein equations for inner (-) region that And 22 equation gives us Vacuum-like 1) a) b) c) 2) 3) a)or Examples. 1) withBR, 2) withNariai 3) string dust

1. Exterior – black hole Goal: smooth gluing (at least, asymtotically) Non-extremal. Types of guing: 1a, 2, 3a. Then in limit 2. Wormholes a)Non-traversable, horizon: BH – N-region – BH, BH – BH Horizon: non-extremal or extremal Even not-traversable WH safe for one-way travel b) Traversable. N-region – WH. Type 1c Extremal. 1b 3. No horizon. Gravastars. Types 1c, 3b

Some properties Example. Case 3a: 2D Rindler x sphere r=const Usually: Rindler coordinates Minlowski infinity Now: r=const, 2D infinity has nothing to do with r-infinity, an observer at r-infinity cannot see what happen inside shell Event (acceleration) hotizon without apparent horizon Black hole: trapped region. Wormhole: antitrapped Now: r=const in both directions, intermediate case

Summary Composite objects interpolating between BH and gravastars: horizon in the limit. Event horizons without apparent ones. Composite objects interpolating between BH and gravastars: horizon in the limit. Event horizons without apparent ones. N-gravastar with infinite tube-like core. N-gravastar with infinite tube-like core. Not-traversable and traversable wormholes with tube inside. Not-traversable and traversable wormholes with tube inside. Mass defect: finite ADM mass but infinite proper mass. Mass defect: finite ADM mass but infinite proper mass. Geometry: N2xS2, where N2=2D Rindler (p=0), AdS (p 0). Geometry: N2xS2, where N2=2D Rindler (p=0), AdS (p 0). Quantum backreaction retains general form of metric. Quantum backreaction retains general form of metric. Black holes with regular core, mass without mass, classical model of elementary particle Black holes with regular core, mass without mass, classical model of elementary particle