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Black holes and black regions, horizons and barriers in Lorentzian manifolds

Cristina Giannotti, Andrea Spiro

TL;DR

This work establishes a general crossing constraint for causal world-lines with time-oriented null hypersurfaces in Lorentzian manifolds, showing that such surfaces act as semi-permeable barriers that separate spacetime into regions with directionally allowed signal flow. It introduces the notion of a dressed barrier (a null hypersurface together with a time-switching vector field) and proves a two-step result: first, that crossing is restricted for barriers, and second, that any time-oriented null hypersurface is locally dressable, giving the semi-permeability property in full generality. By casting barriers as a coordinate-free tool, the paper unifies the treatment of event horizons and black holes, clarifying their role in domain structures and providing a practical route to horizon localization in numerical spacetime data. The authors illustrate the framework with explicit barrier constructions in Kerr-Newman and Myers-Perry spacetimes and extend it to Kerr-type Ricci-flat metrics, highlighting potential numerical applications for horizon finding and the study of black-hole boundaries.

Abstract

We prove that if S is a time-oriented null hypersurface of a Lorentzian n-manifold (M, g), the causal world-lines, which intersect transversally S and are time-oriented in a compatible way, cross the hypersurface all in the same direction, the other being forbidden. Even if it is known that a smooth event horizon (in the sense of Penrose, Hawking and Ellis) is a null hypersurface and has the above semi-permeability property, at the best of our knowledge, it was not known so far that the latter is a mere consequence of the former. Our result leads to the concepts of barriers (= null hypersurfaces separating the space-time into disjoint regions) and black regions (= time-oriented regions bounded by barriers). These objects naturally include (smooth) event horizons and (smoothly bounded) black holes. Since barriers are defined by two simple properties -- the merely local property of "nullity" combined with the global property of "separating the space-time" -- we expect they may be used to simplify computations for locating static and/or dynamic horizons in numerical computations.

Black holes and black regions, horizons and barriers in Lorentzian manifolds

TL;DR

This work establishes a general crossing constraint for causal world-lines with time-oriented null hypersurfaces in Lorentzian manifolds, showing that such surfaces act as semi-permeable barriers that separate spacetime into regions with directionally allowed signal flow. It introduces the notion of a dressed barrier (a null hypersurface together with a time-switching vector field) and proves a two-step result: first, that crossing is restricted for barriers, and second, that any time-oriented null hypersurface is locally dressable, giving the semi-permeability property in full generality. By casting barriers as a coordinate-free tool, the paper unifies the treatment of event horizons and black holes, clarifying their role in domain structures and providing a practical route to horizon localization in numerical spacetime data. The authors illustrate the framework with explicit barrier constructions in Kerr-Newman and Myers-Perry spacetimes and extend it to Kerr-type Ricci-flat metrics, highlighting potential numerical applications for horizon finding and the study of black-hole boundaries.

Abstract

We prove that if S is a time-oriented null hypersurface of a Lorentzian n-manifold (M, g), the causal world-lines, which intersect transversally S and are time-oriented in a compatible way, cross the hypersurface all in the same direction, the other being forbidden. Even if it is known that a smooth event horizon (in the sense of Penrose, Hawking and Ellis) is a null hypersurface and has the above semi-permeability property, at the best of our knowledge, it was not known so far that the latter is a mere consequence of the former. Our result leads to the concepts of barriers (= null hypersurfaces separating the space-time into disjoint regions) and black regions (= time-oriented regions bounded by barriers). These objects naturally include (smooth) event horizons and (smoothly bounded) black holes. Since barriers are defined by two simple properties -- the merely local property of "nullity" combined with the global property of "separating the space-time" -- we expect they may be used to simplify computations for locating static and/or dynamic horizons in numerical computations.

Paper Structure

This paper contains 12 sections, 4 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on orientations, time-orientations and crossing
  3. Oriented curves, oriented hypersurfaces and crossing
  4. Causal world-lines and time-orientations in Lorentzian manifolds
  5. The semi-permeability property of null hypersurfaces
  6. The semi-permeability of the dressed barriers
  7. The semi-permeability property of time-oriented null hypersurfaces
  8. A few (classical and not) examples of black holes and barriers
  9. The inner and outer barriers of Kerr-Newman black holes
  10. Barriers and horizons of Myers and Perry's n-dimensional black holes
  11. Barriers in Ricci flat manifolds with Kerr type optical structures
  12. Penrose, Hawking and Ellis' definitions of black holes and event horizons

Key Result

Lemma 3.2

Let $(\mathscr{S}, \mathbb{T})$ be a dressed barrier in a Lorentzian $n$-manifold $(M, g)$, $n \geq 2$ with mostly minus signature. Let $\mathscr{T}{}$ be the time orientation of $M_+$ determined by $\mathbb{T}|_{M_+}$ and denote by $[E^\perp]$ the orientation of $\mathscr{S}$, given by a vector fie Then there is no time-like world-line $\mathcal{C}$ in $M$, which is $\mathscr{T}{}$-time oriented

Theorems & Definitions (11)

  • Remark 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 3.1
  • Lemma 3.2
  • Corollary 3.3
  • Lemma 3.4
  • Theorem 3.5
  • ...and 1 more