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Energy conditions for non-timelike thin shells

Hideki Maeda

TL;DR

This work provides a general, symmetry-free analysis of energy conditions for non-timelike thin shells in $n\ge 3$ dimensions, showing that the induced $t_{\mu\nu}$ on spacelike $\Sigma$ is Hawking-Ellis type I while null $\Sigma$ yields types I–III. It derives compact, equivalent representations of the standard energy conditions for these shells and shows DEC is inevitably violated on spacelike shells or on null shells with surface current, with type III arising when the null shell has vanishing surface pressure. These results hold in any gravity theory and without symmetry, offering a robust framework for assessing thin-shell constructions. The paper then demonstrates several four-dimensional GR applications, including Schwarzschild-based shells (black-bounce, null impulses, and slowly rotating null shells), a cylindrically symmetric rotating null shell, and cosmological phase transitions, illustrating how the energy-condition constraints guide physically reasonable models and reveal potential pathologies.

Abstract

We study energy conditions for non-timelike thin shells in arbitrary $n(\ge 3)$ dimensions. It is shown that the induced energy-momentum tensor $t_{μν}$ on a shell $Σ$ is of the Hawking-Ellis type I if $Σ$ is spacelike and either of type I, II, or III if $Σ$ is null. Then, we derive simple equivalent representations of the standard energy conditions for $t_{μν}$. In particular, on a spacelike shell or on a null shell with non-vanishing surface current, $t_{μν}$ inevitably violates the dominant energy condition. If the surface pressure on the null shell is vanishing in addition, $t_{μν}$ is of type III and violates all the standard energy conditions. Those fully general results are obtained without imposing a spacetime symmetry and can be used in any theory of gravity. Lastly, several applications of the main results are presented in general relativity in four dimensions.

Energy conditions for non-timelike thin shells

TL;DR

This work provides a general, symmetry-free analysis of energy conditions for non-timelike thin shells in dimensions, showing that the induced on spacelike is Hawking-Ellis type I while null yields types I–III. It derives compact, equivalent representations of the standard energy conditions for these shells and shows DEC is inevitably violated on spacelike shells or on null shells with surface current, with type III arising when the null shell has vanishing surface pressure. These results hold in any gravity theory and without symmetry, offering a robust framework for assessing thin-shell constructions. The paper then demonstrates several four-dimensional GR applications, including Schwarzschild-based shells (black-bounce, null impulses, and slowly rotating null shells), a cylindrically symmetric rotating null shell, and cosmological phase transitions, illustrating how the energy-condition constraints guide physically reasonable models and reveal potential pathologies.

Abstract

We study energy conditions for non-timelike thin shells in arbitrary dimensions. It is shown that the induced energy-momentum tensor on a shell is of the Hawking-Ellis type I if is spacelike and either of type I, II, or III if is null. Then, we derive simple equivalent representations of the standard energy conditions for . In particular, on a spacelike shell or on a null shell with non-vanishing surface current, inevitably violates the dominant energy condition. If the surface pressure on the null shell is vanishing in addition, is of type III and violates all the standard energy conditions. Those fully general results are obtained without imposing a spacetime symmetry and can be used in any theory of gravity. Lastly, several applications of the main results are presented in general relativity in four dimensions.
Paper Structure (16 sections, 2 theorems, 136 equations, 4 figures, 1 table)

This paper contains 16 sections, 2 theorems, 136 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Energy conditions for thin-shells
  3. Hawking-Ellis types
  4. Energy-momentum tensor of thin-shells
  5. Non-null shell
  6. Null shell
  7. Main results
  8. Applications in general relativity in four dimensions
  9. Shells in the Schwarzschild spacetime
  10. Black bounce with a spacelike shell
  11. Lightlike impulse as a null shell
  12. Accretion of a slowly-rotating null shell
  13. Cylindrically symmetric rotating null shell
  14. Cosmological phase transition
  15. Transition at spacelike $\Sigma$
  16. ...and 1 more sections

Key Result

Proposition 1

An induced energy-momentum tensor $t_{\mu\nu}$ on a spacelike hypersurface $\Sigma$ is of the Hawking-Ellis type I and a non-vanishing $t_{\mu\nu}$ violates the DEC. The NEC, WEC, and SEC are all equivalent to that $p_i\ge 0$ are satisfied for all $i(=1,2,\cdots,n-1)$, where $p_i$ are eigenvalues of

Figures (4)

  • Figure 1: A non-null hypersurface $\Sigma$ partitions a spacetime into two regions ${\cal M}_+$ and ${\cal M}_-$.
  • Figure 2: A null hypersurface $\Sigma$ partitions a spacetime into two regions ${\cal M}_+$ and ${\cal M}_-$.
  • Figure 3: A Penrose diagram of the thin-shell black-bounce spacetime constructed by gluing two identical Schwarzschild spacetimes at a spacelike hypersurface $r=r_0(<2M)$.
  • Figure 4: Penrose diagrams of the Schwarzschild spacetime with a lightlike impulse as a null shell for (a) $\epsilon=1$ with $M_+>M_-$ and (b) $\epsilon=-1$ with $M_+<M_-$. These shells satisfy all the standard energy conditions.

Theorems & Definitions (2)

  • Proposition 1
  • Proposition 2