Mirror Symmetry and Double Signature Change

Abstract: The black mirror spacetime proposed by Tzanavaris, Boyle, and Turok [1] connects the two exterior regions of the extended Schwarzschild black hole directly to each other, with no intervening interior region. Using techniques adapted from previous work on signature change, we reexamine the black mirror spacetime as a model of double signature change, and investigate whether there is a surface layer at the horizon, that is, a distributional curvature singularity corresponding to an impulsive gravitational wave. We confirm that the black mirror spacetime does not contain any such singularity, and compare our result with previous claims that the curvature components are analytic. We also discuss the global structure of the black mirror spacetime, and examine what happens to worldlines and curves passing through.

• The paper demonstrates that the black mirror spacetime construction via double signature change yields a smooth metric, free from curvature singularities and surface stress-energy layers.
• Methodologies including distributional computations, variational principles, and unified junction theory verify the continuous curvature and lack of impulsive gravitational waves.
• Implications include altered geodesic propagation and causal structure, challenging classic black hole models and opening new avenues for quantum and topological research.

Mirror Symmetry and Double Signature Change in Black Mirror Spacetime

Overview and Motivation

The paper investigates the geometric and physical structure of the "black mirror" spacetime—a variant of the Schwarzschild solution proposed by Tzanavaris, Boyle, and Turok—which joins the two exterior regions of the maximally extended Schwarzschild/Kruskal spacetime directly across their event horizons, omitting the intermediate interior (black hole/white hole) regions. The analysis is framed in the context of "double signature change," where the metric signature flips across a null surface, raising questions about the presence of curvature singularities, surface layers, and the causal and differentiable structure at the horizon interface.

Construction of the Black Mirror Geometry

The classical Schwarzschild solution, extended via Kruskal--Szekeres coordinates, admits four distinct regions described in the Penrose diagram: two exterior (I, III) and two interior (II, IV). The black mirror construction excises the interior regions and identifies the future horizon of Region I with the past horizon of Region III and vice versa. A coordinate transformation (introducing $\sigma$ with $r = 2m(1 + \sigma^2/16m^2)$) highlights the horizon at $\sigma = 0$, and the metric becomes degenerate at this locus.

The identification is performed in double-null coordinates, with the key mapping $V \mapsto -V$ and $U \mapsto -U$ across the joined horizons, inducing a double signature change. The metric's signature flips two components as one passes through the horizon: timelike directions become spacelike and vice versa for certain indices.

Junction Conditions and Surface Layer Analysis

A primary focus is the proper evaluation of the junction conditions at the horizon to determine if the construction produces any curvature singularity or impulsive gravitational wave—a surface layer as in standard junction theory. The analysis proceeds in three independent approaches:

1. Formal Distributional Computation

Using the step function and distributional calculus, the authors compute the Riemann tensor components and show that no delta-function singularities (impulsive gravitational waves) appear at the horizon. The Ricci tensor remains continuous and vanishes everywhere, confirming the absence of surface stress-energy layers in the distributional sense.

2. Variational Principle and Null Junctions

By applying the Einstein--Hilbert action with null boundaries, the pullbacks of geometric invariants ($\rho_d$, the surface variation quantity) across the junction are shown to be continuous. This confirms via variational methods that no boundary sources are present.

3. Unified Junction Theory (Clarke-Dray Formalism)

Utilizing the Clarke-Dray null hypersurface junction conditions, the authors verify that all relevant discontinuities in second fundamental forms and associated surface terms vanish at the horizon, reaffirming the previous results.

Signature Change and Curve Continuation

A notable consequence of the double signature change is the behavior of geodesics and causality across the horizon interface:

• Double Signature Change (DSC) Continuation: Smooth propagation of worldlines across the horizon forces timelike curves to become spacelike and vice versa. Null geodesics are only preserved in very special directions.
• Lorentzian Kink (LK) Continuation: If one insists on preserving the causal character (timelike remains timelike), a "kink" appears at the horizon, corresponding to a discontinuous change in direction—a feature unsupported physically in the absence of surface layers.

The analysis demonstrates that under the DSC continuation, closed timelike geodesics cannot exist; however, under LK continuation, closed timelike curves can be constructed due to the artificial preservation of causality and a corresponding kink at the interface.

Global Structure and Causal Implications

The paper explores alternative identification schemes of the horizons, yielding fundamentally distinct causal diagrams ("forward" and "reverse" pita-pocket structures). In one case, the horizon acts as a boundary for causal curves; in the other, it admits curve propagation but at the expense of a change in signature and possibly causality.

The structure at infinity ($i^\pm$) also differs from standard Schwarzschild: the joined spacetime alters the geodesic completeness and causal accessibility of future and past null infinity, introducing ambiguities in physical interpretation.

Implications and Future Directions

The absence of surface layers confirms the analytic nature of the curvature across the horizon, in agreement with earlier claims. However, the requirements for smooth propagation of worldlines and the induced signature change denote a fundamental alteration in causal and differentiable structure, raising questions about physical realizability and applicability to black hole information and quantum field theory. The double signature change paradigm extends prior results on classical signature change and could inform future investigations on quantum particle production and nontrivial topological identifications.

The results challenge assumptions about horizon structure, geodesic propagation, and the feasibility of spacetimes constructed by nonstandard identifications. Future work may involve generalizations to other black hole solutions, incorporation of matter, or analysis of quantum effects in the presence of double signature change.

Conclusion

The detailed analysis concludes that the black mirror spacetime—constructed via direct identification of exterior regions across event horizons and analyzed as a model of double signature change—contains no curvature singularity or surface layer at the horizon. While the metric is degenerate at the interface and signature change induces nontrivial propagation properties for curves, the geometry remains a vacuum solution everywhere. The study highlights both the mathematical consistency and the physical ambiguities inherent in such constructions, and suggests caution in assuming physical realization, particularly with respect to causal propagation and geodesic structure. These considerations are pertinent for theoretical models seeking alternative resolutions to black hole information paradoxes or exploring novel spacetime topologies (2606.21805).