Black Mirrors: CPT-Symmetric Alternatives to Black Holes (2412.09558v2)

Abstract: Einstein's equations imply that a gravitationally collapsed object forms an event horizon. But what lies on the other side of this horizon? In this paper, we question the reality of the conventional solution (the black hole), and point out another, topologically distinct solution: the black mirror. In the black hole solution, the horizon connects the exterior metric to an interior metric which contains a curvature singularity. In the black mirror, the horizon instead connects the exterior metric to its own CPT mirror image, yielding a solution with smooth, bounded curvature. We give the general stationary (charged, rotating) black mirror solution explicitly, and also describe the general black mirror formed by gravitational collapse. The black mirror is the relevant stationary point when the quantum path integral is equipped with suitably CPT-symmetric boundary conditions, that we propose. It appears to avoid many vexing puzzles which plague the conventional black hole.

• The paper presents a CPT-symmetric black mirror that replaces the conventional black hole interior with two exterior regions, resolving the information paradox.
• It employs a Euclidean path integral and smooth Eddington-Finkelstein coordinates to yield a singularity-free geometry across the horizon.
• The model upholds charge conservation and links black hole and cosmological entropy via its inherent CPT symmetry.

This paper proposes a novel alternative to the standard black hole solution in general relativity, termed the "black mirror" (2412.09558). It argues that the black mirror arises naturally from CPT symmetry considerations and resolves several conceptual problems associated with black holes, such as curvature singularities and the information paradox.

The core idea stems from re-examining the Euclidean path integral approach to black hole thermodynamics, pioneered by Gibbons and Hawking. For the Schwarzschild metric, $$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2}$$, where $f(r) = 1 - 2m/r$, the standard analysis involves Wick rotating to Euclidean time ($\tau = it$) and transforming the radial coordinate $r$ to a proper distance $\sigma$ near the horizon ($r=2m$). This yields a metric that, near $\sigma=0$, resembles a 2D cone times a 2-sphere: $$ds^{2}\approx\frac{\sigma^{2}}{(4m)^{2}} d\tau^{2}+d\sigma^{2}+(2m)^{2} d\Omega^{2}$$. Requiring smoothness at the tip ($\sigma=0$) fixes the period of Euclidean time to $\beta = 8\pi m$, interpreted as the inverse temperature.

The paper highlights that this standard analysis typically only considers the $\sigma > 0$ region (Fig. 2a, blue cone), corresponding to the black hole exterior (Region I in Fig. 1a). However, the geometry naturally extends to $\sigma < 0$ (Fig. 2a, red cone), creating a double-cone structure. When the conical singularity is removed by identifying $\tau \sim \tau + \beta$, this results in a smooth, two-sided Euclidean manifold (Fig. 2b) representing two exterior regions glued at $\sigma=0$. The paper suggests that the black hole entropy calculated via the one-sided action actually represents the entanglement entropy between these two $(\sigma > 0$ and $\sigma < 0)$ regions.

Wick rotating this complete two-sided Euclidean geometry back to Lorentzian signature yields two exterior regions ($I$ and $I'$ in Fig. 1a). Instead of the standard Kruskal extension which adds interior regions ($II, II'$) and singularities, the black mirror proposes identifying the horizons of $I$ and $I'$ directly. This is achieved by using coordinates that are well-behaved across the horizon, like Eddington-Finkelstein coordinates ($v_\pm, \sigma$), where $dv_{\pm}=dt\pm\frac{dr}{f(r)}$. The resulting black mirror metric (Eq. \ref{BlackMirrorSchwarzschild} near the horizon) describes a spacetime where the two exteriors are glued together (Fig. 1b) with an antipodal map on the $S^2$ sphere $(\theta, \varphi) \to (\pi-\theta, \varphi+\pi)$ across the horizon (detailed in Appendix C).

Key properties of the Schwarzschild black mirror:

• No Curvature Singularities: The metric components and curvature invariants (like the Kretschmann scalar) are finite and smooth everywhere in coordinates like $(v_\pm, \sigma, \theta, \varphi)$.
• No Interior Region: The spacetime consists only of two asymptotically flat exterior regions joined at the horizon surface ($\sigma=0$).
• Mild Horizon Singularity: While curvature is finite, two eigenvalues of the metric tensor $g_{\mu\nu}$ go to zero analytically at $\sigma=0$, while the corresponding eigenvalues of the inverse metric $g^{\mu\nu}$ develop simple poles. This is presented as a genuine, albeit mild, geometric feature inherent to the CPT identification, not a coordinate artifact.
• CPT Symmetry: The construction inherently respects CPT symmetry, mapping one exterior region to the other via the transformation $$(t, \sigma, \theta, \varphi) \to (t, -\sigma, \pi-\theta, \varphi+\pi)$$.

The paper demonstrates that this construction generalizes to stationary charged, rotating black holes in (A)dS spacetime (detailed in Appendix A). The Boyer-Lindquist coordinates are used initially, then Wick rotated to Euclidean signature to find the temperature and horizon angular velocity. Finally, generalized Eddington-Finkelstein coordinates are used to express the metric across the horizon, revealing the same qualitative features: two exteriors joined at the horizon, no curvature singularities, and the characteristic mild singularity in the metric eigenvalues. The general stationary metric in ingoing/outgoing coordinates is given by Eq. \ref{StationaryBlackMirrorMetric} using the tetrad \ref{StationaryBlackMirrorTetrad}.

For time-dependent scenarios like gravitational collapse, the paper proposes a similar picture (Fig. 3). A collapsing star in one spacetime sheet (say, the blue side in Fig. 3b) forms a horizon which is identified with the horizon formed by a CPT-conjugate collapsing configuration (red side). Infalling matter (particle) from one side meets infalling antimatter from the other side at the horizon. The resulting Penrose diagram can be "folded" (Fig. 3c) for visualization. It is conjectured that the relevant surface for this identification in dynamical cases is the apparent horizon, potentially offering observational distinctions from standard black holes via gravitational waves.

Several arguments are presented in favor of the black mirror picture:

1. Path Integral: It's argued to be the relevant saddle point for the gravitational path integral under proposed CPT-symmetric boundary conditions (Fig. 5), avoiding the extra singular boundaries present in the standard black hole solution (Appendix B).
2. Evaporation: It naturally accommodates the finite evaporation time seen by external observers, as objects effectively never cross into a disconnected interior.
3. Information Paradox: Resolved because there is no interior region causally disconnected from future infinity. Information associated with infalling matter/antimatter can potentially re-emerge as radiation produced by annihilation at/near the horizon (Fig. 4c).
4. CPT Symmetry: The final state after collapse naturally settles to a CPT-symmetric configuration, unlike the standard black hole.
5. Global Symmetries: Charge conservation is naturally upheld, as infalling charge on one side is cancelled by infalling anticharge on the CPT-conjugate side.
6. Cosmology: Fits within a proposed CPT-symmetric cosmological model with two spacetime sheets (Fig. 6), providing a unified view of horizon entropy (BH area) and cosmological entropy (volume) as entanglement between the sheets.

In essence, the black mirror replaces the black hole interior and its singularity with a CPT mirror image of the exterior, connected at the horizon. This alternative geometry is singularity-free (in the sense of curvature) and potentially resolves long-standing paradoxes by enforcing CPT symmetry at the level of spacetime geometry.