A Topological Origin of Black Hole Mass

Abstract: We show that the notion of a black hole mass could be superceded by that of a topological charge in a spacetime without matter and curvature singularity. This feature emerges through a set new spacetime solutions of first order gravity in vacuum, named as bubble spacetimes, constructed here by weaving together the degenerate and nondegenerate metric phases. For a static bubble, the boundary surface connecting the two phases is characterized by a universal topological number. Notably, this surface coincides with the photon sphere of the conventional black hole irrespective of the presence of the cosmological constant. In contrast, a phase boundary located at the event horizon is shown to be topologically trivial. Thus, along with the black hole mass, the photon sphere also acquires a topological interpretation.

• The paper shows that black hole mass originates as a universal topological charge using bubble spacetime constructions in first-order gravity.
• It constructs both static and non-static bubbles by gluing nondegenerate and degenerate metric phases, thereby eliminating traditional curvature singularities.
• The analysis redefines the photon sphere as the topologically significant boundary, challenging conventional views centered on the event horizon.

Topological Interpretation of Black Hole Mass

Overview and Motivation

The paper "A Topological Origin of Black Hole Mass" (2604.02895) presents a paradigm shift by proposing that the mass of a black hole, traditionally attributed to matter sources and spacetime curvature, can instead be interpreted as a topological charge in vacuum spacetimes devoid of material or curvature singularities. This study constructs novel "bubble" spacetimes in first-order gravity by gluing together nondegenerate and degenerate metric phases. The resulting configurations eliminate conventional singularities and locate the phase boundary at the photon sphere, providing a universal topological characterization of both the mass and the photon sphere itself. The analysis incorporates Schwarzschild, Schwarzschild-de Sitter, and Schwarzschild-anti-de Sitter geometries, establishing the generality and robustness of the topological mechanism.

Construction of Bubble Spacetimes

The first-order gravity formulation, especially with the Hilbert-Palatini action, is central as it allows for spacetimes with vanishing and non-vanishing metric determinants. The bubble solutions are constructed by consistently joining spherically symmetric black hole exteriors (with $g \neq 0$) to degenerate interiors (with $g = 0$), maintaining continuity of metric, field-strength, and torsion across the dynamical boundary. Crucially, the degenerate phase is torsionless, resulting in real and regular fields throughout.

Two classes of solutions are distinguished:

• Non-static Bubbles: The phase boundary is time-dependent, but cannot coincide with the event horizon ($r_0 = 2M$) and instead manifests at some distinct radius.
• Static Bubbles: The phase boundary is fixed in time and uniquely constrained by equations of motion to coincide with the photon sphere ($r_0 = 3M$ in Schwarzschild; similar analogs for Schwarzschild-dS and Schwarzschild-AdS).

This procedure eliminates curvature singularities inherent in traditional black hole interiors.

Topological Charge as Mass Origin

A conserved current, independent of metric and equations of motion but dependent solely on the boundary topology, is defined within the degenerate phase. The associated topological charge $Q$, evaluated on the boundary (the photon sphere), satisfies $Q = 1$. The black hole mass $M$ is thereby expressed as a function of this topological charge and the boundary radius, $M/r_0 = \frac{1}{2}(1 - Q/3)$ for the Schwarzschild case. The result holds equivalently for Schwarzschild-dS and Schwarzschild-AdS scenarios, demonstrating universality across spherically symmetric vacuum solutions.

By contrast, boundaries at the event horizon are shown to be topologically trivial, thus invalidating previous constructions that focused on this surface.

Numerical Results and Nontrivial Topological Invariants

• Topological Charge: $Q = 1$ (universal for static bubble solutions across vacuum black hole types).
• Photon Sphere Radius: $r_0 = 3M$ for Schwarzschild (static bubble boundary).
• Curvature Scalars: Regular and finite everywhere within the bubble, specifically $g = 0$0, confirming absence of singularities.

An analog of the Chern-Simons density, $g = 0$1, also emerges as a nontrivial topological invariant, proportional to the charge $g = 0$2 and vanishing for event horizon boundaries.

Implications and Theoretical Insights

The work implies that mass and the associated geometry (photon sphere) in black holes are fundamentally topological in origin within first-order gravity in vacuum, dissociating them from explicit matter or geometric torsion contributions. This topological mechanism is achieved without curvature singularities and is independent of the cosmological constant.

The photon sphere, rather than the event horizon, is established as the fundamental geometric surface encoding black hole mass in bubble spacetimes. This suggests practical alternatives to singular black holes and motivates reconsideration of astrophysical observables (ringdown phases, shadow signatures) in terms of photon surface topology.

Theoretically, the results indicate that nontrivial topological features can arise without the need for matter, torsion, or singularities. This approach invites further exploration in higher-dimensional theories, different gauge groups, and may provide novel perspectives on the nature of dark matter and gravitational phenomena where degenerate phases are inaccessible to external observers.

Conclusion

The paper rigorously demonstrates that black hole mass and photon sphere possess a topological origin in vacuum gravity, accomplished via bubble spacetime constructions in first-order gravity. The universal appearance of the topological charge and the regularity of curvature scalars highlight the robustness and regularity of these solutions. The photon sphere is revealed as the unique, topologically nontrivial boundary in all spherically symmetric vacuum black hole spacetimes considered. These findings invite scrutiny of black hole mass and geometry from a topological perspective, with potential implications for quantum gravity, astrophysical observations, and the foundational understanding of gravitational phenomena.