Same-Prompt Bifurcation in MOTS Evolution

• Same-prompt bifurcation is a phenomenon where, under fixed gravitational conditions, the MOTS equation yields multiple solution branches due to a zero mode in the stability operator.
• The analysis organizes MOTS evolution into distinct bifurcation classes—saddle-node, transcritical, and pitchfork—using a nonlinear fixed-point approach and spectral theory.
• Applications in charged, cosmological, and distorted black-hole geometries illustrate that zero eigenvalue crossings drive the emergence of non-unique quasi-local horizons.

Searching arXiv for the cited paper and closely related MOTS references. Same-prompt bifurcation denotes the appearance of multiple nearby branches of marginally outer trapped surfaces (MOTSs) within the same governing gravitational system: the same Einstein equations and the same general geometric setup admit more than one local MOTS continuation when the MOTS stability operator loses invertibility. In the formulation developed in "Black hole evolutions: Lessons from bifurcation theory" (Booth et al., 28 May 2026), MOTSs are treated as fixed points of a nonlinear map, their stability operator is the linearization of the MOTS-defining equations, and a zero eigenvalue of that operator is the precise analogue of a bifurcation point in nonlinear dynamics. This framework organizes non-unique MOTS evolution into familiar bifurcation classes, including saddle-node, transcritical, and pitchfork behavior (Booth et al., 28 May 2026).

1. MOTSs as fixed points of a nonlinear geometric problem

A marginally outer trapped surface is a closed, spacelike $2$-surface $\mathcal S$ in spacetime $(M,g_{ab})$ whose outward null expansion vanishes. With future-directed null normals $(l^+,l^-)$ satisfying $l^+\cdot l^-=-1$, the induced metric is

$q^{ab}=g^{ab}+l_+^a l_-^b+l_-^a l_+^b,$

and the null expansions are

$\theta_{l^\pm}=q^{ab}\nabla_a l^\pm_b.$

Accordingly, $\mathcal S$ is outer trapped when $\theta_{l^+}<0$, marginally outer trapped when $\theta_{l^+}=0$, and outer untrapped when $\mathcal S$0 (Booth et al., 28 May 2026).

In a $\mathcal S$1 decomposition with spacelike slice $\mathcal S$2, unit timelike normal $\mathcal S$3, extrinsic curvature $\mathcal S$4, and outward unit normal $\mathcal S$5 to a $\mathcal S$6-surface $\mathcal S$7, one may take

$\mathcal S$8

so that

$\mathcal S$9

The MOTS condition therefore becomes

$(M,g_{ab})$0

on the slice (Booth et al., 28 May 2026).

The central reinterpretation is to regard the space of closed surfaces as an infinite-dimensional phase space and define a nonlinear map

$(M,g_{ab})$1

MOTSs are then precisely the zeros of $(M,g_{ab})$2, and thus the analogues of fixed points of a nonlinear dynamical system. Introducing a control parameter $(M,g_{ab})$3 produces the constrained equation

$(M,g_{ab})$4

with $(M,g_{ab})$5 representing time, charge, mass, cosmological constant, distortion parameter, or another geometric variable. Same-prompt bifurcation arises when this fixed-point equation admits multiple nearby solution branches for the same underlying setup (Booth et al., 28 May 2026).

2. Stability operator, spectrum, and local uniqueness

For a MOTS $(M,g_{ab})$6 with outward unit normal $(M,g_{ab})$7 and outward null normal $(M,g_{ab})$8, the stability operator in the direction $(M,g_{ab})$9 is defined by

$(l^+,l^-)$0

Using a reference null dyad $(l^+,l^-)$1 with $(l^+,l^-)$2, the variation formula in a slice gives

$(l^+,l^-)$3

where $(l^+,l^-)$4 is the Gauss curvature of $(l^+,l^-)$5, $(l^+,l^-)$6 is the Einstein tensor, $(l^+,l^-)$7 is the shear of the outgoing null congruence, and $(l^+,l^-)$8 is the drifted Laplacian (Booth et al., 28 May 2026). For $(l^+,l^-)$9,

$l^+\cdot l^-=-1$0

This is a second-order linear elliptic operator on the compact surface $l^+\cdot l^-=-1$1. Its spectrum consists of isolated eigenvalues of finite multiplicity, and the principal eigenvalue $l^+\cdot l^-=-1$2 is real and simple, with a strictly positive eigenfunction. A MOTS is strictly stable if $l^+\cdot l^-=-1$3, marginally stable if $l^+\cdot l^-=-1$4, and unstable if $l^+\cdot l^-=-1$5 (Booth et al., 28 May 2026).

The uniqueness result summarized from Andersson, Mars, and Simon states that if $l^+\cdot l^-=-1$6 is not an eigenvalue of $l^+\cdot l^-=-1$7, then a smooth variation of the spacetime metric in a parameter yields a unique nearby smooth marginally outer trapped tube $l^+\cdot l^-=-1$8 through the given MOTS, both to the future and to the past in some interval (Booth et al., 28 May 2026). In the fixed-point formulation, this is the Hilbert-space implicit-function-theorem regime.

The linearized deformation equation is

$l^+\cdot l^-=-1$9

When $q^{ab}=g^{ab}+l_+^a l_-^b+l_-^a l_+^b,$0 is invertible, this equation has a unique solution $q^{ab}=g^{ab}+l_+^a l_-^b+l_-^a l_+^b,$1, and therefore there is exactly one nearby MOTS branch. Same-prompt bifurcation begins precisely where this uniqueness mechanism fails, namely when the operator develops a nontrivial kernel (Booth et al., 28 May 2026).

3. Zero modes and the taxonomy of bifurcations

Loss of invertibility means that there exists a nonzero $q^{ab}=g^{ab}+l_+^a l_-^b+l_-^a l_+^b,$2 such that

$q^{ab}=g^{ab}+l_+^a l_-^b+l_-^a l_+^b,$3

Equivalently, an eigenvalue of the stability operator crosses zero. In the dynamical-systems analogy, the phase space is the space of embedded surfaces, MOTSs are fixed points, the stability operator is the Jacobian of the fixed-point equation in the surface direction, and a zero mode is the bifurcation signal (Booth et al., 28 May 2026).

The paper identifies three generic local normal forms:

Bifurcation type	Normal form	MOTS interpretation
Saddle-node	$q^{ab}=g^{ab}+l_+^a l_-^b+l_-^a l_+^b,$4	Pair creation or annihilation of a stable and an unstable MOTS
Transcritical	$q^{ab}=g^{ab}+l_+^a l_-^b+l_-^a l_+^b,$5	Two MOTS branches intersect and exchange stability
Pitchfork	$q^{ab}=g^{ab}+l_+^a l_-^b+l_-^a l_+^b,$6	A symmetric MOTS branch splits into two symmetry-related MOTSs

In the MOTS dictionary, coordinates $q^{ab}=g^{ab}+l_+^a l_-^b+l_-^a l_+^b,$7 correspond to surface embeddings $q^{ab}=g^{ab}+l_+^a l_-^b+l_-^a l_+^b,$8, fixed points $q^{ab}=g^{ab}+l_+^a l_-^b+l_-^a l_+^b,$9 correspond to $\theta_{l^\pm}=q^{ab}\nabla_a l^\pm_b.$0, eigenvectors in the kernel correspond to eigenfunctions $\theta_{l^\pm}=q^{ab}\nabla_a l^\pm_b.$1, and the sign of eigenvalues of the Jacobian corresponds to the sign of eigenvalues of $\theta_{l^\pm}=q^{ab}\nabla_a l^\pm_b.$2 (Booth et al., 28 May 2026).

The paper further distinguishes between zero crossings of the principal eigenvalue and zero crossings of higher eigenvalues. If $\theta_{l^\pm}=q^{ab}\nabla_a l^\pm_b.$3 crosses zero, a strictly stable branch becomes marginal and then unstable, and the local behavior is typically saddle-node or transcritical. If a higher eigenvalue $\theta_{l^\pm}=q^{ab}\nabla_a l^\pm_b.$4 crosses zero, additional MOTSs may bifurcate from an existing MOTS without directly involving the outermost apparent horizon; pitchfork and transcritical structures arise in this regime as well (Booth et al., 28 May 2026).

This leads to the specific meaning of same-prompt bifurcation: for fixed equations and fixed geometric data, the nonlinear elliptic MOTS equation can have several solutions on the same slice or in the same spacetime neighborhood. The coexistence is not attributed to changing the governing system, but to branching in the solution set at a zero mode of the linearization (Booth et al., 28 May 2026).

4. Explicit realizations in charged, cosmological, and distorted black-hole geometries

The analysis in (Booth et al., 28 May 2026) is developed through spherically symmetric and axisymmetric examples. These examples exhibit the three generic bifurcation types while preserving the same underlying field equations within each family.

Reissner-Nordström-de Sitter

In ingoing Eddington-Finkelstein coordinates,

$\theta_{l^\pm}=q^{ab}\nabla_a l^\pm_b.$5

Choosing

$\theta_{l^\pm}=q^{ab}\nabla_a l^\pm_b.$6

one has

$\theta_{l^\pm}=q^{ab}\nabla_a l^\pm_b.$7

so spherically symmetric MOTSs lie at the positive roots of $\theta_{l^\pm}=q^{ab}\nabla_a l^\pm_b.$8 (Booth et al., 28 May 2026). On a spherical MOTS of radius $\theta_{l^\pm}=q^{ab}\nabla_a l^\pm_b.$9, under the assumptions of spherical symmetry and isolated-horizon conditions, the stability operator reduces to

$\mathcal S$0

with eigenvalues

$\mathcal S$1

For a MOTS at $\mathcal S$2,

$\mathcal S$3

Thus the sign of the principal eigenvalue is the sign of the slope of $\mathcal S$4 at the root; outer black-hole horizons are stable, while inner and cosmological horizons are unstable (Booth et al., 28 May 2026).

The same family displays all three generic bifurcations. With $\mathcal S$5 and varying $\mathcal S$6, the coincidence of inner and outer horizons at extremal Reissner-Nordström gives a saddle-node. With $\mathcal S$7 and a tuned Schwarzschild-de Sitter factorization, black-hole and cosmological horizons exchange roles in a transcritical bifurcation. With a three-parameter factorization in Reissner-Nordström-de Sitter, a pitchfork configuration appears in which three positive horizons meet at a critical value of the parameter $\mathcal S$8 (Booth et al., 28 May 2026).

Axisymmetric bifurcations at the inner Reissner-Nordström horizon

In generalized Painlevé-Gullstrand coordinates,

$\mathcal S$9

with

$\theta_{l^+}<0$0

constant-$\theta_{l^+}<0$1 slices extend smoothly across the horizons (Booth et al., 28 May 2026). Axisymmetric MOTSs are found using the MOTSodesic method by rotating a meridional curve $\theta_{l^+}<0$2 around the symmetry axis. Around the inner spherical horizon $\theta_{l^+}<0$3, the axisymmetric stability eigenvalues are

$\theta_{l^+}<0$4

and vanish at the discrete charge values

$\theta_{l^+}<0$5

At these values, non-principal eigenvalues become zero and axisymmetric non-spherical MOTSs bifurcate from the inner horizon (Booth et al., 28 May 2026).

The paper identifies a saddle-node at $\theta_{l^+}<0$6, a subcritical pitchfork at $\theta_{l^+}<0$7, and a transcritical bifurcation at $\theta_{l^+}<0$8. In the pitchfork case, two axisymmetric MOTSs appear symmetrically above and below the equatorial plane, reflecting the $\theta_{l^+}<0$9-type odd parity of the relevant zero mode. In the transcritical case, a non-spherical MOTS approaching the inner horizon from one side is replaced, after the crossing, by a different non-spherical branch on the other side, with an exchange in the number of negative eigenvalues between the branches (Booth et al., 28 May 2026).

Weyl-distorted Schwarzschild

For a static Weyl-distorted Schwarzschild black hole,

$\theta_{l^+}=0$0

with quadrupole distortion

$\theta_{l^+}=0$1

the horizon $\theta_{l^+}=0$2 remains a null Killing horizon for all $\theta_{l^+}=0$3, and any axisymmetric spacelike slice through it is a MOTS (Booth et al., 28 May 2026). The corresponding axisymmetric stability operator simplifies to

$\theta_{l^+}=0$4

and, after gauge fixing, becomes

$\theta_{l^+}=0$5

The computed zero-eigenvalue values include

$\theta_{l^+}=0$6

(Booth et al., 28 May 2026). At $\theta_{l^+}=0$7, the paper finds a transcritical bifurcation in which a MOTS outside the geometric horizon moves inward, meets the horizon as $\theta_{l^+}=0$8 vanishes, then moves inside while the horizon and off-horizon branch exchange stability classification. At $\theta_{l^+}=0$9, a second eigenvalue crosses zero and a pitchfork bifurcation produces two distinct axisymmetric MOTSs on either side of the horizon (Booth et al., 28 May 2026).

Taken together, these examples show that same-prompt bifurcation is not tied to dynamical spacetimes alone. In the Weyl case the metric is static, and the multiplicity of MOTSs arises entirely from the structure of the horizon stability operator as the distortion amplitude varies. This suggests that non-uniqueness of quasi-local horizons is a structural property of the MOTS equation rather than a byproduct of time dependence.

5. Generalized stability operator and barrier structure

The paper extends the notion of stability to a generalized operator $\mathcal S$00, where $\mathcal S$01 is any normal vector field to $\mathcal S$02. Writing

$\mathcal S$03

one obtains

$\mathcal S$04

(Booth et al., 28 May 2026). Within this formulation, the sign of the principal eigenvalue is invariant under rescaling of $\mathcal S$05 and $\mathcal S$06, and in non-rotating settings the operator can be scaled to be similar to a self-adjoint operator so that the count of negative and zero eigenvalues is robust (Booth et al., 28 May 2026).

This operator supports a precise statement of barrier behavior. Given a two-sided MOTS $\mathcal S$07 with outward normal $\mathcal S$08 and principal eigenfunction $\mathcal S$09, define nearby surfaces by

$\mathcal S$10

Then, for small $\mathcal S$11,

$\mathcal S$12

If $\mathcal S$13, the $\mathcal S$14 side is outer untrapped and the $\mathcal S$15 side is outer trapped. By the maximum principle summarized in the paper, any weakly outer trapped surface in a sufficiently small neighborhood must remain entirely on the $\mathcal S$16 side, while any weakly outer untrapped surface must remain on the $\mathcal S$17 side. A strictly stable MOTS is therefore a local barrier between trapped and untrapped regions (Booth et al., 28 May 2026).

If $\mathcal S$18, the signs reverse, but unstable MOTSs do not possess the same barrier property. Weakly trapped or weakly untrapped surfaces can cross them. When a non-principal eigenvalue vanishes and the corresponding eigenfunction $\mathcal S$19 changes sign, one may construct nearby deformations of the form

$\mathcal S$20

that are trapped, untrapped, or marginal and that straddle the original MOTS (Booth et al., 28 May 2026). This provides a geometric explanation for the appearance of crossing or straddling branches in transcritical and pitchfork diagrams.

A common misconception is that every MOTS behaves as a quasi-local horizon boundary in the same sense. The barrier analysis shows otherwise: strictly stable MOTSs separate trapped from untrapped regions locally, whereas unstable MOTSs can participate in bifurcation structures without acting as such boundaries (Booth et al., 28 May 2026).

6. Scope and conceptual implications

The analysis is presented as not being tied specifically to Einstein’s equations beyond the requirement that the MOTS-defining relation be a local elliptic geometric equation on a surface. The spectral theory of the stability operator and the bifurcation classification derive from linearization and elliptic functional analysis rather than from special features of a particular matter model (Booth et al., 28 May 2026).

Accordingly, the paper argues that any theory of gravity containing MOTSs, or an appropriate generalization of them, should exhibit the same constraint structure on possible bifurcations. Where the relevant stability operator is invertible, local evolution is unique. Where it has a kernel, the same generic bifurcation types—saddle-node, transcritical, and pitchfork—should govern local branching (Booth et al., 28 May 2026).

In this sense, same-prompt bifurcation is a universal mechanism for non-uniqueness of quasi-local horizons. The same field equations, the same background spacetime family, and the same slicing prescription can support multiple coexisting MOTSs because the equation $\mathcal S$21 is nonlinear and its linearization can lose invertibility. A plausible implication is that multiplicity of horizon-like structures in collapse, merger, or distorted stationary settings should be interpreted as organized by the local spectral geometry of the stability operator rather than as an exceptional pathology.

Within the framework of (Booth et al., 28 May 2026), the synthesis is direct: the MOTS equation is a nonlinear fixed-point problem on embedded surfaces; invertibility of the stability operator yields unique continuation; zero modes invalidate the implicit-function-theorem regime; and the resulting local branch structure is controlled by standard bifurcation theory. Same-prompt bifurcation is therefore the gravitational analogue of non-unique fixed-point continuation under parameter variation, realized concretely by MOTSs in black-hole spacetimes (Booth et al., 28 May 2026).