Trapped photon region in the phase space of sub-extremal Kerr-Newman and Kerr-Sen spacetimes

Technical Summary: Trapped Photon Region in Sub-Extremal Kerr-Newman and Kerr-Sen Spacetimes

Problem Statement
The paper investigates the geometric and topological structure of the Trapped Photon Region (TPR) within the domain of outer communication (DOC) of stationary, axisymmetric black hole spacetimes. While the existence of trapped photons (photons with constant radial coordinates) is well-established in Schwarzschild and sub-extremal Kerr spacetimes, the topological nature of the set of all such trapped photons (lifted to the tangent bundle) requires rigorous proof for more general solutions. Specifically, the authors address whether the TPR in the Kerr-Newman (Einstein-Maxwell) and Kerr-Sen (Einstein-Maxwell-Dilaton-Axion) spacetimes forms a specific five-dimensional submanifold with the topology $SO(3) \times \mathbb{R}^2$, a result previously established for the Kerr case by Cederbaum and Jahns.

Methodology
The authors employ a unified analytical framework that adapts the direct approach of Cederbaum and Jahns, avoiding reliance on perturbations from the Schwarzschild limit. The methodology proceeds through the following steps:

1. Metric and Integrability: The study utilizes a general metric form in Boyer-Lindquist coordinates characterized by free functions $K(r)$ and $f(r)$. By setting specific forms for these functions, the metric recovers Kerr-Newman, Kerr-Sen, and Kerr spacetimes. The analysis relies on the complete integrability of geodesic motion, utilizing four first-integrals: energy $E$, angular momentum $L$, the norm of the tangent vector $q$ (set to 0 for null geodesics), and the Carter-like constant $K$ associated with a Killing tensor field.
2. Geometric Characterization (Submanifold Proof):
   • The set of trapped null geodesics $\tilde{P}$ is identified as a subset of the tangent bundle $TN$.
   • The authors construct smooth maps $F$ and $H$ from the tangent bundle to $\mathbb{R}^3$. These maps are defined such that their zero level sets correspond exactly to the trapped photon region (separately handling the region where $\sin^2\theta \neq 0$ and a neighborhood where $\sin^2\theta = 0$).
   • Using the Submersion Theorem, the authors prove that the differential maps $d_zF$ and $d_zH$ are surjective for all points in the trapped region. This surjectivity confirms that the preimage of zero is a submanifold of dimension $8 - 3 = 5$.
   • A critical technical step involves proving that trapped photons in the DOC of these spacetimes cannot have zero energy ($E \neq 0$), allowing for the normalization of first-integrals ($Q_{trap} = K/E^2$ and $\Phi_{trap} = L/E$).
3. Topological Characterization:
   • The authors analyze the topology of the image of $\tilde{P}$ in the cotangent bundle $T^*N$ (denoted $\tilde{P}_Z$) via the musical isomorphism.
   • By restricting attention to a six-dimensional slice defined by time translation invariance and energy scaling ($x_1=0, x_5=-1$), they define a cross-section $X$.
   • The set $X$ is decomposed into three components ($U_E, U_N, U_S$) based on the polar angle $\theta$.
   • Using the Seifert–van Kampen theorem, the fundamental group $\pi_1(X)$ is calculated to be $\mathbb{Z}_2$.
   • Applying classification results for closed three-manifolds, the authors identify $X$ as the lens space $L(2,1)$, which is diffeomorphic to $SO(3)$. Consequently, the full region is shown to have the topology $SO(3) \times \mathbb{R}^2$.

Key Contributions and Results

• Main Theorem: The paper proves that in both sub-extremal Kerr-Newman and Kerr-Sen spacetimes, the image of the trapped photon region under the canonical map to the tangent bundle is a five-dimensional submanifold with topology $SO(3) \times \mathbb{R}^2$.
• Unified Treatment: The work provides a single proof structure applicable to both Einstein-Maxwell and Einstein-Maxwell-Dilaton-Axion theories, generalizing previous results limited to the Kerr metric.
• Technical Verification: The authors explicitly verify the surjectivity of the differential maps for these specific spacetimes, addressing the distinct functional forms of $f(r)$ and $K(r)$ which differ from the Kerr case. They also establish the non-existence of zero-energy trapped photons in the DOC for these metrics, a prerequisite for the normalization of constants used in the proof.

Significance and Applications
The paper positions this result as a foundational contribution to mathematical relativity. The authors state that establishing the precise geometry and topology of the trapped photon region is relevant to:

• Black hole uniqueness theorems.
• The dynamical stability of black holes.
• The study of black hole shadows.
• Gravitational lensing phenomena.

The work serves as an overview of results intended for a full paper, demonstrating that the structural properties of trapped photons in the Kerr spacetime extend to more complex, charged, and dilaton-coupled black hole solutions.