Kerr Generating Functions in Black Hole Physics

• Kerr generating functions are analytic constructs that systematically derive geodesic motion and generate rotating black hole metrics from closed-form equations.
• They employ techniques like the Biermann–Weierstrass theorem, Kerr–Schild ansatz, and Newman–Janis method to achieve explicit solutions and higher-dimensional generalizations.
• Through isohomogeneous transformations, these functions reconcile black hole thermodynamics with the Smarr formula and facilitate mapping to dual quantum field correlators.

Kerr generating functions are central analytic constructs and algorithmic techniques that enable the systematic derivation and characterization of both the geometry and physical observables in rotating black hole spacetimes. They appear in three interconnected domains: (1) the analytic solution of geodesic equations, (2) the generation of Kerr (and higher-dimensional generalizations) metrics, and (3) the construction of black hole thermodynamics. These generating functions are foundational for implementing explicit solutions, formulating correlation functions, and understanding deep symmetries in black hole physics.

1. Analytic Generating Functions in Kerr Geodesics

A key facet of Kerr generating functions is their use in providing closed-form expressions for timelike and null geodesics. The Biermann–Weierstrass theorem, as employed by Słik, Hackmann, and Mach, is applied to the quartic radial and polar equations governing geodesic motion in Kerr spacetime (Bakun et al., 5 Sep 2024). The effective radial potential is written as

$$\tilde{R}(\xi) = a_0\xi^4 + 4a_1\xi^3 + 6a_2\xi^2 + 4a_3\xi + a_4,$$

where the coefficients $a_i$ encapsulate the constants of motion (energy $\epsilon$, angular momentum $\lambda_z$, Carter constant $\kappa$).

Defining Weierstrass invariants,

$$g_2 = a_0 a_4 - 4 a_1 a_3 + 3 a_2^2, \ g_3 = a_0 a_2 a_4 + 2 a_1 a_2 a_3 - a_2^3 - a_0 a_3^2 - a_1^2 a_4,$$

the solution for the radial coordinate $\xi(s)$ is then generated by the Biermann–Weierstrass formula in terms of the Weierstrass $\wp$-function and its derivative. This explicit analytic generating function applies uniformly in both Boyer–Lindquist and horizon-penetrating coordinates, facilitating smooth geodesic evolution across event and Cauchy horizons. The generating function approach extends directly to the polar equation via the substitution $\mu = \cos\theta$ in similar quartic form.

2. Metric-Generating Algorithms: Kerr–Schild Form and Newman–Janis Method

Metric-generating functions are formulaic constructs that generate rotating black hole metrics from static seeds. The Kerr–Schild ansatz,

$g_{ab} = \eta_{ab} + f(r_0) \ell_a \ell_b,$

where $\eta_{ab}$ is the flat metric, $f$ is a scalar “potential” function, and $\ell_a$ is a null vector, is a canonical example (Tavakoli et al., 2020). Rotation is encoded by shifting the seed coordinates through complex or quaternionic transformations.

The Newman–Janis method, and its generalizations, implement this by complexifying the radial parameter and performing coordinate transformations that encode multiple angular momenta for higher-dimensional metrics. For instance, the generating function for the five-dimensional Myers–Perry metric with equal angular momenta is achieved by setting

$$x_i + i y_i = (r + i a) \mu_i e^{i \phi_i}, \quad \sum_i \mu_i^2 = 1,$$

and transforming the potential as $r_0^2 \rightarrow r^2 + a^2$. The generated metric retains Kerr–Schild form, generalizable to arbitrary odd dimensions with equal angular momenta. Thus, Kerr–Schild and Newman–Janis–type generating algorithms are pivotal for the construction of higher-dimensional rotating solutions.

3. Generating Functions for Black Hole Thermodynamics

Thermodynamic generating functions provide systematic frameworks for constructing families of thermodynamic potentials and variables for Kerr–anti–de Sitter (KadS) black holes (Campos et al., 12 Jul 2024). Taking an original potential $M_0$, homogeneous of degree $r$ in its variables, isohomogeneous transformations define a new potential

$M_1 = g(S, J, P) M_0,$

with $g$ homogeneous of degree zero, and corresponding transformed temperature and conjugate variables. The transformation preserves Euler’s theorem and the Smarr formula structure,

$$r M_1 = a_1 S T_1 + a_2 J Y_2' + a_3 P Y_3' + \cdots,$$

by ensuring $$a_1 S (\partial g/\partial S) + a_2 J (\partial g/\partial J) + a_3 P (\partial g/\partial P) = 0$$.

The geometric interpretation is as a rescaling of the Killing field generating the horizon, $K_H \to g K_H$. Isohomogeneous transformations thus act as generators mapping between distinct but scale-invariant thermodynamic theories—rectifying shortcomings in early proposals (e.g., Hawking’s original KadS energy, which satisfies Smarr but not the first law) and enabling alternative consistent thermodynamic formalisms.

4. Kerr Generating Functions in Quantum and Holographic Contexts

In the context of Kerr/CFT correspondence, generating functions appear as analytic constructs mapping gravitational observables to dual conformal field theory (CFT) correlation functions (Becker et al., 2010). Specifically, scalar three-point correlators in the near-NHEK geometry factorize due to extremality $(h_3 = h_1 + h_2)$,

$$\langle O_{h_1} O_{h_2} O_{h_3}\rangle \sim \frac{1}{h_3 - h_1 - h_2} \times (\text{two-point factors}),$$

with bulk integrals over normalizable mode coefficients $B_1$, $B_2$ exhibiting the divergence

$\sim \frac{B_1 B_2}{h_3 - h_1 - h_2},$

necessitating vanishing cubic couplings, $\lambda_{\text{extremal}} \propto h_3 - h_1 - h_2$, to preserve physical finiteness and conformal invariance.

The universal recipe for retarded finite-temperature correlation functions employs bulk-to-boundary propagators and analytic continuation between thermal states. The scalar wavefunctions’ asymptotic expansions act as generating functions for gravitational response, mapping directly onto dual CFT correlator structures, and mirror results in AdS/CFT for other backgrounds.

5. Implications and Applications of Kerr Generating Functions

Kerr generating functions facilitate both analytic and algorithmic advances in black hole physics:

• Geodesic Solutions: They provide explicit analytic solutions for geodesic motion across horizons, critical for gravitational lensing, accretion dynamics, and gravitational wave modeling.
• Metric Generation: They enable efficient construction of rotating black hole metrics and their higher-dimensional analogues, expanding the class of tractable models for general relativity and string theory.
• Thermodynamics: Isohomogeneous transformations serve as engines for generating new black hole thermodynamic descriptions, reconciling geometric and statistical mechanical formulations.
• Quantum Gravity and Holography: They underpin the direct mapping of classical gravity observables to quantum field theoretic correlators, clarifying extremal coupling constraints and anomalies.

The extension of generating function approaches to arbitrary dimensions and backgrounds provides a robust, scalable methodology for both theoretical investigations and concrete model-building in black hole physics and related quantum gravity research.

6. Technical Summary Table

Domain	Generating Function Formulation	Main Outcome
Geodesic Solutions	Weierstrass $\wp$-function applied to quartic equations	Globally analytic geodesic trajectories
Metric Generation	Kerr–Schild potential + coordinate transformation	Rotating black hole metric in arbitrary (odd) dimensions
Thermodynamics	Isohomogeneous transformations $M_1 = g M_0$	Consistent Smarr formula and first law
Quantum/Holography	Bulk-to-boundary propagation & CFT mapping	Structure of correlators, anomaly constraints

Each of these domains leverages specific forms, analytic expansions, or transformation rules as Kerr generating functions for the systematic derivation of physical properties and mathematical structures.

7. Perspectives and Connections

Kerr generating functions unify disparate strands of analytic solution theory, metric generation, and thermodynamic construction under a framework of homogeneity, analytic inversion, and coordinate regularization. Their appearance across geodesic, metric, and thermodynamic contexts demonstrates that generating functions serve as the backbone for explicit, scalable, and physically consistent results in spinning black hole physics. The ability to smoothly interpolate between geometric, analytic, and field-theoretic domains ensures broad applicability and effective integration with contemporary quantum gravity, astrophysics, and holographic methodologies.