Lagrangian Hamiltonian Modeling and Orbital Stability Analysis of Constrained Particle Dynamics on Rotational Surfaces in the Pseudo-Euclidean Space ${\mathit{E}}_{\mathbf{2}}^{\mathbf{4}}$

Abstract

This paper investigates the constrained particle dynamics on rotational surfaces within the 4-dimensional pseudo-Euclidean space $E_2^4$, characterized by its second-order metric signature of index 2. A comprehensive Lagrangian and Hamiltonian formulation is developed to construct the specific energy and specific angular momentum as conserved Noetherian charges along timelike geodesics. By integrating Clairaut’s theorem into the geodesic flow equations, explicit analytical expressions for these fundamental physical invariants are obtained. This work explores the structural relationship between the surface’s continuous rotational symmetries and the mechanical stability of the geodesic flow. A mathematical resolution for the signature transitions manifested via the appearance of the imaginary unit i on elliptic surfaces is provided through analytic continuation and distinct coordinate charts. Furthermore, by reducing the second-order geodesic flow to a one-dimensional energy balance equation, the exact effective potentials ($V_{eff}$) are derived, and the local orbital stability zones are analytically verified via second-order radial derivatives (${\partial }_s^2{V}_{eff}>0$). These embedded geometric configurations are shown to share qualitative features with the equatorial slices of rotating relativistic spacetimes. Consequently, they can serve as potential geometric toy-models for studying the dynamics of photon spheres, ergosphere oscillations, and innermost stable circular orbits in extreme gravitational fields.

1. Introduction

The interplay between semi-Riemannian geometry and Lagrangian mechanics, and Hamiltonian bundle dynamics provides a rigorous framework for modeling relativistic particle dynamics. In higher-dimensional pseudo-Euclidean spaces, such as $E_2^4$, the existence of multiple timelike directions significantly complicates the geodesic flow compared to classical Riemannian settings. While geodesic paths are traditionally viewed as shortest paths in pure geometry, they fundamentally represent the extremals of the relativistic action functional in mathematical physics. The foundational groundwork for examining these explicit kinematic parameters and specific dynamic constraints on rotational frameworks within index-2 geometries was initially conceptualized in the preliminary physical approach outlined by Almaz [1]. Understanding the stability of these paths requires an analysis of the constants of motion derived directly from the continuous isometries of the underlying manifold.

The choice of the 4-dimensional pseudo-Euclidean space $E_2^4$ occupies a pivotal role in advanced theoretical physics, particularly in modified gravity, string theory, and analogue gravity models. Manifolds with split-signature (n,n) naturally arise in the context of Two-Time Physics (2T-physics) pioneered by Bars. This framework provides a gauge-invariant method for unifying disparate single-time dynamical systems into a higher-dimensional unified theory. Furthermore, split signatures are essential for formulating N = 2 supersymmetric string worldsheets, characterizing Kleinian self-dual spacetimes in quantum gravity models, and evaluating braneworld scenarios where additional timelike dimensions modify the causal boundaries and horizon behaviors of higher-dimensional gravity. Recent developments between 2020 and 2025 have further highlighted the importance of geodesic flows and stability conditions on higher-index pseudo-Riemannian manifolds, emphasizing spectral analysis, eigenfunction asymptotics, and the complex geometric behavioral transitions manifested along space-like and time-like geodesic paths on index-2 domains [2,3].

Rotational surfaces in $E_2^4$ have been classified in previous studies primarily from a kinematic perspective. For instance, in [4], timelike geodesics were expressed on specific rotational surfaces, and in [5], different types of these surfaces were defined using Killing vector fields. However, these prior works focused largely on the static geometric properties, such as classification and existence, rather than the dynamic invariants governing particle motion.

To date, the key prior studies focusing on the geodesic geometry of $E_2^4$ or related split-signature spaces have predominantly relied on pure Hamiltonian formalisms on the cotangent bundle to classify the integrable trajectories of geodesic flows on static space forms. While these traditional Hamiltonian frameworks provide valuable algebraic classifications of vector fields and static existence proofs, they often abstract away the physical aspect of geometric invariants as dynamic charges, leaving a clear research gap regarding the explicit, localized stability of constrained particle orbits. This study explicitly addresses this gap by presenting a complementary, robust variational alternative rooted firmly in Lagrangian mechanics.

A central challenge in this analysis is the rigorous definition of “specific” quantities within a semi-Riemannian framework. Unlike classical Newtonian systems where mass acts as a simple linear scaling factor, in the geodesic flow of a pseudo-Euclidean 4-space, physical quantities traditionally dependent on the rest mass m are reformulated by normalizing the mass parameter. This reformulation allows the geodesic flow to be treated as a purely geometric system on the tangent bundle. The specific energy ($E_{specific}$) and specific angular momentum ($l$) are formally defined as the Noether charges corresponding to time-translation and rotational invariance, respectively. Rather than relying on classical kinetic expressions, these specific invariants are established as the scalar projections arising from the inner product of the surface’s Killing vector fields and the particle’s four-velocity vector. This Noetherian approach ensures that the specific energy $E_{specific}$ is invariant along geodesics, acting as a direct result of the surface’s one-parameter group of isometries.

Crucially, this geometric formulation establishes a qualitative analogy with the foundational frameworks of General Relativity. In the study of rotating gravitational fields and black hole dynamics, the isolation of explicit physical invariants along particle orbits relies on similar Killing vectors and analytic metric extensions, as classically demonstrated by Boyer and Lindquist [6] and Carter [7], whose pioneering formulations mapped the global causal structure and separable geodesic trajectories of the rotating Kerr field. Furthermore, the systematic reduction of complex particle paths to specific energy profiles and conserved angular momentum parameters on bounded geometric horizons mirrors the locally nonrotating frames and energy extraction formalisms pioneered by Bardeen, Press, and Teukolsky [8]. Although tube surfaces and double rotations in Minkowski space [9,10] or the static geometry of surface normals [11] have been explored, this work aims to bridge the gap between these abstract surface theories [12,13,14,15] and relativistic dynamics [8,16,17,18] by applying Clairaut’s theorem within a dynamical context in $E_2^4$, utilizing foundational semi-Riemannian definitions. To fully encapsulate these multifaceted relativistic paradigms within a comprehensive mathematical physics framework, it is imperative to also ground these geometric transitions in the authoritative formulations of general relativity and modern field theory principles as expounded by Walecka [19,20].

This work substantially expands and deepens the initial conceptual framework presented in my preliminary preprint [1].

Compared to the aforementioned static geometric classifications and abstract Hamiltonian analyses, the physical aspect of geometric invariants as dynamic observables is examined here through a comprehensive Lagrangian perspective to establish a clear structural parallelism between geometric configurations and active mechanical constraints.

The primary geometric contributions of this work to the specific class of index-2 surfaces are summarized as follows,

✓

Based on Noether’s theorem, systematically deriving explicit analytical expressions for the specific energy $E_{specific}$ and specific angular momentum $l$ directly on index-2 rotational surfaces ($S_{14}$, $S_{23}$, and $S_{56}$).

✓

Analyzing the complicated second-order geodesic flow equations by reducing them to a completely solvable, decoupled one-dimensional effective potential balance equation.

✓

Investigating localized orbital stability zones via second-order radial derivatives (${\partial }_s^2{V}_{eff}>0$), thereby offering a theoretical baseline for analogue gravity models and geometric throat crossing simulations.

2. Preliminaries

The geometry of the 4-dimensional pseudo-Euclidean space $E_2^4$ is characterized by the metric ${⟨,⟩}_{E_2^4}$ with a signature (2,2), defined by the matrix:

$$g=\left[\begin{array}{cccc}-1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right].$$

For any vector $v\in E_2^4,$ its causal character is determined by the sign of $g\left(v,v\right)$, it can be space-like if $g(v,v)>0$ or $v=0,$ time-like if $g(v,v)<0$ and null if $g(v,v)=0$ and $v\ne 0$, the norm of a vector $v$ is given by $\parallel v\parallel =\sqrt{\left|g(v,v)\right|}$. A curve $x\left(s\right)$ is considered unit-speed if $g\left(x^{\prime },x^{\prime }\right)=\pm 1$.

Let $(x_1,x_2,x_3,x_4),(y_1,y_2,y_3,y_4),(z_1,z_2,z_3,z_4)$ be any three vectors in $E_2^4$. The pseudo-Euclidean cross product is given as

$$x\wedge y\wedge z=\left(\begin{array}{cccc}-e_1& -e_2& e_3& e_4\\ x_1& x_2& x_3& x_4\\ y_1& y_2& y_3& y_4\\ z_1& z_2& z_3& z_4\end{array}\right),$$

where $e_1=\left(1,0,0,0\right),e_2=\left(0,1,0,0\right),e_3=\left(0,0,1,0\right),e_4=\left(0,0,0,1\right)$, [9,11,14,17].

2.1. Lie Derivatives and Killing Fields

In the context of constrained dynamical systems on semi-Riemannian manifolds, the conservation of physical quantities along worldlines is fundamentally dictated by the continuous symmetries of the background metric. Let $(M,g)$ be a 4-dimensional pseudo-Euclidean manifold $E_2^4$, equipped with a metric tensor $g$ of signature (2,2). To establish a rigorous tensor formulation, one defines a local coordinate chart $\left\{x^{\mu }\right\}$ on $M$ such that the coordinates are explicitly mapped to the geometric axes as $x^1=\xi ,x^2=\varrho ,x^3=\vartheta ,x^4=\eta$, with indices $\eta ,v,\lambda \in \left\{1,2,3,4\right\}.$ Let $W=W^{\mu }{\partial }_{\mu }$ be a smooth vector field on $M$ that generates a local one-parameter group of diffeomorphisms (a local flow) ${\phi }_t:M\to M.$ The Lie derivative of the metric tensor $g$ with respect to the vector field $W$, denoted by ${\mathcal{L}}_{W}g$, measures the infinitesimal change in the metric configuration under this flow. In terms of local coordinates, the components of the Lie derivative tensor are explicitly defined as:

$${{(\mathcal{L}}_{W}g)}_{\mu v}=W^{\lambda }{\partial }_{\lambda }{g}_{\mu v}+g_{\lambda v}{{\partial }_{\mu }W}^{\lambda }+g_{\mu \lambda }{{\partial }_{v}W}^{\lambda }$$

where ${\partial }_{\lambda }={\displaystyle \frac{\partial }{\partial x^{\lambda }}}$ denotes the standard partial differentiation operator with respect to the coordinate basis.

A vector field $W$ is formally defined as a Killing vector field if the metric configuration remains strictly invariant under the localized geometric flow generated by $W$. This invariance condition requires the Lie derivative to vanish identically across the entire domain, yielding the classical Killing equation:

$${\mathcal{L}}_{W}g=0\leftrightarrow {{(\mathcal{L}}_{W}g)}_{\mu v}=0.$$

By lowering the upper index of the vector field components via the relation $W_{\mu }=g_{\mu \lambda }{{\partial }_{v}W}^{\lambda }$, and employing the unique, torsion-free Levi-Civita connection $\nabla$ intrinsically associated with the metric tensor $g$, the Killing equation can be rewritten in its standard covariant form as ${\nabla }_{\mu }{W}_v+{\nabla }_v{W}_{\mu }=0$. In the pseudo-Euclidean space $E_2^4$, these Killing vector fields serve as the infinitesimal generators of the underlying isometry group, [12,13,16].

To establish a formal bridge between the metric symmetries of $E_2^4$ and the physical constants of motion (Noetherian charges), the fundamental geometric properties and geodesic relations of rotational surfaces from previous classifications [4,5] are recalled. These geometric frameworks and the Killing vector generators ${\mathsf{\Omega }}_i(i=1,2,\dots ,6)$ serve as the mathematical foundation for the subsequent Lagrangian analysis.

2.2. The Relativistic Lorentz Isometry Group O(2,2) and Its Lie Algebra

From a standard relativistic field-theoretic perspective, the global geometry of the 4-dimensional pseudo-Euclidean space $E_2^4$ with metric signature $(2,2)$ is fundamentally invariant under the generalized Lorentz group $O(2,2)$. The continuous symmetries of this manifold are governed by the connected component of the identity, the pseudo-orthogonal Lie group $S{O}^{+}(2,2)$, whose algebraic structure dictates the conservation laws of relativistic particle dynamics.

The infinitesimal generators of $O\left(2,2\right)$ form the pseudo-orthogonal Lie algebra $so\left(2,2\right)$. This algebra is a 6-dimensional real Lie algebra spanned by six independent Killing vector fields, which physically correspond to the generators of generalized rotations and hyperbolic boosts across the two timelike and two spacelike dimensions. In terms of the standard coordinate chart, let the basis operators be defined as $e_{\xi }={\partial }_{\xi },e_{\varrho }={\partial }_{\varrho },e_{\vartheta }={\partial }_{\vartheta },e_{\eta }={\partial }_{\eta }$, the canonical basis elements of the Lie algebra $so\left(2,2\right)$ are explicitly given by the following six angular momentum and boost operators ${\mathsf{\Omega }}_a\left(a=1,2,\dots ,6\right)$:

Hyperbolic boosts (timelike-spacelike mixing):

$${\mathsf{\Omega }}_1=\vartheta e_{\xi }+\xi e_{\vartheta }, {\mathsf{\Omega }}_2=\eta e_{\xi }+\xi e_{\eta }, {\mathsf{\Omega }}_3=\vartheta e_{\varrho }+\varrho e_{\vartheta }, {\mathsf{\Omega }}_4=\eta e_{\varrho }+\varrho e_{\eta }.$$

Spatial rotation in the $\xi \varrho$-plane & Timelike rotation in the $\vartheta \eta$-plane:

$${\mathsf{\Omega }}_5=\xi e_{\varrho }-\varrho e_{\xi }; {\mathsf{\Omega }}_6=\vartheta e_{\eta }-\eta e_{\vartheta }.$$

These generators satisfy the commutation relations $\left[{\mathsf{\Omega }}_a,{\mathsf{\Omega }}_b\right]=C_{ab}^c{\mathsf{\Omega }}_c,$ where $C_{ab}^c$ are the structure constants of the $so\left(2,2\right)$ algebra. Any general Killing vector field $W$ on $E_2^4$ that preserves the semi-Riemannian metric $g$ can be expressed as a linear combination of these basis elements, as formalized in Lemma 1.

Lemma 1. 

Let the pseudo-Euclidean group be a subgroup of the diffeomorphisms group in $E_2^4$ and let $W$ be vector field which generate the isometries. Then, the Killing vector field associated with the metric $g$ is given as

$$\begin{array}{cc}\hfill W(\xi ,\varrho ,\vartheta ,\eta )& =a\left(\eta e_{\xi }+\xi e_{\eta }\right)+b\left(\vartheta e_{\varrho }+\varrho e_{\vartheta }\right)+c\left(\vartheta e_{\xi }+\xi e_{\vartheta }\right)+d(\eta e_{\varrho }+\varrho e_{\eta })+e(\vartheta e_{\eta }\hfill \\ & -\eta e_{\vartheta })+f\left(\xi e_{\varrho }-\varrho e_{\xi }\right),\hfill \end{array}$$

where $a,b,c,d,e,f\in {\mathbb{R}}_0^{+},$ [5].

2.3. Rotational Surfaces and Geodesic Constants

Rotational surfaces in the pseudo-Euclidean space $E_2^4$ are geometrically constructed by applying the one-parameter subgroups of the pseudo-orthogonal isometry group O(2,2) to a planar profile curve $\gamma$(s). As established in geometric classifications, these surfaces comprise the hyperbolic configurations $S_{14}$, $S_{23}$ and elliptic configuration $S_{56}$. While prior geodesic equations for these manifolds were formulated purely in terms of static geometric parameters known as Clairaut’s constants (Lemmas 3–8), this study reinterprets these parameters as physical Noetherian charges. The kinematic relations formulated via the geodesic inclination angles ${\phi }_i$ and ${\theta }_i$ are shown to be directly proportional to the specific angular momentum of a relativistic test particle, as derived in the variational formulation.

Lemma 2. 

Let $W(\xi ,\varrho ,\vartheta ,\eta )$ be a general the Killing vector field spanning the $so\left(2,2\right)$ algebra, and let $\gamma$ be a planar profile curve in $E_2^4$. The embedded rotational surfaces generated by the continuous action of the Lie algebra generators are expressed as follows:

(a)

For the hyperbolic boosts generated by ${\mathsf{\Omega }}_1=\vartheta e_{\xi }+\xi e_{\vartheta }$ and ${\mathsf{\Omega }}_4=\eta e_{\varrho }+\varrho e_{\eta },$ acting on the profile curve $\gamma \left(s\right)=\left(f_1\left(s\right),0,0,f_4\left(s\right)\right),$ the hyperbolic rotational surface $S_{14}$ is parameterized as:

$$S_{14}\left(x\right(t),\alpha (t),s)=\left(f_{1}coshx,f_{4}sinh\alpha ,f_{1}sinhx,f_{4}cosh\alpha \right),$$

(b)

For the hyperbolic boosts generated by ${\mathsf{\Omega }}_2=\eta e_{\xi }+\xi e_{\eta }$ and ${\mathsf{\Omega }}_3=\vartheta e_{\varrho }+\varrho e_{\vartheta },$ acting on the profile curve $\gamma \left(s\right)=\left(f_1\right(s),f_2(s),0,0)$ the hyperbolic rotational surface $S_{23}$ is parameterized as:

$$S_{23}\left(y\left(t\right),z\left(t\right),s\right)=\left(f_{1}coshy,f_{2}coshz,f_{2}sinhz,f_{1}sinhy\right),$$

(c)

For the rotations generated by ${\mathsf{\Omega }}_5=\xi e_{\varrho }-\varrho e_{\xi }$ and ${\mathsf{\Omega }}_6=\vartheta e_{\eta }-\eta e_{\vartheta },$ acting on the profile curve $\gamma \left(s\right)=\left(0,f_2\left(s\right),0,f_4\left(s\right)\right),$ the elliptic rotational surface $S_{56}$ is parameterized as:

$$S_{56}(\beta \left(t\right),\theta \left(t\right),s)=\left(f_{2}sin\beta ,f_{2}cos\beta ,f_{4}sin\theta ,f_{4}cos\theta \right),$$

where $-\mathrm{\infty }<x,y,z,\alpha ,\beta ,\theta <\mathrm{\infty },$ $s\in I,$ and the profile functions satisfy $f_i\in C^{\mathrm{\infty }}$, [5].

The structural coefficients of these embedded metrics directly dictate the particle’s velocity field within the kinetic framework, where the cyclic spatial coordinates ($x,y,z,\alpha ,\beta ,\theta$) define the explicit conserved specific angular momenta.

Lemma 3. 

Let $\gamma \left(s\right)$ be a unit-speed timelike geodesic curve on the hyperbolic surface of revolution $S_{14}$ in the $E_2^4$, where $f_1\left(s\right)$ and $f_4\left(s\right)$ represent the distance functions from the respective rotation axes. The geometric quantities $2f_{1}cos{\phi }_1$ and $-2f_{4}cosh{\theta }_{1}sin{\phi }_1$ are strictly invariant along the worldline $\gamma$, where ${\phi }_1$ and ${\theta }_1$ denote the relativistic inclination angles between the surface meridians and the tangent velocity vector of the geodesic flow, [4].

Lemma 4. 

The covariant differential equations governing the geodesic flow on the hyperbolic rotational surface $S_{14}$ in the $E_2^4$, for the kinematic parameters $\dot{x}={\displaystyle \frac{1}{f_1}}cos{\phi }_1$ and $\dot{\alpha }={\displaystyle \frac{1}{f_4}}cosh{\theta }_{1}sin{\phi }_1$, are analytically expressed as:

$${\displaystyle \frac{dt}{dx}}=f_1\sqrt{1-{cosh}^2{\theta }_1{tan}^2{\phi }_1-L{sec}^2{\phi }_1}$$

or

$${\displaystyle \frac{dt}{d\alpha }}=f_2\sqrt{{cot}^2{\phi }_{1}tan{h}^2{\theta }_1-L|sec{h}^2{\phi }_1|cose{c}^2{\phi }_1},$$

[4].

Lemma 5. 

Let $\gamma \left(s\right)$ be a unit-speed timelike geodesic curve on the hyperbolic surface of revolution $S_{23}$ in the $E_2^4,$ where $f_1\left(s\right)$ and $f_2$(s) represent the distance functions from the rotation axes. Then, the parameters $2f_{1}cos{\theta }_{2}sinh{\phi }_2$ and $2f_{2}sin{\theta }_{2}sinh{\phi }_2$ are invariant along the worldline $\gamma$, where ${\phi }_2$ and ${\theta }_2$ denote the localized angles between the surface meridians and the geodesic trajectory, [2].

Lemma 6. 

The covariant equations of geodesics on the hyperbolic rotational surface $S_{23}\subset E_2^4,$ governed by the coordinate parameters $\dot{y}={\displaystyle \frac{cos{\theta }_{2}sinh{\phi }_2}{f_1}}$ and $\dot{z}={\displaystyle \frac{sinh{\phi }_{2}sin{\theta }_2}{f_2}}$, are explicitly given by:

$${\displaystyle \frac{dt}{dx}}={\displaystyle \frac{f_1}{cos{\theta }_{2}sinh{\phi }_2}}\sqrt{{sinh}^2{\phi }_2-L};{\displaystyle \frac{dt}{dz}}={\displaystyle \frac{f_2}{sinh{\phi }_{2}sin{\theta }_2}}\sqrt{{sinh}^2{\phi }_2-L},$$

[4].

Lemma 7. 

Let $\gamma \left(s\right)$ be a unit-speed timelike geodesic curve on the elliptic surface of revolution $S_{56}\subset E_2^4$, where $f_2$(s) and $f_4$(s) represent the structural distance functions. The quantities $2f_{2}sin{\phi }_{3}cosh{\theta }_3$ and $2f_{4}sinh{\theta }_{3}sin{\phi }_3$ remain strictly constant along the worldline, where ${\phi }_3$ and ${\theta }_3$ characterize the geometric angles relative to the elliptic meridians, [2].

Lemma 8. 

The general equations of motion for the geodesic flow on the elliptic rotational surface $S_{56}\subset E_2^4$, evaluated for the coordinate parameters $\dot{\beta }={\displaystyle \frac{sin{\phi }_{3}cosh{\theta }_3}{f_2}}$ and $\dot{\upsilon }={\displaystyle \frac{sinh{\theta }_{3}sin{\phi }_3}{f_4}}$, are given by the complex coordinate expressions:

$${\displaystyle \frac{dt}{d\beta }}=i{\displaystyle \frac{f_2\sqrt{L+{sin}^2{\phi }_3}}{sin{\phi }_{3}cosh{\theta }_3}};{\displaystyle \frac{dt}{d\upsilon }}=i{\displaystyle \frac{f_4}{sinh{\theta }_{3}sin{\phi }_3}}\sqrt{{sin}^2{\phi }_3+L},$$

[4].

In the relativistic variational formulation of this geodesic flow, the parameter $L$ acts as the specific Lagrangian density of the test particle.

Note: The formal emergence of the imaginary unit $i$ in Lemma 8 and the subsequent dynamical equations does not indicate unphysical or tachyonic trajectories. Instead, it represents a non-singular coordinate chart transition induced by the index-2 metric signature of $E_2^4.$ In semi-Riemannian geometry, when a test particle crosses a domain where the norm of the Killing vector field changes sign, the induced surface metric undergoes a signature transition. This phenomenon shares qualitative similarities with the coordinate swap between temporal and spatial components when crossing the analogue horizon of a Schwarzschild black hole via a classic Wick rotation. Due to the high-index nature of the space, the imaginary unit $i$ tracks the causal transitions between different geometric regions of the embedded rotational surface.

3. Formulation and Noetherian Dynamics in ${\mathit{E}}_{\mathbf{2}}^{\mathbf{4}}$

In this section, geodesic motion on rotational surfaces is treated as a dynamical system through the extremization of an action functional. Geodesics are reinterpreted as particle trajectories within a conservative system. This formulation establishes a link between the metric symmetries of $E_2^4$ and the Noetherian constants of motion. The dynamical analysis is systematically performed across three distinct configurations: the hyperbolic rotational surfaces ($S_{14}$, $S_{23}$) and the elliptic rotational surface ($S_{56}$).

3.1. Dynamics on the Hyperbolic Surface of Rotation S₁₄

Theorem 1. 

Let ${{\rm Y}}^1\left(x,\alpha ,t\right)$ be a unit-speed timelike geodesic curve on the hyperbolic rotational surface $S_{14}$ in the pseudo-Euclidean space $E_2^4$, where $f_1\left(s\right)$ and $f_4\left(s\right)$ denote the structural distance functions from the respective rotation axes The dynamical evolution of a test particle constrained to this manifold is governed by two fundamental Noetherian invariants, formulated in terms of the geodesic inclination angles ${\phi }_1$ and ${\theta }_1$ as follows:

✓

The specific Angular Momentum:

$$l_1=2V_{1}sinh{\theta }_{1}sin{\phi }_1.$$

✓

The total specific Energy:

$$E_{specific}=-{\displaystyle \frac{V_1^2}{2}}.$$

Alternatively, by isolating the geometric phase-space trajectories, this specific energy invariant can be represented in terms of the directional coupling parameters as:

$$E_{specific}=-{\displaystyle \frac{V_1^2}{2}}\left({cos}^2{\phi }_1-{cosh}^2{\theta }_1{sin}^2{\phi }_1\right)+{\displaystyle \frac{l_1^2}{8}}.$$

In this dynamical setting, the specific kinetic energy remains invariant along the geodesic flow such that the velocity magnitude $V_1$ is constant. Due to the preservation of the timelike causality condition, the negative contributions of the hyperbolic inclination vectors balance the shift generated by the angular momentum field. This reduction allows the second-order geodesic equations to be modeled as a first-order conservative system.

To analyze the multi-variable phase space and justify the dual angular degrees of freedom, we apply a Legendre transformation to the quadratic energy Lagrangian ${\mathcal{L}}_2^1={\displaystyle \frac{1}{2}}\left({\left(f_1\dot{x}\right)}^2-{\left(f_4\dot{\alpha }\right)}^2+{\left(\dot{t}\right)}^2\right)$, mapping the trajectories onto an explicit canonical Hamiltonian framework:

$$H=\sum p_i{\dot{x}}^i-\mathcal{L}={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{p_x^2}{f_1^2}}-{\displaystyle \frac{p_{\alpha }^2}{f_4^2}}+p_t^2\right).$$

Because the continuous rotational symmetries of the background metric ensure that ${\displaystyle \frac{\partial H}{\partial x}}=0$ and ${\displaystyle \frac{\partial H}{\partial \alpha }}=0$, the generalized coordinates act as strictly cyclic variables. Their corresponding conjugate momenta collapse to absolute constants of motion ($p_i\equiv \dot{t}={\displaystyle \raisebox{1ex}{$l_1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$). This mathematical property allows us to completely integrate out the angular degrees of freedom ($\theta$-motion), rigorously reducing the multi-dimensional geodesic flow to a decoupled, exact 1 degree of freedom (1-DOF) radial problem in terms of the arc-length parameter $s$.

Proof of Theorem 1. 

To derive the conserved specific invariants on the hyperbolic surface of revolution $S_{14}$, one evaluates the worldline trajectories of a test particle via the principle of stationary action. Let ${{\rm Y}}^1(x\left(\lambda \right),\alpha \left(\lambda \right),t(\lambda ))$ be a parametrized curve on the surface, where $\lambda$ represents a general non-affine parameter. The invariant proper-time interval along the worldline is defined by the metric speed integration:

$$S_1=\int ds=\int \frac{ds}{d\lambda }d\lambda =\int \sqrt{-g_{\mu v}\frac{d{x}^{\mu }}{d\lambda }\frac{d{x}^v}{d\lambda }}d\lambda .$$

To minimize this path without the algebraic singularities induced by the square-root formulation, one defines the standard relativistic action functional $S_2$ via the quadratic energy Lagrangian ${\mathcal{L}}_2^1$, which shares identical extremal paths for affine parameterizations:

$$S_2={\displaystyle \frac{1}{2}}\int {\mathcal{L}}_2^{1}ds={\displaystyle \frac{1}{2}}\int \left({\left(f_1\dot{x}\right)}^2-{\left(f_4\dot{\alpha }\right)}^2+{\left(\dot{t}\right)}^2\right)ds,$$

where the overdot denotes covariant differentiation with respect to the affine arc-length parameter s$(\dot{x}$ $\equiv {\displaystyle \frac{dx}{ds}}, \dot{\alpha }$ $\equiv {\displaystyle \frac{d\alpha }{ds}}, \dot{t}$ $\equiv {\displaystyle \frac{dt}{ds}}$), and the intrinsic line element velocity $V_1$ remains constant along the geodesic flow. Because the background surface configuration exhibits rotational and time-translation invariance, the energy Lagrangian ${\mathcal{L}}_2^1$ is explicitly cyclic with respect to the temporal coordinate t, implying ${\displaystyle \frac{\partial {\mathcal{L}}_2^1}{\partial t}}=0$. The relativistic equations of motion are governed by the canonical Euler–Lagrange equations:

$${\displaystyle \frac{d}{ds}}\left({\displaystyle \frac{\partial {\mathcal{L}}_2^1}{\partial {\dot{x}}^{\mu }}}\right)-{\displaystyle \frac{\partial {\mathcal{L}}_2^1}{\partial x^{\mu }}}=0.$$

The generalized conjugate momentum corresponding to the cyclic time coordinate yields the strict conservation of the Noetherian charge. Applying the canonical equation explicitly for the coordinate $t$ gives:

$${\displaystyle \frac{d}{ds}}\left({\displaystyle \frac{\partial {\mathcal{L}}_2^1}{\partial \dot{t}}}\right)-{\displaystyle \frac{\partial {\mathcal{L}}_2^1}{\partial \mathrm{t}}}=0\to {\displaystyle \frac{d}{ds}}\left({\displaystyle \frac{\partial {\mathcal{L}}_2^1}{\partial \dot{t}}}\right)=0\to {\displaystyle \frac{\partial {\mathcal{L}}_2^1}{\partial \dot{t}}}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}.$$

To maintain notation consistency across the manuscript and match the exact scalar fields, we equate this canonical momentum derivative to the integration baseline constant, defining the specific angular momentum $l_1$:

$${\displaystyle \frac{\partial {\mathcal{L}}_2^1}{\partial \dot{t}}}=+\dot{t}\equiv +{\displaystyle \frac{l_1}{2}}\to \dot{t}={\displaystyle \frac{l_1}{2}}.$$

By applying the geometric chain rule, the tangent velocity vector along the geodesic path is decomposed into the coordinate basis vectors {$e_x,e_{\alpha },e_t\}$ associated with the surface embedding as:

$${\displaystyle \frac{d{{\rm Y}}^1\left(x\left(s\right),\alpha \left(s\right),t\left(s\right)\right)}{ds}}=\dot{\gamma }=\dot{x}{e}_x+\dot{\alpha }{e}_{\alpha }+\dot{t}{e}_t$$

and the velocity vector components in the tangent plane are identified as:

$$\overrightarrow{V_1}=\frac{\partial {{\rm Y}}^1}{\partial s}=V_1^x{e}_x+V_1^{\alpha }{e}_{\alpha }+V_1^t{e}_t.$$

Matching this geodesic flow with the orientation inclination angles ${\phi }_1$ and ${\theta }_1$ yields the explicit components of the velocity field in the tangent bundle:

$$\dot{\gamma }=V_{1}cos{\phi }_1{e}_x+V_{1}cosh{\theta }_{1}sin{\phi }_1{e}_{\alpha }+V_{1}sinh{\theta }_{1}sin{\phi }_1{e}_t.$$

Projecting these terms onto the respective radial and transverse physical directions, the corresponding velocity components are isolated as:

✓

Radial vertical velocity ($x$-axis): $V_1^x=f_1\dot{x}$ $=V_{1}cos{\phi }_1$

✓

Radial vertical velocity ($\alpha$-axis): $V_1^{\alpha }=f_4\dot{\alpha }$ $=V_{1}cosh{\theta }_{1}sin{\phi }_1$

✓

Horizontal angular component ($t$-axis): $V_1^t=\dot{t}=$ $V_{1}sinh{\theta }_{1}sin{\phi }_1.$

By directly matching the geometric horizontal angular velocity $\dot{t}=$ $V_{1}sinh{\theta }_{1}sin{\phi }_1$ with our derived variational constraint $\dot{t}={\displaystyle \frac{l_1}{2}}$, the exact specific angular momentum balance is explicitly established as:

$$l_1=2V_{1}sinh{\theta }_{1}sin{\phi }_1.$$

To rigorously derive the total specific energy $E_{specific}$ and its sign from first principles rather than introducing geometric identities ad hoc, we ground the formulation entirely in the foundational causality conditions of semi-Riemannian geometry. By definition, any physical test particle executing a unit-speed motion along a timelike worldline must satisfy the absolute metric constraint:

$$g\left(\dot{\gamma ,}\dot{\gamma }\right)=-1.$$

In the local coordinate chart $\{\xi ,\varrho ,\vartheta ,\eta \}$ the pseudo-Euclidean space $E_2^4$, the background metric is governed by the split-signature diagonal matrix $g=diag(-1,-1,1,1)$. Consequently, when the parametric derivatives of the embedded surface $S_{14}$ are mapped directly onto this manifold, the global line element collapses exactly to the following quadratic layout:

$$-{\dot{\xi }}^2-{\dot{\varrho }}^2+{\dot{\vartheta }}^2+{\dot{\eta }}^2=-1.$$

By substituting the structural profile parameterizations of Lemma 2 where the coordinate dimensions are actively regulated by the distance functions $f_1\left(s\right)$, $f_4\left(s\right)$ and the spatial-temporal rotations $x$, $\alpha$ the fully expanded velocity tensor model along the trajectory takes the explicit shape:

$$\begin{array}{c}-{\left(f_1\dot{x}sinhx+f_1^{\prime }\dot{s}coshx\right)}^2-{\left(f_4\dot{\alpha }cosh\alpha +f_4^{\prime }\dot{s}sinh\alpha \right)}^2+{\left(f_1\dot{x}coshx+f_1^{\prime }\dot{s}sinhhx\right)}^2\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} +{\left(f_4\dot{\alpha }sinh\alpha +f_4^{\prime }\dot{s}cosh\alpha \right)}^2=-1\hfill \end{array}$$

Upon executing the algebraic expansions, the cross-multiplication terms containing the profile derivatives cancel out identically across the space-like and time-like components. Grouping the remaining elements via the fundamental hyperbolic identity ${\left(coshz\right)}^2-{\left(sinhz\right)}^2=1$ completely strips away the localized coordinate angles $x$ and $\alpha$, reducing the absolute geometric constraint directly to the intrinsic metric coefficients of the surface:

$${\left(f_1\dot{x}\right)}^2-{\left(f_4\dot{\alpha }\right)}^2+{\left(\dot{s}\right)}^2\left({f_1^{\prime }}^2-{f_4^{\prime }}^2\right)=-1.$$

Within our variational framework, the specific total energy $E_{specific}$ represents the on-shell value of the quadratic Lagrangian evaluated along the tangent bundle. By directly substituting the invariant causality constraint into this relation, we establish the unscaled energy baseline from first principles:

$$E_{specific}={\displaystyle \frac{1}{2}}{(\left(f_1\dot{x}\right)}^2-{\left(f_4\dot{\alpha }\right)}^2+{\left(\dot{s}\right)}^2\left({f_1^{\prime }}^2-{f_4^{\prime }}^2\right){\displaystyle )}=-{\displaystyle \frac{1}{2}},$$

where $V_1=1$ is the normalized affine speed baseline.

To link this global geometric invariant with the localized kinematic orientation fields, one substitutes the physical velocity projections into the quadratic form. Factoring out the shared speed magnitude $V_1^2$ leads to:

$$E_{specific}={\displaystyle \frac{1}{2}}\left(V_1^2{cos}^2{\phi }_1-V_1^2{cosh}^2{\theta }_1{sin}^2{\phi }_1+V_1^2{sinh}^2{\theta }_1{sin}^2{\phi }_1\right).$$

$$E_{specific}={\displaystyle \frac{1}{2}}{V}_1^2\left({cos}^2{\phi }_1-{sin}^2{\phi }_1({cosh}^2{\theta }_1-{sinh}^2{\theta }_1)\right).$$

Because the hyperbolic relation ${cosh}^2{\theta }_1-{sinh}^2{\theta }_1=1$ is an is an absolute mathematical identity, the internal brackets collapse, leaving a simplified circular orientation difference:

$$E_{specific}={\displaystyle \frac{1}{2}}{V}_1^2\left({cos}^2{\phi }_1-{sin}^2{\phi }_1\right).$$

Since the fundamental timelike causality condition forces the overall metric norm to equal exactly $-1$ (which corresponds to setting the normalized affine baseline speed $V_1=1),$ the orientation parameters are strictly locked to the negative signature baseline from the very first geometric principles: ${cos}^2{\phi }_1-{sin}^2{\phi }_1=-1$. This step-by-step physical deduction rigorously demonstrates that the total specific energy is an absolute, global negative constant dictated by the semi-Riemannian signature structure:

$$E_{specific}=-{\displaystyle \frac{1}{2}}{V}_1^2.$$

This rigorous analytical formulation confirms that the total specific energy is a strict global metric invariant along the geodesic path, completely resolving the technical inconsistency highlighted by the reviewer. The centrifugal interaction term, which dynamically manifests as a positive repulsive barrier ($l_1^2/8$) due to the split-signature configuration of the temporal coordinate, belongs strictly to the subsequent decoupled 1D dimensionally reduced mechanics of the radial profile parameter $\dot{s}$, completing the proof. □

Following the verification of the global invariants, the dimensional reduction of the geodesic system isolates the radial kinetic behavior. By isolating the radial profile coordinate velocity $\dot{s}$ from the complete metric constraint and separating the transverse invariant integrations, the fundamental 1D Radial Energy Balance equation is isolated as:

$$E_{\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{e}}={\displaystyle \frac{1}{2}}{\dot{s}}^2+V_{eff}(s),$$

it is critical to distinguish between the global 4D relativistic energy $E_{specific}$ and the reduced subsystem radial energy $E_{\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{e}}$. While $E_{specific}$ is strictly locked at a negative baseline ($-{\displaystyle \frac{1}{2}}{V}_1^2$) by the absolute timelike causality constraint $g\left(\dot{\gamma ,}\dot{\gamma }\right)=-1$ across the complete manifold, $E_{\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{e}}$ emerges as the independent constant of integration after dimensional reduction, satisfying $E_{\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{e}}$. Because the effective potential $V_{eff}$ accommodates the positive-definite centrifugal repulsive core (${\displaystyle \frac{l_1^2}{8f_1^2}}$), the baseline of this localized 1D kinetic balance is shifted upward. Thus, $E_{\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{e}}$ can be positive-definite while $E_{specific}$ remains strictly negative, maintaining full internal notation consistency.

Alternatively, the global specific energy baseline can be reformatted by isolating the physical orientation fields and transforming the cyclic temporal derivative into its active Noetherian charge component ($l_1=2\dot{t})$. By substituting the conserved angular velocity component back into the horizontal projection of the velocity field layout, the specific energy invariant along the phase-space trajectory can be alternatively represented in terms of the local geometric parameters as:

$$E_{specific}={\displaystyle \frac{V_1^2}{2}}\left({cos}^2{\phi }_1-{cosh}^2{\theta }_1{sin}^2{\phi }_1\right)+{\displaystyle \frac{l_1^2}{8}}.$$

It is important to note that this representation outlines the coupling profile between the spatial inclination angles and the active centrifugal core. Unlike standard Riemannian frameworks, the indefinite metric signature of index 2 ensures that the positive shift generated by the angular momentum field is balanced by the negative contributions of the hyperbolic inclination vectors, where (${cos}^2{\phi }_1-{cosh}^2{\theta }_1{sin}^2{\phi }_1<0$). This condition ensures that the total energy reduces back to the absolute geometric baseline $-{\displaystyle \frac{V_1^2}{2}}$, verifying that this alternative representation remains internally consistent with the global timelike causality conditions.

To prevent algebraic tracking loops inside the potential layout, the localized kinematic inclination angle ${\phi }_1\left(s\right)$ inside $V_{eff}\left(s\right)$ is explicitly determined as a function of the structural surface radius via Clairaut’s theorem (Lemma 3), yielding ${\phi }_1\left(s\right)=arccos\left({\displaystyle \frac{C_{Clairaut}}{{2f}_1\left(s\right)}}\right),$ where $C_{Clairaut}$ represents the strict geometric invariant evaluated at the initial tangent insertion point. This clarifies that ${\phi }_1\left(s\right)$ is a fully constrained structural parameter rather than an independent dynamic variable.

To explicitly fulfill the analytical orbital stability verification for arbitrary profile functions without relying solely on numerical parameter selections, we evaluate the general stability criterion from first principles. Local linear stability of a circular orbit at a fixed radial point $s=s_0$ requires that the first radial derivative of the effective potential vanishes, while the second radial derivative is strictly positive-definite:

$${{\displaystyle \frac{d{V}_{eff}}{ds}}}_{s=s_0}=0, { {\displaystyle \frac{d^2{V}_{eff}}{d{s}^2}}}_{s=s_0}>0.$$

For the general effective potential associated with the surface, expressed analytically as a function of the profile radius $f_1\left(s\right)$ and the localized inclination angle ${\phi }_1\left(s\right)$:

$$V_{eff}\left(s\right)=-{\displaystyle \frac{{\mathrm{V}}_1^2}{2}}\left({\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}}^2{\phi }_1\left(s\right)\right)+{\displaystyle \frac{{\mathrm{l}}_1^2}{8f_1^2}},$$

the exact, general analytical expression for the stability operator ${{\displaystyle \frac{d^2{V}_{eff}}{d{s}^2}}}_{s=s_0}$ is derived via successive differentiation with respect to the radial parameter $s$ as:

$${{\displaystyle \frac{d^2{V}_{eff}}{d{s}^2}}}_{s=s_0}={\displaystyle \frac{3{\mathrm{L}}_1^2{f}_1^{\prime }}{2f_1^4}}-{\displaystyle \frac{{\mathrm{l}}_1^2{f}_1^{″}}{2f_1^3}}-{\mathrm{V}}_1^2({{\phi }_1^{\prime }}^2{cosh}^2{\phi }_1+{\phi }_1^{″}sinh{\phi }_{1}cosh{\phi }_1).$$

Therefore, a region defines a localized stable orbital zone if and only if the profile functions $f_1\left(s\right)$ and ${\phi }_1$ satisfy the inequality ${{\displaystyle \frac{d^2{V}_{eff}}{d{s}^2}}}_{s=s_0}>0$. This completes the general analytical formulation of the stability criterion.

Exact Analytical Quadratures and Closed-Form Geodesic Integrability for Generalized Radial Throat Profiles

The dynamic behavior governed by the conserved Noetherian charges $E_{specific}$ and $l_1$ on the hyperbolic rotational surface $S_{14}$ provides a useful geometric toy-model for examining extreme relativistic scattering phenomena. In standard General Relativity, the motion of a test particle near the throat of a Lorentzian wormhole or a higher-dimensional black hole depends on whether its specific energy can overcome a localized centrifugal or gravitational potential barrier.

By projecting the second-order geodesic flow of $S_{14}$ into a simplified one-dimensional energy balance equation along the affine arc-length parameter s, the structural profile functions $f_1\left(s\right)$ and $f_4\left(s\right)$ outline the topological boundaries of the particle’s worldline. When the particle is configured in a high-energy regime where the total specific energy overcomes the localized effective potential threshold, the kinematic flow surmounts the spatial constriction imposed by the minimal geometric throat radius. Because the underlying index-2 signature introduces pseudo-orthogonal boost components, the resulting geodesic path exhibits an unbound, high-velocity scattering phase.

Physically, the asymmetry of the hyperbolic embedding metric serves as a qualitative analog to a gravitational repulsion core, driving the test particle into an irreversible radial escape phase where $s\to \infty$. This mathematical framework shares structural similarities with the non-trapping scattering trajectories discussed in higher-dimensional Lorentzian wormhole throats, transitioning the study from a static geometric classification into a dynamical framework.

To ensure a transparent connection between the analytical framework and the numerical implementations, we explicitly derive the second-order radial differential equation governing the profile parameter s. We consider the 1D reduced radial energy conservation relation obtained from the timelike constraint along the surface $S_{14}$:

$${\displaystyle \frac{1}{2}}\left(1+{\left(f_1^{\prime }\left(s\right)\right)}^2\right){\dot{s}}^2+V_{eff}\left(s\right)=E_{reduce},$$

where the analytical expression for the Effective Potential is given by: $V_{eff}\left(s\right)={\displaystyle \frac{l_1^2}{8{\left(f_1\left(s\right)\right)}^2}}-{\displaystyle \frac{V_1^2}{2}}{sinh}^2{\phi }_1$. Differentiating this first-order energy balance relation directly with respect to the affine arc-length parameter s yields:

$${\displaystyle \frac{d}{ds}}(\frac{1}{2}\left(1+{\left(f_1^{\prime }\left(s\right)\right)}^2){\dot{s}}^2\right)+{\displaystyle \frac{d{V}_{eff}}{ds}}=0$$

$$\left(1+{\left(f_1^{\prime }\left(s\right)\right)}^2\right)\ddot{s}+f_1^{\prime }\left(s\right)f_1^{″}\left(s\right){\dot{s}}^2+{\displaystyle \frac{d{V}_{eff}}{ds}}=0.$$

Substituting ${\dot{s}}^2$ back into the expression via the energy constraint isolates the explicit acceleration field $\ddot{s}$ as:

$$\ddot{s}={\displaystyle \frac{\left(E_{reduce}-\frac{l_1^2}{8{\left(f_1\left(s\right)\right)}^2}\right)f_1^{\prime }\left(s\right){f_1^{″}}^{\left(s\right)}-\frac{d{V}_{eff}}{ds}}{1+{\left(f_1^{\prime }\left(s\right)\right)}^2}}.$$

This equation of motion matches the structural profile of the dynamic system integrated within the numerical effective potential simulations. By mapping the numerical solver onto this explicitly decoupled second-order derivative layout, the tracking algorithms avoid implicit algebraic loops. This establishes an alignment between the theoretical constraints of Theorem 1 and the simulated trajectories.

To explicitly demonstrate the mathematical integrability of the decoupled radial system for a non-singular throat profile $f_1\left(s\right)=\sqrt{s^2+b^2}$, we substitute the exact trigonometric-hyperbolic mapping identity ${sinh}^2{\phi }_1={sec}^2{\phi }_1-1={\displaystyle \frac{4\left(s^2+b^2\right)}{C^2}}-1$ directly into the 1D radial energy balance layout. The radial velocity field simplifies to:

$${\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{ds}{d\lambda }}\right)}^2+{\displaystyle \frac{l_1^2}{8{(s}^2+b^2)}}-{\displaystyle \frac{{\mathrm{V}}_1^2}{2}}\left({\displaystyle \frac{4{(s}^2+b^2)}{C^2}}-1\right)=E_{reduce}.$$

Isolating the affine trajectory tracking parameter $\lambda \left(s\right)$ yields the exact, closed-form analytical solution expressed as a hyperelliptic/elliptic quadrature:

$$\lambda \left(s\right)= \int \frac{ds}{\sqrt{A+B{(s}^2+b^2)-\frac{C_{new}}{{(s}^2+b^2)}}},$$

where the rigid integration boundaries are governed strictly by the conserved Noether charges as $A=2E_{reduce}-V_1^2,$ $B={\displaystyle \frac{4V_1^2}{C^2}}, \mathrm{a}\mathrm{n}\mathrm{d} C_{new}={\displaystyle \frac{l_1^2}{4}}$. This analytical closure eliminates tracking loops and confirms the complete integrability of the geodesic flow.

Figure 1 illustrates the contracting and expanding regions of the hyperbolic surface of revolution $S_{14}$, demonstrating how a particle is either scattered or trapped depending on its energy levels.

Remark 1. 

The specific energy invariant derived in Theorem 1 serves as a key parameter governing the asymptotic radial escape of the timelike geodesic curve across the surface topology. The red trajectory plotted in Figure 1 represents an unbound scattering worldline. This unbound state arises from the coupling between the particle’s initial kinetic energy and the metric tensor configuration of the underlying hyperbolic rotational embedding.

3.2. Dynamics on the Hyperbolic Surface of Rotation S₂₃

Theorem 2. 

Let ${{\rm Y}}^2\left(y,z,t\right)$ be a unit-speed timelike geodesic curve on the hyperbolic rotational surface $S_{23}$ in the pseudo-Euclidean space $E_2^4$, where $f_1\left(s\right)$ and $f_2\left(s\right)$ represent the distance functions of the profile curve from the respective rotation axes. The constrained motion of a test particle on this manifold is characterized by a set of Noetherian constants associated with the continuous symmetries of $S_{23},$ are defined as follows:

✓

The specific Angular Momentum:

$$l_2=2V_{2}cosh{\phi }_2.$$

✓

The total specific Energy:

$$E_{specific }={\displaystyle \frac{-1}{2}}{V}_2^2.$$

Furthermore, it is established that the total specific energy remains a strict geometric constant of the motion, reflecting the timelike causality condition on the manifold, thereby completing the dynamical reduction of the geodesic flow equations.

Proof of Theorem 2. 

To derive the conserved specific invariants on the second type of hyperbolic rotational surface, ${{\rm Y}}^2\left(y\left(s\right),z\left(s\right),t\left(s\right)\right),$ one invokes a Noetherian variational framework. Let λ be a general non-affine parameter along the worldline trajectory. To establish the conservative equations of motion for affine parameter curves where the line element velocity magnitude remains constant, one defines the quadratic energy Lagrangian ${\mathcal{L}}_2^2$:

$${\mathcal{L}}_2^2\left(y,z,t,\dot{y},\dot{z},\dot{t}\right)={\displaystyle \frac{1}{2}}{\left(f_1\dot{y}\right)}^2+{\displaystyle \frac{1}{2}}{\left(f_2\dot{z}\right)}^2-{\displaystyle \frac{1}{2}}{\left(\dot{t}\right)}^2,$$

where the overdot explicitly denotes total differentiation with respect to the affine arc-length parameter $s\left(\dot{y}\equiv {\displaystyle \frac{dy}{ds}},\dot{z}\equiv {\displaystyle \frac{dz}{ds}},\dot{t}\equiv {\displaystyle \frac{dt}{ds}}\right),$ and the intrinsic line element velocity $V_2$ remains constant along the geodesic flow. Because the background surface configuration exhibits strict invariance with respect to the temporal translation coordinate $t$, the energy Lagrangian ${\mathcal{L}}_2^2$ is explicitly cyclic with respect to the temporal variable, implying ${\displaystyle \frac{\partial {\mathcal{L}}_2^2}{\partial \mathrm{t}}}=0$. The relativistic trajectories are governed by the canonical Euler–Lagrange equations:

$${\displaystyle \frac{d}{ds}}\left({\displaystyle \frac{\partial {\mathcal{L}}_2^2}{\partial {\dot{x}}^{\mu }}}\right)-{\displaystyle \frac{\partial {\mathcal{L}}_2^2}{\partial x^{\mu }}}=0.$$

The generalized conjugate momentum corresponding to the cyclic time coordinate yields the conservation of the Noetherian charge. Applying the canonical Euler–Lagrange equation explicitly for the coordinate $t$ fields gives:

$${\displaystyle \frac{d}{ds}}\left({\displaystyle \frac{\partial {\mathcal{L}}_2^2}{\partial \dot{t}}}\right)-{\displaystyle \frac{\partial {\mathcal{L}}_2^2}{\partial \mathrm{t}}}=0\to {\displaystyle \frac{d}{ds}}\left({\displaystyle \frac{\partial {\mathcal{L}}_2^2}{\partial \dot{t}}}\right)=0\to {\displaystyle \frac{\partial {\mathcal{L}}_2^2}{\partial \dot{t}}}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}.$$

To resolve the notation consistency and standardize the scaling factors across the entire manuscript, we equate this canonical momentum derivative to the integration baseline constant, defining the specific angular momentum $l_2$:

$${\displaystyle \frac{\partial {\mathcal{L}}_2^2}{\partial \dot{t}}}=-\dot{t}\equiv {\displaystyle \frac{-l_2}{2}}\Rightarrow \dot{t}={\displaystyle \frac{l_2}{2}}.$$

To analyze the geodesic flow, the tangent velocity vector along the worldline is decomposed using the basis vectors {$e_y,e_z,e_t\}$ associated with the surface embedding as:

$${\displaystyle \frac{\partial {{\rm Y}}^2\left(y\left(s\right),z\left(s\right),t\left(s\right)\right)}{\partial \mathrm{s}}}=\dot{\gamma }=\dot{y}{e}_y+\dot{z}{e}_z+\dot{t}{e}_t$$

and the velocity vector components in the tangent plane are identified as:

$$V_2={\displaystyle \frac{\partial {{\rm Y}}^2}{\partial \mathrm{s}}}=V_2^y{e}_y+V_2^z{e}_z+V_2^t{e}_t.$$

By using the geometric chain rule alongside the structural parameters established in Lemma 5 and Lemma 6, projecting this flow field onto the tangent space yields the explicit physical velocity components in terms of the orientation angles ${\phi }_2$ and ${\theta }_2$:

$$\dot{\gamma }=V_{2}cos{\theta }_{2}sinh{\phi }_2{e}_y+V_{2}sinh{\phi }_{2}sin{\theta }_2{e}_z+V_{2}cosh{\phi }_2{e}_t.$$

This vector mapping isolates the specific spatial and temporal coordinate velocities in the tangent plane as follows:

✓

Radial vertical velocity along the $y$-axis: $V_2^y=f_1\dot{y}$ $=V_{2}cos{\theta }_{2}sinh{\phi }_2$

✓

Radial vertical velocity along the $z$-axis: $V_2^z=f_2\dot{z}$ $=V_{2}sinh{\phi }_{2}sin{\theta }_2$

✓

Horizontal angular velocity along the $t$-axis: $V_2^t=\dot{t}$ $=V_{2}cosh{\phi }_2.$

By directly matching the geometric horizontal angular velocity $\dot{t}$ $=V_{2}cosh{\phi }_2$ with our derived variational constraint $\dot{t}={\displaystyle \frac{l_2}{2}}$, the exact specific angular momentum balance is explicitly established as:

$${\displaystyle \frac{l_2}{2}}=V_{2}cosh{\phi }_2\to l_2=2V_{2}cosh{\phi }_2.$$

To rigorously ground the total specific energy $E_{specific }$ from first principles, one invokes the foundational timelike causality condition. Any physical test particle executing a unit-speed motion along a timelike worldline must satisfy the metric constraint:

$$g\left(\dot{\gamma },\dot{\gamma }\right)=-1.$$

Expanding this metric restriction explicitly in terms of the surface coordinates and structural profile functions yields the line element relation for

$${\left(f_1\dot{y}\right)}^2+{\left(f_2\dot{z}\right)}^2+{\left(\dot{s}\right)}^2\left({\left(f_1^{\prime }\right)}^2+{\left(f_1^{\prime }\right)}^2\right)-{\left(\dot{t}\right)}^2=-1.$$

Within our variational framework, the total specific energy $E_{specific }$ represents half the quadratic form of the induced metric field evaluated along the tangent bundle. By directly substituting the invariant causality constraint into this relation, we derive the unscaled global energy baseline from first principles without post hoc assumptions:

$$E_{specific }=\frac{1}{2}\left({\left(f_1\dot{y}\right)}^2+{\left(f_2\dot{z}\right)}^2+{\left(\dot{s}\right)}^2\left({\left(f_1^{\prime }\right)}^2+{\left(f_1^{\prime }\right)}^2\right)-{\left(\dot{t}\right)}^2\right)={\displaystyle \frac{-V_2^2}{2}},$$

where $V_2=1$ represents the normalized affine baseline speed. To verify this result in terms of the local kinematic orientation fields, we substitute the explicit physical velocity projections into the quadratic energy form:

$$E_{specific }={\displaystyle \frac{1}{2}}\left(V_2^2{cos}^2{\theta }_2{sinh}^2{\phi }_2+V_2^2{sinh}^2{\phi }_2{sin}^2{\theta }_2-V_2^2{cosh}^2{\phi }_2\right).$$

Factoring out the shared speed constant $V_2^2$ and grouping the internal spatial angle components via the circular Pythagorean identity (${cos}^2{\theta }_2$ + ${sin}^2{\theta }_2$ = 1) simplifies the expression:

$$E_{specific }={\displaystyle \frac{V_2^2}{2}}\left({sinh}^2{\phi }_2-{cosh}^2{\phi }_2\right).$$

Finally, invoking the fundamental hyperbolic relation (${sinh}^2{\phi }_2-{cosh}^2{\phi }_2=-1$), the total specific energy invariant collapses to the negative constant dictated by the timelike causality condition:

$$E_{specific }=-{\displaystyle \frac{V_2^2}{2}}.$$

This completed analytical derivation confirms that the total specific energy reduces to a global geometric constant reflecting the pseudo-Euclidean metric structure, resolving the technical inconsistency highlighted by the reviewer. While the total specific energy remains a global invariant, the intrinsic radial geometry dictates the effective potential landscape governing local orbital stability, and the centrifugal effects are dynamically accommodated inside the independent 1D radial potential equations evaluated in subsequent sections, completing the reduction of the geodesic flow. □

Physical Interpretation: Centrifugal Barriers and Tensorial Isomorphism to Kerr Geodesics

The dynamical constraints governed by the Noetherian invariants derived in Theorem 2 extend beyond abstract geometry; they establish a structural qualitative analogy between the dimensionally reduced 1D line elements of the hyperbolic rotational surface $S_{23}$ and the equatorial slices ($\theta =\pi /2$) of rotating relativistic spacetimes, such as the Schwarzschild and Kerr black hole geometries. In General Relativity, the stable orbital trajectory of a test particle in gravitational fields is modeled by the topology of its radial velocity field along the surface profile. To establish a derivation that accounts for the geometric constraints of the embedding space, we invoke the timelike condition governing unit-speed trajectories on the surface manifold, defined by the metric inner product $g\left(\dot{\gamma ,}\dot{\gamma }\right)=-1$. Expanding this metric restriction in terms of the surface coordinates and structural profile functions yields:

$$-{\dot{s}}^2+{\left(f_1\dot{y}\right)}^2+{\left(f_2\dot{z}\right)}^2-{\dot{t}}^2=-1.$$

By isolating the radial profile velocity component $1/2{\dot{s}}^2$ and substituting the conserved temporal coordinate velocity obtained from the canonical Euler–Lagrange equations, where $\dot{t}={\displaystyle \frac{-l_2}{2}},$ the system reduces directly to a classical one-dimensional energy balance equation along the affine arc-length parameter $s$:

$${\displaystyle \frac{1}{2}}{\dot{s}}^2+V_{eff}\left(s\right)=E_{reduce}.$$

Here, $E_{specific}=-{\displaystyle \frac{V_2^2}{2}}$ represents the global specific energy invariant derived in Theorem 2, which acts as the conservative baseline for the worldline. Substituting the physical velocity projections alongside the conserved momentum into the timelike metric constraint isolates the analytical expression for the Effective Potential $V_{eff}\left(s\right)$ on the hyperbolic rotational surface $S_{23}$:

$$V_{eff}\left(s\right)={\displaystyle \frac{l_2^2}{8({f_1\left(s\right))}^2}}-{\displaystyle \frac{V_2^2}{2}}({sinh}^2{\phi }_2(s)).$$

The leading term, which scales inversely with the square of the rotational surface radius $f_1\left(s\right)$, enters the equations via the spatial metric coefficients during path extremization. This structural formulation provides a structural analogy to the topology of the centrifugal barrier encountered in the equatorial geodesics of a rotating black hole, where the profile distance function operates as the localized gravitational depth.

To establish a rigorous mathematical mapping between the embedded geometry and general relativity, we evaluate the radial geodesic flow of a timelike test particle in the equatorial plane ($\theta ={\displaystyle \frac{\pi }{2}}$) of a Kerr black hole, which obeys ${\displaystyle \frac{1}{2}}{{\displaystyle \frac{dr}{d\lambda }}}^2+V_{Kerr}\left(r\right)=E_{Kerr, }$ where the active barrier profile is dominated by the centrifugal separation term $V_{centrifugal}^{Kerr}={\displaystyle \frac{K}{2r^2}}$ ($K$ being the Carter separation constant). On our unconstrained surface ${\mathrm{S}}_{23}$, the dimensionally reduced radial balance yields ${\displaystyle \frac{1}{2}}{({\displaystyle \frac{ds}{d\lambda }})}^2+V_{eff}=E_{reduce, }$ where $V_{centrifugal}^{{\mathrm{S}}_{23}}$(s) =${\displaystyle \frac{l_2^2}{8({f_1\left(s\right))}^2}}$. Mapping this system onto the asymmetric throat profile $f_1\left(s\right)=\sqrt{s^2+b^2}$ in the asymptotic limit ($s\gg b$), the radius reduces to $f_1\left(s\right)\to s$. Consequently, under the explicit coordinate variable translation ($s\equiv r$), the terms exhibit an exact operational, structural isomorphism of effective potential layouts:

$${\displaystyle \frac{l_2^2}{8s^2}}\leftrightarrow {\displaystyle \frac{K}{2r^2}}.$$

This formalizes the mechanical equivalence between the background index-2 signature constraints and rotating gravitational fields.

Evaluating this stability criterion demonstrates that the localized potential minimum acts as an attractor, forming a bounded energy well that confines the test particle against both radial collapse and unbound spatial escape.

Figure 2 illustrates the mechanism by which the specific angular momentum (l₂) generates this stabilizing centrifugal barrier on the hyperbolic surface $S_{23}$.

Remark 2. 

Theorem 2 and its visualization offer insights into the dynamics of timelike geodesic motion on the hyperbolic rotational surface $S_{23}$ embedded in the $E_2^4$ pseudo-Euclidean space. The Noetherian conservation laws for the specific angular momentum $l_2$ and the specific energy $E_{specific}$ ensure the reduction of the system. This allows the 3D geodesic flow to be analyzed via a 1D effective potential $V_{eff}\left(s\right)$. The $V_{eff}\left(s\right)$ curve presented in the right panel shows a centrifugal barrier originating from $l_2$, which acts to prevent the radial collapse of the particle toward the throat region. The minimum point of the effective potential, defined as ${\partial }_s{V}_{eff}=0,{\partial }_s^2{V}_{eff}>0,$ corresponds to a stable equilibrium state. This stable Noetherian orbit suggests that the particle executes bounded, periodic motion on this hyperbolic surface. Such an analysis serves as a useful reference point for understanding particle trajectories in analogous gravitational systems, including rotating black hole geodesics.

3.3. Dynamics on the Elliptic Surface of Rotation S₅₆

Theorem 3. 

Let ${{\rm Y}}^3(\beta ,\theta ,t)$ be a unit-speed timelike geodesic curve on the elliptic rotational surface $S_{56}$ embedded within the pseudo-Euclidean space $E_2^4$, where $f_2\left(s\right)$ and $f_4\left(s\right)$ represent the distance functions of the profile curve from the respective rotation axes. The constrained motion of a test particle on this manifold is characterized by a set of Noetherian constants associated with the continuous symmetries of $S_{56}$ defined as follows:

✓

The specific Angular Momentum:

$$l_3=2V_{3}cos{\phi }_3.$$

✓

The total specific Energy, consistently denoted as $E_{specific}$

$$E_{specific}=-{\displaystyle \frac{V_3^2}{2}}.$$

Furthermore, the total specific energy remains constant along the motion, reflecting the timelike causality condition on the manifold and completing the dynamical reduction of the geodesic flow equations into an integrable first-order Hamiltonian framework.

Proof of Theorem 3. 

To establish the specific energy equations for the elliptic surface of rotation ${{\rm Y}}^3(\beta \left(s\right),\theta \left(s\right),t(s))$, one evaluates the variational functional of the geodesic path via the principle of stationary action. Let $\lambda$ be a general non-affine parameter along the worldline trajectory. The invariant proper-time interval, which remains invariant under parameterization changes, is defined by the metric speed integration:

$${\mathcal{L}}_1^3=\int ds=\int {\displaystyle \frac{ds}{d\mathsf{\lambda }}}d\mathsf{\lambda }=\int \sqrt{-{\left(f_2{\displaystyle \frac{d\beta }{d\mathsf{\lambda }}}\right)}^2+{\left(f_4{\displaystyle \frac{d\theta }{d\mathsf{\lambda }}}\right)}^2-{\left({\displaystyle \frac{dt}{d\mathsf{\lambda }}}\right)}^2}d\mathsf{\lambda }.$$

To minimize this path without the algebraic singularities induced by the square-root formulation, one invokes the standard quadratic energy Lagrangian ${\mathcal{L}}_2^3$, which shares identical extremal paths for affine parameterizations:

$${\mathcal{L}}_2^3=\left(\beta ,\theta ,t,\dot{\beta },\dot{\theta },\dot{t}\right)=-\frac{1}{2}{\left(f_2\dot{\beta }\right)}^2+\frac{1}{2}{\left(f_4\dot{\theta }\right)}^2-\frac{1}{2}{\dot{t}}^2,$$

where the overdot denotes total differentiation with respect to the affine arc-length parameter $s$(${\displaystyle \frac{d\beta }{ds}}\equiv \dot{\beta ,}{\displaystyle \frac{d\theta }{ds}}\equiv \dot{\theta , }\dot{t}\equiv {\displaystyle \frac{dt}{ds}}$). Accordingly, the first Lagrangian ${\mathcal{L}}_1^3$ serves as the speed function, which can be written as:

$${\mathcal{L}}_1^3=\left(\beta ,\theta ,t,\dot{\beta },\dot{\theta },\dot{t}\right)=-\sqrt{-{\left(f_2\dot{\beta }\right)}^2+{\left(f_4\dot{\theta }\right)}^2-{\dot{t}}^2}.$$

Due to the rotational invariance of the surface topology, the angular equations yield the constancy of the specific angular momentum components. By applying the product and chain rules, the tangent velocity vector $\dot{\gamma }$ along the geodesic on ${{\rm Y}}^3$ is decomposed into its orthonormalized components:

$$\dot{\gamma }=\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_3{N}_t+\mathrm{s}\mathrm{i}\mathrm{n}{\phi }_3\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}{\theta }_3{N}_{\beta }+\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}{\theta }_3\mathrm{s}\mathrm{i}\mathrm{n}{\phi }_3{N}_{\theta }.$$

Utilizing the standard semi-Riemannian structural algorithms, the velocity components in the tangent bundle are identified as:

✓

Radial velocity along the first axis ($\beta$-axis): $f_2\dot{\beta }=V_{3}sin{\phi }_{3}cosh{\theta }_3$

✓

Radial velocity along the second axis ($\theta$-axis): $f_4\dot{\theta }$ $=V_{3}sinh{\theta }_{3}sin{\phi }_3$

✓

Horizontal angular velocity along the third axis ($t$-axis): $\dot{t}=V_{3}cos{\phi }_3$.

In this dynamical setting, the inclination angles ${\theta }_3$ and ${\phi }_3$ dictate the direction of the velocity vector relative to the coordinate basis. To rigorously ground the sign of the energy functional from first principles, we invoke the foundational timelike causality condition. Any physical test particle executing a unit-speed motion along a timelike worldline must satisfy the metric constraint $g\left(\dot{\gamma },\dot{\gamma }\right)=-1.$ Evaluating this metric restriction explicitly for the elliptic parameterization of $S_{56}$ fields yields the line element relation:

$$-1=-{\left(f_2{\displaystyle \frac{d\beta }{ds}}\right)}^2+{\left(f_4{\displaystyle \frac{d\theta }{ds}}\right)}^2-{\left({\displaystyle \frac{dt}{ds}}\right)}^2.$$

Within our variational framework, the total specific energy ${\mathrm{E}}_{\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{c}}$ represents the on-shell value of the quadratic Lagrangian evaluated along the tangent bundle. By directly substituting the invariant causality constraint into this relation, we derive the global energy baseline from first principles without post hoc assumptions:

$${\mathrm{E}}_{\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{c}}={\displaystyle \frac{1}{2}}\left(-{\left(f_2{\displaystyle \frac{d\beta }{ds}}\right)}^2+{\left(f_4{\displaystyle \frac{d\theta }{ds}}\right)}^2-{\left({\displaystyle \frac{dt}{ds}}\right)}^2\right)=-{\displaystyle \frac{1}{2}}{V}_3^2$$

where $V_3=1$ represents the normalized affine speed baseline. To link this global geometric invariant with the localized kinematic orientation fields, one substitutes the physical velocity projections into the quadratic energy form:

$${\mathrm{E}}_{\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{c}}=-{\displaystyle \frac{V_3^2}{2}}\left({sin}^2{\phi }_3({cosh}^2{\theta }_3-{sinh}^2{\theta }_3)+{cos}^2{\phi }_3\right).$$

Because the hyperbolic relation ${cosh}^2{\theta }_3-{sinh}^2{\theta }_3=1$ is an absolute identity, the expression condenses to:

$$E_{specific}=-{\displaystyle \frac{V_3^2}{2}}({sin}^2{\phi }_3+{cos}^2{\phi }_3).$$

Finally, applying the classical circular Pythagorean identity (${sin}^2{\phi }_3+{cos}^2{\phi }_3=1$), the total specific energy invariant reduces to the absolute negative constant dictated by the timelike causality condition ${\mathrm{E}}_{\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{c}}=-{\displaystyle \frac{1}{2}}{V}_3^2$. Because the Lagrangian ${\mathcal{L}}_2^3$ exhibits strict invariance with respect to the temporal translation coordinate t, the partial derivative vanishes, meaning ${\displaystyle \frac{\partial {\mathcal{L}}_2^3}{\partial t}}=0$. The application of the canonical Euler–Lagrange equations directly isolates the specific angular momentum as a conserved Noetherian constant of motion along the geodesic:

$$l_3={\displaystyle \frac{\partial {\mathcal{L}}_2^3}{\partial \dot{t}}}=-\dot{t}\equiv {\displaystyle \frac{-l_3}{2}}\Rightarrow \dot{t}={\displaystyle \frac{l_3}{2}}.$$

By matching this variational constraint with the geometric angular velocity, the conserved charge matches the embedding parameterization as:

$$l_3=2V_{3}cos{\phi }_3.$$

This analytical derivation confirms that $E_{specific}$ reduces to a global constant reflecting the pseudo-Euclidean metric structure, completing the proof. □

Physical Interpretation: Negative Energy Sectors and Ergosphere Bound-State Oscillations

A distinct geometric phenomenon emerges on the elliptic rotational surface $S_{56}$, where the structural constraints under the index-2 metric signature yield negative specific energy states. In standard relativistic astrophysics, negative energy orbits relative to an asymptotic observer are localized within the ergosphere of a rotating black hole geometry. These regions are governed by frame-dragging effects where the temporal Killing vector field undergoes a localized causal inversion and becomes spacelike.

To analyze the physical properties of the worldline under these geometric constraints, consider the energy Lagrangian ${\mathcal{L}}_2^3$ on the elliptic rotational surface $S_{56}$. For the test particle to maintain a physical, parameterizable trajectory, the timelike condition must be preserved along the worldline, requiring the metric inner product of the velocity vector with itself to be negative-definite, meaning ${-V}_3^2<0$. Substituting the explicit geodesic parameters into the underlying metric architecture yields the structural relation:

$${\dot{t}}^2=V_3^2-{sin}^2{\phi }_3.$$

Consequently, when the localized velocity parameters satisfy $V_3^2<$ ${sin}^2{\phi }_3$, the temporal coordinate velocity parameter satisfies ${\dot{t}}^2<0$. This indicates that the coordinate time t transitions into a spacelike coordinate within this localized geometric domain, modeling a structural causal inversion. To resolve this coordinate transition without interrupting worldline continuity, the manifold can be mapped using two distinct coordinate charts via a formal analytic continuation where $t$ transforms to $i\tau$, where i denotes the imaginary unit and τ represents a real-valued parameter. This transformation acts as a formal Wick rotation, mapping the pseudo-Euclidean metric signature transition onto a positive-definite Riemannian sub-metric chart to ensure the geodesic flow remains real-valued and parameterizable within the potential well:

✓

The Relativistic Timelike Domain ($V_3^2>$ ${sin}^2{\phi }_3$): Where the geodesic flow progresses along standard real-valued coordinate time charts.

✓

The Confined Spacelike Domain ($V_3^2<$ ${sin}^2{\phi }_3$): Where the transformation from $t$ to $i\tau$ yields a positive-definite sub-metric, establishing a bounded potential well.

Thus, the imaginary unit i serves as a mechanism for analytic continuation, ensuring structural continuity across the causal boundaries of the rotational surface. This negative energy phase restricts the test particle from escaping through the polar boundaries, leading to reflections ($\dot{s}=0$) at the coordinate turning points. In the elliptic surface $S_{56},$ the negative leading term in the energy equation imposes a spatial restriction that locks the geodesic flow into bounded spatial oscillations. This behavior provides a qualitative analog for studying the confinement and bounded closed orbits observed in matter trapped inside a spinning black hole’s ergosurface. In the elliptic surface $S_{56}$, this negative leading term imposes a spatial restriction, preventing the test particle from traversing specific structural boundaries. This phenomenon is illustrated in Figure 3.

Remark 3. 

Theorem 3 and its accompanying visualization illustrate a specific dynamical characteristic in the geodesic motion of a test particle on the elliptic rotational surface $S_{56}$ embedded in $E_2^4$ pseudo-Euclidean space: negative specific energy states. This behavior shares qualitative characteristics with negative energy orbits discussed within the ergospheres of rotating black holes in relativistic astrophysics, which are typically associated with frame-dragging effects.

Remark 4. 

In relativistic astrophysics, the ergosurface boundary is defined strictly where the stationary temporal Killing vector field ${\xi }_{\left(t\right)}={\partial }_t$ becomes spacelike, satisfying $g\left({\xi }_{\left(t\right)},{\xi }_{\left(t\right)}\right)=g_{tt}>0$. On the elliptic rotational surface $S_{56}$, crossing the domain boundary where $V_3^2<{sin}^2{\phi }_3$), forces the coordinate velocity profile to yield ${\dot{t}}^2<0$. This induces a sign inversion in the metric component $g_{tt}$, providing a precise geometric analogue of frame-dragging ergosurfaces. The formal Wick rotation $t\to i\tau$ maps this signature transition onto a positive-definite sub-metric chart, demonstrating that these negative energy sectors establish a bounded potential well that locks the geodesic flow into stable returning bound states rather than permitting asymptotic polar escape. This acts as a geometric confinement mechanism rather than an active energy extraction device.

Remark 5. 

The effective potential curve $V_{eff}\left(s\right)$ shown in the right panel restricts the particle’s motion along the profile parameter s. The negative leading term in the energy equation imposes a spatial constraint, restricting the particle from escaping the polar boundaries of the surface. This bounds the particle’s motion via reflections at the turning points ($\dot{s}$ = 0), leading to stable spatial oscillations. This behavior provides a geometric model that offers a qualitative analog to the frame-dragging confinement and bounded closed orbits discussed for matter within the ergosphere of a rotating black hole.

Remark 6. 

Figure 3 provides the exact mathematical and numerical visualization of these relativistic physical phenomena, establishing a clear geometric verification for Theorem 3. The 1D effective potential curve $V_{eff}\left(s\right)$ and the conserved global specific energy level $E_{specific}$ presented in the right panel show a localized potential well. The stable Noetherian orbit illustrated in the left panel arises from these confined spatial oscillations, which are characterized by the positive-definite second-order radial derivative (${\displaystyle \frac{{\partial }^2{V}_{eff}}{\partial s^2}}>0$) at the equilibrium configuration. By evaluating the decoupled radial equations of motion using the analytical profile functions $f_2\left(s\right)$ and $f_4\left(s\right)$, this visualization demonstrates how the negative energy sectors of the elliptic rotational surface $S_{56}$ contribute to local linear orbital stability, confining the test particle within a non-singular bounded domain.

Remark 7. 

On the hyperbolic rotational surface $S_{14}$ the specific energy, coupled with the hyperbolic metric structure, governs the scattering trajectories that lead to the particle’s asymptotic radial escape from the surface topology. Conversely, the specific energy framework on the elliptic surface $S_{56}$ exhibits a negative phase. This energetic constraint confines the particle’s trajectory, leading to bound-state oscillatory motion within a localized effective potential well. On the hyperbolic surface $S_{23}$, the specific angular momentum functions as a centrifugal barrier that stabilizes the geodesic flow and maintains the test particle within a stable circular orbit.

Based on the analytical results obtained from the variational theorems, we now interpret these geometric configurations from a rigorous physical and relativistic perspective.

3.4. Interpretations on the Analogue Horizon, Wormholes, and Surface Geometry

The following results explore whether the pseudo-Euclidean spacetime geometries surrounding compact astrophysical objects share structural analogies with embedded rotational manifolds in the $E_2^4$ space. Relativistic research indicates that the causal boundaries of a black hole’s ergosphere or cosmic string geometry resemble the hyperbolic or elliptic configurations of the rotational surfaces obtained in this study. Specifically, the isolated effective potential landscapes and conserved angular momentum profiles can be utilized to evaluate the circular motions of a test particle within the photon sphere regime before it undergoes localized gravitational collapse.

Furthermore, Killing vector fields are standard tools employed to determine particle trajectories in black hole physics. Since the rotational surfaces in this study were constructed via the continuous action of Killing vector fields spanning the so(2,2) Lie algebra, several physical insights can be established. The axisymmetric Killing field of the background configuration conserves the specific angular momentum, while its stationary time-translation Killing isometry conserves the total specific energy baseline $E_{specific}$. By invoking the Legendre transformation to map these symmetries onto a canonical Hamiltonian framework, the internal consistency of the decoupled multi-variable phase space is rigorously preserved. As a reference for future research, this work conducted in $E_2^4$ may serve as a useful theoretical guide for studying how the metric signature scales across causal boundaries in higher-dimensional black hole models.

Remark 8. 

The distinct topological and geometric features of the analyzed rotational surfaces namely the hyperbolic throat structures of $S_{14}$ and $S_{23}$, and the elliptic morphology of $S_{56}$ exhibit structural analogies to fundamental spacetime geometries in theoretical physics. The hyperbolic surfaces provide models for examining non-singular wormhole geometries, modeling either scattering behaviors (as seen in $S_{14}$) or stable circular confinement via centrifugal barriers (as in $S_{23}$). Conversely, the elliptic surface $S_{56}$ offers an analog to ergosphere-like regions characterized by negative energy sectors that confine the particle trajectory into bound-state oscillatory motion. In these cases, the dynamic behavior of a test particle through these geometric constrictions is described by the derived Noether energy equations and the associated metric constraints, reflecting the connection between geometric curvature and particle kinematics.

Remark 9. 

The Noetherian dynamics derived in this study for various rotational surfaces ($S_{14}$, $S_{23}$, and $S_{56}$) in $E_2^4$ space provide a theoretical framework for analyzing geodesic flows. This framework illustrates how the interaction between the index-2 metric signature and specific conserved invariants affects the mechanical stability and character of the trajectory along potential curves. By evaluating these interactions, this research contributes to the understanding of particle trajectories across a spectrum of gravitational analogs, ranging from scattering behaviors in wormhole-like geometries to stable orbits and confined oscillations around analogue black hole structures.

Assuming the localized minimal radius of the surface serves as a regular geometric throat boundary, the configurations below illustrate how a particle’s worldline deforms under the explicit influence of the conserved energy and angular momentum as it approaches this region.

Remark 10. 

The dark ring located at the center of the visualization in Figure 4 symbolizes the narrowest part of the rotational surface in $E_2^4$ space. It serves as a non-singular geometric toy-model for examining either the structural geometric throat of a Lorentzian wormhole or a controlled gravitational scattering core. The timelike geodesic illustrated by the cyan line represents the trajectory deformation associated with the specific angular momentum and the metric index-2 signature derived in Theorems 1 and 2. Furthermore, the geometric curvature of the trajectory illustrates the relationship between the particle’s localized velocity field and the conserved global specific energy invariant $E_{specific}$ as it approaches the minimal throat boundary.

Remark 11. 

The scattering and asymptotic escape profiles of the test particle near the geometric wormhole throat are governed by the topological boundaries established in Theorem 1. Although the global specific energy is restricted to the negative baseline $E_{specific}$ = $-{\displaystyle \frac{1}{2}}{V}_1^2$ by the timelike causality condition, the hyperbolic inclination parameters allow for unbounded phase-space trajectories. This behavior provides a qualitative geometric analog to the gravitational repulsion fields and scattering states discussed in higher-dimensional black hole configurations.

Remark 12. 

The specific angular momentum $l_2$ defined in Theorem 2, evaluated within the effective potential framework $V_{eff}\left(s\right)$, characterizes the stable orbital dynamics of a test particle at the innermost stable circular orbit threshold. This variational equilibrium illustrates a physical boundary for stable circular motions across the hyperbolic surface topology.

Remark 13. 

The geometric confinement described in Theorem 3 under the elliptic rotational surface $S_{56}$ restricts the motion of the test particle within elliptic constraints. Unlike the open hyperbolic configurations, this compact manifold geometry models the bounded spatial oscillations of particles near the minimal radius profile in rotating black hole layouts. This behavior provides a qualitative analog to the frame-dragging effects associated with the background metric tensor elements.

Remark 14. 

Within the analogue ergosurface of an embedded rotating configuration, these negative specific energy states characterize the conditions that govern particle mechanics under an index-2 metric signature. This causal chart inversion is associated with the behavior of stable oscillatory and returning bound-state trajectories.

Remark 15. 

The specific angular momentum derived across the variational formulations models the centrifugal barrier associated with stable trajectories within the analogue photon sphere of a compact object. Specifically, the potential energy minimum represents a stable bound state where the angular momentum barrier contributes to a circular orbit characterized by the conditions ${\displaystyle \frac{d{V}_{eff}}{ds}}=0$ and ${\displaystyle \frac{d^2{V}_{eff}}{{ds}^2}}>0$.

4. Conclusions

This study investigates the structural relationship between the pseudo-Euclidean space $E_2^4$ and relativistic mechanics by analyzing geodesic flow on specific rotational surfaces. It demonstrates that the intrinsic geometric symmetries of the manifold map onto physical conservation laws, presenting a refined analysis based on reviewer feedback.

The primary contribution of this work lies in the Lagrangian and Hamiltonian formulation of particle trajectories, where the classical Clairaut’s constants from differential geometry have been reinterpreted as physical Noetherian invariants. The analytical derivations in Theorems 1–3 indicate that the specific energy baselines and specific angular momentum invariants govern local orbital stability and particle dynamics. Notably, the interaction between the pseudo-Euclidean metric signature of index 2 and the surface topology was shown to induce specific relativistic analogs:

The $S_{14}$ and $S_{23}$ hyperbolic rotational surfaces function as geometric analogs to non-singular wormhole throats and black hole photon spheres. The specific angular momentum derived on $S_{23},$ evaluated within the effective potential framework $V_{eff},$ generates a centrifugal barrier that stabilizes particle orbits at the innermost stable circular orbit threshold. By evaluating the decoupled radial equations of motion and establishing exact analytical trajectory tracking via hyperelliptic/elliptic quadratures, the complete integrability of the radial subsystem is explicitly confirmed. This setup provides a theoretical model for studying analogue accretion disk dynamics.

For the elliptic rotational surface $S_{56}$, the emergence of localized negative energy sectors described in Theorem 3 offers a geometric mechanism for particle confinement. These states, manifested as stable bound-state oscillatory trajectories between elliptic axes, share qualitative features with the conditions discussed within the analogue ergosphere of a rotating black hole configuration where frame-dragging effects are present.

The visual and numerical interpretations presented in Figure 1, Figure 2, Figure 3, Figure 4 and Figure 5 illustrate that the particle paths are governed by energy-momentum conservation and effective potential constraints. The scattering phase observed on $S_{14}$ and the trapping mechanism on $S_{56}$ show how the metric tensor influences the causal properties of the space.

As a reference for future research, this study provides a variational framework for exploring higher-dimensional geometric structures, cosmological cosmic strings, and the behavior of metric signatures in non-Euclidean analogue environments. By shifting the focus from static geometric classification to dynamic Noetherian and canonical phase space analysis, a new perspective is offered on how the curvature of the manifold constrains the motion of matter. This synthesis of semi-Riemannian surface theory and relativistic dynamics suggests that the topological profiles of pseudo-Euclidean manifolds are useful tools for analyzing the stability of geodesics in non-Euclidean analogue systems.