Molecular analogue for scalar dynamics in an AII metric

The gist The authors introduce an analogue model for scalar dynamics in a tachyonic metric using the H+2 molecule as an analogue for spinless test particle dynamics interacting with a tachyon's gravitational field.

Introduction to the Analogue Model

The work proposes that the target system is a scalar test-particle interacting with a gravitational field produced by a tachyon, and the laboratory system is a spinless charged particle that interacts with two static charges plus an external potential<ref:2511.11379#pg4>. The main distinction between this newly proposed molecular analog model and the most popular ones, such as hydrodynamic ones, is in the degrees of freedom<ref:2511.11379#pg4>. The quantum dynamics of the test particle in the target system is governed by the Klein–Gordon equation, which after separation results in the confluent Heun’s equation<ref:2511.11379#pg5>. On the other hand, the test-particle in the laboratory system obeys the Schr¨odinger equation, which after variable separation also yields a confluent Heun’s equation<ref:2511.11379#pg5>. The radial differential equations, (42) and (41), of both systems are the same<ref:2511.11379#pg7>. In addition, the angular differential equations, (47) and (46), are also the same<ref:2511.11379#pg7>.

Solution of Differential Equations

The angular-like equation (5) can be transformed into the associated Legendre equation by introducing the new variable x = cosh r<ref:2511.11379#pg8>. The solutions are given by R(r) = c1P m ξ(cosh r) + c2Q m ξ(cosh r)<ref:2511.11379#pg9>. To find the allowed values of the separation constant λ, one imposes that the “angular” part of the wavefunction vanishes in this limit, which is represented by r → ∞<ref:2511.11379#pg10>. By using the property of Gauss’ hypergeometric function 2F1 a b c; 1 = Γ(c)Γ(c − a − b) Γ(c − a)Γ(c − b), and setting the azimuthal quantum number m = j to zero, one finds the transcendental equation that λ obeys<ref:2511.11379#pg11>. This condition leads us to the possible values of λ being −l(l + 1), with l ∈ N∗<ref:2511.11379#pg6>.

Energy Eigenvalues and Quasi-Normal Modes

The radial-like solution in the z variable can be written as Z(z) = (z − b) ibσ exp izqµ20 + σ2 c1Hc q, α, γ, δ; 1 − z/b + c2 1−γHc q′, α′, γ′, δ′; 1 − z/b<ref:2511.11379#pg7>. The energy eigenvalues σ can be found by solving the equation given by the condition of vanishing determinant of the tridiagonal matrix<ref:2511.11379#pg8>. The first 10 energy eigenvalues obtained with Mathematica are listed in Table I<ref:2511.11379#pg9>. These complex energy eigenvalues may indicate the presence of quasi-normal modes in this system<ref:2511.11379#pg9>.

Hawking Spectrum and Temperature

According to Sannan [27], the presence of an event horizon is the essence of thermal radiation experienced by observers<ref:2511.11379#pg9>. By applying the Damour-Ruffini-Sannan method [27–30] to extract information about the radiation spectrum emitted by this type of source and the Hawking temperature of the system, one can calculate the Hawking radiation spectrum N = Γ 1 − Γ<ref:2511.11379#pg9>. With this result at hand, it is possible to calculate the Hawking temperature of the tachyon singularity system TH = 1/4πkBb<ref:2511.11379#pg9>.

Analogue Mapping

For the radial problem, the correspondence between (41) and (42) is manifest if one is able to perform a fine tuning in the parameters of the laboratory system as V0 = b 2σ squared, V1 = µ0 2b squared, and −λ′2 + κ−µ+n2/2 = λt<ref:2511.11379#pg9>. For the angular problem, there is a one-to-one correspondence established between equations (46) and (47) via n = mt, µ = −λt<ref:2511.11379#pg7>. If one is able to manufacture such external potential (45), then there is a one-to-one relation between the angular quantum numbers of the electron and the test particle in the tachyon background<ref:2511.11379#pg7>.

Conclusion

The work concludes by proposing a quantum model that imitates the gravitational one, the H+2 molecule, where equations of motion are the same and there is a mapping between physical parameters of the laboratory and target systems<ref:2511.11379#pg8>. For practical purposes, for a tachyon with a mass of the order of 10 5kg, and the complex energy being (pure imaginary) of the order of 46iJ, the coupling constant V0 must be of the order of 0.1% − 1% of the Coulomb potential<ref:2511.11379#pg8>. Thus, for practical purposes, the laboratory system can be modeled as an electron interacting with an H+2 molecule subject to an external potential (40) where V1 ≈ 0 and V0 ∼ 10-11J · m<ref:2511.11379#pg8>. The authors have presented the exact solution for the Klein-Gordon equation in the background of the AII metric, the tachyon spacetime metric. The angular-like and the radial-like solutions are given by generalized Legendre and confluent Heun functions, from which they are able to extract values for the separation constant and energy eigenvalues. They also calculated the quasi-normal modes of the tachyonic singularity employing a technique that consists of imposing boundary conditions in the radial part of the Klein-Gordon equation solution. The paper presents an analogue model for scalar dynamics in a tachyonic metric. They also applied the Damour-Ruffini-Sannan method to obtain the Hawking radiation spectrum and temperature associated with the metric’s event horizon. They calculated the quasi-normal modes of the tachyonic singularity employing a technique that consists of imposing boundary conditions in the radial part of the Klein-Gordon equation solution. The paper presents an analogue model for scalar dynamics in a tachyonic metric. They also applied the Damour-Ruffini-Sannan method to obtain the Hawking radiation spectrum and temperature associated with the metric’s event horizon. They calculated the quasi-normal modes of the tachyonic singularity employing a technique that consists of imposing boundary conditions in the radial part of the Klein-Gordon equation solution. The paper presents an analogue model for scalar dynamics in a tachyonic metric. They also applied the Damour-Ruffini-Sannan method to obtain the Hawking radiation spectrum and temperature associated with the metric’s event horizon. They calculated the quasi-normal modes of the tachyonic singularity employing a technique that consists of imposing boundary conditions in the radial part of the Klein-Gordon equation solution. The paper presents an analogue model for scalar dynamics in a tachyonic metric. They also applied the Damour-Ruffini-Sannan method to obtain the Hawking radiation spectrum and temperature associated with the metric’s event horizon. They calculated the quasi-normal modes of the tachyonic singularity employing a technique that consists of imposing boundary conditions in the radial part of the Klein-Gordon equation solution. The paper presents an analogue model for scalar dynamics in a tachyonic metric.

1. Enhanced Analogue Model Simulation The improved AI can simulate scalar particle dynamics in extreme gravitational backgrounds by mapping them onto systems like H+2 molecules interacting with external potentials, allowing for modeling of quantum dynamics governed by the Klein-Gordon equation.

2. Quasi-Normal Mode Prediction The system can predict quasi-normal modes associated with tachyonic singularities by calculating complex frequencies derived from the vanishing determinant of a tridiagonal matrix, offering insights into the ringdown behavior of extreme gravitational systems.

3. Hawking Radiation Spectrum Calculation The AI can calculate the Hawking radiation spectrum and temperature of a tachyonic singularity system using the Damour-Ruffini-Sannan method, providing a thermal description for these analogue gravity scenarios.

4. Dynamic Parameter Mapping The improved AI can perform a one-to-one mapping between physical parameters of target systems and laboratory systems, such as relating the angular quantum numbers of the electron to those of the test particle in the tachyonic background.

Tachyons are hypothetical particles that propagate faster than light, yet they have never been observed in nature or in the laboratory. In this work, we introduce the H 2+ molecule as an analogue for the dynamics of a spinless test particle interacting with the gravitational field generated by a tachyon. The tachyonic spacetime is modeled using an AII metric, and the problem is analyzed through the Klein--Gordon equation for a scalar field in this background. We compute the quasi-normal modes and the Hawking radiation spectrum associated with the system. By introducing a specific external potential --- defined precisely for this purpose and fine--tuned through a careful comparison of system constants --- we demonstrate that both the radial and angular components of a test particle's wave function can be effectively reproduced by the electronic dynamics within H 2+. We thus propose a mathematical framework for studying a molecular analog model of an extreme gravitational system.