Charged particle dynamics in singular spacetimes: hydrogenic mapping and curvature-corrected thermodynamics

TL;DR

This work analyzes charged test-particle dynamics in a horizonless, massless spacetime generated by electric charge $Q$ within the Einstein-Maxwell-Scalar framework. By exploiting exact first integrals, it derives the effective potential, circular-orbit criteria, epicyclic frequencies, and periastron precession, revealing a hard boundary at the outer singular shell $r_*$ and a transition from Coulombic to curvature-dominated dynamics. In the weak-field regime the motion maps to a hydrogenic system with curvature-induced energy shifts, while near $r_*$ strong confinement arises from diverging potentials. The authors also couple these dynamics to a curvature-corrected spectral thermodynamics, showing systematic increases in free and internal energies and subtle changes to entropy and heat capacity, thereby connecting microscopic orbital structure to macroscopic thermodynamics in a charge-driven spacetime.

Abstract

We analyze the dynamics of charged test particles in a singular, horizonless spacetime arising as the massless limit of a charged wormhole in the Einstein--Maxwell--Scalar (EMS) framework. The geometry, sustained solely by an electric charge $Q$, features an infinite sequence of curvature singularity shells, with the outermost at \( r_* = \frac{2|Q|}π \) acting as a hard boundary for nonradial motion, while radial trajectories can access it depending on the particle charge-to-mass ratio \( |q|/m \). Exploiting exact first integrals, we construct the effective potential and obtain circular orbit radii, radial epicyclic frequencies, and azimuthal precession rates. In the weak-field limit (\( r \gg |Q| \)), the motion reduces to a Coulombic system with small curvature-induced retrograde precession. At large radii, the dynamics maps to a hydrogenic system, with curvature corrections inducing perturbative energy shifts. Approaching \( r_* \), the potential diverges, producing hard-wall confinement. Curvature corrections also modify the spectral thermodynamics, raising energies and slightly altering entropy and heat capacity. Our results characterize the transition from Newtonian-like orbits to strongly confined, curvature-dominated dynamics.

Figures (5)

• Figure 1: Curvature invariants (see Appendix A) of the massless spacetime with discrete singular shells. Left: Ricci scalar $R(r)$, Right: Kretschmann scalar $K(r)$, both plotted in log-log scale as functions of radial coordinate $r$. Vertical dashed lines indicate the positions of singular shells $r_n = Q/(\pi/2 + n\pi)$ for $n=0,1,2,3,4...$, with the outermost shell $r_\ast = Q/(\pi/2) \approx 0.6366$ highlighted in black. Both scalars diverge at each singular shell and at the origin, representing true curvature singularities. Away from the singularities, the log-log representation reveals the asymptotic decay $R(r)\sim r^{-4}$ and $K(r)\sim r^{-8}$ for $r \gg r_\ast$, highlighting the fall-off of curvature in the weak-field region. Values used: $Q=1$, radial range $r \in [0.01\,Q,10\,Q]$. The log-log scale emphasizes the power-law divergence near each singular shell and the central singularity, as well as the asymptotic behavior far from the shells. The outermost shell $r_\ast$ acts as the first hypersurface of infinite curvature, demonstrating the nontrivial geometric structure of this massless spacetime. The sequence of singular shells constrains geodesic motion and defines regions of extreme tidal forces.
• Figure 2: Effective radial potentials $V_{\rm eff}(r)$ for different angular momentum states $L = 1, 3, 5$ of a particle with charge $q=-1$ and unit mass $m=1$ in the presence of a singular shell located at $r_* = 2|Q|/\pi$ ($Q = 10$). Colored curves represent the effective potentials for each $L$ state, with the corresponding colored stars indicating the classical turning points for energies $\mathcal{E} = 3, 5, 10$. Shaded regions highlight the classically allowed radial motion ($\mathcal{E} > V_{\rm eff}(r)$), and the dashed vertical line marks the outermost singular shell position $r_{\ast}$. This visualization illustrates how the effective potential and the allowed regions depend on angular momentum and energy levels (see Table \ref{['tab:dims']}).
• Figure 3: Dynamics of a charged particle with unit mass $m = 1$ and charge $q = -1$ around a central Coulomb charge $|Q| = 2.4$ for angular momenta $L = 1, 2, 3$. (a) Radial energy $\mathcal{E}_+(r)$ showing contributions from the rest mass (black dashed line), Coulomb interaction, and angular momentum, with circular orbits $r_c$ indicated by vertical dashed lines and the outermost barrier $r_{\ast}$ highlighted in purple. (b) Effective potential $V_{\rm eff}(r)$ illustrating the radial dependence of the potential energy, with circular orbits and $r_{\ast}$ similarly marked (purple). (c) Radial oscillations $r(t)$ around circular orbits with shaded envelopes representing the oscillation amplitude, and $r_{\ast}$ shown as a purple dashed line. (d) Two-dimensional precessing orbits in the $xy$ plane, exhibiting retrograde precession around the central charge (black dot), with maximum and minimum radial excursions, and the outermost barrier $r_{\ast}$ shown as a dashed purple circle (see also Table \ref{['tab:dims']}).
• Figure 4: Hydrogenic energy levels (left) and curvature-induced shifts (right). The Coulomb potential is shown in blue, with unperturbed hydrogenic energies for $n = 1, 2$ depicted as solid lines. Curvature-perturbed energies are indicated by dashed black lines. The bar plot quantifies curvature-induced shifts, highlighting that the ground state ($n=1$) experiences the largest shift.
• Figure 5: Thermodynamic properties of the truncated hydrogenic spectrum with curvature corrections. Subplots (a--d) display the absolute canonical quantities: Helmholtz free energy $F(T)$, internal energy $U(T)$, entropy $S(T)$, and heat capacity $C_V(T)$ for $n_{\max}=200$, with solid black lines for the unperturbed energies $E_n^{(0)}$ and dashed blue lines including curvature shifts $\Delta E_n^{(1)}$. Subplots (e--h) present the curvature-induced differences $\Delta F$, $\Delta U$, $\Delta S$, and $\Delta C_V$. Subplots (i--l) show convergence for $n_{\max}=100,200,300$, illustrating the stability of canonical sums. Residuals $F-(U-TS)$ are smaller than $10^{-14}\,\mathrm{eV}$, confirming numerical consistency. All quantities are in eV or eV/K; the temperature axis is logarithmic to emphasize low- and high-temperature regimes.