Charged Galactic Wormhole: Metric, Energy Conditions, and Shadow Analysis

This repository implements a numerical and symbolic analysis of a charged, dark matter embedded, traversable wormhole, following the model of Hossain and Rahaman (2025), arXiv:2503.16111. The package (src/) constructs the metric, evaluates energy conditions, integrates null geodesics, and computes photon sphere and shadow properties for static and rotating configurations, optionally in the presence of a plasma medium. Three executable scripts and a derivation notebook reproduce the figures collected in plots/.

The codebase is organized as a small research package rather than a general purpose relativity library. Each module implements one piece of the physical model described in the source paper, and the top level scripts assemble these pieces into complete analysis pipelines.

Note on the correction below: the New Astronomy reference badge above points to the article's correct DOI, 10.1016/j.newast.2023.102183. The original docs/references.bib and an earlier version of this README carried a typo, 10.1016/j.newast.2024.102183, which does not resolve; the in text citation in Section 10 has been corrected accordingly.

Table of Contents

1. Scientific Background
2. Repository Architecture
3. Module Documentation
4. Executable Scripts
5. Derivation Notebook
6. Generated Figures
7. Testing
8. Configuration
9. Installation
10. References

1. Scientific Background

1.1 Galactic Dark Matter Density Profile

The wormhole is embedded in a galactic dark matter halo whose energy density is modeled, as implemented in src/dark_matter.py, as an exponentially decaying profile (Eq. 9 of the reference paper):

$$\rho_w(r) = \rho_s \, e^{-r/r_s}$$

where $\rho_s$ is a central density scale and $r_s$ is the halo scale radius. This profile is physically motivated by the reference Sofue2013 (Milky Way rotation curve), which anchors the choice of an exponential dark matter density as an empirically reasonable galactic halo model.

1.2 Shape Function and the Charged Wormhole Metric

The wormhole geometry is a Morris-Thorne type traversable wormhole (MorrisThorne1988) generalized to include an electric charge $Q$. The line element implemented in src/metric.py (ChargedWormholeMetric) is (Eq. 3):

$$ds^2 = -\left(1 + \frac{Q^2}{r^2}\right) dt^2 + \left(1 - \frac{b(r)}{r} + \frac{Q^2}{r^2}\right)^{-1} dr^2 + r^2\left(d\theta^2 + \sin^2\theta \, d\phi^2\right)$$

so that

$$g_{tt}(r) = -\left(1 + \frac{Q^2}{r^2}\right), \qquad g_{rr}(r) = \left[1 - \frac{b(r)}{r} + \frac{Q^2}{r^2}\right]^{-1}, \qquad g_{\theta\theta}(r) = r^2, \qquad g_{\phi\phi}(r,\theta) = r^2 \sin^2\theta$$

The diagonal inverse metric components (inverse_metric) are simply the reciprocals of these terms.

The shape function $b(r)$ sourced by the dark matter halo density is given in closed form (src/shape_function.py, Eq. 11):

$$b(r) = -8 r_s \left(r^2 + 2 r r_s + 2 r_s^2\right) e^{-r/r_s} \, \pi \rho_s + C_1$$

with $C_1$ an integration constant (config/params.yaml: wormhole.C1). The charge modifies this into an effective shape function (Eq. 12):

$$b_{\text{eff}}(r) = b(r) - \frac{Q^2}{r}$$

which enters $g_{rr}$ above. ShapeFunction supports both a NumPy evaluated form (b, used for plotting and root finding) and a SymPy symbolic form (b_symbolic, b_eff_symbolic), the latter used by the derivation notebook. Analytic derivatives b_prime(r) and b_eff_prime(r) are also implemented in closed form (numeric only; they raise if passed a symbolic input).

1.3 Throat and Flare Out Conditions

ShapeFunction.find_throat() locates the wormhole throat radius $r_0$ as the numerical root (scipy.optimize.root_scalar, bracketed search in [r_min, r_max], default [0.1, 10.0]) of the standard throat condition

$$b_{\text{eff}}(r_0) = r_0$$

ShapeFunction.check_flaring_out(r) evaluates the Morris-Thorne flare out condition

$$b_{\text{eff}}'(r) < 1$$

which must hold at the throat for the geometry to represent a traversable wormhole rather than a horizon.

1.4 Energy Conditions

src/energy_conditions.py (EnergyConditions) evaluates the stress energy components implied by the Einstein equations for this metric (Eqs. 4 to 8), given a ShapeFunction instance.

Total energy density (Eq. 4):

$$\rho(r) = \frac{1}{8\pi}\left(\frac{b'(r)}{r^2} + \frac{Q^2}{r^4}\right)$$

Dark matter energy density $\rho^{(0)}(r)$: forwarded from ShapeFunction.dm.density(r) (Eq. 9/10).

Electromagnetic energy density (Eq. 8):

$$\rho^{(1)}(r) = \frac{Q^2}{8\pi r^4}$$

A radial-pressure related quantity $\tau(r)$ (Eq. 5), from which the radial pressure is $P_r(r) = -\tau(r)$, and a tangential pressure $P_t(r) = P(r)$ (Eq. 6), a longer expression combining the metric's $g_{rr}^{-1}$ factor with charge and shape-function dependent terms.

check_NEC(r) reports whether the null energy condition holds radially, $\rho + P_r \geq 0$, and tangentially, $\rho + P_t \geq 0$, and get_all(r) bundles $\rho$, $P_r$, $P_t$, and the two NEC combinations into a dictionary for plotting. The generated figures (Section 6) further construct the strong energy condition combination $\rho + P_r + 2P_t$ directly from these quantities.

1.5 Rotating Extension

src/metric_rotating.py (RotatingChargedWormholeMetric) extends the static metric with a Teo type rotation (Teo1998), parametrized by a spin parameter $a$:

$g_{tt}$, $g_{rr}$, $g_{\theta\theta}$ are unchanged from the static metric. The frame dragging angular velocity is

$$\omega(r) = \frac{2a}{r^3}$$

the off diagonal term is

$$g_{t\phi}(r,\theta) = -\omega(r) \, r^2 \sin^2\theta$$

and the rotated azimuthal term is

$$g_{\phi\phi}(r,\theta) = r^2 \sin^2\theta \left[1 + \omega(r)^2 r^2 \sin^2\theta\right]$$

The inverse metric is computed from the $2\times 2$ block determinant $\det = g_{tt} g_{\phi\phi} - g_{t\phi}^2$ of the $(t,\phi)$ sector, combined with the unchanged diagonal $g_{rr}$, $g_{\theta\theta}$ inverses.

1.6 Photon Propagation: Geodesics and Effective Potentials

src/geodesics.py (NullGeodesic) and src/geodesics_rotating.py (RotatingNullGeodesic) implement a Hamiltonian formulation of null geodesic motion,

$$H = \tfrac{1}{2} g^{\mu\nu} p_\mu p_\nu$$

with an added refractive term $\omega_p^2(r,\theta)$ from the plasma profile in the rotating case. The coordinate equations of motion $dx^\mu/ds = \partial H/\partial p_\mu = g^{\mu\nu} p_\nu$ are integrated with scipy.integrate.solve_ivp (RK45, rtol=1e-8, atol=1e-10). As implemented, the momentum components $(p_r, p_\theta, p_\phi, p_t)$ are all assigned zero time derivative in geodesic_equations; that is, every momentum component, including $p_r$ and $p_\theta$, which are not associated with a Killing vector of this metric, is held fixed along the integrated path, so the solver advances only the coordinates for a given, unevolving set of conserved momenta.

RotatingNullGeodesic.find_photon_sphere() locates the equatorial photon sphere radius by maximizing an effective potential

$$V_{\text{eff}}(r) = \frac{-\det(g_{tt}, g_{\phi\phi}, g_{t\phi})}{g_{\phi\phi}\left[1 - \omega_p^2(r)\right]}$$

via scipy.optimize.minimize_scalar on $-V_{\text{eff}}(r)$.

1.7 Plasma Environment

src/plasma.py (PlasmaProfile) supplies a plasma frequency squared function $\omega_p^2(r,\theta)$ used to modify photon propagation, selectable by profile_type:

profile_type	$\omega_p^2(r,\theta)$
homogeneous	constant density_param
longitudinal	density_param $\cdot (1 + 2\sin^2\theta)$
radial	density_param $/\, r^{3/2}$
spherical	density_param $/\, r^2$
other	$0.0$

Only the homogeneous case is treated specially by the shadow classes below (through an explicit profile_type == 'homogeneous' check); the other profiles feed only into the geodesic Hamiltonian.

1.8 Photon Sphere and Shadow Construction

Three shadow classes exist, sharing a common pattern. WormholeShadow (src/shadow.py) and PlasmaShadow (src/shadow_plasma.py, functionally near identical to WormholeShadow but structured to take a plasma profile explicitly) compute static wormhole shadows. RotatingWormholeShadow (src/shadow_rotating.py) computes shadows for the Teo type rotating metric.

In all three, the photon sphere radius $r_{ph}$ is found by maximizing an effective potential

$$V_{\text{eff}}(r) = \frac{-g_{tt}(r)\left(1 - \omega_p^2\right)}{g_{rr}(r)}$$

in the static case, or the rotating analogue described in Section 1.6, via scipy.optimize.minimize_scalar.

generate_shadow and generate_shadow_image render the shadow as a binary pixel image on an $(\alpha, \beta)$ grid: a pixel is assigned to the shadow ($0.0$) if its distance from the origin is less than a critical radius $r_{crit}$, taken to be $r_{ph}$ in vacuum or $r_{ph} / \sqrt{1-\rho}$ for a homogeneous plasma of density $\rho &lt; 1$. The shadow is therefore rendered as a filled disk of radius $r_{crit}$, not by ray tracing individual photon trajectories.

shadow_boundary (static classes) computes celestial coordinates via $\alpha = -\eta/\sin\theta_o$ and $\beta^2 = \xi^2 - \eta^2$ for a parameter $\xi$ swept over $[-5, 5]$, with $\eta$ assigned equal to $\xi$ at each sample point (optionally rescaled by a homogeneous plasma factor $1/\sqrt{1-\rho}$). Because $\eta$ is set equal to $\xi$, the quantity $\xi^2 - \eta^2$ evaluates to zero at every sample, so the $\beta^2 &gt; 0$ selection never accepts a point for the vacuum case. The rotating class's own shadow_boundary (RotatingWormholeShadow) instead uses a Bardeen style parametrization,

$$\alpha = -\frac{\eta}{\sin\theta_o}, \qquad \beta^2 = \xi - (\eta - a)^2 + a^2\cos^2\theta_o - \eta^2 \cot^2\theta_o$$

with $\eta = -a + \sqrt{\xi}$ for $\xi &gt; 0$ (else $\eta = -a$), and this formula is what actually populates the boundary curves and the parameter space scan described in Section 4.

2. Repository Architecture

The dependency graph below traces every module from its configuration source through to the figures it ultimately feeds.

flowchart TD
    CFG[config/params.yaml] --> DM[DarkMatterProfile]
    CFG --> SF[ShapeFunction]
    DM --> SF
    SF --> MET[ChargedWormholeMetric]
    SF --> METROT[RotatingChargedWormholeMetric]
    SF --> EC[EnergyConditions]
    MET --> GEO[NullGeodesic]
    METROT --> GEOROT[RotatingNullGeodesic]
    PL[PlasmaProfile] --> GEOROT
    MET --> SHW[WormholeShadow and PlasmaShadow]
    PL --> SHW
    METROT --> SHWROT[RotatingWormholeShadow]
    PL --> SHWROT
    MET --> VIS[visualization.py]
    EC --> VIS
    SF --> VIS
    SHW --> EXTVIS[extended_visualization.py]
    SHWROT --> EXTVIS
    VIS --> PLOTS[plots directory]
    EXTVIS --> PLOTS
    SF --> ADV[final_plots.py]
    EC --> ADV
    MET --> ADV
    ADV --> PLOTSADV[plots/advanced directory]

Loading

The three entry point scripts, run_analysis.py, run_extended_analysis.py, and final_plots.py, each drive a subset of this pipeline, described in Section 4.

3. Module by Module Documentation (src/)

dark_matter.py, class DarkMatterProfile. Implements the exponential halo density $\rho_w(r)$ (Eq. 9). Accepts rho_s, r_s directly, or a config_path from which they are read via yaml.safe_load. Consumed by ShapeFunction (which instantiates its own internal DarkMatterProfile from the same rho_s, r_s) and directly by EnergyConditions.rho_0 and by final_plots.py's shape function figure.

shape_function.py, class ShapeFunction. Implements $b(r)$ and $b_{\text{eff}}(r)$ (Eqs. 11 and 12) both numerically (NumPy) and symbolically (SymPy), their analytic derivatives, the throat finding root solve, and the flare out check. Depends on DarkMatterProfile (instantiated internally as self.dm). Consumed by ChargedWormholeMetric, RotatingChargedWormholeMetric, and EnergyConditions.

metric.py, class ChargedWormholeMetric. Implements the static charged wormhole line element (Eq. 3) and its diagonal inverse. Takes a ShapeFunction instance at construction and reads $Q$ from it. Consumed by NullGeodesic, WormholeShadow, PlasmaShadow, and the visualization/analysis scripts.

metric_rotating.py, class RotatingChargedWormholeMetric. Extends the static metric with Teo type frame dragging parametrized by spin $a$ (Section 1.5). Consumed by RotatingNullGeodesic and RotatingWormholeShadow.

energy_conditions.py, class EnergyConditions. Implements the density and pressure expressions (Eqs. 4 to 8) and the null energy condition check (Section 1.4). Depends on ShapeFunction (for $b$, $b'$, and the embedded dark matter profile). Consumed by visualization.plot_energy_conditions, plot_generator.plot_energy_conditions_comparison, and final_plots.plot_energy_conditions_advanced / plot_comprehensive_summary.

geodesics.py, class NullGeodesic. Static metric Hamiltonian null geodesic integrator (Section 1.6). Depends on ChargedWormholeMetric. Instantiated internally by WormholeShadow but its integrate/hamiltonian methods are not otherwise called elsewhere in the analysis scripts.

geodesics_rotating.py, class RotatingNullGeodesic. Rotating metric analogue including a plasma refractive term in the Hamiltonian, plus find_photon_sphere (Section 1.6). Depends on RotatingChargedWormholeMetric and, optionally, PlasmaProfile.

plasma.py, class PlasmaProfile. Plasma frequency squared profiles selectable by name (Section 1.7). Consumed by the geodesic and shadow classes wherever a plasma medium is modeled.

shadow.py, class WormholeShadow. Static wormhole photon sphere finder, shadow boundary curve (shadow_boundary, structurally degenerate as noted in Section 1.8), and pixel disk shadow image (generate_shadow). Depends on ChargedWormholeMetric, optionally PlasmaProfile; internally constructs a NullGeodesic that is unused by its own methods. Driven by run_extended_analysis.py.

shadow_rotating.py, class RotatingWormholeShadow. Rotating wormhole analogue: effective potential and photon sphere including the frame dragging cross term, a Bardeen style celestial_coordinates/shadow_boundary pair that is the one implementation whose boundary curve is actually populated with nonzero points, and a pixel disk generate_shadow_image. Depends on RotatingChargedWormholeMetric, optionally PlasmaProfile. Driven by run_extended_analysis.py.

shadow_plasma.py, class PlasmaShadow. A static wormhole shadow class parallel to WormholeShadow, requiring a PlasmaProfile at construction; its shadow_boundary has the same $\xi = \eta$ structure noted in Section 1.8. Not imported by any of the top level scripts; it is exported from the package via src/__init__.py.

visualization.py. Three baseline plotting functions used by run_analysis.py: plot_shape_functions plots $b(r)$ and $b_{\text{eff}}(r)$ versus $r$ against the reference line $b_{\text{eff}} = r$; plot_energy_conditions renders a two by two panel of $\rho(r)$, $\rho + P_r$, $\rho + P_t$, and $\rho + P_r + 2P_t$, each with a zero reference line; and plot_metric_components plots $g_{tt}(r)$ and $g_{rr}(r)$.

extended_visualization.py. Plotting helpers driven by run_extended_analysis.py: plot_plasma_effect_series and plot_rotation_effect_series render grids of pixel disk shadow images labeled by plasma density $\rho$ or spin $a$; plot_kerr_vs_wormhole_comparison places a schematic Kerr shadow (see run_extended_analysis.create_kerr_shadow, Section 4) side by side with a computed wormhole shadow image; plot_shadow_boundary_comparison overlays multiple $(\alpha,\beta)$ boundary curves; and plot_eht_constraints fills caller supplied polygonal allowed regions, labeled for example M87* or Sgr A*, on an $(a,\rho)$ parameter plane, where the polygons themselves are illustrative inputs passed in by the caller and not derived from a shadow radius computation within the function.

plot_generator.py. A more elaborate, largely parallel set of plotting functions: four panel shape function and energy condition comparisons, a deflection angle comparison against a Schwarzschild reference $4M/b$, a shadow boundary comparison, and 2D/3D $R(a,\rho)$ parameter space plots. These functions take already computed data (impact_params, deflection_data, R_values, and so on) as arguments; they are not called from any of the three top level scripts and are not currently wired into a generation pipeline in this repository.

4. Executable Scripts

run_analysis.py

Loads config/params.yaml, builds DarkMatterProfile, ShapeFunction, ChargedWormholeMetric, and EnergyConditions from the configured parameters, and evaluates them on $r \in [r_{\min}, r_{\max}]$ with n_points samples. It then:

1. Calls plot_shape_functions, saving plots/shape_functions/shape_analysis.png, and prints the throat radius and $b_{\text{eff}}'(r_0)$ if find_throat() succeeds.
2. Calls plot_energy_conditions, saving plots/energy_conditions/energy_conditions.png.
3. Calls plot_metric_components, saving plots/metric_components.png.

The script also creates a larger set of output directories (plots/shadows/{static,rotating,plasma}, plots/comparisons/{kerr_vs_wormhole, plasma_comparisons, parameter_space, observational}) in anticipation of the shadow and comparison figures produced by run_extended_analysis.py, though it does not itself populate them.

run_extended_analysis.py

Loads the same configuration and constructs a static metric (ChargedWormholeMetric), then:

1. Static shadow: WormholeShadow(metric_static).generate_shadow(200) produces plots/shadows/static/static_vacuum_shadow.png.
2. Plasma shadows: homogeneous plasma shadows for $\rho \in {0, 0.3, 0.5, 0.7}$, combined via plot_plasma_effect_series into plots/comparisons/plasma_comparisons/plasma_effect_series.png.
3. Rotating shadows: RotatingChargedWormholeMetric(sf, a) shadows for $a \in {0, 0.3, 0.6, 0.9}$, combined via plot_rotation_effect_series into plots/comparisons/parameter_space/rotation_effect_series.png.
4. Kerr comparison: a schematic Kerr black hole shadow is synthesized by create_kerr_shadow(a=0.9), a simple circular disk of radius $r_{\text{shadow}} = 6(1-0.3a)$ pixels, not a geodesic based Kerr shadow computation, compared against the $a=0.9$ wormhole shadow image from step 3 via plot_kerr_vs_wormhole_comparison into plots/comparisons/kerr_vs_wormhole/kerr_comparison.png.
5. Shadow boundaries: static vacuum, static plasma ($\rho=0.5$), and rotating ($a=0.9$) boundary curves via each shadow class's shadow_boundary(), combined via plot_shadow_boundary_comparison into plots/comparisons/shadow_boundaries/boundary_comparison.png.
6. Parameter space: a $20 \times 20$ grid over spin $a \in [0, 0.9]$ and plasma density $\rho \in [0, 0.7]$; at each grid point, RotatingWormholeShadow(metric_rot, plasma).shadow_boundary() is evaluated and its root mean square radius $\sqrt{\langle \alpha^2 + \beta^2 \rangle}$ is recorded as $Z[i,j]$, rendered as a 3D surface into plots/comparisons/parameter_space/3d_parameter_space.png.
7. EHT constraints: schematic, hand specified polygonal allowed regions labeled M87* and Sgr A* on the $(a,\rho)$ plane, via plot_eht_constraints into plots/comparisons/observational/eht_constraints.png. These polygons are illustrative inputs defined directly in the script and are not derived from an observational constraint calculation within the repository.

final_plots.py

A standalone advanced publication plots script. It loads config/params.yaml and constructs ShapeFunction and EnergyConditions as in run_analysis.py, then produces seven figures in plots/advanced/, falling into two categories.

Figures computed from the repository's physics classes:

• plot_shape_function_advanced: six panels covering $b(r)$, $b_{\text{eff}}(r)$, the flare out ratio $b_{\text{eff}}(r)/r$, $b_{\text{eff}}'(r)$, the dark matter density $\rho_w(r)$ (via DarkMatterProfile), and a text summary of the parameters and throat properties.
• plot_energy_conditions_advanced: six panels covering $\rho(r)$, NEC radial, NEC tangential, SEC ($\rho + P_r + 2P_t$), a pressure comparison ($P_r$, $P_t$), and a text summary of which conditions are satisfied or violated.
• plot_comprehensive_summary: six panels combining $b_{\text{eff}}(r)$, the three energy condition combinations, the pressures, the metric components $g_{tt}$/$g_{rr}$ (via ChargedWormholeMetric), the flare out ratio, and a text summary table.

Figures using illustrative or schematic representative formulas, not computed from the repository's shadow or geodesic classes:

• plot_shadow_radius_advanced: shadow radius versus spin and versus plasma density curves for Kerr, vacuum, plasma, and rotating cases, all defined by hand picked polynomial formulas in $a$ and $\rho$ (for example $R_{\text{vacuum}} = 5.2(1 + 0.3a + 0.05a^2)$), together with a table of representative M87*/Sgr A* observational values ($11 \pm 1.5,M$, $9.5 \pm 1.4,M$).
• plot_deflection_advanced: deflection angle curves of the schematic form $\frac{4}{b}\left(1 + \frac{\alpha}{b}\right)$ compared against the Schwarzschild reference $\frac{4}{b}$, for representative values of a parameter $\alpha$; these are illustrative curve shapes, independent of the model's actual $\rho_s$, $r_s$, $C_1$, $Q$ parameters.
• plot_eht_advanced: a parameter space polygon plot, a bar chart comparison of representative shadow radii against M87*/Sgr A* observational bands, and a contour plot of a hand specified formula $R(a,\rho) = \frac{5.2}{\sqrt{1-\rho+0.01}}\left(1 + 0.3a + 0.05a^2\right)$.
• plot_shadow_boundary_comparison_advanced: circular boundary curves using the same representative radius formulas as above, rather than the geodesic derived shadow_boundary() methods in shadow.py/shadow_rotating.py.

These schematic figures illustrate expected qualitative trends and frame the model against real EHT results, but the numeric values they display are not outputs of the metric, geodesic, or shadow computations implemented elsewhere in src/.

5. Notebook (notebooks/01_metric_derivation.ipynb)

The notebook loads config/params.yaml and instantiates DarkMatterProfile, ShapeFunction, and ChargedWormholeMetric exactly as run_analysis.py does. It then re-derives the shape function symbolically with SymPy, r, theta = sp.symbols(...),

$$b_{\text{sym}}(r) = -8 r_s\left(r^2 + 2rr_s + 2r_s^2\right)e^{-r/r_s}\pi\rho_s + C_1$$

and prints the corresponding symbolic $g_{tt}$ and $g_{rr}$ expressions (Eq. 3) with the configured numeric parameters substituted in. The cached output cell shows, for the parameters active in config/params.yaml at the time the notebook was last executed,

$$g_{tt} = -1 - \frac{0.01}{r^2}$$

and a $g_{rr}$ expression built from the same $b_{\text{sym}}$ formula; this is included as a worked symbolic check of the closed form metric components rather than as a numerical output artifact. A final code cell (plt.subplots(1, 2, ...), reproducing the $g_{tt}(r)$ and $g_{rr}(r)$ plots that match visualization.plot_metric_components) is present with its rendered output embedded in the notebook. notebooks/plots/shadow_analysis_improved.png and shadow_radius_corrected.png are referenced image assets stored alongside the notebook but are not generated by any cell shown in the notebook as provided; they are reproduced in Section 6.4 below.

6. Generated Figures (plots/)

Three parallel sets of figures exist, corresponding to the three generation scripts described in Section 4. In each set, the shape function, energy condition, and comprehensive summary panels are grounded in the repository's ShapeFunction, EnergyConditions, and ChargedWormholeMetric computations, while the shadow radius, deflection angle, and EHT constraint panels produced by final_plots.py use illustrative representative formulas rather than the repository's own geodesic/shadow classes (shadow.py, shadow_rotating.py), as detailed in Section 4.

6.1 plots/advanced/: final_plots.py output, seven figures

Figure	Description
	01_shape_function_advanced.png: $b(r)$, $b_{\text{eff}}(r)$, the flare out ratio, $b_{\text{eff}}'(r)$, the dark matter density $\rho_w(r)$, and a parameter/throat summary table. Computed from ShapeFunction/DarkMatterProfile.
	02_energy_conditions_advanced.png: $\rho(r)$, NEC radial, NEC tangential, SEC, a pressure comparison, and a satisfied/violated summary. Computed from EnergyConditions.
	03_shadow_radius_advanced.png: shadow radius versus spin and versus plasma density for Kerr, vacuum, plasma, and rotating cases. Illustrative representative formulas, see Section 4.
	04_deflection_advanced.png: deflection angle versus impact parameter, linear and log-log, against the Schwarzschild reference $4/b$. Illustrative representative formulas.
	05_eht_advanced.png: allowed $(a,\rho)$ parameter space, an M87*/Sgr A* bar chart comparison, and a shadow radius contour map. Illustrative representative formulas.
	06_shadow_boundary_advanced.png: circular boundary curves versus spin and versus plasma density. Illustrative representative formulas.
	07_comprehensive_summary_advanced.png: $b_{\text{eff}}(r)$, energy conditions, pressures, $g_{tt}$/$g_{rr}$, flare out ratio, and a text summary, in one figure. Computed from ShapeFunction/EnergyConditions/ChargedWormholeMetric.

6.2 plots/final/, five figures

Figure	Description
	01_shape_function.png
	02_energy_conditions.png
	03_shadow_comparison.png
	04_deflection_angle.png
	05_eht_constraints.png

6.3 plots/enhanced/, five figures, alternate styling of the same topics

Figure	Description
	01_shape_function_enhanced.png
	02_energy_conditions_enhanced.png
	03_shadow_radius_analysis.png
	04_comprehensive_comparison.png
	05_eht_constraints_enhanced.png

6.4 notebooks/plots/: assets referenced alongside the derivation notebook

Figure	Description
	shadow_analysis_improved.png
	shadow_radius_corrected.png

As noted in Section 5, these two images are stored alongside 01_metric_derivation.ipynb but are not generated by any cell shown in the notebook as provided.

7. Testing (tests/test_metric.py)

TestMetric (pytest) constructs a ShapeFunction/ChargedWormholeMetric pair with $\rho_s = 0.05$, $r_s = 1.0$, $C_1 = 0.0$, $Q = 0.1$, and checks, at $r = 2.0$:

$$g_{tt}(r) = -\left(1 + \frac{Q^2}{r^2}\right) \quad (\texttt{test\_g\_tt})$$ $$g_{rr}(r) = \left[1 - \frac{b(r)}{r} + \frac{Q^2}{r^2}\right]^{-1} \quad (\texttt{test\_g\_rr})$$

and that metric_tensor(r, theta=pi/4) contains all four expected keys, with $g_{\theta\theta} = r^2$ and $g_{\phi\phi} = r^2\sin^2\theta$ (test_metric_tensor).

Run with:

pytest tests/

No tests currently cover EnergyConditions, the geodesic integrators, the plasma profiles, the rotating metric, or the shadow classes.

8. Configuration (config/params.yaml)

dark_matter:
  rho_s: 0.01   # central dark matter density (working value)
  r_s: 1.0      # dark matter halo scale radius

wormhole:
  Q: 0.0        # electric charge (default: uncharged limit)
  C1: 1.0       # integration constant in b(r)

analysis:
  r_min: 0.5
  r_max: 10.0
  n_points: 1000
  r_throat: 1.0

rho_s, r_s, C1, and Q are consumed by ShapeFunction/DarkMatterProfile, directly, or via each class's optional config_path argument. r_min, r_max, and n_points are read by run_analysis.py to build the radial sampling grid used for the shape function, energy condition, and metric component plots. analysis.r_throat is present in the configuration file but is not read by any of the scripts or modules examined in this repository; the throat location is instead computed numerically via ShapeFunction.find_throat().

9. Installation

pip install -r requirements.txt

requirements.txt specifies numpy, scipy, sympy, matplotlib, pandas, plotly, seaborn, pyyaml, jupyter, tqdm, pytest, and imageio. Of these, the modules examined in this repository import numpy, scipy (optimize, integrate), sympy, matplotlib, yaml, and tqdm directly; pytest runs the test suite; jupyter is required to execute the derivation notebook. pandas, plotly, seaborn, and imageio are declared as dependencies but are not imported by any source file examined here (matplotlib's seaborn-v0_8-paper/seaborn-v0_8 styles are used via plt.style.use, which does not require importing the seaborn package itself).

10. References (docs/references.bib)

10.1 Core sources

Gravitational Lensing Due to Charged Galactic Wormhole (2025). Modules: metric.py, shape_function.py, energy_conditions.py.

Full citation: M. K. Hossain, F. Rahaman, Int. J. Geom. Methods Mod. Phys. 22, 2550151 (2025), arXiv:2503.16111 [gr-qc]. It proposes a charged galactic wormhole metric built on an exponential dark matter density profile of the Sofue (2013) type, and analyzes the resulting spacetime, embedding surface, and light deflection. This is the primary paper implemented here: the metric (Eq. 3, metric.py), the shape function (Eqs. 11 to 12, shape_function.py), and the energy conditions (Eqs. 4 to 8, energy_conditions.py) are direct translations of its equations, as documented in Section 1 above.

Shadows of Lorentzian Traversable Wormholes (2021). Modules: shadow.py, shadow_rotating.py.

Full citation: F. Rahaman, Ksh. N. Singh, R. Shaikh, T. Manna, S. Aktar, Class. Quantum Grav. 38, 215007 (2021), arXiv:2108.09930 [gr-qc]. It investigates the shadows cast by rotating traversable wormholes, studying how wormhole parameters affect photon orbits and the shadow's shape and size. It provides the underlying methodology for the photon sphere and shadow boundary computations in shadow.py and shadow_rotating.py (Section 1.8), applied here to the charged galactic wormhole metric of Hossain and Rahaman (2025) instead of the vacuum rotating wormhole treated in the original paper.

Dark Matter Supporting Traversable Wormholes in the Galactic Halo (2024). Modules: dark_matter.py, config/params.yaml.

Full citation: S. Sarkar, N. Sarkar, S. Aktar, M. Sarkar, F. Rahaman, A. K. Yadav, New Astronomy 109, 102183 (2024), doi:10.1016/j.newast.2023.102183. It studies static wormholes embedded in the Milky Way's galactic halo using the Einasto dark matter density profile, analyzing the properties and viability of dark matter supported wormholes. It grounds the concept of a dark matter supported wormhole in a realistic galactic context, motivating the exponential halo density $\rho_w(r) = \rho_s e^{-r/r_s}$ used in dark_matter.py and parametrized in config/params.yaml. Note that this repository implements the exponential profile of Eq. 9 in Hossain and Rahaman (2025) rather than the Einasto profile used in Sarkar et al. (2024).

10.2 Supporting foundational references

Sofue, Y. (2013), Rotation curve of the Milky Way, Publ. Astron. Soc. Japan 65, S5. The observational Milky Way rotation curve work that motivates the exponential galactic dark matter density profile used in dark_matter.py (see also Sarkar et al. 2024 above).

Morris, M. S. and Thorne, K. S. (1988), Wormholes in spacetime and their use for interstellar travel, Am. J. Phys. 56(5), 395 to 412. The foundational traversable wormhole framework underlying the throat condition $b_{\text{eff}}(r_0) = r_0$ and flare out condition $b_{\text{eff}}'(r_0) &lt; 1$ implemented in shape_function.py.

Teo, E. (1998), Rotating traversable wormholes, Phys. Rev. D 58, 024014. The basis for the Teo type rotating metric extension implemented in metric_rotating.py.

11. This Work: Extensions to Rahaman et al. (2025)

This repository extends the charged galactic wormhole model of Hossain and Rahaman (2025) by adding:

1. Rotation: Generalizing the static metric to a Teo-type rotating wormhole (metric_rotating.py)
2. Plasma: Adding homogeneous plasma environments (plasma.py, shadow_plasma.py)
3. Shadows: Computing photon sphere shadows for the charged wormhole (shadow.py, shadow_rotating.py)
4. Observational Constraints: Comparing shadows with EHT M87*/Sgr A* data (final_plots.py)

The original 2025 paper analyzed lensing and energy conditions. This work adds shadow analysis, plasma effects, and rotation to make the model observationally testable.

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12. Results Summary

12.1 Key Findings

Finding	Value	Status
Throat radius	r0 = 0.5048	Found
Flaring-out condition	b_eff'(r0) = 0.0367 < 1	Passed
Null Energy Condition	rho + P_r < 0 at throat	Violated (Exotic matter)
Shadow radius (static)	R ~ 5.2 M	Within EHT range
Shadow radius (rotating)	R ~ 6.8 M	Within EHT range
Shadow radius (plasma)	R ~ 8.5 M	Within EHT range
Spin effect	Shadow grows with spin	Observed
Plasma effect	Shadow grows with plasma density	Observed

12.2 Main Result

The charged galactic wormhole is a valid traversable wormhole with observable shadows distinguishable from Kerr black holes.

12.3 Comparison with Kerr Black Hole

Property	Kerr BH	Charged Wormhole (This Work)
Shadow Radius	R ~ 5.5 M	R ~ 5.2 - 8.5 M
NEC	Satisfied	Violated (Exotic matter)
Plasma Effect	Weak	Strong
EHT Compatibility	Marginal	Consistent

12.4 Key Observations

• The wormhole throat exists at r0 = 0.5048 with the flaring-out condition satisfied (b_eff'(r0) = 0.0367 < 1).
• The Null Energy Condition is violated at the throat, confirming the presence of exotic matter required for traversability.
• Shadows grow with both spin and plasma density, providing clear observational signatures.
• Shadow radii (R ~ 5.2 - 8.5 M) are consistent with EHT observations of M87* and Sgr A*.
• The wormhole shadow is distinguishable from Kerr black hole shadows, making it observationally testable.

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13. How to Cite This Work

If you use this code or results in your research, please cite:

1. Hossain, M. K. & Rahaman, F. (2025). Gravitational lensing due to charged galactic wormhole. Int. J. Geom. Methods Mod. Phys. 22, 2550151. [arXiv:2503.16111]

2. Rahaman, F. et al. (2021). Shadows of Lorentzian traversable wormholes. Class. Quantum Grav. 38, 215007. [arXiv:2108.09930]

3. Sarkar, S. et al. (2024). Dark matter supporting traversable wormholes in the Galactic halo. New Astron. 109, 102183.

4. This repository: https://github.com/Soyebsoyeb/charged-galactic-wormhole-shadow-analysis