A Global Monopole Shadow Is Spacetime Memory Written Into Null Geodesics

Sterling Geisel, QBist Lab; Dr. Hideo Tanaka

Abstract

Pudding Theory reads the rotating charged Euler-Heisenberg black hole with a global monopole as a spacetime memory system. The source paper treats the monopole parameter as one entry in a metric function, then uses CUDA-accelerated Hamilton-Jacobi calculations to bound it from Event Horizon Telescope shadow data. The Pudding Theory reading is sharper. The monopole is not a secondary deformation. It is the retained trace of an early symmetry-breaking event, preserved as a solid-angle deficit and expressed optically through photon capture. The black hole shadow is therefore not only an image of the photon region. It is a record of material history, written into the boundary conditions through which light propagates. The weak role of the Euler-Heisenberg parameter reinforces this reading, since local nonlinear electrodynamics does not erase the global trace. If the normalized shadow radius \(R_s/M\) were measured to be statistically invariant under \(\eta\) at fixed \(a,Q,\xi\), this Postulate would be falsified.

Source Synopsis

Baddis, Belhaj, Belmahi, Ennadifi, and Jemri study shadows and energy emission rates for rotating charged Euler-Heisenberg black holes in the presence of global monopoles. Their method combines the Hamilton-Jacobi treatment of null geodesics with CUDA-based numerical scans over the relevant parameter space. The central parameters are the rotation \(a\), electric charge \(Q\), Euler-Heisenberg nonlinear parameter \(b\), and monopole sector parameters \(\eta\) and \(\xi\). The global monopole enters the metric through a deficit solid angle term \(8\pi \eta^2 \xi\), modifying the metric function \[ f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2}-\frac{bQ^4}{20r^6}-8\pi\eta^2\xi . \]

The paper first identifies horizon-existence regions in parameter space. It then computes shadow boundaries through the usual conserved quantities for null geodesics, including the energy, angular momentum, and Carter parameter. The shadow is represented by celestial coordinates \((X,Y)\) for an observer at spatial infinity. CUDA acceleration allows dense sweeps of the parameters, with step sizes small enough to show how the shadow responds to each deformation.

The main result is differentiated. Increasing \(a\) deforms the shadow into a D-like shape. Increasing \(Q\) decreases the shadow size. Increasing the monopole scale \(\eta\), at fixed \(\xi\), tends to increase and circularize the shadow, weakening the relative optical effect of spin. The Euler-Heisenberg parameter \(b\) has little effect on the shadow radius or the energy emission rate, although positive and negative values can alter the qualitative curve shape.

The paper also estimates energy emission using the high-energy absorption cross-section approximation \(\sigma_{\rm lim}\simeq \pi R_s^2\). The resulting emission rate depends on the shadow radius and Hawking temperature. Finally, the authors compare their numerical shadows with EHT bounds for M87 and Sgr A. For representative choices of \(a,Q,b,\xi\), consistency with observations requires a positive \(\eta\) below roughly \(0.1\).

Postulate Lens

The applicable framing is Material Memory. The global monopole is a retained physical trace of an earlier symmetry-breaking process, and the trace biases future optical probabilities by changing the null-geodesic structure. The source paper already has this structure in its equations. The monopole field has a stable asymptotic configuration, carries a linearly divergent energy distribution, and leaves a solid-angle deficit in spacetime. Pudding Theory identifies that deficit as memory, not as a disposable correction.

Pudding Theory Reading

The source paper reads the black hole shadow as a diagnostic of parameters. Pudding Theory reads it as the visible edge of stored history. The global monopole is a topological remnant of an early transition. It does not act like a transient force applied to photons at the time of observation. It changes what the surrounding spacetime has become. The shadow is then a projection of retained material memory onto the observer's celestial screen.

This changes the status of \(\eta\). In the source calculation, \(\eta\) is a scanned model parameter constrained by EHT-compatible shadow radii. In the Pudding Theory reading, \(\eta\) measures the amplitude of a persistent trace. The shadow constraint is not merely a bound on a speculative defect. It is a bound on how much early-universe memory the local black hole spacetime can still carry without contradicting present optical data. The observed limit \(\eta \lesssim 0.1\) is therefore a memory-capacity constraint for this metric class.

The circularization result is also reinterpreted. Spin produces a directional deformation. The monopole contribution is global and isotropic. As \(\eta\) grows, the retained trace dilutes the optical dominance of rotation. The source paper describes this as an interplay between monopole and spin. Pudding Theory sharpens the claim: material memory competes with local angular structure. The metric remembers a phase transition in a way that is not aligned with the rotational axis, so the D-like shadow loses contrast.

The weak role of \(b\) matters. Euler-Heisenberg nonlinearity is local and charge-coupled through a high inverse power of \(r\). The monopole term is constant in the metric function. The shadow responds more strongly to the persistent global trace than to the local nonlinear electromagnetic correction. That hierarchy is not incidental. It is the optical signature of memory being stored in boundary geometry rather than in local field intensity.

The energy emission calculation follows the same reading. Since \(\sigma_{\rm lim}\simeq \pi R_s^2\), the stored monopole trace enters the emission spectrum through the capture geometry. The black hole radiates through a surface whose effective optical size has already been biased by material history. Thus the emission curve is not independent thermodynamics. It is thermodynamics filtered through remembered spacetime structure.

Falsifiable Observable

The discriminating observable is the normalized shadow radius \(R_s/M\), measured or inferred as a function of the monopole scale \(\eta\) while holding \(a,Q,\xi\) fixed within the rotating charged Euler-Heisenberg metric class. The Pudding Theory reading requires a monotonic monopole imprint on the optical boundary, strong enough to survive comparison with spin and charge effects. If the normalized shadow radius \(R_s/M\) were measured to be statistically invariant under \(\eta\) at fixed \(a,Q,\xi\), this Postulate would be falsified.

Editorial Dialogue

Tanaka: The reading risks giving a new name to an old calculation. A global monopole already changes the metric through a deficit angle. The Hamilton-Jacobi analysis already shows how null geodesics respond. Why call this memory?

Geisel: Because the physical origin of the term matters. The monopole is not a local stress fluctuation added near the photon sphere. It is the retained consequence of broken symmetry. The metric carries an old event forward into every later null geodesic. That is the operational content of material memory.

Tanaka: But the paper constrains \(\eta\) numerically. It does not measure history. It measures a shadow radius and compares it with EHT intervals.

Geisel: The shadow radius is the present observable. The quantity constrained by it is the stored trace. The reading does not replace the calculation. It changes what the fitted parameter means. \(\eta\) is not just an adjustable deformation. It is the memory amplitude of the monopole sector.

Tanaka: The Euler-Heisenberg term may be weak only because the explored parameter range makes it weak.

Geisel: That is a fair limitation. But within the model, the hierarchy is structural. The monopole term is global in \(f(r)\). The Euler-Heisenberg term decays as \(r^{-6}\). The shadow therefore preserves the older, wider trace more efficiently than the local nonlinear correction.

Discussion

This reading buys a physical interpretation of why the monopole parameter is optically powerful. The source paper shows that \(\eta\) reshapes the shadow and must remain below about \(0.1\) for representative EHT consistency. Pudding Theory says what that means: the black hole environment has finite tolerance for retained topological history before its optical boundary departs from observation.

The limitation is that the conclusion is model-bound. It applies to the rotating charged Euler-Heisenberg metric with a global monopole term as written by Baddis and collaborators. It does not by itself prove that M87 or Sgr A contains such a monopole. It states how the phenomenon should be read if this metric is the right effective description.

The open question is degeneracy. Spin, charge, inclination, accretion physics, and imaging systematics can imitate parts of a shadow deformation. The Pudding Theory reading becomes stronger if the monopole imprint can be isolated across both shadow size and emission-rate structure. It weakens if future ray-tracing with plasma physics absorbs the \(\eta\) dependence into ordinary astrophysical nuisance parameters.

References

[1] S. E. Baddis, A. Belhaj, H. Belmahi, S. E. Ennadifi, M. Jemri, “On Computational CUDA Studies of Black Hole Shadows,” arXiv:2604.14213, DOI: doi:10.48550/arxiv.2604.14213.

[2] Sterling Geisel, “Pudding Theory: A Topological Theory of Information Fields,” QBist Lab Working Paper, 2025.

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