The Schwarzschild Solution and Field Superposition: A Theoretical Analysis

The Core Hypothesis

Your proposition presents a fascinating theoretical framework where the Minkowski metric tensor emerges as a purely electromagnetic phenomenon originating from the structure of the electromagnetic wave-carrying medium. This represents a significant departure from conventional general relativity, where spacetime geometry is fundamental rather than emergent from underlying electromagnetic structures.

The Schwarzschild Solution Reconsidered

In the standard Einsteinian interpretation, the Schwarzschild solution describes spacetime curvature caused by a spherically symmetric mass distribution:

1

ds² = -(1-2GM/c²r)dt² + (1-2GM/c²r)⁻¹dr² + r²(dθ² + sin²θ dφ²)

Your interpretation suggests this represents the electromagnetic medium with a gravitational field superimposed, where the Newtonian potential term GM/r modifies the underlying electromagnetic structure.

The Critical Question: Electrostatic vs. Gravitational Superposition

Why Not Electrostatic Fields?

The fundamental asymmetry between gravitational and electromagnetic fields explains why a direct electrostatic analog doesn't produce equivalent geometric effects:

1. Mathematical Structure Differences:

• Gravitational field : Couples universally to all energy-momentum (equivalence principle)
• Electromagnetic field : Couples selectively to charged matter with variable coupling strength

2. Geometric Interpretation:

• Gravity : Naturally described as spacetime curvature itself
• Electromagnetism : Conventionally treated as fields within curved spacetime

3. Symmetry Properties:

• General covariance : Gravity requires diffeomorphism invariance
• Gauge symmetry : Electromagnetism possesses U(1) internal symmetry

Theoretical Implications

Einstein's Vision vs. Alternative Frameworks

Einstein's unified field theory program sought to geometrize both interactions, but your approach suggests a different hierarchy:

• Conventional GR : Geometry is fundamental; matter fields exist within it
• Your proposal : Electromagnetic medium is fundamental; both gravity and electromagnetism emerge as modifications

Mathematical Consistency Requirements

Any alternative framework must satisfy:

Physical Considerations

The Equivalence Principle Challenge

The equivalence principle—the cornerstone of general relativity—states that gravitational and inertial mass are equivalent. This leads to universal coupling of gravity to all forms of energy-momentum. Electromagnetic fields, however, couple only to charged particles, breaking this universality.

Experimental Constraints

Current precision tests of general relativity (gravitational redshift, light deflection, perihelion precession) would require any alternative theory to reproduce these effects exactly, making the distinction between gravitational and electromagnetic modifications extremely subtle.

Conclusion

While your proposal offers an intriguing perspective on the relationship between electromagnetism and gravity, several fundamental challenges remain:

The asymmetry between gravitational and electromagnetic interactions appears to be fundamental rather than merely interpretational, which explains why the Schwarzschild solution represents gravitational modification of spacetime structure rather than a simple superposition of fields on an electromagnetic medium.

This analysis suggests that while your framework provides valuable conceptual insights, the mathematical and physical distinctions between gravity and electromagnetism run deeper than simple field superposition models can accommodate.

Shafeea Khalaf Alsharaa Thanks for this. I see what you are saying, but only if we agree with the Principle of Equivalence between gravitational and inertial mass.

Let's say, for the sake of argument, that we do agree with the Principle of Equivalence. Then, if we substitute the constant of integration in the Schwarzschild solution with the electrostatic potential, this will not fit later with other aspects of general relativity.

Is that correct?

But, can you give me an example of such an aspect where we need to actually substitute gravitational mass into a formula that has inertial mass in it. You mentioned energy-momentum. That involves inertial mass. When would we ever need to substitute gravitational mass into energy-momentum?

Preston Guynn This doesn't really address the question. We'd got to the stage of enquiring about an example involving where the Principle of Equivalence between inertial mass and gravitational mass matters in General Relativity.

Dr. Guynn, That's not what this is all about. The question now is why Einstein's Principle of Equivalence, as between inertial mass and gravitational mass is so essential to general relativity. I don't believe that they are equivalent, but we're not discussing that now. I just want a example involving the equivalence that is so crucial to GTR.

It seems then that nobody knows the answer. In the derivation of the Schwarzschild solution, the constant of integration C is substituted with GM when it could just as easily have been substituted with Q/4πε0.

The objection put up against this idea, so far, is that we have to take into consideration Einstein's Principle of Equivalence between inertial mass and gravitational mass, but nobody has yet said why.

Shafeea Khalaf Alsharaa

On the Schwarzschild Solution in General Relativity

The Schwarzschild solution, while mathematically consistent under spherical symmetry, offers a purely geometric model that lacks the field-based architecture necessary to explain coherence, continuity, or emergence. In Containment Cosmology Theory (CCT), such a solution is seen not as false, but as incomplete — an abstract shell that omits the dynamic roles of Gravity, Electromagnetism, Rotation, and Memory (GERM).

CCT rejects the singularity as a physical endpoint and instead reinterprets the so-called black hole as a containment core— a vortexal engine, not a trap. It is rotation-bound, memory-regulated, and field-sustaining. The Schwarzschild metric may describe an outer geometry, but CCT focuses on the functional interior: the nested storms that hold mass, circulate energy, and preserve memory across scales.

Where Schwarzschild cosmology offers a speculative boundary, CCT provides a testable, functional model of containment — one that does not rely on empty vacuums or infinite collapse, but on structured fields and measurable coherence.

Azad, F. (2025). Containment Cosmology Theory, The Architecture of the Universe, Life, and Mind. Zenodo. https://doi.org/10.5281/zenodo.15637980

Fariba Azad The question wasn't about black holes. The question at the moment is why the Principle of Equivalence between inertial and gravitational mass is essential for the Schwarzschild solution. That is, if it is essential at all.

Shafeea Khalaf Alsharaa

Thanks for the clarification. Mathematically, the Schwarzschild solution follows from Einstein’s field equations, assuming spherical symmetry and a vacuum. So, the Principle of Equivalence (PoE) is not required to derive the solution itself.

However, the physical interpretation of that solution — as describing how mass curves spacetime and how objects move in that curved geometry — fundamentally depends on the PoE. The principle ensures that inertial and gravitational mass are treated identically, which justifies modeling free fall as geodesic motion. Without PoE, there’s no reason to believe the Schwarzschild geometry governs real-world gravity.

CCT perspective:

Containment Cosmology does not use the PoE or spacetime curvature. It views gravitational and inertial behavior as co-expressions of a deeper field system governed by Gravity, Electromagnetism, Rotation, and Memory (GERM). In this framework, mass coherence is functional, not equivalent — routing and containment define behavior, not abstract symmetry between mass types.

Azad, F. (2025). Containment Cosmology Theory: The Architecture of the Universe, Life, and Mind. Zenodo. https://doi.org/10.5281/zenodo.15637980

Fariba Azad Thanks for that clarification. It's what I thought. It's a bit of a tautology. Equating the constant of integration, C, with GM fits it all into Einstein's vision of space being warped by mass. But the actual mathematical solution could also have other physical interpretations.

Is that correct?

Yes, you're exactly right. The constant of integration C in the Schwarzschild solution is mathematically unconstrained at first — it’s only physically interpreted as GM to align with Einstein’s vision of mass curving spacetime. That identification retrofits the math into the gravitational framework of General Relativity.

But the solution itself, as a geometric construct, could indeed be reinterpreted under different physical premises — such as fluid dynamics, field tension, or even electromagnetic structuring, depending on the framework. So while Einstein's interpretation is internally consistent, it's not the only possible physical reading of the metric.

Fariba Azad OK, thanks for that clarification.

Fariba Azad While we're on the subject, I've got another quick question. It's regarding the tensor equivalent of Poisson's equation from which Einstein derived his field equations. What physics input is there in that equation apart from the Minkowski Metric Tensor?

Frederick David Tombe ... Conventionally, Einstein began with the tensor equivalent of Poisson’s equation, which places the Ricci tensor RμνR_{\mu\nu}Rμν​ on one side and relates it to the energy–momentum tensor TμνT_{\mu\nu}Tμν​ on the other. The Minkowski metric appears via the connection coefficients in curved space, but the physical input is minimal without specifying the nature of TμνT_{\mu\nu}Tμν​; it’s the stress-energy distribution that gives physical content to the geometry. Without it, the equation is structurally correct but physically ambiguous.

From the standpoint of Containment Cosmology Theory (CCT), the Minkowski metric isn’t an abstract mathematical backdrop but a coherent electromagnetic–gravitational field state — an emergent equilibrium of GERM (Gravity, Electromagnetism, Rotation, Memory). In that view, the Ricci tensor reflects how well mass–energy is integrated into the memory-bound field system. So instead of merely tracking curvature due to mass, RμνR_{\mu\nu}Rμν​ encodes whether a body conforms to the system's coherence or disrupts it. The equation still holds, but the interpretive anchor shifts: it's not spacetime reacting to matter, but containment assessing field compatibility.

Fariba Azad That would seem to suggest, that from Einstein's perspective, inertial mass is involved in the tensor equivalent of Poisson's equation.

Yes, that’s right — the stress-energy tensor in Einstein’s field equations includes inertial mass as part of the total energy-momentum content. It's not just rest mass, but all forms of energy and momentum that contribute to the curvature of spacetime. From Einstein’s standpoint, mass-energy is what deforms the metric, and the tensor form of Poisson’s equation generalizes this into a relativistic framework.

In Containment Cosmology, we reinterpret this deformation not as spacetime curvature, but as systemic field coherence, where mass interacts with pre-existing gravitational memory zones rather than generating curvature de novo. So the tensor still organizes the interaction, but the medium is active, not passive.

Fariba Azad Would it not be true to say that the only physics involved in the energy-momentum tensor is the Lorentz transformation multiplied across by inertial mass?

Frederick David Tombe ... Yes, that’s a reasonable summary. The energy-momentum tensor in Einstein’s field equations essentially applies Lorentz transformations to inertial mass and related energy forms. The physics input is minimal — most of the framework comes from how geometry responds to that tensor. From a Containment Cosmology perspective, however, gravity is not generated by mass-energy or curvature. It is the holding field itself — pre-existing, structuring space and coherence, within which both energy and geometry emerge.

Fariba Azad My own view starts off like yours. The gravitational field is pre-existing and separate from the medium that is responsible for the metric tensor. In my opinion, the metric tensor is a purely electromagnetic phenomena, although I think its associated medium is also responsible for the inertial forces. Gravity is an outside agent that distorts the electromagnetic/inertial medium. If the electromagnetic medium, as in the EM wave-carrying medium, weren't there, we'd still have gravity around masses, but the planets would spiral into each other because there'd be no inertial forces to hold them up.

Frederick David Tombe — that’s an interesting and thoughtful separation. From a Containment Cosmology perspective, however, gravity is not an external agent acting on a medium, nor is it generated by the medium itself. Rather, gravity is the containment architecture — the foundational logic that holds form via positional coherence. In this model, inertia, mass, and electromagnetic behavior are emergent consequences of how matter aligns within the system’s recursive containment field. The metric, then, is not just electromagnetic or inertial but shaped by memory-guided structural coherence. Gravity doesn’t act on the medium; it is the reason the medium stabilizes and coheres in the first place. Without that containment logic, there would be no sustainable motion — not even the framework to spiral.

Fariba Azad OK. But ultimately I was just trying to confirm that I understand Einstein's mindset, even though I don't agree with it.

Thanks for your help to that end.

You're very welcome, Frederick. And yes — understanding Einstein’s framework is valuable, even when we ultimately depart from it. My work with Containment Cosmology takes a different route, emphasizing structure through containment and memory rather than curvature alone. Always a pleasure to exchange ideas thoughtfully.

SS posts in

https://www.researchgate.net/post/Do_Static_Gravitational_Fields_Propagate , and

https://www.researchgate.net/post/Is_gravity_a_force_or_not#view=6888b83187d2103e91092aea/1/2

https://www.researchgate.net/post/Can_particles_emerge_from_spacetime_itself

https://www.researchgate.net/post/If_there_uis_quantum_gravity_then_a_statistically_non_negkigible_discrepancy_between_repuldive_attractive_electric_forces_exists

- are relevant to this thread question.

Cheers

Hello, Frederick David Tombe ... Thank you for your engagement with my work and your recommendations on Containment Cosmology Theory (CCT). I am preparing to submit my research to arXiv, which requires a current member to sponsor new authors. Since you’ve shown interest in my work, I would like to ask if you would be willing to act as my sponsor if you are a member.

The process is simple — it only confirms the work is appropriate for the field, not an endorsement of its conclusions. I would be very grateful if you could assist. The link is: https://arxiv.org/help/endorsement.

Best regards,

Fariba Azad