And Still, Spheres Gravitate

I. The Photon Cannot Explain Coulomb

Before solving the gravity problem, consider an identical problem that was solved 150 years ago in electromagnetism.

A photon has two transverse polarizations. That is the complete physical content of the electromagnetic field: two propagating degrees of freedom per wavevector. No longitudinal photon. No timelike photon. Just two transverse modes.

Now place two static charges in space. They attract (or repel) with Coulomb's law: $F = q_1 q_2 / (4\pi\varepsilon_0 r^2)$. But no transverse photon is exchanged between static charges. Zero. The force is instantaneous in Coulomb gauge. There is no propagating mode carrying the interaction.

So where does the Coulomb potential come from?

In QED, the scalar potential $A_0$ is not a dynamical degree of freedom. It has no conjugate momentum. It does not propagate. But it is determined instantaneously by Gauss's law:

This constraint is not an additional input to the theory. It follows automatically from the dynamical equations (Maxwell's equations for the transverse sector) plus charge conservation ($\partial_\mu J^\mu = 0$). The logic:

1. The dynamical equations govern the transverse modes: $\square A^T_i = J^T_i$
2. Charge conservation constrains: $\partial_0 \rho + \nabla \cdot \mathbf{J} = 0$
3. Consistency forces the constraint: $\nabla^2 A_0 = -\rho/\varepsilon_0$
4. The Coulomb potential emerges — carried by a non-dynamical field, not by photons

Two transverse polarizations. No longitudinal photon. And yet the full Coulomb potential is there, because constraints are not optional — they are consequences of dynamics plus symmetry.

This has been understood since Maxwell. It is textbook physics. And it is exactly what happens with gravity.

II. The Graviton Cannot Explain Newton

The graviton — the quantized perturbation of the spacetime metric — has exactly two physical polarizations: $h_+$ and $h_\times$. These are transverse-traceless modes. They propagate at $c$. They are the gravitational waves detected by LIGO.

But the Newtonian potential $\Phi = -GM/r$ is not carried by gravitational waves. It is an instantaneous (in harmonic gauge: retarded, but static for static sources) relationship between mass density and the metric. In the canonical (Hamiltonian) formulation of linearized gravity, $h_{00} = 2\Phi/c^2$ is a constrained variable, not a dynamical one. It has no independent equation of motion. It is determined by the Hamiltonian constraint:

The analogy is exact:

In both cases, the static potential is real, physical, and measurable — but it is not carried by propagating quanta. It is carried by the constraint sector.

III. From TT Dynamics to Full Gravity

Now apply this to Bath-TT. The framework claims TT coupling produces Unimodular Gravity. The no-go theorem (verified in Lean 4) proves TT coupling cannot reproduce full GR. The resolution is that UG's traceless field equations are the dynamical content — and the constraint sector (including the Newtonian potential) follows automatically.

Here is the derivation. Start from the traceless Einstein equations (Unimodular Gravity):

These are the traceless part of Einstein's equations. Taking the trace gives $0 = 0$ — a tautology, not an equation. The trace sector appears absent.

Step 1. Take the covariant divergence of both sides. The left side, by the contracted Bianchi identity:

Step 2. The right side, using matter conservation $\nabla^\mu T_{\mu\nu} = 0$ (assumption A1 of the framework):

Step 3. Equating gives:

Step 4. Substitute $R = -\kappa T - 4\Lambda$ back into the traceless equations. After algebra:

The Newtonian limit of this equation is:

For any system smaller than cosmological scales, $\Lambda c^2 \sim 10^{-35}\,\text{m/s}^2$ is negligible compared to $4\pi G\rho$. The Poisson equation holds. Spheres gravitate.

IV. Why the Full Propagator Is Forced

A natural question: why does TT coupling produce Unimodular Gravity — with its full constraint structure, including the Newtonian potential — rather than just a TT wave equation in vacuum?

The standard answer invokes the Fierz-Pauli theorem (1939): the unique Lagrangian for a free massless spin-2 field is linearized GR. But the Bath-TT framework has no Lagrangian. It is an open quantum system described by Lindblad dynamics. The Fierz-Pauli argument does not directly apply.

There is a deeper argument that does not require a Lagrangian.

Weinberg's soft graviton theorem (1964)

In any Lorentz-invariant theory, the amplitude for emitting a soft (low-energy) massless particle of spin $s$ is proportional to a conserved charge of the external particles. For spin 1, this charge is electric charge — which gives the universality of electromagnetism. For spin 2, this charge is the stress-energy tensor $T^{\mu\nu}$ — which gives the universality of gravity.

This theorem requires only three inputs:

1. Lorentz invariance (assumption A3 of the framework)
2. Unitarity of the tree-level S-matrix (the classical/mean-field limit is unitary)
3. A massless spin-2 particle exists (the Lindblad mean-field dynamics produces $\square h^{TT}_{ij} = -16\pi G\, T^{TT}_{ij}$ — a massless spin-2 field with two TT polarizations)

It does not require a Lagrangian, an action principle, or a fundamental gravitational field.

The consequence: the full propagator is unique

Weinberg's theorem forces the full exchange amplitude between any two sources to be:

This amplitude decomposes in the 3+1 (ADM) split into two sectors:

The TT sector is what the Lindblad dynamics directly produces. The constraint sector is the complement — the unique remainder within the full amplitude. It is not an additional coupling. It is not put in by hand. It is forced by the spin-2 nature of the emergent graviton, via Weinberg's theorem.

For a static sphere ($T^{TT}_{ij} = 0$):

This is exactly what happens in QED. The full photon propagator splits into a transverse sector (propagating photons) and a constraint sector (Coulomb potential). Nobody adds the Coulomb interaction by hand — it follows from U(1) gauge invariance, which is itself forced by the spin-1 nature of the photon (Weinberg, 1964, for $s = 1$). The gravitational constraint sector follows from diffeomorphism invariance, forced by spin-2.

V. What the No-Go Theorem Actually Shows

The no-go theorem (Entry EG-NOGO-008, verified in Lean 4) computes the gap between the GR and Bath-TT direct exchange amplitudes:

This is correct and machine-verified. But it requires careful interpretation. This gap is the difference in the direct coupling amplitude — what you get from tree-level exchange of the dynamical modes alone.

In the full theory (with constraints imposed), this gap is exactly compensated by the constraint contribution — the gravitational Gauss's law — leaving only the cosmological constant as a genuine physical difference between Bath-TT and GR.

The no-go theorem is not a weakness. It is a precision tool. It tells you exactly what TT coupling misses at the direct level (the trace) and exactly what it gains at the constraint level (a cosmological constant that doesn't couple to vacuum energy). The 120-orders-of-magnitude cosmological constant problem dissolves — not by fine-tuning, but by architecture.

VI. The Aharonov-Bohm Structure

There is a name for this pattern — a physical effect carried entirely by a non-dynamical sector, invisible to propagating modes, with zero local force but real measurable consequences. It is the Aharonov-Bohm effect.

In 1959, Aharonov and Bohm showed that a charged particle acquires a quantum phase from the electromagnetic potential $A_\mu$ even in regions where the field strength $F_{\mu\nu} = 0$. No electric field, no magnetic field, no local force — but a real, measurable shift in the interference pattern:

The potential $A_\mu$ is not a propagating degree of freedom. In Coulomb gauge, it has no independent dynamics. But it has physical content beyond what transverse photons carry. The AB effect proved that the gauge potential is not merely a mathematical convenience — it is the deeper object.

The constraint sector in Bath-TT has exactly this structure, operating at three levels:

Level 1: The Newtonian potential as gravitational Coulomb

The Newtonian potential $\Phi = -GM/r$ is the gravitational analogue of the Coulomb potential. Both live in the constraint sector. Neither is carried by propagating quanta. Both are determined instantaneously by their respective Gauss's laws. This is the content of Sections I–III above.

Level 2: The COW experiment — gravitational AB in the laboratory

In 1975, Colella, Overhauser, and Werner sent neutrons through an interferometer tilted in Earth's gravitational field. The two arms of the interferometer traversed different heights. No gravitational wave was present. No TT mode was exchanged. But the neutrons accumulated a gravitational phase:

They observed interference fringes that shifted with the tilt angle, exactly as predicted. This is the Newtonian potential producing a quantum phase shift — real physics from the constraint sector, with no propagating graviton involved. The COW experiment is, structurally, a gravitational Aharonov-Bohm experiment. And the blog post's Bianchi argument guarantees that Bath-TT reproduces it identically.

Level 3: The cosmological constant as topological phase

The AB phase is topological — it depends on the enclosed magnetic flux, not on the path taken. It has no local force, but it is physically real.

The cosmological constant $\Lambda$ that emerges from the Bianchi argument (Section III, Step 3) has the same character. It is an integration constant — a global quantity, not sourced by any local field. Its local force contribution ($\Lambda c^2 \sim 10^{-35}$ m/s$^2$) is negligible at every scale below cosmological. But it is physically real: it drives the accelerated expansion of the universe.

The no-go theorem's trace gap $\left(\frac{1}{d} - \frac{1}{2}\right)(\text{Tr}\,T)^2$ produces zero local consequences (the constraint compensates it exactly) and one global consequence ($\Lambda$ as an integration constant, decoupled from vacuum energy). That is the Aharonov-Bohm structure: the physically real residue of a sector that has no local force.

VII. What Remains Testable

If the classical gravitational predictions are identical to GR for local physics, what can we actually test?

The quantum sector. The framework predicts that the decoherence rate depends on geometry:

A dumbbell decoheres faster than a sphere of equal mass. This is a quantum prediction, invisible to classical gravity measurements. A Cavendish experiment sees no difference between a sphere and a dumbbell — both produce the same Newtonian field (from the constraint sector). But an interferometric experiment at the quantum level should see a difference in coherence times.

The constraint sector handles the classical world. The dynamical sector handles the quantum world. The framework gives the same classical gravity as GR (it must — the Bianchi identity is a mathematical identity, not a physical assumption) but different quantum behaviour. The test is quantum, not classical.

VIII. The Entanglement Test

The geometry-dependent decoherence rate (Section VII) gives a unique quantum signature. But there is a sharper test — one that distinguishes Bath-TT from both standard quantum gravity and classical gravity in a single experiment.

The BMV experiment

In 2017, Bose, Marletto, and Vedral proposed a landmark experiment: place two masses in spatial superposition, let them interact gravitationally, and check whether they become entangled.

The standard predictions are clean:

• Quantum gravity (the gravitational field is quantized): the Newtonian interaction creates entanglement. The two masses end up in a non-separable state.
• Classical gravity (gravity is fundamentally classical): no entanglement is generated. A classical channel cannot create quantum correlations — this is a theorem of quantum information (LOCC cannot produce entanglement).

Bath-TT predicts neither. It predicts something no other framework does:

Why shape matters

The Bath continuously monitors $T^{TT}_{ij}$. This monitoring extracts information about the matter state and decoheres it. But the amount of information extracted depends on the TT content — which depends on geometry.

Consider two scenarios:

The transition between these regimes is controlled by a single dimensionless ratio:

When $\eta \ll 1$ (spheres, low $Q_\ell$): decoherence is slow, entanglement wins. When $\eta \gg 1$ (dumbbells, high $Q_\ell$): decoherence is fast, entanglement is destroyed before it forms. The crossover depends on the quadrupole moment of the test mass — a geometric property, not a mass property.

This is a single experiment with three possible outcomes, and only Bath-TT predicts the shape-dependent one. If BMV is run with spherical test masses and finds entanglement, it rules out classical gravity but not Bath-TT. If it is then repeated with elongated masses and finds less entanglement, that is a signature no other framework produces.

IX. The Lesson

The sphere objection felt devastating because it seemed to show TT coupling missing the most basic gravitational phenomenon: a ball attracting another ball. The resolution is that Newton's law was never about propagating modes. It lives in the constraint sector — the gravitational Gauss's law — which is forced by Weinberg's theorem and the Bianchi identity, not by the Lindblad dynamics directly.

This is the same structure QED has used since the 1940s. The Coulomb potential is not a photon. The Newtonian potential is not a graviton. Both are constraints — the unique complement of the propagating sector within the full spin-$s$ propagator, forced by Lorentz invariance alone. The propagating modes carry radiation and decoherence. The constraints carry the static forces.

But the sphere is not merely a problem that gets solved. It is a probe. Because the Bath cannot see a sphere's TT content, a sphere in superposition remains coherent — and the constraint-sector potential remains quantum. This makes the sphere the ideal test mass for gravitational entanglement experiments. The very geometry that seemed to break the framework becomes the geometry that tests it most sharply.