Technical Appendix | Emergent Gravity Framework

Section 0

Gravity emerges as feedback in open quantum systems. Matter couples to a large-N Bath via transverse-traceless (TT) stress-energy projection. The Wiseman-Milburn theorem implies a compensating feedback Hamiltonian; locality, no-signaling, and energy conservation uniquely fix it as gravitational attraction with G = 4π/(λ²N²). The framework predicts shape-dependent decoherence and correlated force noise. A no-go theorem shows TT-coupling yields Unimodular Gravity — naturally decoupling vacuum energy from geometry.

Status: Theoretical framework · Not experimentally validated · Not peer-reviewed

Keywords: emergent gravity · quantum decoherence · Lindblad dynamics · transverse-traceless · unimodular gravity

Section 1 — Mathematical Setup

1.1 Assumptions Input

A1 (Matter): All observable degrees of freedom are characterized by a conserved stress-energy tensor T^μν(x) satisfying ∂_μT^μν = 0.
A2 (Bath): There exists a quantum system B with Hilbert space H_B of dimension scaling as N → ∞, initially in a stationary state ρ_B (vacuum or KMS).
A3 (Lorentz Invariance): The Bath dynamics and matter-Bath coupling respect Poincaré symmetry.
A4 (TT Coupling): The unique matter-Bath interaction is H_int = λ ∫d³x T^TT_ij(x) Ξ^ij(x), where Ξ^ij are Bath operators with TT symmetry.

No spacetime metric is assumed as fundamental. The TT projection is defined on a flat background via the spatial projector P_ij = δ_ij - ∂_i∂_j/∇².

1.2 TT Projection Standard

The transverse-traceless projection of the spatial stress tensor:

T^TT_ij = (P_ikP_jl − ½P_ijP_kl) T_kl

where P_ij = δ_ij − ∂_i∂_j/∇² is the transverse projector.

Physical interpretation: T^TT_ij captures quadrupole and higher multipole deformations — shape-changing modes that carry gravitational information.

Section 2 — Derivation

2.1 Reduced Dynamics Theorem

Tracing out the Bath under assumptions A1-A4 yields the Lindblad master equation:

dρ/dt = −i[H_eff, ρ] − ½ ∫d⁴x d⁴x' N_TT(x−x') [T^TT(x), [T^TT(x'), ρ]]

where N_TT(x−x') is the noise kernel (Bath TT correlator) and H_eff includes Lamb-shift corrections.

2.2 Measurement-Feedback Equivalence Wiseman-Milburn

By the Wiseman-Milburn theorem, any dynamics of the above form is operationally equivalent to:

Continuous weak measurement of T^TT
Conditioned on a feedback Hamiltonian H_fb

This is a theorem of open quantum systems theory, not an interpretive choice.

2.3 Uniqueness of Gravitational Feedback Central Result

Imposing constraints on H_fb:

Constraint	Mathematical Form
Locality	H_fb = ∫d³x h(x)
No-signaling	[H_fb(t), H_fb(t')] = 0 for spacelike separation
Universality	Couples to T^μν only
Energy conservation	⟨dE/dt⟩ = 0 statistically

uniquely determines:

H_fb = −G ∫d³x d³x' T^TT_ij(x) T^TT_ij(x') / |x−x'|

with Newton's constant:

G = 4π / (λ²N²)

2.4 No-Go Theorem (Trace Obstruction) Theorem

The full covariant GR propagator includes trace-trace coupling:

D^GR_μνρσ ∝ (η_μρη_νσ + η_μση_νρ − η_μνη_ρσ) / k²

The TT projection removes the trace sector T = η^μνT_μν, which is an independent Lorentz scalar inaccessible to TT measurement.

Consequence: TT-only coupling yields Unimodular Gravity (trace-free field equations), not full GR. This automatically decouples vacuum energy from geometry.

Section 3 — Predictions

3.1 Decoherence Rates Testable

Γ_TT ~ (GM²/ℏc) · ω₀ · Q_ℓ² · f_ℓ(Δx/R)

where Q_ℓ is the ℓ-th multipole moment and ω₀ = c/R is the characteristic frequency.

Key signature: shape-dependent, not mass-only.

3.2 Correlated Force Noise Testable

Nearby masses experience correlated force fluctuations:

⟨δF(x) δF(x')⟩ ∝ N_TT(x−x') ≠ 0

Distinguishes from independent-fluctuation models (Penrose-Diósi, graviton emission).

3.3 Experimental Discriminator Proposed

Geometry-correlated torque noise in torsion balance experiments:

S_τ(f, θ) = S₀(f) + α(f) · Q²(θ)

GR predicts α = 0. Bath-TT predicts α > 0, locked to quadrupole moment.

Section 4 — Known Limitations

L1 · Unobservable Bath

The Bath is traced out by construction. It cannot be directly probed — only its effects on matter are observable. This is a feature (emergence) not a bug, but limits testability.

L2 · Flat Background

The TT projection is defined on Minkowski space. Extension to curved backgrounds requires care — the framework is currently limited to weak-field / perturbative regimes.

L3 · No Full GR

The trace sector is inaccessible. Full GR requires additional structure (scalar channel, constrained variable) beyond pure TT coupling. The framework yields Unimodular Gravity.

L4 · Microscopic Bath Unknown

While holographic CFTs satisfy the abstract Bath requirements, no unique microscopic realization is established. The Bath is operationally defined, not identified.

L5 · Not Experimentally Tested

All predictions await experimental verification. The framework is falsifiable but unconfirmed.

Section 5 — References

Foundational Results (Established)

Lindblad, G. (1976). "On the generators of quantum dynamical semigroups." Commun. Math. Phys. 48, 119.
Wiseman, H.M. & Milburn, G.J. (1993). "Quantum theory of optical feedback via homodyne detection." Phys. Rev. Lett. 70, 548.
Zurek, W.H. (2003). "Decoherence, einselection, and the quantum origins of the classical." Rev. Mod. Phys. 75, 715.
Weinberg, S. (1989). "The cosmological constant problem." Rev. Mod. Phys. 61, 1.

Related Approaches

Penrose, R. (1996). "On gravity's role in quantum state reduction." Gen. Rel. Grav. 28, 581.
Diósi, L. (1987). "A universal master equation for the gravitational violation of quantum mechanics." Phys. Lett. A 120, 377.
Jacobson, T. (1995). "Thermodynamics of spacetime: The Einstein equation of state." Phys. Rev. Lett. 75, 1260.

This Work

December 2025 · Not peer-reviewed · Available at emergent-gravity.com