Why TT-Only Coupling Cannot Reproduce General Relativity
A Rigorous Derivation of Emergent Unimodular Gravity
// CLASSIFICATION: OBSTRUCTION_PROOF
// ORIGIN: SECTOR 7G (2045)
// CONTEXT: This document details the 2025 discovery that TT-only coupling cannot reproduce GR. Initially considered a failure — later recognized as the key insight enabling the Unimodular pivot.
"TT-only coupling implies trace-decoupled (unimodular-like) gravity, unless the Bath is modified to include an additional scalar/trace channel."
A Hamiltonian interaction that couples matter only through the spatial transverse-traceless (TT) stress tensor:
Because it projects out the trace sector.
The exchange amplitude that defines gravitational interaction.
For linearized GR (Fierz-Pauli in de Donder gauge), the tree-level exchange between conserved sources is:
where T = ημνTμν is the trace
The -½T² term is essential:
Depends only on spatial TT projection of Tij.
Bath exchange is invariant under adding:
Therefore insensitive to the 4D trace T.
Temporal components are redundant — but the trace is not.
The Bath coupling ignores T00 and T0i. But conservation shows these are not independent for exchange at momentum k.
From kμTμν = 0:
For ω ≠ 0:
Conservation can "restore" the temporal components as dependent variables, mitigating the frame paradox.
But it does not address the trace.
Conservation kμTμν = 0 imposes 4 constraints on the 10 components of Tμν, leaving 6 independent components.
The trace T = ημνTμν is an additional independent scalar datum not determined by those constraints.
A proof by counterexample that TT-only cannot reproduce GR.
There is no choice of scalar function G₂(ω,k) such that, for all conserved sources:
Take a conserved source with no spatial shear, only energy density:
(A stationary nonrelativistic source — the spatial TT part vanishes)
(Note: This is a static source where ω = 0. The conservation reconstruction above applies only to dynamical modes with ω ≠ 0.)
Zero — no spatial TT content.
Non-zero — produces Newtonian gravity!
Trace-decoupled gravity — aligned with Unimodular theory.
Because the coupling is traceless and transverse in space, the emergent mediator behaves like a pure spin-2 field coupled to the traceless part of stress.
The traceless combination — exactly what Unimodular Gravity couples to.
What seems like a bug is actually the central feature.
In standard GR, the vacuum energy of the Bath would act as a massive Cosmological Constant, curving the universe into a singularity.
In Unimodular Gravity, vacuum energy (pure trace: Tμν ∝ ημν) decouples completely.
This pattern is not new. Physics has known it since 1959.
In the Aharonov-Bohm effect, an electron traverses a region where the magnetic field is zero. The field is confined entirely inside a solenoid — a region the electron never enters. Yet the electron acquires a measurable phase shift. It feels the enclosed flux without ever touching the source.
The trace obstruction has the same structure:
The 10120 discrepancy asks: "Why doesn't the observed cosmological constant match the calculated vacuum energy?" This is the wrong question — the same way asking "why doesn't the interference pattern depend on the detailed field distribution inside the solenoid?" is the wrong question. The coupling is structurally blind to it. The TT-projector kills the trace for the same reason the exterior region has B = 0: the connection, not the curvature, is what mediates the interaction.
"The corpus claim 'Hamiltonian TT coupling ⇒ emergent GR' is not proven and in fact generically false without an additional ingredient. But this 'failure' is precisely what resolves the cosmological constant problem."
The No-Go Theorem leads to Unimodular Gravity. But what does that mean for the framework?