The Trace No-Go Theorem

Why TT-Only Coupling Cannot Reproduce General Relativity

A Rigorous Derivation of Emergent Unimodular Gravity

The Central Result

"TT-only coupling implies trace-decoupled (unimodular-like) gravity, unless the Bath is modified to include an additional scalar/trace channel."

No-Go Theorem (Trace Mismatch)

A Hamiltonian interaction that couples matter only through the spatial transverse-traceless (TT) stress tensor:

  • Can reproduce the radiative, spin-2 part of gravity
  • Yields the correct Newtonian 1/r potential in the static limit
  • Cannot reproduce the fully covariant Fierz-Pauli/GR exchange kernel

Because it projects out the trace sector.

The Logical Chain

TT Coupling → Projects Out Trace → Missing -½T² Term → Unimodular, Not GR

What General Relativity Requires

The exchange amplitude that defines gravitational interaction.

The GR Exchange Amplitude

For linearized GR (Fierz-Pauli in de Donder gauge), the tree-level exchange between conserved sources is:

GR Exchange
AGR ~ (1/k²) [ TμνTμν - ½T² ]

where T = ημνTμν is the trace

The Critical Structure

The -½T² term is essential:

  • Encodes the correct spin-2 projector in 4D
  • Ensures the right relative coupling to pressure/stress vs energy density
  • Cannot be generated by a purely TT spatial kernel

What Bath TT Produces

ABath ~ G₂(ω,k) Tij ΠTTij,kl Tkl

Depends only on spatial TT projection of Tij.

The Mismatch

Bath exchange is invariant under adding:

  • Pure trace: δij
  • Longitudinal: kivj + kjvi

Therefore insensitive to the 4D trace T.

The Conservation Reconstruction

Temporal components are redundant — but the trace is not.

The Frame Paradox Resolution

The Bath coupling ignores T00 and T0i. But conservation shows these are not independent for exchange at momentum k.

Conservation Equations

From kμTμν = 0:

ωT00 = kiTi0
ωT0j = kiTij

The Reconstruction

For ω ≠ 0:

T0i = (kj/ω) Tji
T00 = (kikj/ω²) Tij

Interim Conclusion

Conservation can "restore" the temporal components as dependent variables, mitigating the frame paradox.

But it does not address the trace.

Why The Trace Is Independent

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.

The Trace Obstruction

A proof by counterexample that TT-only cannot reproduce GR.

Proof by Counterexample

Claim (No-Go)

There is no choice of scalar function G₂(ω,k) such that, for all conserved sources:

Tij ΠTTij,kl Tkl ∝ TμνTμν - ½T²

The Counterexample

Take a conserved source with no spatial shear, only energy density:

T00 ≠ 0
T0i ≈ 0
Tij ≈ 0

(A stationary nonrelativistic source — the spatial TT part vanishes)

Bath TT Exchange

ABath ∝ Tij ΠTT Tkl = 0

Zero — no spatial TT content.

GR Exchange

AGR ∝ (T₀₀² - ½T₀₀²)/k² = ½T₀₀²/k²

Non-zero — produces Newtonian gravity!

The two kernels disagree on a conserved class of sources.
No multiplicative choice of G₂ can fix a structural zero enforced by ΠTT.

What Actually Emerges

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.

Effective Source
T̃μν = Tμν - ¼ ημν T

The traceless combination — exactly what Unimodular Gravity couples to.

What TT-Only Selects

  • The radiative graviton (2 polarizations)
  • Correct long-range forces in special limits
  • 1/r potential from TT shadows

What TT-Only Misses

  • The trace coupling (-½T² term)
  • Full covariant constraint structure
  • How vacuum energy gravitates

The Profound Implication

What seems like a bug is actually the central feature.

Why This Is A Success

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.

The framework naturally explains why the huge energy of the Bath does not gravitate.

The Full Picture

TT-Only Coupling → No-Go Theorem → Unimodular Gravity → Λ Problem Solved

Open Questions

Technical

  • Exact IR form of G₂(ω,k)?
  • Static limit subtleties at ω → 0?
  • Nonlinear completion stability?

Phenomenological

  • Coupling to relativistic pressure?
  • Cosmological perturbations?
  • Gravitational lensing vs dynamical mass?

Summary

"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."

Key Results

4 Conservation Constraints
6 Independent Components
1 Independent Trace
0 Vacuum Energy Coupling

Continue the Journey

The No-Go Theorem leads to Unimodular Gravity. But what does that mean for the framework?

The Unimodular Bath Synthesis

See how the "failure" to reproduce GR becomes the framework's greatest strength — automatically solving the cosmological constant problem.

Explore the Unimodular Pivot →