What question did this study set out to answer?

The research aims to extend the TUCSL framework into a tensorial stochastic gravity context while addressing issues of unimodular closure.

April 14, 2026Open Access

TUCSL Theory: Bogoliubov–Dressed Horizon Noise, Unimodular Tensorial Completion, and a Projected Einstein–Langevin Correspondence

Key Points

The research aims to extend the TUCSL framework into a tensorial stochastic gravity context while addressing issues of unimodular closure.
Developed a Bogoliubov-dressed null stress correlator in Rindler space.
Constructed a Fourier transform of the horizon kernel with thermal fluctuation properties.
Incorporated transverse and full q-smearing to derive effective collapse scale.
Addressed the limitations of the scalar horizon kernel in deriving a tensorial noise kernel.
Projected onto an unimodularly admissible subspace for tensorial completion.
Demonstrated the closed relation between collapse strength and horizon-focusing noise.
Developed a new tensorial noise kernel that remains symmetric and positive.
Established a projected Einstein–Langevin correspondence that guides further exploration in stochastic gravity.

Abstract

We develop a controlled extension of the Transactional Unimodular Continuous Sponta-neous Localization (TUCSL) programme from the previously established horizon-projectednoise sector to a canonical tensorial stochastic-gravity completion compatible with unimodularclosure. The starting point is the already fixed covariant kernel backbone selected by bandage ultralocality, strong Markov coarse-graining, covariance, positivity, and no-signalling. We first construct the Bogoliubov-dressednull stress correlator in Rindler space, derive the explicit Wick-contracted form of the horizonkernel, and show that its Fourier transform obeys a thermal fluctuation–dissipation relation. We then incorporate transverse smearing and full q-smearing (rc, ℓ₀), obtaining an explicitlow-frequency amplitude and an induced effective collapse scale λₑff (a). This yields a closedrelation between collapse strength and stationary horizon-focusing noise. The main new conceptual step is a no-go and a replacement theorem. We show that thescalar horizon kernel Cf alone cannot determine a full tensorial Einstein–Langevin noisekernel. A naive lifting based on the scalar heat kernel fails the correct tensorial consistencytest. We then construct a canonical minimal completion on the trace-free symmetric tensorbundle over the Euclidean q-domain by projecting onto the unimodularly admissible subspace. The corresponding semigroup yields a tensorial noise kernel that issymmetric, positive, compatible with unimodular closure in the precise Hodge-projectedsense, and reduces to the previously obtained horizon kernel in the local near-horizonlimit. The outcome is not a derivation of the full nonlinear Einstein equations from RTI/QDATalone. Rather, it is a mathematically controlled “projected Einstein–Langevin correspondence” andits canonical unimodular tensorial completion. We formulate precisely what is proved, what remains conjectural, and which further steps would be needed for a full tensorialstochastic-gravity completion of the TUCSLprogramme.

TUCSL Theory: Bogoliubov–Dressed Horizon Noise, Unimodular Tensorial Completion, and a Projected Einstein–Langevin Correspondence

Key Points

Abstract

Cite This Study