Oriented Volume Spacetime Theory: Dual Spacetime Branches, Signed Volume Operator, and Singularity Resolution via the Planck Brane (Version 3.0)

Author Information **Jinchuan Liu** 

Independent Researcher, Yibin, Sichuan, China ORCID: https://orcid.org/0009-0003-7488-3862

Contact: 21605566@qq.com

Overview This preprint presents Version 3.0 of the **Oriented Volume Spacetime Theory (OVST)**, a minimal conservative extension of Loop Quantum Gravity (LQG) built from the scale self-consistency principle. OVST postulates that scale inversion $L \leftrightarrow \ell_P^2/L$ is a fundamental symmetry at the Planck scale, accompanied by a sign reversal of the spacetime volume density. From this single postulate, we derive a dual-branch spacetime structure $\mathcal{M}_{\text{tot}} = \mathcal{M}_+ \cup \mathcal{M}_-$, where $\mathcal{M}_+$ is the observable positive-volume spacetime, and $\mathcal{M}_-$ is its scale-dual negative-volume counterpart. The two branches are connected exclusively at the *Planck brane*, a null hypersurface (in the semiclassical approximation) that resolves the classical black hole singularity at the Planck scale. This work focuses on the rigorous kinematical foundation of OVST, with complete mathematical proofs of core operators, and avoids overclaiming unresolved dynamical results. The framework is fully background-independent, compatible with all well-established results of LQG, and reduces exactly to general relativity in the low-energy classical limit.

Core Rigorous Results

1. A consistent dual-branch spacetime structure derived from the scale self-consistency principle, with a clear distinction between kinematical sign flip of the densitized triad and dynamical scale inversion symmetry.
2.  A rigorously defined **self-adjoint signed volume operator** with relaxed positive-definiteness requirement, which reduces exactly to the standard LQG volume operator on the positive-volume branch, with a symmetric spectrum about zero.
3. A physically consistent mechanism for classical black hole singularity resolution via the Planck brane, with a clear semiclassical definition and interpretation in the full discrete quantum geometry of LQG, where all curvature invariants are bounded by the Planck scale.
4. Full compatibility with the core kinematical results of standard LQG, including discrete area/volume spectra, background independence, diffeomorphism invariance, and spin network foundations.

Version Update Notes (3.0) This version fixes critical mathematical and logical issues in previous releases, following peer review feedback: - Abandoned the unproven claim of $\hat{H}_- = -\hat{H}_+$ and identically vanishing total Hamiltonian, focusing only on rigorously proven kinematical results. - Clarified the definition of the Planck brane, explicitly stating it is an effective description in the semiclassical approximation, with a corresponding interpretation in the full quantum spin network framework. - Fixed the lack of rigor in the unidirectionality proof of the Planck brane, marking heuristic arguments and acknowledging that a full rigorous proof requires the branch coupling Hamiltonian (currently an open question). - Clearly distinguished the kinematical sign flip of the triad (via the duality map $\mathcal{D}$) from the dynamical scale inversion symmetry of the theory. - Corrected LaTeX formula errors and missing numbering in the appendix, aligning the volume operator definition with standard LQG literature.

Keywords: Oriented Volume Spacetime Theory, OVST, Loop Quantum Gravity, LQG, scale self-consistency principle, dual spacetime branches, signed volume operator, Planck brane, black hole singularity resolution, quantum gravity, canonical quantum gravity

License This work is licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0). You are free to share and adapt the material, provided appropriate credit is given, a link to the license is provided, and changes are indicated. 

Related Links - Previous Version of OVST: https://doi.org/10.5281/zenodo.19640806