A Mathematical and Physical Analysis On the Origin of Objects (or Matters) in Space (3)

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3.2 Topological Defects as Precursors of Matter

In TVT, the essence of an object is a topological vortex in the spatial superfluid. Considering a vortex in a two-dimensional cross-section with a winding number \(n \in \mathbb{Z}\):

\[\oint_C \nabla \theta \cdot d\mathbf{l} = 2\pi n\]

This vortex possesses:

1)A core: \(\Psi = 0\), corresponding to a localized region of spatial "excitation," potentially manifesting as an elementary particle;

2) Quantized circulation: Stable flow around the core, endowing the vortex with angular momentum and inertia;

3) Topological stability: The vortex cannot be continuously dissolved into the homogeneous background unless crossing an energy barrier.

3.3 Possible Pathways for Topological Phase Transitions to Generate Matter

1) Spontaneous symmetry changing and vortex lattice formation: When the superfluid velocity exceeds a critical value (or temperature falls below a critical temperature), the system can spontaneously generate a network of topological defects via the Kibble-Zurek mechanism [3, 6]. These defects can be regarded as primordial material seeds.

2) Vortex interactions and composite structures:

  1. Multiple vortices can form knots through intertwining and linking, corresponding to different particle types [4];
  2. Vibrational modes of vortex rings can be interpreted as internal degrees of freedom (e.g., electric charge, color charge);
  3. Vortex dynamics described by topological current equations may be isomorphic to Yang-Mills equations.

3) Unification of spacetime geometry and matter: According to TVT, the matter stress-energy tensor \(T_{\mu\nu}\) in Einstein's field equations may stem from the mean-field representation of topological defects in the spatial superfluid. In other words, matter is a kind of "hardening" of the curvature modes of spacetime [1, 5].

4. Interface with Existing Physical Theories and Predictions

4.1 Correspondence with Quantum Field Theory

1) The vortex core scale may correspond to the Planck scale or some equivalent cutoff scale;

2) Different topological quantum numbers of vortices may correspond to particle quantum numbers (e.g., electric charge, baryon number);

3) The vortex excitation spectrum might reproduce the Standard Model particle spectrum [5].

4.2 Integration with General Relativity

1) The sound cone structure of the superfluid background naturally gives rise to the light cone, i.e., the relativistic causal structure [1];

2) The stress-energy of vortices can modify the background geometry, realizing the coupling between matter and geometry.

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