Meta Theory of the Zeckendorf-Hilbert Universe

8.2 Time is Matter: Proportional Relationship Between Time Flow Rate and Local Density of States

In Section 8.1, we established the unified time identity $\kappa (E)={\rho }_{\text{rel}}(E)$. This equality is not just a mathematical coincidence but a profound reconstruction of physical ontology. In traditional physics, “time” is viewed as a background parameter independent of matter (Newton’s absolute time or Minkowski’s geometric time), while “matter” is viewed as entities filling spacetime. However, the unified time identity strongly suggests that the rate of time flow is directly proportional to the density of microscopic states contained in a physical system.

This section will explore the physical implications of this relationship in depth, propose the thesis “Time is Matter,” and prove that macroscopic time dilation effects are essentially caused by congestion of microscopic information processing channels.

8.2.1 Duality of Physical Reality: Time as Process and Matter as State

In the discrete ontology of QCA, there is no continuously flowing background time, only discrete unitary update steps. For a subsystem at energy $E$, the “experienced” time is measured by the number of interactions with the environment (or between its own parts).

According to Wigner-Smith time delay theory (Chapter 6), the “coordinate time” $\Delta t$ required for a wave packet to pass through an interaction region is given by the trace of the time delay operator:

$$\Delta t={\tau }_{\text{dwell}}\approx \text{Tr}[\mathsf{Q}(E)]$$

And according to the unified time identity, this strictly equals:

$$\Delta t=2\pi \hslash \cdot {\rho }_{\text{rel}}(E)$$

Here, ${\rho }_{\text{rel}}(E)$ is the local density of states (LDOS) that the system adds relative to the free vacuum.

Definition 8.2.1 (Time-Matter Duality Principle)

The “time flow rate” of a physical system in a certain region (defined as the ratio of process duration measured by external observers to the number of internal process steps) is strictly proportional to the effective density of states in that region.

$$\frac{d{t}_{\text{obs}}}{d{N}_{\text{step}}}\propto {\rho }_{\text{eff}}(E)$$

This means: matter (high density of states regions) are regions where time flows slowly. The more massive an object, the denser the quantum states it contains internally, and the more “computational steps” external information needs to spend to pass through it or interact with it, manifesting as macroscopic time lag.

8.2.2 Microscopic Information Interpretation of Time Flow Rate $\kappa$

To more intuitively understand the physical meaning of $\kappa (E)$, we can re-examine the scattering process from an information-theoretic perspective.

Consider a particle (probe) trying to pass through a medium.

• Vacuum Case: Density of states ${\rho }_0(E)$ is low (determined only by geometric volume). Particles barely scatter, phase accumulates linearly $\varphi \sim kx$, group delay ${\tau }_0$ is minimal. This corresponds to “light-speed” propagation, fastest time flow.

• Matter Case: There exists an interaction potential $V$, causing energy level rearrangement, density of states significantly increases $\rho (E)={\rho }_0(E)+\Delta \rho (E)$. When particles pass through this region, they must “handshake” or resonate with these new quantum states. Each new quantum state is like a “traffic light” or “information processing node,” forcing particles to dwell here for $\sim \hslash /\Gamma$ time (where $\Gamma$ is the level width).

Therefore, $\kappa (E)$ can be interpreted as information processing load per unit energy interval.

• Large $\kappa (E)$ $⟹$ high information density $⟹$ high processing load $⟹$ slow time flow (large delay).

• Small $\kappa (E)$ $⟹$ low information density $⟹$ low processing load $⟹$ fast time flow (small delay).

Corollary 8.2.2 (Locality of Time)

Since density of states $\rho (\mathbf{x},E)$ is a function of spatial position (depending on local potential $V(\mathbf{x})$), the rate of time flow must be local. Absolute unified time does not exist, because the distribution of density of states throughout the universe is non-uniform. This conclusion independently derives the core principle of general relativity from the perspective of quantum statistics.

8.2.3 Spectral Geometric Definition of Matter Density

From the perspective of “Time is Matter,” what is “matter”?

Traditionally, we define matter through mass $M$ or energy density $T_{00}$. But in spectral geometry, matter is defined as spectral perturbation.

Recalling Friedel’s sum rule (Section 7.3):

$$N_{\text{matter}}={\int }_{-\infty }^{E_F}\Delta \rho (E)\mathrm{d}E$$

This shows that the “number of particles” or “total amount of matter” contained in an object equals the integral of its density of states increment over energy.

Combining with the unified identity $\Delta \rho =\frac{1}{2\pi \hslash }{\tau }_{\text{delay}}$, we obtain a surprising equivalence:

$$\text{Total Matter}\propto \int \text{Time Delay}\mathrm{d}E$$

Physical entities (Matter) are merely dense clusters of “time delay” in spacetime structure. When we say “there is a stone here,” in the underlying QCA language, this means “the lattice update rules here cause extremely high time delay density.” Gravitational mass $M$ is actually a measure of the total time impedance that this region imposes on all passing test particles.

8.2.4 Case: Resonant States and Particle Lifetime

To verify this viewpoint, we examine an unstable particle (resonant state).

In scattering cross-sections, resonances appear as Breit-Wigner peaks, with phase derivative (time delay):

$$\tau (E)\approx \frac{\hslash \Gamma }{(E-E_0{)}^2+{\Gamma }^2/4}$$

The corresponding density of states increment $\Delta \rho (E)$ has exactly the same Lorentzian line shape.

• At Peak ($E=E_0$): Maximum density of states, maximum time delay (${\tau }_{max}=4\hslash /\Gamma$). Particles are “almost stagnant” here, manifesting as a long-lived material entity.

• Far from Peak: Density of states tends to zero, time delay tends to zero. Particles manifest as free radiation, without “materiality.”

This case clearly demonstrates: Materiality (Mass/Substance) is the localization of time delay on the energy axis. If $\Gamma \to 0$ (lifetime tends to infinity), density of states becomes a $\delta$ function, time delay tends to infinity, this is a stable elementary particle (bound state).

8.2.5 Conclusion

This section eliminates the binary opposition between “time” and “matter” through the unified time identity $\kappa =\rho$.

1. Time is the inverse effect of density of states (or direct effect of delay): the higher the density of states, the longer the physical time required for microscopic processes to complete one step.

2. Matter is the local accumulation of density of states: it is a “knot” that hinders free propagation of information.

This picture provides a completely new microscopic perspective for understanding gravity: gravity is no longer a geometric effect of curved spacetime, but rather statistical pressure of quantum information in high density of states regions. In the next section, we will use this principle to directly derive gravitational redshift effects from scattering mechanisms, proving that Einstein’s curved spacetime metric $g_{\mu \nu }$ can be reconstructed from microscopic $\kappa (E)$.