Meta Theory of the Zeckendorf-Hilbert Universe

26.3 Atomic Clock Gravitational Redshift Experiment: Testing Local Density of States Rescaling Effects

In Sections 26.1 and 26.2, we respectively used microwave cavities and quantum dots to verify validity of unified time identity $\kappa (E)=\rho (E)$ in boson and fermion systems. Now, we turn to macroscopic gravitational fields, using most precise measurement tool currently available to humans—Optical Lattice Atomic Clocks—to test microscopic interpretation of gravitational redshift in this book.

In standard general relativity, gravitational redshift is described as modulation of metric $g_{00}$ on proper time passage rate. But in QCA discrete ontology, as proven in Section 8.3, this effect originates from rescaling of Local Density of States (LDOS) by gravitational potential. This section will propose an experimental scheme based on atomic clock comparison, aiming to prove: reason atomic clocks run slow in deep gravitational potential is because vacuum microscopic density of states there is higher, causing electron transitions to experience longer Wigner-Smith time delays.

26.3.1 Scattering Picture of Atomic Transitions

Usually we view atomic clocks as ideal oscillators. But in holographic QCA framework, atomic energy level transitions are essentially resonant scattering processes.

Consider a two-level atom (ground state $|g⟩$, excited state $|e⟩$).

• Energy Levels: Correspond to two density of states spikes ($\delta$-functions or Lorentz peaks) of atomic internal Hamiltonian $H_{atom}$.

• Transition: Electron scatters from $|g⟩$ to $|e⟩$ (or vice versa).

• Clock Frequency: Determined by energy difference between two levels: ${\omega }_0=E_e-E_g$.

According to unified time identity, energy $E$ and time delay $\tau$ are conjugate. “Tick” rate of atom is actually determined by phase precession speed of electron between energy levels:

$$\frac{d\varphi }{dt}={\omega }_0⟹T_{period}=\frac{2\pi }{{\omega }_0}$$

26.3.2 Squeezing Effect of Gravitational Potential on Density of States

When atom is placed in gravitational potential $\Phi (\mathbf{x})$, its Hamiltonian is rescaled by redshift factor (see Section 8.3.1):

$$H(\mathbf{x})=(1+\Phi /c^2)H_0$$

This means energy level spacing is compressed:

$$\Delta E(\mathbf{x})=(1+\Phi /c^2)\Delta E_0$$

According to inverse relationship between density of states and level spacing $\rho (E)\sim 1/\Delta E$, local density of states is amplified:

$$\rho (\mathbf{x},E)\approx (1-\Phi /c^2){\rho }_0(E)$$

(Note $\Phi <0$, so $1-\Phi >1$).

Prediction:

Atomic clock located at low potential (such as ground) faces a higher density state space for internal electrons.

According to $\tau =h\rho$, time required for electron to complete one phase cycle (Wigner delay) increases:

$$T_{ground}=(1-\Phi /c^2)T_{free}>T_{free}$$

Frequency decreases:

$$f_{ground}=(1+\Phi /c^2)f_{free}<f_{free}$$

This completely agrees with predictions of general relativity, but physical picture is completely different: not time itself slows down, but timing process (scattering) slows down due to congestion of density of states.

26.3.3 Experimental Scheme: Frequency Comparison at Centimeter-Level Height Differences

Precision of modern optical clocks has reached $10^{-18}$ level, sufficient to sense centimeter-level height differences.

Experimental Setup:

1. Dual Clock System: Set up two identical optical lattice clocks (e.g., strontium-87 atomic clocks), one fixed on platform A, other fixed on platform B.

2. Height Modulation: Precisely adjust height difference $\Delta h$ of platform B relative to A.

3. Fiber Link: Connect two clocks through phase-stabilized optical fiber, use heterodyne detection to measure their frequency difference $\Delta f=f_B-f_A$.

Verification Target:

Test whether frequency difference strictly follows formula derived from density of states rescaling:

$$\frac{\Delta f}{f}=\frac{g\Delta h}{c^2}$$

More crucially, by changing atom types (e.g., comparing Sr clock and Yb clock), verify whether this redshift is universally independent of internal structure of atoms.

Characteristic Fingerprint of QCA Theory:

If at extremely high precision ($10^{-20}$ or higher), different types of atomic clocks (with different fine structure constant sensitivities) exhibit tiny redshift differences, this would suggest violation of equivalence principle.

In QCA theory, this means rescaling factors $Z_i$ of density of states for different matter sectors (Flavor Sectors) may have Planck-scale corrections:

$$\frac{\Delta f_i}{f_i}=(1+{\eta }_i\frac{E}{M_P})\frac{\Delta \Phi }{c^2}$$

where ${\eta }_i$ depends on topological structure of particles (Section 17.4). Current experiments (such as NIST, PTB) have not found such violations, providing strong constraints on “geometric-information coupling” of QCA—gravity must be universal entropy force.

26.3.4 Interpretation: Spacetime as Inhomogeneous Medium

This experiment would provide most direct evidence that gravitational redshift is not geometric effect of spacetime curvature, but physical effect of vacuum density of states modulation.

In QCA discrete ontology, spacetime is not empty stage, but a medium with variable refractive index. Gravitational potential $\Phi$ changes local “optical density” of vacuum, causing all physical processes (including atomic transitions) to slow down proportionally.

This interpretation unifies gravitational redshift with other “medium effects” in physics (such as light slowing in glass), revealing that gravity is not a force, but a property of information-carrying vacuum itself.