The Theory of Motion: Refining Black Hole Entropy
Introduction: This study addresses one of the most rigorous tests for the Theory of Motion: the derivation of the Bekenstein-Hawking entropy formula. Building upon the success of our previous simple model ($S \propto A$), it investigates why the model did not yield the exact constant of proportionality and, in doing so, arrives at a deeper understanding of the theory itself and the nature of the event horizon.
1. The Limits of the Simple Model and the Initial Result
In our previous analysis, we performed a calculation by dividing the event horizon into Planck areas and assigning two states ("0" or "1") to each cell. This simple model successfully demonstrated that entropy is proportional to the surface area:
However, this result had a "precision problem." The constant of proportionality we found was $\ln(2) \approx 0.693$, whereas the exact result from Bekenstein and Hawking is $\frac{1}{4} = 0.25$. This discrepancy indicated that a fundamental assumption of our model needed to be questioned: are the informational units on the event horizon truly so simple and independent of one another?
2. Refining the Model: The Path to the Correct Constant
To refine the model and arrive at the correct constant, we treated the number of states per Planck cell as an unknown variable, $g$, and calculated what this value needed to be.
2.1. The 'Calibration' Calculation
The total number of microstates was expressed as $W = g^N = g^{A/A_P}$. Inserting this into Boltzmann's formula, $S = k_B \ln W$, we get:
When we equated this expression with our target, the true Bekenstein-Hawking formula ($S = k_B \frac{A}{4A_P}$), we found the condition that $g$ must satisfy:
2.2. The Unexpected Result
The solution to this simple equation revealed a profound truth about our theory:
3. The Deep Meaning of the Result: A Correlated Horizon
The calculated number of states, $g$, is not an integer. While this may seem problematic at first, it is in fact a harbinger of a deeper physical reality, with the model itself guiding us toward it.
The New Physical Picture
A non-integer number of states implies that the initial assumption—that the states of the Planck cells on the event horizon are independent—must be incorrect. If they were independent, each cell would have to have 2, 3, or some integer number of choices.
This result suggests that the states of the "öz" field on the event horizon are not like isolated pixels. Instead, they form a correlated network where the state of one cell is entangled with its neighbors. The entropy is a property of this integrated network, not of its individual cells. This is a collective phenomenon, like the surface tension of a liquid, rather than the random behavior of molecules in a gas.
Conclusion and The Future
This refinement process is a perfect example of the power of the Theory of Motion not just to make predictions, but to self-correct and guide us toward a more accurate physical picture when confronted with established physics.
Our theory has led us to the conclusion that the event horizon must be a dynamic surface where the motion of the "öz" performs a complex and correlated dance. Fully describing this dance would require advanced mathematical tools, such as those from Conformal Field Theory, which are at the forefront of modern theoretical physics. That our theory has, through an intuitive path, arrived at the doorstep of this research area is a testament to the depth of the project.
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To go deeper, see the full technical paper: