GSC Model: Derivations from First Principles

Problem Statement: Stephen Hawking showed that black holes radiate thermally ("Hawking radiation"), causing them to shrink and eventually evaporate. If the radiation is truly thermal, it contains no information about what fell in. This violates the principle of unitarity in quantum mechanics, which states that information can never be destroyed.

The GSC Model Approach: In the GSC model, information is never destroyed; it is merely transformed and redistributed. The paradox is resolved by showing how the information of infalling matter is shunted from local, classical degrees of freedom into non-local, multiverse degrees of freedom, preserving unitarity.

Proposed Derivation Steps:

1. Modeling Infalling Matter in the GSC Hilbert Space:

   The full GSC Hilbert space $\mathcal{H}_{GSC}$ is a tensor product of the Hilbert space for our local universe branch, $\mathcal{H}_{local}$, and the space for all other potential branches, $\mathcal{H}_{multi}$.

   $$\mathcal{H}_{GSC} = \mathcal{H}_{local} \otimes \mathcal{H}_{multi}$$

   An electron's state $|\psi_{e}\rangle$ is not just its local wavefunction, but a state in this larger space. Initially, it is unentangled with the multiverse:

   $$|\psi_{e}\rangle_{initial} = |\psi_{local}\rangle \otimes |\psi_{multi}^{0}\rangle$$
2. Information Shunting at the Event Horizon:

   The interaction with the black hole's event horizon is a unitary transformation $U_{BH}$ that acts as an "information shunt". It entangles the local state with the multiverse state, effectively transferring the information from $\mathcal{H}_{local}$ to $\mathcal{H}_{multi}$. The combined state of the system (BH + electron) evolves as:

   $$|\Psi'_{BH+e}\rangle = U_{BH} (|\Psi_{BH}\rangle \otimes |\psi_{e}\rangle_{initial})$$

   The result is a new state where the electron's initial local information $|\psi_{local}\rangle$ is now encoded in the correlations between the black hole and the multiverse structure.

3. Conservation of Informational Freedom:

   We can model the total information (or "entropic freedom") of the electron state as $S_{total}$. This is conserved, but its components change. Let $S_{local}$ be the entropy associated with the local degrees of freedom (position, momentum in our universe) and $S_{multi}$ be the entanglement entropy with the multiverse branches.

   $$S_{total} = S_{local} + S_{multi} = \text{constant}$$

   As the electron falls into the black hole, its local entropic freedom collapses ($S_{local} \to 0$). To conserve total entropy, its multiverse entanglement must increase correspondingly: $S_{multi}$ increases, preserving the information.

4. Hawking Radiation as Information Leakage:

   Hawking radiation is generated by quantum fluctuations at the horizon. In the GSC model, these fluctuations are coupled to the full GSC state, including the multiverse entanglement. The emitted radiation quanta $|\psi_{rad}\rangle$ are therefore subtly entangled with the multiverse state that holds the original information. This provides a channel for the information to "leak" back into our local universe.

5. Deriving the Page Curve:

   The Page Curve describes the entanglement entropy of the emitted radiation. Our model must reproduce it. The entropy of the radiation, $S(\rho_{rad})$, will initially grow as it becomes entangled with the BH-multiverse system. However, as the BH evaporates, the information-carrying multiverse degrees of freedom are returned to our universe via radiation, causing the entanglement of the *remaining* system to decrease. Thus, the radiation's entropy must also decrease after the Page time, returning to zero for the final state.

   $$S(\rho_{rad}) = -\text{Tr}(\rho_{rad} \log \rho_{rad})$$