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 undergoes a transformation from being locally accessible to being encoded in the non-local correlational structure of the multiverse. This happens via a continuous process of causal history imprinting during infall.

Proposed Derivation Steps:

1. Modeling the State in the GSC Hilbert Space:

   The full Hilbert space is a tensor product of the local branch and the multiverse: $\mathcal{H}_{GSC} = \mathcal{H}_{local} \otimes \mathcal{H}_{multi}$. An infalling electron begins in a state largely unentangled with the multiverse:

   $$|\psi_{e}\rangle_{initial} = |\psi_{local}\rangle \otimes |\psi_{multi}^{0}\rangle$$
2. Continuous Information Transfer via Relativistic Branching:

   As the electron falls, it continues to interact with the local quantum vacuum. This interaction, governed by an interaction Hamiltonian $H_{int}(t)$, causes a continuous branching of its causal history into the multiverse. This is not a single event, but a gradual process where the electron's state evolves unitarily:

   $$|\Psi(t)\rangle = \mathcal{T} e^{-i \int_{t_0}^{t_f} H_{int}(t') dt'} |\Psi(t_0)\rangle$$

   The operator $\mathcal{T} e^{-i \int H_{int}(t') dt'}$ represents the time-ordered evolution that progressively entangles the local state with $\mathcal{H}_{multi}$.

3. Conservation of Informational Freedom during Infall:

   The total information, $S_{total}$, is conserved. As the electron falls, its ability to be in a superposition of states locally diminishes ($S_{local} \to 0$). This loss of local freedom is precisely compensated by an increase in its entanglement with the multiverse ($S_{multi}$ increases), as its causal history is imprinted across many branches.

   $$\frac{dS_{local}}{dt} = -\frac{dS_{multi}}{dt}$$

   The process concludes when no "superpositional invariance" remains—the particle has fully decohered with the local system and its information is now entirely non-local.

4. Hawking Radiation and the Concept of Multiversal Time:

   The imprinted causal history is not static. The parameter tracking the evolution of this branching and entanglement can be understood as a form of "multiversal time." Hawking radiation is the mechanism by which this process runs in reverse: quantum fluctuations at the horizon couple to the imprinted history, slowly untangling it and leaking the information back into $\mathcal{H}_{local}$ as correlated radiation.

5. Deriving the Page Curve:

   The Page Curve emerges naturally from this model. The "turn" of the curve happens at the Page time—the point where the rate of information being returned to our universe via Hawking radiation (the untangling of causal histories) becomes greater than the rate of information being imprinted by new infalling matter. The entropy of the radiation $S(\rho_{rad})$ must then decrease, ensuring the total process is unitary.

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