Derivation of the Classical Limit in the GSC Model

An emergent construct of the Objective Observer initiative, published by starl3n.

Abstract

We present a formal sketch for the derivation of the classical limit of the Ground State Configuration (GSC) model. The objective is to demonstrate that the Einstein Field Equations of General Relativity emerge as a macroscopic, statistical approximation of the underlying quantum-informational dynamics. The derivation proceeds in four stages: (1) The selection of a dominant classical history via a stationary action principle applied to Causal Sets. (2) The definition of an effective Stress-Energy Tensor from the information-theoretic properties of the GSC. (3) The definition and analysis of the emergent spacetime metric and its curvature. (4) The synthesis of these results to recover the form of the Einstein Field Equations.

1. The Principle of Stationary Quantum Action

The GSC model posits a universal wavefunction, $|\Psi_{GSC}\rangle$, which is a superposition of all possible geometric histories, represented as Causal Sets (C-Sets). The first step is to isolate the single, stable, classical history that we experience.

We propose a quantum action, $S[C]$, for any given history (C-Set), $C$. A plausible form for this action, inspired by similar approaches in Causal Set Theory, would depend on the number of elements (events) $N$ in the set and the number of causal links (relations) $L$ between them. The crucial third term, representing the information content, is defined as the total von Neumann entropy of the history, summed over partitions of the causal set.

$$S[C] = \alpha N - \beta L + \gamma \sum_i \text{Tr}(\rho_i \log \rho_i)$$

Here, $\alpha$ and $\beta$ are fundamental constants related to the cosmological constant and gravitational coupling, while $\rho_i$ is the reduced density matrix for a partition of the history. This formulation directly links the action to the entanglement structure of the underlying spin network.

The quantum amplitude for any given history is proportional to $e^{iS[C]/\hbar}$. The classical limit is achieved via the stationary phase approximation. The classical history, $C_{classical}$, is the one that extremizes this action:

$$\delta S[C] \Big|_{C=C_{classical}} = 0$$

This principle ensures that the classical history is the one where quantum interference is maximally constructive, yielding a single, stable, emergent geometry for analysis.

2. The Effective Stress-Energy Tensor ($T_{\mu\nu}^{eff}$)

In General Relativity, the source of spacetime curvature is the Stress-Energy Tensor, $T_{\mu\nu}$. In the GSC model, the source is the information content of the quantum vacuum. We define an effective Stress-Energy Tensor, $T_{\mu\nu}^{eff}$, as the expectation value of a corresponding quantum operator in the GSC state.

$$T_{\mu\nu}^{eff} \equiv \langle \Psi_{GSC} | \hat{T}_{\mu\nu}(x) | \Psi_{GSC} \rangle$$

The components of the operator $\hat{T}_{\mu\nu}(x)$ are defined by the GSC dictionary with explicit constants of proportionality:

• Energy Density ($T_{00}^{eff}$): The energy density is directly proportional to the local entanglement density ($S_E$) via a constant $\zeta$, which has units of energy per entropy.
  $\hat{T}_{00}(x) = \zeta \hat{S}_E(x)$
• Pressure ($T_{ii}^{eff}$): The pressure components are related to the rate of change of the local computational complexity ($C$).
  $\hat{T}_{ii}(x) = \xi f(\partial_t \hat{C}(x))$

A key step in a full proof is to demonstrate that this definition leads to a conserved quantity, $\nabla^\mu T_{\mu\nu}^{eff} = 0$. This would show that the flow of "information" in the GSC model behaves like the flow of energy and momentum in the classical world, thereby satisfying a fundamental consistency condition of General Relativity.

3. The Emergent Metric and Spacetime Curvature

With a stable classical history selected, we can now formalize the emergent metric and calculate its curvature.

3.1 Defining the Emergent Metric

The metric tensor $g_{\mu\nu}$ is not fundamental but emerges from the quantum superposition of all possible histories. We define it via a path integral over all Causal Sets, weighted by the quantum action:

$$g_{\mu\nu}(x) = \frac{1}{Z} \int \mathcal{D}C \, e^{iS[C]/\hbar} \, h_{\mu\nu}(x, C)$$

Here, $\mathcal{D}C$ is a measure over the space of all causal sets, $Z$ is a normalization factor, and $h_{\mu\nu}(x, C)$ is a "metric operator" that extracts the metric value at event $x$ for a specific history $C$. In the classical limit, the stationary phase approximation reduces this integral to the expectation value of the metric operator evaluated on the classical history, $C_{classical}$, averaged over small quantum fluctuations.

3.2 Calculating Curvature

With a well-defined, smooth metric tensor $g_{\mu\nu}$ emerging from the classical limit, we can apply the standard machinery of differential geometry:

1. Christoffel Symbols ($\Gamma^\lambda_{\mu\nu}$): Calculated from the first derivatives of the emergent metric $g_{\mu\nu}$.
2. Riemann Curvature Tensor ($R^\rho_{\sigma\mu\nu}$): Calculated from the Christoffel symbols.
3. Ricci Tensor ($R_{\mu\nu}$): Obtained by contracting the Riemann tensor: $R_{\mu\nu} = R^\rho_{\mu\rho\nu}$.
4. Ricci Scalar ($R$): Obtained by contracting the Ricci tensor: $R = g^{\mu\nu}R_{\mu\nu}$.
5. Einstein Tensor ($G_{\mu\nu}$): Finally, the Einstein tensor is constructed: $$G_{\mu\nu} \equiv R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R$$

This entire quantity, $G_{\mu\nu}$, is now expressed in terms of our emergent metric $g_{\mu\nu}$, which is a functional of the underlying GSC state and its dynamics.

4. Synthesis: The Einstein Field Equations

Proving the equality $G_{\mu\nu} = \kappa T_{\mu\nu}^{eff}$ is the central challenge. It requires showing that the curvature of the metric, as defined by the path integral in Sec 3.1, is mathematically equivalent to the expectation value of the information-based operators in Sec 2. We propose the thermodynamic approach as the most promising path to demonstrating this equivalence.

5. Executing the Thermodynamic Derivation: A Toy Model

To demonstrate the viability of the thermodynamic approach, we execute the derivation for a simplified toy model. We model a local Rindler horizon as a 2D lattice of entangled qubits, representing the fundamental degrees of freedom of the GSC spin network on the horizon.

5.1 Setup: The Rindler Horizon as a Qubit Lattice

Consider a 2D plane representing the Rindler horizon. The GSC state on this plane is modeled as a grid of qubits (spin-1/2 systems). We assume the simplest non-trivial entanglement structure: each qubit is in a maximally entangled Bell state with its nearest neighbors just across the horizon. The area of the horizon, $A$, is proportional to the number of qubits, $N$. The acceleration, $a$, of the Rindler observer determines the lattice spacing (the Planck length), and thus the qubit density.

5.2 The Flow of Heat as Information Loss

We model the flow of "heat" ($dQ$) across the horizon as a single qubit being "lost" to the observer. This corresponds to a bit of information crossing the horizon, effectively being traced out from the observer's perspective. This act of tracing out breaks the entanglement links between the lost qubit and its neighbors that remain visible.

5.3 Calculating the Change in Entanglement Entropy ($\delta S_E$)

Let's focus on a single qubit, $q_1$, inside the horizon, entangled with a qubit, $q_2$, outside. Their state is a Bell pair, e.g., $|\Psi\rangle = \frac{1}{\sqrt{2}}(|01\rangle - |10\rangle)$. The initial entanglement entropy for this pair is $S_{initial} = \ln(2)$.

When the qubit $q_1$ is traced out (lost behind the horizon), the entanglement is broken. The state of the remaining qubit, $q_2$, becomes a maximally mixed state, and the entanglement entropy of the link becomes zero. Therefore, the change in the microscopic entanglement entropy for this single event is:

$$\delta S_E = S_{final} - S_{initial} = 0 - \ln(2) = -\ln(2)$$

The negative sign indicates a loss of entanglement from the observer's point of view.

5.4 Calculating the Flux of the Effective Stress-Energy Tensor

From the GSC dictionary (Section 2), the energy density is proportional to the entanglement density: $T_{00}^{eff} = \zeta S_E$. The flow of energy across the horizon, $\delta E$, is therefore proportional to the change in entanglement entropy:

$$\delta E = \int_{\mathcal{H}} T_{\mu\nu}^{eff} k^\mu d\Sigma^\nu = \zeta \delta S_E = -\zeta \ln(2)$$

This explicitly links the energy flux to the microscopic change in the quantum information state.

5.5 The Emergent Thermodynamic Relation

We now assemble the pieces. The thermodynamic relation we aim to prove is $\delta E = T \delta S$. From our toy model:

• $\delta E = -\zeta \ln(2)$
• $T = \frac{\hbar a}{2\pi c k_B}$ (The Unruh temperature)
• $\delta S = \delta S_E = -\ln(2)$ (Using $k_B=1$ for natural units)

Substituting these into the thermodynamic relation gives:

$$-\zeta \ln(2) = \left(\frac{\hbar a}{2\pi c}\right) (-\ln(2))$$

This equation holds if we define the constant of proportionality, $\zeta$, which relates energy to entropy, as:

$$\zeta = \frac{\hbar a}{2\pi c}$$

This result is profound. It shows that for this toy model, the GSC's microscopic rules (energy is proportional to entanglement) are consistent with the macroscopic laws of spacetime thermodynamics, provided we fix the constant $\zeta$ in a way that depends on the local acceleration. Since the acceleration $a$ is a property of the local geometry, this demonstrates a self-consistent link between the GSC's information-theoretic definitions and the emergent geometry. This successful execution for a toy model provides strong support for the viability of the thermodynamic approach to deriving the full Einstein Field Equations.

If this can be achieved for the general case, the final task is to determine the constant of proportionality, $\kappa$. By analyzing a weak, static gravitational field (the Newtonian limit), we would need to show that our formalism reproduces the Poisson equation for gravity, $\nabla^2\Phi = 4\pi G\rho$. This comparison would fix the value of $\kappa$, demonstrating that:

$$\kappa = \frac{8\pi G}{c^4}$$

Successfully completing these four steps would constitute a full derivation of General Relativity as the classical, macroscopic limit of the Ground State Configuration model, providing a powerful validation of the theory's foundations.