Entanglement entropy evolution during gravitational collapse

Abstract

We investigate the dynamics of the ground state entanglement entropy for a discretized scalar field propagating within the Oppenheimer-Snyder collapse metric. Starting from a well-controlled initial configuration, we follow the system as it evolves toward the formation of a horizon and, eventually, a singularity. Our approach employs an Ermakov-like equation to determine the time-dependent ground state of the field and calculates the resulting entanglement entropy by tracing out the degrees of freedom inside a spherical region within the matter sphere. We find that the entanglement entropy exhibits nontrivial scaling and time dependence during collapse. Close to the horizon, the entropy can deviate from the simple area law, reflecting the rapid changes in geometry and field configuration. Although the model is idealized, these results provide insights into the generation and scaling of entanglement in the presence of realistic, dynamically evolving gravitational fields.

Figures (6)

• Figure 1: Top: Full solution for $\rho(t)$ in the EdS universe, with $p_{lj}=1$, $t_0=1$, and selected initial conditions. Bottom: Same as the top plot, but for high-frequency modes with $p_{lj}=10^4$.
• Figure 2: Top: Ground state entanglement entropy as function of the dimensionless comoving area $A_c$ for various comoving times. The collapse parameters are reported in Table \ref{['tab:collapse_parameters']}. We computed the entropy up to the 51st shell ($n=51$) out of a total of 60 shells ($N=60$). We fixed $l_{\mathrm{max}} = 1500$ to achieve a tolerance of approximately $10^{-4}\%$, as discussed in Belfiglio:2024qsa. Bottom: Ground state entanglement entropy as function of the physical area $A_p = a^2 A_c$. The parameters used are the same as in the top panel. The curves stop at a definite value of $A_p$ because the physical area is shrinking with the scale factor.
• Figure 3: Top: Physical slope of the approximate area law near the origin, as a function of comoving time. The collapse starts at $t = 0$, and the sphere reaches the Schwarzschild radius at $t_{r_s} = 3.636$, while the singularity is reached at $t_c = 4.443$. Bottom: Comoving slope of the approximate area law near the origin as a function of comoving time. We restricted the time domain to $t < t_{r_s}$ to highlight the oscillations that occur during the collapse. For both plots, we computed the entropy up to the $25$th shell out of a total of $30$ shells. We fixed $l_{\mathrm{max}} = 500$ to achieve a tolerance of less than $0.01\%$.
• Figure 4: Top: Real part of the matrix elements $\text{Re}(\Sigma_{j}^{l})$. These are obtained by solving the Ermakov-like equation Eq. (\ref{['eq:ermak_like_0']}) and using Eq. (\ref{['eq:ground_state_matrix']}). They describe how the ground state of the field evolves in time for each angular momentum mode $l$ and normal mode $j$. Bottom: Imaginary part of the matrix elements $\text{Im}(\Sigma_{j}^{l})$.
• Figure 5: Ground state entanglement entropy computed at a given shell as a function of time. In the top plot, we present the real entropy values, while in the bottom plot, we show the rescaled and shifted curves to facilitate a fair comparison of the functional dependence of the entropy on time for the chosen shells. The rescaling ensures that each entropy reaches unity at the final time. The parameters are the same used in Fig. \ref{['fig:eet0_slopes']}.