Black Hole Interior and Time-like Entanglement Entropy

We establish time-like entanglement entropy (TEE) as a novel tool to characterize the black hole interior from a single-boundary perspective. In the Schwarzschild-AdS black hole, we show that TEE of time-like boundary strips exhibits linear growth as a function of temporal width in the limit of large temporal width, and that its imaginary part carries physical significance rather than being a constant. By analyzing charged, scalar-hairy black holes, we present evidence that TEE detects a hidden "causal phase transition" separating Type-I and Type-II interiors -- distinguished by singularity structure. We identify a critical temporal width $\tau_c$ that acts as the order parameter for this transition: for strips narrower than $\tau_c$, the system enters a distinct "time-like entanglement phase" dominated purely by time-like contributions, up to a regulator effect; conversely, for strips wider than $\tau_c$, space-like entanglement re-emerges. Notably, the existence of a Cauchy horizon drives the $\tau_c$ to infinity, leading to pure time-like entanglement. These results suggest that the TEE may supply a novel boundary quantum-information measure to detect structure hidden inside the black hole and suggests a deep connection between TEE and cosmic censorship.