  In this paper, we develop a general effective theory for two copies of the
Sachdev-Ye-Kitaev (SYK) model with a time-dependent bilinear coupling. For a
quantum quench problem with an initial state of the thermofield double state,
we show how the evolution of the system is described by a complex
reparametrization field with a classical Hamiltonian. We study correlation
functions in this system and compare the large-$q$ theory with the bulk low
energy effective theory. In particular, we study the special case of a
``rescued black hole'', which describes how a time-evolved thermofield double
state can evolve to the ground state of a coupled SYK model by a carefully
tuned time-dependent coupling. In the low energy region, there is a holographic
dual interpretation, which is a geometry that crosses over from an eternal
black hole to a global AdS$_2$ vacuum. This family of geometries allow us to
access the bulk region that would be the black hole interior without the rescue
process. By comparing the large-$q$ and low energy theory, we find that even in
the low energy region the deviation from the low energy theory cannot be
neglected if the rescue process starts late. This provides evidence that the
low energy effective theory of the bulk fails near the inner horizon of the
black hole. We note the possibility of a connection to a two-dimensional analog
of the higher-dimensional black hole singularity.
