  In the context of the Bank-Fishler-Shenker-Susskind Matrix theory, we analyze
a spherical membrane in light-cone M theory along with two asymptotically
distant probes. In the appropriate energy regime, we find that the membrane
behaves like a smeared Matrix black hole; and the spacetime geometry seen by
the probes can become non-commutative even far away from regions of Planckian
curvature. This arises from non-linear Matrix interactions where fast matrix
modes lift a flat direction in the potential -- akin to the Paul trap
phenomenon in atomic physics. In the regime where we do have a notion of
emergent spacetime, we show that there is non-zero entanglement entropy between
supergravity modes on the membrane and the probes. The computation can easily
be generalized to other settings, and this can help develop a dictionary
between entanglement entropy and local geometry -- similar to Ryu-Takayanagi
but instead for asymptotically flat backgrounds.
