If I surrounded a black hole with gravitometers and observed the gravitational field as mass passed the event horizon, what would I observe? By surrounding a black hole with gravitometers I would be able to get a 3-D map of the gravitational field. If I observed this field as an object approaches the event horizon, what would I see?
Would the gravitational field allow me to track the mass on its journey from the event horizon to the singularity? Would this allow me to observe what is going on inside the black hole?
To clarify: At some point in time there are two distinct gravitational fields, one for the approaching object and one for the singularity. At a later time there is just the gravitational field for the singularity. It is the nature of the transition in the gravitational field that I am specifically interested in.
 A: This problem can be (and has been) studied numerical. One can simulate the gravitational field of a small massive object dropping radially into a (much larger) Schwarzschild black hole. This was done, for example, in arXiv:1012.2028 by Mitsou. (This is simply the first one I found, there are more, and probably much earlier.) 
The easiest way to represent the gravitational field is in the form of the Weyl scalar $\psi_4$, which contains (almost) all gauge invariant dynamical information about the gravitational field. Further more, it is convenient to write the field as a sum of (spin-weighted) spherical harmonics $Y_{lm}$. (You can think of this as the analog of a Fourier series on the sphere.) This conveniently captures the angular dependence of the gravitational field. Moreover, if one picks coordinates such that particle falls along the coordinate axis, the field is axisymmetric, meaning that all but the $m=0$ modes are zero.
With this setup one can plot the time dependence of the remaining $l$ modes. The plot below (from arXiv:1012.2028) plots the time dependence of the $l=2$ mode. As the particle approaches the black hole the field grows monotonically (mostly not shown in this plot) until particle reaches the horizon, after which the field "ringsdown" decaying exponentially while oscillating with a characteristic frequency known as a quasinormal mode (or QNM).

For higher $l$ modes the picture is qualitatively similar except that the amplitude is (much) smaller and the frequencies of the QNM ringdown higher. For example here is the $l=6$ mode from the same paper.

Note that the time scale for this to happen is quite short. In the plots, the units of time are scaled by the natural time scale for the black hole. For a 10 solar mass black hole one such unit is about 50 microseconds.
