Quote by George Jones
You haven't been specific enough. Are the particles falling radially? Falling initially from rest (with respect to Schwarzschild coordinates)? Falling form rest at infinity?
Distance is tricky and ambiguous in general relativity. What does "separated a distance d from each other" mean?

Yes they are falling radially, you can take any condition you like: falling from rest at infinity (problem then would be how to define any notion of distance between them) or initially falling from rest. In other words the two particles are exactly equivalent (the Schwarzschild solution is symmetrical after all). With regards to d, you are free to choose. I understand that talking about distance is tricky. However one could perhaps envision the definition of a function (valid for the Schwarzschild solution) that translates a given d at the initial position to a d at the event horizon. Alternatively you can express the situation with an angle or if everything else fails just pick some coordinate distance that you prefer. Also consider the extreme case, two test particles opposite to each other with the black hole
exactly in the middle (this one could in principle be expressed in terms of 'falling from rest at infinity' using symmetry).
However if it is not possible to express any kind of distance then the question becomes: how can you prove they do
not meet at the event horizon.