Can I put a 6" hole in this ceiling joist?
That joist is carrying something, even if it the weight of the material that make up that part of the house. I would say it is more than that since the new 2X with the framing anchor is attached to it, is telling me it is passing the load from the new 2X onto the one you wish to drill.
Now onto drilling dimensional lumber. Code only allows you to drill a hole that is no larger than 1/3 the total height of the material you plan to drill. 12&quot; material, 4&quot; hole maximum. There are metal brackets available online that will allow larger holes to be drilled.
10-23 edit
If I might add, and it should have been obvious earlier...That doubled joist is actually a beam, not a joist. Those follow a more stringent rule for drilling. I do not know what the largest allowable hole is, short of a 3/4&quot; hole for electrical wiring, but that is all I do know. The mention by the OP about the outside roof and inside the house, that beam is carrying the outside wall above it with possibly the roof portion as well