B747 Captain reported landing gear would not retract after take off; with associated gear tilt; gear door; and gear position messages. The flight crew performed an air turn back and landing at departure airport.

Narrative

After takeoff the landing gear would not retract. A Gear Tilt message was displayed on the Engine Indicating and Crew Alerting System. Also gear door and gear lever disagree messages became displayed. We informed departure of the problem and were given vectors while we ran the appropriate checklists. The airplane never exceeded 265 knots during the flight. We consulted with dispatch; the Duty Pilot; Maintenance Control and the mechanic onboard the aircraft. We came to the conclusion that the best course of action was to divert to ZZZ.Fuel was dumped in the ZZZ area; while in coordination with air traffic control. We proceeded to land uneventfully in ZZZ. Upon arrival in ZZZ Maintenance informed us that a body gear door had separated from the aircraft. I believe it was on the same gear as the tilt message. We did experience some vibration in the airframe at all airspeeds after departure. The vibration remained constant at all airspeeds. The vibration did cease and it is my belief that that is when the door departed the airframe.Gear would not retract after normal takeoff. Gear tilt message. After debrief with other crew members we agreed that the event was handled fairly well. All of us had minor suggestions as to what we do differently if the same event were to occur again; but overall I/we feel the event was handled okay.