Is it possible to use the Heathrow Express to transfer from Terminal 4 to Terminal 3? Would I have to purchase a ticket to London and then just get off the train at Terminal 3 or can I ride this distance at a reduced rate? If not, is it possible to take the Piccadilly Line between the terminals? It seems that the normal airport bus transfer may take too long. I cannot use the Flight Connections center because I understand BA won't check my bags through on VS so I must collect bags and check in again anyway.