A little help with AA

[Yahweh]

The game is PL Holdem, 50cent ante. It's a full table and the game is Very Loose-Passive. I have AA in middle position and bet the pot($5) and get 3 callers. Flop comes JT3 rainbow. I bet $15 and get two to fold around to a loose-aggressive player who raises me $30. I think for a sec and make the call. The turn came a 6 and I check and he bets $30. I contimplate raising but just call never really thinking that I was beat because of his style of play. I was thinking maybe AJ. The river comes another 3, he bets $20 and I call and he shows me pocket TT's.

Where were my mistakes and at what point in this hand do I know for sure that my pocket AA's are beat. Remember that this is an older guy with alot of money and he doesn't mind pushing it around.

Appreciate the help