I am new to this Doctor Who world so bear with me here, please.
If the tenth Doctor had already met River Song, why didn't the eleventh Doctor recognize her when he first met her?
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He does recognize her, at least as far as recognizing her as being the same person he met in The Library.
As far as we see on screen, the first time The Doctor meets River after his regeneration from Ten to Eleven is "The Time of Angels" when she leaves him a message. Even then, he's not surprised to find a message from her, and he seems readily prepared to rescue her when she gives him coordinates.
River, of course, is near the end of her sentence, meaning she's known the Doctor for much of her life. The Doctor does remember her as well, though he addresses her as a professor, which means he likely has not interacted with her much beyond the Library or he would have been more careful. He does seem to have met her in between "Library" and "Angels" (or, at least, has very little contact) but still remembers that she is somehow special to him, as he already knows enough to explain to Amy why their relationship is complicated.
The Doctor's timeline and River's timeline are working away from each other. Her past is the Doctor's future and vice versa. That is why she is always saying "spoilers". That's also why he isn't allowed to look in her diary, or he would know his own future. She also explained in one episode that every time they see each other he knows her less and less.
I believe there's an episode early in season 6 - I believe when they first meet the silence - they're going back to 1969 - The Doctor questions who River is. River doesn't answer other than he has to trust her. As he says he doesn't know her - Amy steps in - when she says the one thing she'd bet on is: fish fingers and custard.
We find out later this season who River is - but prior to this, as we can see - the Doctor says he doesn't know who River is.
My curiosity goes as to why the Doctor asks and questions who she is - when it seems he knows who she is - River told the 10th doctor his name. Who would the Doctor tell that to? Who would the Doctor give his sonic screwdriver to? Yet - these things River had. Why does he almost freak out wondering who she is?
The reason why is because their time streams are going in opposite directions. Every time they meet she knows him more and he knows her less.