What is the reason that Linda Hamilton didn't reprise the character of Sarah Connor in Terminator 3: Rise of the Machines?
Was she busy on another film or had she stated no interest in appearing in the third film?
Linda Hamilton claimed it was because she felt that her character arc was complete and that the new script didn't offer her much.
Q: Your character went through an amazing arc in the sequel...
LH: How lucky am I? I loved it. I turned down "T3" because I really did feel that it was so complete, that the character arc was completely there and could not be improved upon. And the script that I read proved that to me! But I'm retiring a champ.
"They offered me a part. I read it and I knew my character arc was so complete in the first two, and in the third one it was a negligible character," she explained. "She died halfway through and there was no time to mourn her. It was kind of disposable, so I said no thank you."
Apparently she was also undergoing treatment at this point for her (pre-existing) bipolar disorder and it was only a couple of years after a messy divorce so it's certainly possible that she had some personal issues that prevented her from accepting the role. To deal with her absence, the writers dropped a bridge on her by giving her fatal leukaemia.