Carter sustained the injury during his first match back for the champions, a 36-11 win over the Stormers in Christchurch last weekend.
Crusaders coach Robbie Deans said: "Dan jammed his neck a little bit in that game. Initially we thought it was an injury they call a stinger, which is painful at the time but comes right pretty quickly.
"It turned out that it wasn't, but our medical team were optimistic all week that he would still be ready to play, which is why we named him in the team.
"We don't think he's far away, certainly that's what we're hoping, and the medical opinion supports that. But we were not about to compromise the player or the team by taking an unnecessary risk.
"For that reason, once it became clear he was still feeling some discomfort passing to one side, we ruled him out prior to even attempting to try him in contact. Hopefully he will be ready to go again next week."
Stephen Brett will return at number 10, having scored 72 points during Carter's absence.
Former New Zealand international Caleb Ralph comes on to the bench.