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Williamson, who sustained the injury in the second test in Wellington, will be replaced by Central Districts batsman Will Young, who is set to make his international debut in the last test of the summer.
Tim Southee, who captained the Black Caps in Williamson's absence in the latter stages of the second test victory, will become New Zealand's 30th test captain.
Black Caps coach Gary Stead said Williamson had improved since the second test, but it wasn't worth the risk of playing him at Hagley Oval in Christchurch.
"Kane is obviously a really important player for us and with the nature of the injury we couldn't be sure he wouldn't aggravate it.
"He's improving every day and will continue his rehabilitation as planned.
"Will has been waiting in the wings for a while now and he'll get a great opportunity batting at number three in a settled line-up."
Black Caps wicketkeeper BJ Watling has overcome his hamstring injury suffered in warm-ups on day five in Wellington, and has passed a fitness test today to be available for selection.