After rain washed out the whole of the morning session, England announced an unchanged side following their 124-run win in the first Test at Lord's on Monday that put them 1-0 up in this two-match series.
By opting to field under increasingly bright skies, Cook gave spearhead seamer James Anderson an early chance to take the one wicket he needed to become the first England bowler, and 12th overall, to reach 400 Test wickets.
Cook himself needs just 32 more runs to break batting mentor Graham Gooch's England Test runs record of 8,900.
New Zealand gave a Test debut to one-day wicket-keeper Luke Ronchi, who took over the gloves from BJ Watling, suffering with a knee injury sustained at Lord's.
Watling retained his place in the side as a batsman only, with all-rounder Corey Anderson ruled out because of back trouble.
England: Adam Lyth, Alastair Cook (capt), Gary Ballance, Ian Bell, Joe Root, Ben Stokes, Jos Buttler (wkt), Moeen Ali, Stuart Broad, Mark Wood, James Anderson
New Zealand: Tom Latham, Martin Guptill, Kane Williamson, Ross Taylor, Brendon McCullum (capt), BJ Watling, Luke Ronchi (wkt), Mark Craig, Tim Southee, Matt Henry, Trent Boult
Umpires: S Ravi (IND), Rod Tucker (AUS)
TV umpire: Marais Erasmus (RSA)
Match referee: David Boon (AUS) – AFP
Original source: Cook elects to field first