Revealing mistake: When he kicks Colonel Green away during the first fight, Kirk splits his pants. Fortunately, they mend themselves a few shots later. (00:28:00)
Revealing mistake: When Kirk & Spock beam down with Lincoln to the supposedly newly-created non-volcanic landscape, there are footprints all over the ground. They disappear a few shots later, though.
Revealing mistake: When Abraham Lincoln shoves the Klingon away from him during their fight, the Klingon collides with a heavy boulder which moves several inches at impact.
Revealing mistake: About 4/5 through the show, while the Vulcan is being killed, if you look at Lincoln's wrist, you can see his makeup line.
Answer: Kirk was getting his physical and Dr. McCoy probably turned off communications, because if he hadn't, Kirk would have left and headed straight for the bridge, leaving McCoy irritated.