You can make your statement more accurate by saying that a clock will measure vanishingly little time as you get sufficiently close to the 45 degree line. But t

You can make your statement more accurate by saying that a clock will measure vanishingly little time as you get sufficiently close to the 45 degree line. But the 45 degree line isn't achievable. (Note I've made the same inaccuracy further up the page. Sometimes it's just easier to drop the epsilons and speak informally.) Physicists are precise people :)