Does time always slow down as gravity increases?

We consider gravitational time dilation between stationary observers and present examples which contradict the statement that ‘time slows down as gravity increases’. We furthermore show that this statement cannot be true in general, if gravity strength is defined in an observer independent manner. We also provide a pedagogical introduction to gravitational time dilation, and discuss aspects of this phenomenon, which are often omitted in textbooks on general relativity. [ABSTRACT FROM AUTHOR]