1. The problem statement, all variables and given/known data I'm reading the book Relativity, Gravitation, Cosmology by Ta-Pei Cheng. I'm in the part where he derived the gravitational time dilation formula for static gravitational field, τ1=[1+(Φ1-Φ2)/c2]τ2. This implies that clocks at a higher gravitational potential will run faster. He stated that, for two positions r1 = r and r2 = ∞, with r2 being the reference point Φ(∞) = 0, while τ(r) is the local proper time, the clock at r = ∞ gives the coordinate time t ≡ τ(∞). By the equation above this then implies, dτ=[1+(Φ(r)/c2]dt My first question is why set r2 = ∞ as Φ=0, it is more natural in mechanics to set the position r1 = r (say, near earth) at Φ=0. I know this is just the same but my point is other books use the other way such that dτ=[1-(Φ(r)/c2]dt ← minus sign instead of plus. My second question is he gave an example of the case of a black hole and uses the above formula to expound. He said that, "for an observer, with the time t measured by clocks located far from the gravitational source (taken to be the coordinate time), the velocity of the light appears to this observer to slow down. A dramatic example is offered by the case of black holes. There, as a manifestation of an infinite gravitational time dilation, it would take an infinite amount of coordinate time for a light signal to leave a black hole. Thus, to an outside observer, no light can escape from a black hole, even though the corresponding proper time duration is perfectly finite." 2. Relevant equations τ1=[1+(Φ1-Φ2)/c2]τ2 dτ=[1+(Φ(r)/c2]dt 3. The attempt at a solution For the first question, I think it will just be a matter of sign difference but I'm not sure if there are other things involved. For the second question, I understand that an observer far from the black hole sees that his clock ticks faster since he is in a higher gravitational potential but he sees the clock of someone near the black hole as ticking very slowly, also someone near the black hole will see his own clock to run slow. My problem was when I tried to correspond this concept to the formula, I'M CONFUSED. If it requires an infinite coordinate time for light to escape a black hole and the gravitational potential Φ(r) is also infinite, doesn't that give an infinite dτ? Any help?