phinds said:
I think WallaceHS has a good question. I too am confused. If the NET gravitational effect is zero at some point, why would that point act like it was in a deep gravity well and exhibit gravitational time dilation relative to somewhere a million miles away from the center of the binary system? What are he and I missing in this regard?
You are missing that time dilation is not a local effect-- you can't, like the proverbial groundhog, just peak at your own shadow to see what your time dilation should be. Time dilation is a comparison between two points, and everything that goes on between those points contributes to that time dilation effect. So that's why it's the gravitational potential, not the Newtonian local gravity, that matters for time dilation, because to get a gravitational potential difference between two points, you have to integrate over all the gravitational effects along a path that connect the two points.
To see this, imagine you are stationary deep in the center of some well, and some distant clock is stationary far from the well, and you reckon, by exchanging light signals, that your clock is ticking more slowly than theirs. Let's say you want to account for this difference, you want to understand what is responsible for it by figuring out where the difference appeared. So you connect yourself, and that distant point, by a chain of hovering stationary observers, and you notice that you get a smoothly varying answer for where the time dilation is happening. The only observation you can make of your own local environment to tell you this time dilation is happening is you could measure the tidal stresses of gravity in your vicinity, but that will only explain why you get a tiny time dilation between yourself and the nearby observers in that chain you set up. To get the full time dilation between you and that distant point, you have to add up the tidal effects all along that path, so it is an integral over the path that determines the total time dilation. It's just not a local effect, it's a global effect. The only local effect of gravity is a tidal stress that curves inertial paths in spacetime, and that has to be added up over the whole path to get the cumulative effect.
The same is actually true of the speed a particle acquires if you drop it from some distant observer and it falls to the center of the well, where you are-- the local Newtonian gravity around you (which is essentially zero except for the tidal stresses) won't tell you that either, you have to integrate the gravitational acceleration over the whole path. So just think of time dilation as being more analogous to the speed a falling particle acquires, than to the local gravitational acceleration. (And by the way, we should repeat for clarity
marcus' point above-- a particle at the center of mass of a binary is not generally in a place of zero net gravity, but we could interpret your question in the case of two stars of equal mass, and then it is.)