Is curved space cancelled between two massive bodies?

[Buckethead]

Considering two equally massive stars stationed at some small distance from each other, would a clock stationed between the two stars equadistance from both tick more slowly due to the proximity to the massive stars, or would the effective cancellation of the gravitational attraction also cancel the time dilation? Or another way to ask this: Is time dilation related to the total gravity of both stars or to the apparent lack of gravity because of equal but opposite pull of each mass?

 

[elfmotat]

Well, the weak field metric is given by:

$$ds^2=-(1+2\phi)dt^2+(1-2\phi)(dx^2+dy^2+dz^2)$$

Where $\phi$ is the Newtonian potential. Since the potentials between the masses add, so does time dilation.

 

[Matterwave]

There's only 1 sign in the curvature, so you can't actually "cancel curvature" like maybe you could if you had some kind of a geometric theory with 2 different mass charges. You can think of it more like "curved in such a way that that there is a straight-line geodesic which must remain equidistant from the two stars at all times".

If that makes sense.

 

[Bill_K]

No, I'm afraid it doesn't. The curvature tensor in empty space has 10 components and each of these can either be positive or negative. And while it's true that mass is always positive, curvature components resulting from one source can certainly cancel ones from another.

The two effects mentioned in the OP, time dilation and gravitational 'pull', do not arise from the curvature anyway.

 

[PAllen]

There is only one case I know of where curvature can be canceled in a finite region in GR. That is the case of spherical shell of matter in an otherwise empty universe. Inside the shell, the metric is a flat Minkowski metric.

 

[Passionflower]

Considering two equally massive stars stationed at some small distance from each other, would a clock stationed between the two stars equidistant from both tick more slowly due to the proximity to the massive stars, or would the effective cancellation of the gravitational attraction also cancel the time dilation?

More slowly compared to what?

 

[Matterwave]

No, I'm afraid it doesn't. The curvature tensor in empty space has 10 components and each of these can either be positive or negative. And while it's true that mass is always positive, curvature components resulting from one source can certainly cancel ones from another.

The two effects mentioned in the OP, time dilation and gravitational 'pull', do not arise from the curvature anyway.

Guess I should have thought about it more, whoops.