Is curved space cancelled between two massive bodies?

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Discussion Overview

The discussion revolves around the effects of gravitational time dilation between two equally massive stars and whether the gravitational influences of both stars cancel each other out in terms of time dilation experienced by a clock positioned equidistant from both. The scope includes theoretical considerations of general relativity and gravitational effects.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants question whether a clock equidistant from two massive stars would tick more slowly due to gravitational time dilation or if the gravitational effects would cancel each other out.
  • One participant presents the weak field metric and suggests that since the gravitational potentials add, time dilation would also add.
  • Another participant argues that curvature cannot be canceled in the way suggested, emphasizing that curvature is a single sign and cannot be treated as having opposing effects like charges in a geometric theory.
  • There is a mention of the curvature tensor in empty space having multiple components that can cancel each other, indicating that gravitational effects do not arise solely from curvature.
  • A participant notes a specific case in general relativity where curvature can be canceled, referring to a spherical shell of matter in an empty universe.
  • One participant seeks clarification on what the clock's time dilation would be compared to, indicating a need for further specification in the original question.

Areas of Agreement / Disagreement

Participants express differing views on whether gravitational effects can cancel each other out and how time dilation is influenced by the presence of multiple massive bodies. The discussion remains unresolved with multiple competing perspectives presented.

Contextual Notes

Participants highlight the complexity of gravitational effects and the nuances of curvature in general relativity, pointing out that assumptions about cancellation may not hold in all scenarios.

Buckethead
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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?
 
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Well, the weak field metric is given by:

[tex]ds^2=-(1+2\phi)dt^2+(1-2\phi)(dx^2+dy^2+dz^2)[/tex]

Where [itex]\phi[/itex] is the Newtonian potential. Since the potentials between the masses add, so does time dilation.
 
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.
 
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.
 
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.
 
Buckethead said:
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?
 
Bill_K said:
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.
 

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