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View Full Version : Cancellation of gravitational t i m e dilation at isogravitic points?

mugaliens
2009-Aug-25, 08:41 PM
While looking at a countour plot of the Lagrangian points (http://en.wikipedia.org/wiki/Lagrangian_point), in order to formulate a response in this thread (http://www.bautforum.com/against-mainstream/91534-cogitations-twin-paradox-post1555152.html), I realized I didn't know the answer to the following question:

Given:

1. Two BHs of 1 solar mass each are in space

2. They are near one another, but otherwise alone.

3. They have identical mass

4. They are orbiting one another (circular orbit)

5. No other mass exists except two observers

6. One is positioned directly between the bodies and undergoes no acceleration.

7. The other is positioned at a distance approaching infinity and

Ignoring Hawking and CMBR radiations for a minute, frame-drag-induced spiral orbit decay, and the rest of those miniscule effects for a moment...

Does a clock with the observer between the black holes run slower than the distant observer, or does it run at the same rate?

Key point: While the gravitational attraction of each BH cancels out that from the other so far as acceleration of the mid-point observer is concerned, I am trying to ascertain whether this cancellation extends to time dilation as well.

grant hutchison
2009-Aug-25, 08:48 PM
Gravitational time dilation goes with the potential, not the force. Although the potential gradient is flat at the balance point between your two black holes, giving a point of zero net force, your observer at that point is still at a lower gravitational potential than the one at infinity (by virtue of being within the merged gravity wells of the two black holes), and so will experience gravitational time dilation relative to that distant observer.
This is related to the fact that, when two black holes merge, their event horizons bulge towards each other, meeting at the zero-force balance point. So your observer at that point will have a near-infinite time dilation, just before the horizons merge around him.

Grant Hutchison

mugaliens
2009-Aug-26, 06:40 PM
Gravitational time dilation goes with the potential, not the force. Although the potential gradient is flat at the balance point between your two black holes, giving a point of zero net force, your observer at that point is still at a lower gravitational potential than the one at infinity (by virtue of being within the merged gravity wells of the two black holes), and so will experience gravitational time dilation relative to that distant observer.

I see - it's the potential difference between the midpoint and the distant point. I admit I was caught off guard by something as simple as a 2D gradient map! (sigh) If it'd been a 3D gradient/potential map, or even a colored 2D map, with colors depicting a log of the potential.

So your observer at that point will have a near-infinite time dilation, just before the horizons merge around him.

:eek:

Thanks, Grant