Posted: Feb 27, 2011 12:49 pm
Xaihe wrote:Since gravity is stronger at the surface of a planet than at the center of a planet, does that also mean spacetime is curved less at the center of the planet, or just that the gravitational effect is canceled? In the same effect, does time go slower or faster at the center of the earth than at the earth's surface? If spacetime is curved less due to this cancellation, what does this tell us about the goings on inside black holes?
Imagine you're in space and you watch a clock drifting towards the earth. You would see it ticking slower and slower as it approached the earth. If it carried onwards, sinking into the earth, then it would continue to tick slower and slower until it reached the centre. One way to think of it is that effectively it's sinking deeper and deeper into the potential well, and photons would have to work harder and harder to climb out of it, hence get more redshifted. The ticks of the clock are equivalent to the cycles of the photons.
The strength of the gravity is more related to the gradient of this potential, whereas for comparing clocks we want to compare the values of the potential at two separate points. So the local strength of the gravitational force is not relevant.
To do this quantitatively, you'd have to look at the Schwarzschild Interior solution and look at the behaviour of g00
Xaihe wrote:
In the same spirit, what happens to the shape and position of the event horizons of two black holes as they approach each other? I'd imagine an extending of the event horizons outward and a decrease of the event horizon between the two masses.
My own understanding of GR is too limited to figure this out, nor could I find the answer elsewhere. I suspect I'm employing some kind of naive view of spacetime curvature here.
If you google something like "merging black holes" you should find loads of computer simulations and animations of this.