at the center of a planet/star/etc
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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?
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.
twistor59 wrote: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.
Xaihe wrote:CdesignProponentsist's answer seems to be in conflict with your answer. Is it too forward to ask you two to fight it out?
Xaihe wrote:twistor59 wrote: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.Xaihe wrote:
It makes sense, but isn't the redshift only dependent on the gradient of the potential? For instance, in the case of 2 equal mass bodies, one smaller than its Schwarzschild radius, the other larger. Would then the redshift be different for both objects, while they have the same total potential?
twistor59 wrote:Well, it's 25 years since I did any relativity, but I'm pretty certain that at the centre of a massive body, clocks would be running slower compared to ones at a distance from the body.
hackenslash wrote:twistor59 wrote:Well, it's 25 years since I did any relativity, but I'm pretty certain that at the centre of a massive body, clocks would be running slower compared to ones at a distance from the body.
Indeed, that would be my understanding as well. The effect that a body would feel would be equivalent to having been cancelled out, because the gravitational attraction is coming from all directions rather than being unidirectional, but the immersion in a gravitational field would be precisely the same as it was at the surface. All that has changed is the direction of mass from the perspective of the observer, because now the observer is surrounded by it as opposed to being outside it. The relativistic time dilation would be the same though, I think.
hackenslash wrote:It might be worth explaining why '-'dx2, etc, in other words explaining why the minus solution is employed rather than the plus. That makes the math more understandable, I reckon. It's how I got how Minkowski spacetime works.
hackenslash wrote:Actually, thinking about it, it isn't really relevant to the question at hand, although it could possibly firm up how it works. I'd be happy to explain it if any thinks it worth while. It does have very profound implications for causality, though.
hackenslash wrote:
If anybody wants to see this explained in greater detail, I recommend Why Doe E=mc2, by Brian Cox and Jeff Forshaw, from which this example is borrowed, as are the diagrams.
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