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Suppose a planet is orbiting a black hole, is it true that the time elapsed at the side that face to black hole is slower? Because I think the near side is nearer to the black hole, so it experienced stronger gravity and hence time dilation occurs more obviously, is it true?

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Right, time would pass slower. But unless the planet was very close to the black hole (BH) horizon, you wouldn't notice it. See the calculations of the approximate numbers below.

If for instance, for a solar mass BH, the planet was at the same distance from the BH as the earth is from the Sun, it would be no different than it is now on one side and the other of the earth. We would not notice the difference. If the solar mass was a few tens of solar masses,we still would not notice it, though it would be more.

The distance to the horizon for a Sun mass BH is about 3 Km. For 10, 100, and 1000 solar masses, it's 30, 300, and 3000 Kms. For a planet the size of the earth to be orbiting at any of those distances from the center of the BH half or most of the earth would already have merged into the BH, with the rest to get merged at a speed close to the speed of light, so not important what the time differences are. If we orbited at distances 1000 times larger, for each of those, the different time dilations in the near side and far side of the earth would be for each, approx.

$t_1/t_2= 10^{-6} d_1/d_2$

With the t's ratios being the time dilation differences, and the d's ratios the distance for each side from the center of the BH. For 1, 10, and 100 solar masses, and an earth sized planet, those are

For 1 solar mass BH: half the earth already merged, not a good question

For 10: about $10^{-7}$

For 100: about $10^{-8}$

For 1000: about $10^{-9}$

And so on.

It gets more and more insignificant, but still measurable, for those sizes.

If it orbits another factor of 1000 further away multiple by another factor of 10^{-6}$, and so on. An astronomical unit away where we are from the Sun, for those sized BH's, it's not measurable.

The equation above was derived from the time dilation equation by taking derivatives. I used the equation at Wikipedia at

https://en.m.wikipedia.org/wiki/Schwarzschild_radius#In_gravitational_time_dilation

I may have made a computational error, but the trend is right. And the numbers small unless the planet is close to the BH horizon.

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Yes. Time even passes more slowly on the surface of the earth than on a mountain, right here on Earth, for the same reason. Deeper in the gravity well == slower passing time.

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Yes, but insignificantly.

The same happens while in orbit around a "normal" object (not black hole), also insignifiacantly.

If you would like to increase the difference between time dilatations, you would have to bring the planet closer. But then, much before the difference becomes significant, the planet would cross the Roche limit and disintegrate, forming a ring around the black hole.

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