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There are a bunch of stars orbiting the black hole in the center of our galaxy.

These stars move at huge speed. Why do we see this? Why do the black hole not impose any noticeable time dilation on these stars?

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Great question Lucas. The velocity of an object in orbit around a massive body can be expressed roughly as

$v(R)\sim \sqrt{ \frac{GM}{R}}$

The closer you are to the mass (e.g. the black-hole), the bigger v(r) becomes. It turns out, for a black-hole like the one at the galactic center, with stars about 100 AU away.... they travel at about 500 km/s---fast!

enter image description here

Now, the effects of general relativity are only significant when you're near the event horizon. In this case, even though the stars are relatively 'close' (about $10^{15}$ cm), they're still almost about 1000 times further away than the event-horizon! And so the effects of general relativity (e.g. time dilation) are very very small (in this case, currently unobservable at all).

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Thank you Zhermes. Great answer! –  Lucas D. Luengo Apr 10 '12 at 18:32

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