The speed of the outer stars of galaxies The stars on the outer perimeter of galaxies rotate faster than expected. Could this be because they are traveling faster in time than the center where the black hole is? I mean due to extreme gravitation in the galaxy center time slows down and as u go further away time speeds up. Maybe we didn't estimate correctly how time behaves in spacetime? 
 A: If gravitational time dilation was in any way connected with the velocities of stars in the Milky Way, we should, in general, see a relatively smooth, monotonic curve. Gravitational time dilation scales as
$$\sqrt{1-\frac{r_s}{r}}$$
where $r_s$ is the black hole's Schwarzschild radius and $r$ is the distance from its center. However, we don't see that. What we do see is a sharp increase in velocity from relatively low values near the center of galaxies to some peak, and then a gradual decline.
You've probably seen this occur on galactic rotation curves before; I decided to grab some data from this page. There's raw data for stars in the Milky Way here, given in kilometers per second relative to radius in kiloparsecs. You can see the peak appear in the data within about 20 kiloparsecs, and then a gradual decline.
I also plotted select data points:

Here they are with a  polynomial trendline1:

Dark matter density profiles can explain the curve well; gravitational time dilation can't.
This is actually a very non-quantitative way of looking at it. I've just given a weak argument by stating that any effects from the supermassive black hole should vary differently with an increase in $r$. For a quantitative demonstration, I'd recommend looking at some of the answers in the linked question and links therein, including this one by Rob Jeffries and these two by John Rennie, which cover both types of time dilation.


1 A polynomial trendline like this isn't what would be used to model the velocity dispersion of stars in a galactic disk. However, I'm just using a polynomial fit (12th-order, because that's what's needed to reduce bumps) to better show the overall curve from the data points.
A: Yes, Time will be observed to flow differently on outer stars observed from center of our galaxy, but speed of stars will be observed to be the same from both stars and center of galaxy frames of reference due to gravitational length contraction.
Consider one person Bob at the center of our galaxy and another person Nick at outer reaches of our galaxy and assume their relative velocity is 0, Time for Bob will go faster relative to Nick, so their clocks will disagree about time, So one might think that since Bob observes that time goes faster for Nick, Bob will see Nick going faster since $v_{Nick}=\frac{\Delta x}{\Delta t}$ and $\Delta t$ is observed to be smaller from Bobs frame of reference, But in actuality their relative velocity is 0 as assumed earlier and from both Nicks and Bobs frame of reference relative velocity is observed to be 0. This caused due to phenomenon of length contraction, Bob will observe that time for Nick goes faster but he also observes that Nick is "contracted" so velocity remains 0 as assumed earlier.
Same can be observed with stars their velocity will remain the same relative to center of the galaxy no matter from which frame of reference you observe it.
From Newtonian mechanics stars that are more distant from the center of our galaxy require less velocity to remain in orbit compared to stars that are closer in, this is represented by following equation:
$$F_{g}=G\frac{Mm}{r^2}$$
But we observe that this is not the case in a galaxy system. Following graph represents velocities of stars as a function of distance from the center of our galaxy:



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*This is explained by introducing another kind of matter (dark matter) which doesn't interact with light but interacts with gravitational field, But dark matter wasn't observed directly as an elementary particle, But it's effects on the universe have been observed for example the following picture is light changing its trajectory due to gravitational influence caused by dark matter





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*This phenomenon is also "explained" by Modified Newtonian Dynamics
