I understand that time is relative for all but as I understand it, time flows at a slower rate for objects that are either moving faster or objects that are near larger masses than for those that are slower or further from mass.
So, the illustrative example I always see is that if I were to leave earth and fly around at near light speed for a while or go into orbit around a black hole the time I experience would be substantially shorter than for those I left behind at home on earth and I'd come back to find that I've only aged however long I felt I was gone by my own clock but that on Earth substantially more time would have elapsed.
Following through on this model, the stars in orbit around the black hole at the center of the Milky Way are aging much much slower (relative to us), right? So does it not follow that the center of the galaxy is some appreciable (I have no idea how to go about putting this in an equation so will avoid guessing at the difference) amount "younger" than the stuff further away from the center?
If this is not true, could someone please explain why not, and if it is true, can someone please point me to where I can calculate the age of the center of the galaxy :-)
And to be clear... what I'm asking is... if there was an atomic clock that appeared at the center of the galaxy when the center was first formed, and we brought it through a worm hole to earth today - how much time would have elapsed on that clock vs. the age we recon the galaxy currently is? (13.2 billion years)
Answer
The gravitational potential of the disk of the Milky Way can be approximated as:
$$ \Phi = -\frac{GM}{\sqrt{r^2 + (a + \sqrt{b^2 + z^2})^2}} \tag{1} $$
where $r$ is the radial distance and $z$ is the height above the disk. I got this equation from this paper, and they give $a$ = 6.5 kpc and $b$ = 0.26 kpc.
In the weak field approximation the time dilation is related to the gravitational potential by:
$$ \frac{\Delta t_r}{\Delta t_\infty} = \sqrt{1 - \frac{2\Delta\Phi}{c^2}} \tag{2} $$
At the centre of the galaxy $r = z = 0$ and equation (1) simplified to:
$$ \Phi = -\frac{GM}{a + b} \tag{3} $$
No-one really knows the mass of the Milky Way because we don't know how much dark matter it contains, but lets guesstimate it at $10^{12}$ Solar masses. With this value for $M$ and using $a$ + $b$ = 6.76 kpc equation (3) gives us:
$$ \Phi = 6.4 \times 10^{11} \text{J/kg} $$
Feeding this into equation (2) gives:
$$ \frac{\Delta t_r}{\Delta t_\infty} = 0.999993 $$
So over the 13.7 billion year age of the universe the centre of the Milky Way will have aged about 100,000 years less than the outskirts.
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