Why isn't the center of the galaxy “younger” than the outer parts?

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)

• A more interesting question would be, looking at the recent movie Interstellar as a guide, would any planets/stars extremely close to Sagittarius A* be younger, as this question suggests, even if the 2 current answers are correct and most of the ones only somewhat close to it are not much younger? Sorry if this comment spoiled anything in the movie. – trysis Jan 25 '15 at 21:55
• @trysis, could you elaborate? I'm unable to parse from your comment what your "more interesting" question actually is - I didn't see the movie (no worries about the spoiler :-). – Genia S. Jan 26 '15 at 0:16
• Well, in the movie, there is a planet that the protagonists go to that is extremely close to a black hole. You can be close enough to the planet to orbit it and still experience time "normally", but when you get really close, and, in particular, when you are on the planet, you experience 1 hour for every 7 years that passes away from the planet. I don't know how realistic this is, but I have heard that the creators tried to make it realistic. I suppose by "more interesting" I meant is it possible :). – trysis Jan 26 '15 at 4:45
• @trysis: the star nearest to Sagittarius A$^*$ is S2. I make its time dilation (including the effect of orbital speed) about 0.999 i.e. time runs about 0.1% slower. – John Rennie Jan 26 '15 at 6:59
• @John Rennie: S2 gets to about 1% of the speed of light in perihelion. Adding the time dilation from Sagittarius A*'s gravity field, wouldn't the total be more than 0.1%? – Blackscale Mar 31 '15 at 11:47

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.

• Probably a stupid question, but… we have a tangential velocity that the centre of the galaxy doesn't have, haven't we? Does this fact reduce the ratio you've calculated, or is the impact negligible? – Blackhole Jan 25 '15 at 23:15
• @Blackhole: no, that's a good question and as it happens I answered it recently. The time dilation due to our orbital velocity is about an order of magnitude less than the time dilation due to the gravitational potential. So if you include this the centre will have only aged about 90,000 years less than the outskirts. However take this figure with a pinch of salt as we don't know the dark matter distribution. – John Rennie Jan 26 '15 at 6:27
• @Blackhole while the "center" doesn't have a tangential velocity, the stars (and everything else) near the center still orbit, and I think they'd have velocities higher than ours, so the velocity difference would actually add to the gravitational one rather than subtract. – Rick Jan 26 '15 at 15:44
• It might be worth taking that comment (re 90,000) and making it an endnote to the question itself. – David Conrad Jan 26 '15 at 15:52

The centre of the galaxy will indeed appear to pass through time more slowly than the edges, but the effect will not be great.

Because the Einstein field equations are very difficult to solve, it is not possible to calculate the exact magnitude of the time dilation, but we can make an approximation. By assuming that the black hole at the centre of the galaxy is electrically neutral and non-rotating, and ignoring the effects of all other mass/energy, we can calculate the time dilation at a distance $r$ from the galactic centre, as seen by an observer at infinity.

The formula for this time dilation is $\Delta t_0 = \Delta t_\infty \sqrt{1 - \frac{r_S}{r}}$, where $t_0$ is the proper time at a distance of $r$ from the galactic centre; $t_\infty$ is the proper time measured at infinity, and $r_S$ is the Schwarzschild radius of the black hole that lives at the centre of the galaxy. Because $r_S$ is many times smaller than $r$ (except for any unlucky stars finding themselves being eaten by the black hole), we would not see any appreciable difference in the rate at which time passes between stars close to the centre and those far away.

All of this analysis assumes that Sagittarius A* is exactly at the centre of the Milky Way, which is not exactly true. The distance between the two will cause the actual centre to be slowed by the gravity of the black hole, just like anything else. This will be highly dependant on the proper distance between the centre and the hole, but could be calculated - with some approximation - by the above formula.

• The mass of the Milky Way is about $10^{12}$ Solar masses and the mass of Sagittarius A$^∗$ is around $10^6$ to $10^7$ Solar masses. Ignoring the mass of the Milky Way seems a poor approximation to me ... – John Rennie Jan 25 '15 at 20:21