# Time according to the gravity of Sagittarius A*?

This may be a really dumb question (I'm more of a Biologist than an Astronomer) so I apologize in advance for my little knowledge pertaining to Astronomy, but, if I'm not mistaken, time is effected by gravity, right? So what is Sagittarius A*'s time compared to ours since it has so much stronger gravity? Do we know specifically the difference?

Not at all a dumb question. As you have heard, it is true that time is affected by gravity. The stronger the gravitational field, the slower time passes. If you're far from any gravitating matter, time passes "normally".

But to answer your question, we must specify what is meant by "the black holes's time" (let's call the black hole $$\mathrm{BH}_\mathrm{Sgr\,A^*}$$; see note below on the nomenclature), since it depends on how far from Sgr A* we are talking. The time pace at a distance $$r$$ from the center of a BH is given by $$t = t_\infty \sqrt{1 - \frac{r_\mathrm{S}}{r}},$$ where $$t_\infty$$ is the time "at infinity", i.e. far from the BH, and $$r_\mathrm{S} \equiv \frac{2GM}{c^2} \simeq 3\,\mathrm{km}\,\times \left( \frac{M}{M_\odot}\right)$$ is the so-called Schwarzschild radius (the "surface" of the BH), which is where not even light can escape. Here, $$G$$ is the gravitational constant, $$M$$ is the mass of the BH, $$c$$ is the speed of light, and $$M_\odot$$ is the mass of the Sun.

The last equality shows that a BH with the mass of the Sun would have a radius of 3 km. The mass of $$\mathrm{BH}_\mathrm{Sgr\,A^*}$$ is some 4.1 million Solar masses, so its radius is $$r_\mathrm{S} = 12.1$$ million km.

Plugging in the other numbers, we can see that at a distance from $$\mathrm{BH}_\mathrm{Sgr\,A^*}$$ of

1. 1 lightyear, time runs slower by a factor of 1.00000064, i.e. unnoticeably.
2. 1 astronomical unit (the distance from Earth to the Sun), time runs 4% slower.
3. 1 million km from the surface, time runs slower by a factor of 3.6.
4. 1000 km from the surface, time runs slower by a factor of 110.
5. 1 km from the surface, time runs slower by a factor of ~3500.
6. 1 m from the surface, time runs more than a 100,000 times slower.
7. At the surface, time stops.

Note that this time dilation is what a distant observer (i.e. the guy with the $$t_\infty$$ time) would measure for an observer at the distance $$r$$. The person at $$r$$ would just measure his/her own time as usual. For instance, according to point 5 above, if you were hovering 1 km from the surface, waving your hand every second, then I, choosing to stay at a safe distance of 1 lightyear but with a magically powerful telescope, would see you wave approximately once every hour. And when you run out of fuel and plummet into the BH, then when you cross the surface you wouldn't notice anything particular, but I would see you frozen in time. This is the concept of relativity.

Finally, let me use this chance to clarify something that people, including myself, often have gotten wrong: Sagittarius A (without an asterisk) is a radio source in the center of the Milky Way. It consists of three parts: Sagittarius A East (a supernova remnant), Sagittarius A West (dust and gas clouds), and Sagittarius A*, or Sgr A*, which is a very bright and compact radio source believed to be formed by a supermassive BH. Sgr A* isn't actually the BH itself. I think the BH doesn't really have a name, so I'll call it $$\mathrm{BH}_\mathrm{Sgr\,A^*}$$. Maybe that's a bad name…

• @user6760: Time stops only for an "outside" observer. The person close to the BH doesn't notice anything. I clarified in the text. Thanks for prompting me to do this. – pela Apr 14 '15 at 13:08
• Great answer and very illustrating numbers about the gradual time dilatation with distance from a SMBH. Could throw in the effect for a GPS satellite there too. As for the naming issue, maybe we should call it the "A-Star Sagittarius Hole", or for short the AS... no I won't type that. I'm afraid that the next IAU meeting might buy it, though. – LocalFluff Apr 14 '15 at 13:51
• :D @LocalFluff. And thanks for the link, I didn't know about the "mis"-etymology of dilation. Also thanks for the encouragement about the GPS, but I think I'll stick with the time stuff. – pela Apr 14 '15 at 19:24
• For a hypothetical Sgr A*-massed Schwarzschild black hole, the tidal forces across $1.8\,\mathrm{m}$-tall human near the horizon should be on the scale of $10^{-4}\, \mathrm{gee}$ or so. Supermassive black holes don't spaghettify until well past the horizon. The time dilation calculations are misleading because the Schwarzschild radial coordinate does not straightforwardly correspond to a radial distance. For example, if $r_\text{ft} = r_\text{S}+1\,\mathrm{m}$, then $r_\text{hd} = r_\text{ft} + 16\,\mathrm{\mu m}$ for the human. That's one way to think about why the tidal forces are small. – Stan Liou Apr 15 '15 at 10:31
• Excellent comment, @StanLiou, I hadn't thought of that. I'll remove the part about time dilation difference. – pela Apr 15 '15 at 11:57