How long would it take?

Yep, that's right

 

Even crazier things happen if you don't return. Say your friends back home are sending you messages. How many years would you see go by in their messages?

Depends on your means of getting close to lightspeed, but say you're accelerating at 1 gee. Then once you've travelled more than a few light-years, the time-difference between when you left and the messages from home you receive on coming to a halt at the Galactic Core, or wherever, is just 1.9 years. That's the difference between light's travel time and yours, regardless of how far you go, if you accelerate constantly the whole way.

 

think about russ' response above for a moment - from the perspective of someone on earth watching you, it would take you just over a year to get there - BUT from your own perspective, you will have traveled that "one light year" in far less that a year - which brings up the question of "just how far away is that star?" from your perspective, you have either traveled faster than the speed of light, or the star is not "really" 1 LY away.

from earth's perspective, again, it is 1 LY, but from your ship's perspective, it is perhaps only a few million miles (depending on how close to C you are traveling). if you could actually travel at C, people on earth would say it took you exactly one year to arrive, but from your perspective, you would be there instantaneously, since time does not exist at C. so, how far away is that star, really?

 

FYI, it is "only" about 50,000 light-years to the center of our galaxy, not 50 million. But even that is an excruciatingly long distance / travel time.

 

50,000 ly to the Core and back. The current best estimate puts the Core about ~26,000 ly away. But if one was flying to the very heart of the Virgo Super-Cluster then it'd be about 50 million ly. Would be quite a challenging trip.

Ursula LeGuin's Nearly-as-Fast-as-Light (NAFAL) starships, from her Hainish stories, can do 256 ly in 10.65 hours of subjective time. If acceleration was (near) instantaneous then the time-distortion factor (TDF) would be ~210,000. Constant acceleration would need to be at 25,000 g and a peak TDF of ~2,700,000. For constant acceleration the ratio of cosmic time to subjective time is ln(2*TDF)/(TDF) for high TDF (above about 10.)

BTW what I'm calling TDF is normally referred to as γ <gamma>, and is equal to ~1/√[1 - (v/c)²] so the speed is incredibly close to lightspeed, a gnat's whisker per century difference in speed for that trip to Virgo...

 

they say that if you could continually accelerate at one g (from your perspective I guess) then you could go virtually anywhere in the universe in one lifetime. I've always found that to be a very surprising result. I think you have to use 'rapidity' to calculate it.

 

See

https://www.physicsforums.com/showthread.php?p=1193451#post1193451.

 

around 2.4 months from your perspective. 1 year to the rest of us.