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If we would achieve 99.9999 of light speed, we could get there in just under a year (288 days).

https://www.emc2-explained.info/Dilation-Calc/

I hope it would look more like the first picture instead of like the second.

well, as much i wish we will not acheive said speed to propel human to that planet.
I am afraid of that as well. The acceleration to such a speed would probably take quite some time if you don't want to become a pancake.
It would take a little less than 13 years to reach speed of light if acceleration is at 1 G (Earth gravity).

Of course, we could probably do that much sooner at 5G or so if in cryogenic stasis tank.

Are you off by an order of magnitude here? The closest star to the Sun would be around 4 light years
No, I meant this particular planet. If you travel aboard a ship going that fast, the time dilation would ensure it would just take a bit under a year.

For everyone else it would take longer than 558 years of course.

Can you please ELI5 why it would only take a year? Asking for a friend..
Essentially the closer you move at the speed of light, time flows slower for you. So, if you are in a rocket which is moving at close to c, for you the journey to Alpha Centauri which is 4 light years away may be like just an year. But for us outside of your rocket, it would likely be more than 4 years.

If you were to goto Alpha Centauri and then back to Earth, you may have spent just 2 years in your rocket. But when you get out of your rocket, you will find that 8 years have gone by on Earth.

Taking a lot of liberties with my explanation, but hope this ELI5 works.

stop time with coldsleep and get there instantaneously?
A lot of people seem outraged about the speed of light limitation and demand to know when we're going to get around it... as if our current speed limit weren't <.0001% of the speed of light.

You don't need to go faster than light to get anywhere you want, in any time period you like. The speed of light is not the limiting factor.

The amount of energy available to you, and the tyranny of the Rocket Equation, those are the constraining factors. Those are perfectly ordinary engineering issues, rather than magically hoping to circumvent the laws of physics.

If you were hoping to come back and explain it to your spouse/sibling/Nobel Prize Committee, then yes, you'd need new physics. But if you just wanted to go, that's a problem that you could actually make progress on by studying actual science, rather than sci-fi.