Transit time for a round trip to alpha centuri...

[oracle99]

Yes, I suppose its more than a perception, its how the traveller experiences time.

If we also place earth in the gravitational well of a black hole, could it be arranged that time elapses at the same rate for the folks back home?

 

[DaveC426913]

If we also place earth in the gravitational well of a black hole, could it be arranged that time elapses at the same rate for the folks back home?

"Back home"? On Earth?

 

[Filip Larsen]

If we also place earth in the gravitational well of a black hole, could it be arranged that time elapses at the same rate for the folks back home?

You can always "balance" the observers depth in a gravity potential so that the accumulated Doppler shift between observer and traveler gets zeroed out.

The relative rate at any given time between an observer in a well and his observation of the traveler (i.e. the observed Doppler shift) is of the order $\tfrac{1}{2}\beta^2+\frac{\Delta U}{c^2}$, with $\Delta U$ being the potential difference. Assuming, for simplicity, that $\Delta U$ to be constant for the trip mentioned this can be easily be integrated over the round-trip time to find a $\Delta U$ that match the accumulated age difference between observer and traveler. If the travel starts and end at same potential as the observer (which is technically needed in order to compare clocks after travelling) the calculations become more involved, but the principle remain the same.

 

[PeroK]

No it doesn't. It shows that 3.5 years pass. You're assuming (incorrectly) that 4 light years is the distance traveled.

The quantity $\dfrac{dx}{d\tau}$ can be greater than $c$. You have to be able to deal with that!

In the traveller's rest frame, they have travelled no distance. That's not very useful.

It's perfectly legitimate, IMO, for the space traveller to say that in the approximate rest frame of our part of the galaxy, they have travelled 4 light years in $t$ coordinate time and $\tau$ proper time, where $\tau < 4$ years.

Trying to avoid that "paradox" seems like the wrong approach to learning the physics.

 

[PeroK]

Yes, I suppose its more than a perception, its how the traveller experiences time.

There is coordinate time and proper time. That is physics. Perception is for the philosophers!

 

[Ibix]

It's perfectly legitimate, IMO, for the space traveller to say that in the approximate rest frame of our part of the galaxy, they have travelled 4 light years in $t$ coordinate time and $\tau$ proper time, where $\tau < 4$ years.

I agree there's nothing wrong with $dx/d\tau$ as a quantity - it's the x component of four velocity for a start. It's calling it "faster than light" that's problematic, because it isn't. The more or less equivalent measure to $dx/d\tau$ for light is $c\,dx/ds$ which is infinite.

However, I think there's a reason you don't find celerity much outside textbooks, which is that it doesn't do anything that velocity doesn't do, it can't express superluminal "speeds" like the old laser pointer swept across the moon, and it's yet another frame-dependant quantity that we won't really use. Even the case you're making is (IMO) better phrases as "I travelled 4ly in 3.5y by my clocks" rather than "I travelled at an average celerity of 1.14c".

 

[OmCheeto]

Delightful thread.
I worked on this problem over 20 years ago. (except for the relativity part, as I'm a bit slow.)
I found it most interesting that you can get to Mars in about a day or two at 1g.

 

[PeroK]

I agree there's nothing wrong with $dx/d\tau$ as a quantity - it's the x component of four velocity for a start. It's calling it "faster than light" that's problematic, because it isn't. The more or less equivalent measure to $dx/d\tau$ for light is $c\,dx/ds$ which is infinite.

However, I think there's a reason you don't find celerity much outside textbooks, which is that it doesn't do anything that velocity doesn't do, it can't express superluminal "speeds" like the old laser pointer swept across the moon, and it's yet another frame-dependant quantity that we won't really use. Even the case you're making is (IMO) better phrases as "I travelled 4ly in 3.5y by my clocks" rather than "I travelled at an average celerity of 1.14c".

That's not the point at all. The point at issue is that if we were visited by an alien intelligence from a star, say, 20 light years away. Then that is what they would say: "we've come from a star 20 light years away". It wouldn't make sense for us to say "No, you haven't ... in your reference frame, the Earth has travelled 15 light years to your rocket."

This is back to one of my pet peeves about the way relativity is taught. There is an inference that you must analyse things from your rest frame. Which is not true at all.

 

[oracle99]

For those interested in the practical side of interstellar travel, the Universe Today have a 5 part series covering the history of proposed projects, from using atom bombs in the 50s (!) through to solar/magnetic sails, and "space warps".

https://www.universetoday.com/articles/interstellar-travel-the-space-age-and-nuclear-rockets

and sails, you don't have to carry the fuel!

https://www.universetoday.com/articles/interstellar-travel-iv-solar-magnetic-directed-energy-sails

BTW: The New Horizons spacecraft is still in transit and working, and looking for another deep space object within its remaining delta-v capability to fly past, having already visited Pluto and Arrokoth. If it finds one, it will be the best effort so far (in terms of visiting a distant world), but only a tiny fraction of whats needed for interstellar travel.