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.

 

[oracle99]

I wonder what would be the peak speed in this scenario, as a % of light speed?

 

[DaveC426913]

I wonder what would be the peak speed in this scenario, as a % of light speed?

Are you referring specifically to planetary slingshot maneuvers? The peak will be an very small fraction of c.

It is a self-limiting technique. The faster you go, the less you can extract from a fly-by.
You have to pass ever closer to gain more speed. At some point, the fly-by will risk grazing the atmo. It cannot extract any more energy beyond that point.


I wont dignify it by copy-pasting, since I don't trust it, but ChatGPT suggests that NASA did such a study examining many combinations of multiple fly-bys and determined that the max boost attainable was about 26km/s.

That's one 14,000th, or 0.014% of c.



And, in case you're wondering, you cannot slingshot around the sun to gain escape speed. Simplistically, any gain in infall speed is lost on outclimb. It's complicated. Maneuvers within the solar system are a different animal than trying to escape the solar system.

 

[Sagittarius A-Star]

I wonder what would be the peak speed in this scenario, as a % of light speed?

I assume you mean the scenario, which you described in the OP.

The peak speed is reached in the middle between Earth and Alpha centauri, because then you begin to decelerate.
Constant proper acceleration and rapidity are related by
$\alpha=c{\frac {\Delta \eta }{\Delta \tau }}$

Rapidity and speed $v/c$ are related by
$\eta =\tanh ^{-1}\left({\frac {v}{c}}\right)$

According to the calculation of @Ibix in posting #2, after half the distance
$\tau \approx 1.76 \ \text{years}$.

It follows
$v_{\text{peak}}/c = \tanh \eta=\tanh (\alpha \frac {\tau}{c}) \approx 0.9425$

Source:
https://en.wikipedia.org/wiki/Proper_acceleration#Acceleration_in_(1+1)D

 

[Ibix]

I wonder what would be the peak speed in this scenario, as a % of light speed?

Alternative to @Sagittarius A-Star's rapidity method, you can use $v/c=\frac{at/c}{\sqrt{1+a^2t^2/c^2}}$. Using years and light years (so $c=1\mathrm{ly\,y^{-1}}$) and approximating $1g\approx 1\mathrm{ly\,y^{-2}}$ gave us $t=\sqrt{8}\mathrm{y}$, which yields $v/c=\sqrt{8}/3\approx 0.94$.

(Edit: corrected the messed up the factors of c.)

 

[oracle99]

Thanks, I was not very clear, yes I meant the threads first post, rather than my last one. .94 c is quite a speed, I think there are practical problems such as the heat generated hitting extremely small particles or atoms at that velocity.

One of the articles above mentions using light sails, and a massive solar power laser inside the orbit of mercury to drive a craft, which saves carrying fuel. One point I didn't realize was that they say they can apply deceleration too with that set up. Most of these "practical" schemes travel propose about 0.01 c as max, traveling for hundreds of years.

On solar "slingshot", not that, but I think what is possible is to fly very close to the sun and accelerate at perihelion. One idea proposed using direct solar heat to provide power, but not providing relativistic speed.

 

[Filip Larsen]

I think what is possible is to fly very close to the sun and accelerate at perihelion.

I assume you refer to the Fermi Explorer which has a mission design that (if I read their mission design right and their AI didn't mess it up) gain around 7 km/s (~40%) over a more than a "traditional" outwards spiral. Not much relative to interstellar distances, but 40% is huge considering a planned 80'000 year transit time (not that I understand why that matters for anyone currently alive).