Twins Paradox in a Quasi-Uniform Gravitational Field | A Sci-Fi Exploration

[Al68]

We have a huge spaceship at some distance from a huge black hole cluster where the region of space has a quasi-uniform gravitational field of 1G. This ship is keeping it's thrusters firing in order to stay at rest relative to the black hole cluster. And, for centuries, this ship has been at rest with the cluster, with it's inhabitants at 1G proper acceleration.

An observer on this ship notices an alien space station approach from the direction of the cluster, apparently in freefall, since it is slowing down at a rate of 9.8 m/s^2, and a few years later, a small alien ship approaches from the same direction, also slowing down at 9.8 m/s^2, and it looks like the alien ship will stop and start falling toward the cluster about the time it reaches the large ship, but just as the alien ship comes to a stop, it fires its engines and stays at rest with the big ship. A short while later, a second space station in freefall approaches both ships, slowing to a stop as it reaches them, then starts falling toward the cluster. Simultaneously, in the large ship's rest frame, the first space station comes to rest several light years ahead and reverses direction to start falling toward the cluster. A short while after that, the alien ship cuts its engines off and starts falling toward the cluster. A few years later, the first space station comes flying back past the big ship, and the crew notes that it will catch up with the small ship before long. The big ship's scientist remarks to the captain, "Oh, they must be doing the twins paradox".

How's this for a third observer?

 

[Dale]

Just integrate the metric over each ship's path.

 

[RandallB]

Not a Twins Problem

Twins Paradox is a SR problem.
This is a spaceships acceleration problem (without the string), looking at "equivalency".
You can do the same with clocks on and moving next to high tower on Earth but the intervals and differences would be small.

 

[Al68]

Twins Paradox is a SR problem.
This is a spaceships acceleration problem (without the string), looking at "equivalency".
You can do the same with clocks on and moving next to high tower on Earth but the intervals and differences would be small.

Should I make my point less subtle?

Al

 

[RandallB]

Should I make my point less subtle?

what point?

 

[Al68]

what point?

The scenario I described is the standard twins paradox, except as observed by a third observer, one accelerating at 1G the entire time, who considers himself at rest.

My point was only that this frame is a legitimate frame of reference, one that is ignored in the standard resolutions (except Einstein's), and wrt this frame the Earth accelerates (coordinate) relative to the ship to "turnaround".

Thanks,
Al

 

[George Jones]

one that is ignored in the standard resolutions (except Einstein's)

Do you really think that the tens of thousands of people that have understood relativity could all have overlooked something so basic?

Consider twins Ted and Alice that are coincident both at event A and at some later event B. If the twins take different spacetime paths (worldlines) from A to B, which twin is older?

Prescription for obtaining the answer:

1) pin down the spacetime involved, i.e., give the metric;

2) for each twin, integrate the metric along the worldline; (as DaleSpam has said)

3) compare results.

END OF STORY

This method works for all spacetimes, and all timelike worldlines. In particular, this method works for all observers in Minkowski spacetime, and for all observers in Schwarzschild spacetime.

Sometimes both twins will be the same age, sometimes a twin that follows a geodesic will be older than a twin that accelerates, and sometimes a twin that follows a geodesic will be younger than a twin that accelerates.

For examples, see

https://www.physicsforums.com/showthread.php?p=1836071#post1836071.

 

[Al68]

Do you really think that the tens of thousands of people that have understood relativity could all have overlooked something so basic?

Consider twins Ted and Alice that are coincident both at event A and at some later event B. If the twins take different spacetime paths (worldlines) from A to B, which twin is older?

Prescription for obtaining the answer:

1) pin down the spacetime involved, i.e., give the metric;

2) for each twin, integrate the metric along the worldline; (as DaleSpam has said)

3) compare results.

END OF STORY

This method works for all spacetimes, and all timelike worldlines. In particular, this method works for all observers in Minkowski spacetime, and for all observers in Schwarzschild spacetime.

Sometimes both twins will be the same age, sometimes a twin that follows a geodesic will be older than a twin that accelerates, and sometimes a twin that follows a geodesic will be younger than a twin that accelerates.

For examples, see

https://www.physicsforums.com/showthread.php?p=1836071#post1836071.

Huh? Just getting the right answer is the end of story? And a method which obtains the right answer is all anyone should ever care about? I'm sure you don't mean this the way it sounds.

And no, I don't think my point has been overlooked, Einstein addressed it directly 90 yrs ago. But it's ignored in textbooks, and seems to be avoided at all costs on this board.

Al

 

[Dale]

seems to be avoided at all costs on this board.

It is the approach preferred by many on this board, including myself. Look for the word "metric" or "geometric" in many twins threads.

 

[Al68]

OK, I guess nobody is going to notice the obvious flaw in my initial post. I thought I worded it in a way to make it obvious. Maybe I'm on the wrong forum?

Al

 

[JesseM]

The scenario I described is the standard twins paradox, except as observed by a third observer, one accelerating at 1G the entire time, who considers himself at rest.

My point was only that this frame is a legitimate frame of reference, one that is ignored in the standard resolutions (except Einstein's), and wrt this frame the Earth accelerates (coordinate) relative to the ship to "turnaround".

It is not a legitimate inertial frame of reference in which you can use the standard SR equations for time dilation and so forth (it is impossible to find any valid inertial coordinate system to cover a large region of curved spacetime like in your example). However, the method that DaleSpam mentioned will work in any possible coordinate system, not just an inertial one--find the correct metric in that coordinate system, integrate it over the spacetime path in that coordinate system, and you'll get the elapsed time on a clock. This is a satisfactory resolution to the "paradox" because it gives the same answer for the proper time no matter which coordinate system you choose, and the supposed paradox was that different frames would get different predictions for the elapsed time (though the reasoning behind this conclusion was incorrect, since it was based on the assumption that the standard SR equation for time dilation still works in non-inertial coordinate systems, which is false).

 

[Al68]

It is not a legitimate inertial frame of reference in which you can use the standard SR equations for time dilation and so forth (it is impossible to find any valid inertial coordinate system to cover a large region of curved spacetime like in your example). However, the method that DaleSpam mentioned will work in any possible coordinate system, not just an inertial one--find the correct metric in that coordinate system, integrate it over the spacetime path in that coordinate system, and you'll get the elapsed time on a clock. This is a satisfactory resolution to the "paradox" because it gives the same answer for the proper time no matter which coordinate system you choose, and the supposed paradox was that different frames would get different predictions for the elapsed time (though the reasoning behind this conclusion was incorrect, since it was based on the assumption that the standard SR equation for time dilation still works in non-inertial coordinate systems, which is false).

Hi Jesse,

I agree that this method is a satisfactory resolution for the same reason you stated, it can be used for any frame, including the accelerated frame. This is untrue of the textbook resolutions I've seen.

It seems obvious to me that, from the ship's point of view during the acceleration, Earth's clock "runs" faster than his. We can split hairs and say that there is a shift in simultaneity for each of many sequential co-moving inertial frames, but if we take that in the limit, it amounts to the same thing. Why not just say that the acceleration causes Earth's clock to run faster than the ship's from the ship's point of view? Why is Einstein's own resolution the only one that makes this clear?

Thanks,
Al

 

[JesseM]

It seems obvious to me that, from the ship's point of view during the acceleration, Earth's clock "runs" faster than his.

That's only true if you use a specific type of coordinate system to represent the ship's "point of view", one where the definition of simultaneity at every point on the ship's worldline in the accelerating coordinate system is made to match up with the definition of simultaneity in the ship's instantaneous inertial rest frame at that moment. But unlike with inertial frames, there is no compelling physical reason to construct a non-inertial frame in any particular way, with any particular definition of simultaneity. You could choose all sorts of other non-inertial coordinate systems where the ship has a constant coordinate position throughout the trip and yet the Earth's clock doesn't run faster during the acceleration. When you use the GR method of calculating the elapsed time on each clock using the metric expressed in that coordinate system, all coordinate systems are on equal footing.

Why not just say that the acceleration causes Earth's clock to run faster than the ship's from the ship's point of view? Why is Einstein's own resolution the only one that makes this clear?

This isn't Einstein's resolution at all, he would agree that the ship does not have any single "point of view", that any coordinate system you choose to call the ship's point of view is fundamentally no better than any other.

 

[Al68]

That's only true if you use a specific type of coordinate system to represent the ship's "point of view", one where the definition of simultaneity at every point on the ship's worldline in the accelerating coordinate system is made to match up with the definition of simultaneity in the ship's instantaneous inertial rest frame at that moment. But unlike with inertial frames, there is no compelling physical reason to construct a non-inertial frame in any particular way, with any particular definition of simultaneity. You could choose all sorts of other non-inertial coordinate systems where the ship has a constant coordinate position throughout the trip and yet the Earth's clock doesn't run faster during the acceleration.

I don't quite see how, at least not one that makes any sense.

This isn't Einstein's resolution at all, he would agree that the ship does not have any single "point of view", that any coordinate system you choose to call the ship's point of view is fundamentally no better than any other.

Well, he chose that one for his resolution for some reason. And he just refers to it as the one in which the ship is "stationary".

Thanks,
Al

 

[Al68]

You could choose all sorts of other non-inertial coordinate systems where the ship has a constant coordinate position throughout the trip and yet the Earth's clock doesn't run faster during the acceleration.

I just noticed this part of your post. That would be one weird coordinate system.

Al

 

[JesseM]

I just noticed this part of your post. That would be one weird coordinate system.

Al

Huh? Your comments about the Earth-clock running very fast while the ship is accelerating only make sense if you are talking about a non-inertial coordinate system (since in any inertial system, clocks on Earth can't tick any faster than one second per second of coordinate time), and I presumed this would be a non-inertial coordinate system where the ship is at rest if you are calling it the ship's "point of view" (since in the context of inertial observers, we always use an observer's 'point of view' as a shorthand for his inertial rest frame). If that presumption was wrong, please explain what type of coordinate system you meant when you talked about the "point of view" of the ship.

Well, he chose that one for his resolution for some reason. And he just refers to it as the one in which the ship is "stationary".

I assume that means Einstein was talking about a non-inertial coordinate system where the ship was at rest throughout the journey, so I don't understand your puzzlement above. Anyway, what specific writing of Einstein are you referring to? Do you see anything in there that suggests he'd say this coordinate system is the only one that can be described as the ship's "point of view" (or words to that effect), rather than just using the coordinate system as a single example of analyzing the twin paradox from the perspective of a non-inertial coordinate system?

 

[Al68]

Huh? Your comments about the Earth-clock running very fast while the ship is accelerating only make sense if you are talking about a non-inertial coordinate system (since in any inertial system, clocks on Earth can't tick any faster than one second per second of coordinate time), and I presumed this would be a non-inertial coordinate system where the ship is at rest if you are calling it the ship's "point of view" (since in the context of inertial observers, we always use an observer's 'point of view' as a shorthand for his inertial rest frame). If that presumption was wrong, please explain what type of coordinate system you meant when you talked about the "point of view" of the ship.

I assume that means Einstein was talking about a non-inertial coordinate system where the ship was at rest throughout the journey, so I don't understand your puzzlement above. Anyway, what specific writing of Einstein are you referring to? Do you see anything in there that suggests he'd say this coordinate system is the only one that can be described as the ship's "point of view" (or words to that effect), rather than just using the coordinate system as a single example of analyzing the twin paradox from the perspective of a non-inertial coordinate system?

Well, the ship is considered to be in inertial motion for the journey except for the turnaround. I think I misinterpreted what you meant.

As far as the rest, what suggests to me that Einstein would consider a certain frame as the ship's point of view is that he defined it as the system in which the non-inertial clock is at rest.
I'm referring to Einstein's twins paradox resolution. Here's a link to an English translation: http://en.wikisource.org/wiki/Dialog_about_objections_against_the_theory_of_relativity