How much time passes on Earth?

[DaveC426913]

TL;DR

We sometimes talk about millions of years passing back on Earth during a relativistic journey but how is that possible without actually traveling 3 million light years?

The recent threads about relativity and the (ersatz) twin paradox got me flummoxed (again).

I referenced a story (A World Out of Time) where the man character came back to an Earth that was 3 million years older than when he left. The reason is that he journeyed to the galactic centre and back at very near light speed - the time dilation was such that only 150 years passed on board his ship - a factor of 20,000.

Question: The only way three million years could have passed on Earth is if he actually travelled 3 million light years (at 0.99999999875c), right? i.e. he would have had to loop around the core and back again, like, 57 times!

* there's mention of a trip around a black hole in the story but let's just ignore that for now and concentrate on a generic scenario

 

[PeroK]

TL;DR Summary: We sometimes talk about millions of years passing back on Earth during a relativistic journey but how is that possible without actually traveling 3 million light years?

The recent threads about relativity and the (ersatz) twin paradox got me flummoxed (again).

I referenced a story (A World Out of Time) where the man character came back to an Earth that was 3 million years older than when he left. The reason is that he journeyed to the galactic centre and back at very near light speed - the time dilation was such that only 150 years passed on board his ship - a factor of 20,000.

Question: The only way three million years could have passed on Earth is if he actually travelled 3 million light years (at 0.99999999875c), right? i.e. he would have had to loop around the core and back again, like, 57 times!

Yes.

* there's mention of a trip around a black hole in the story but let's just ignore that for now and concentrate on a generic scenario

There was a thread about this a while ago. There is a minimum stable circular orbit around a non-spinning black hole, where the time dilation is not that large. In order to come back a lot younger, you have find some clever tricks, although I think a factor of 20,000 is probably unrealistic.

 

[Ibix]

In flat spacetime yes, he'd have to travel 3 million years as measured by Earth. The black hole could have a material effect depending what he did with it.

 

[PeterDonis]

there's mention of a trip around a black hole in the story but let's just ignore that for now

If the ship is capable of rocket thrust for a long enough period, it could hover close enough to the hole's horizon to achieve the required time dilation factor. But "close enough" is extremely close: 1.0000000025 times the horizon radius for a time dilation factor of 20,000, which for a 3 million solar mass black hole means about 22 meters above the horizon for a hole whose horizon radius is 9 million kilometers. You would want to be extremely confident in your navigation.

 

[DaveC426913]

...about 22 meters above the horizon for a hole whose horizon radius is 9 million kilometers. You would want to be extremely confident in your navigation.

There would be detectable time dilation within the ship!

You could literally observe the twin paradox play out in real-time as your buddy walked* from the cockpit to the loo.

* fell. And died.

 

[DrGreg]

But "close enough" is extremely close: 1.0000000025 times the horizon radius for a time dilation factor of 20,000, which for a 3 million solar mass black hole means about 22 meters above the horizon for a hole whose horizon radius is 9 million kilometers.

At that distance the required acceleration would be enormous, too much for humans to bear.

For a large black hole and a short distance, we can approximate it using the Rindler horizon formula $c^2/22$, about $4 \times 10^{15} \text{ m/s}^2$.

 

[PAllen]

Depends what you mean by distance traveled. A rocket could never leave the solar system, yet when the traveler returns to earth, it is 3 million years ahead of when they left, while they are still alive. All you need to do is follow a circular loop e.g. between Jupiter and Saturn orbit, at sufficient speed.

 

[DaveC426913]

Depends what you mean by distance traveled. A rocket could never leave the solar system, yet when the traveler returns to earth, it is 3 million years ahead of when they left, while they are still alive. All you need to do is follow a circular loop e.g. between Jupiter and Saturn orbit, at sufficient speed.

Yes. And that Jupiter-Saturn loop would have to cover 3 million light years.

I pointed this out in the OP - a loop around the core and back is "only" 52,000ly, so you'd have to do that loop about 57 times.

 

[PAllen]

Yes. And that Jupiter-Saturn loop would have to cover 3 million light years.

I pointed this out in the OP - a loop around the core and back is "only" 52,000ly, so you'd have to do that loop about 57 times.

Yeah, but the core central BH complicates things. Also, you wouldn’t cover 3 million ly from the traveler perspective. If, for example, the traveler only aged 50 years, then an odometer used by the traveler would show slightly less than 50 ly traveled. In the case of a loop within the solar system, earth and traveler would agree on number of loops, but would wildly disagree on distance around each loop.

 

[jbriggs444]

odometer used by the traveler would show slightly less than 50 ly traveled

Measuring the movement of a hideously length-contracted and shape-shifting galaxy against his accelerating, but always at rest self.