Is energy conserved in general relativity?

[Marc76Law]

I've been reading about the possibility of time travel by accelerating one end of the wormhole and then returning it to its original position giving you a wormhole with an entrance in the present and exit in the past.

If time travel using wormholes as described above is possible, can anyone explain whether this is a paradox or an error in my thinking.

Hypo: A wormhole is created that is equal in diameter to the Earth and the wormhole exit is accelerated so that it is 24 hours in the past. If such a wormhole were dropped into the Earth's orbit, then every 24 hours the Earth would pass through the wormhole with the effect being that the Earth never aged because with at the end of every revolution around the sun, the Earth would age 24 hours, fall into the wormhole, and become 24 hours younger again, ad infinitum so that the Earth would cease to age.

If I go to the moon and watch as the 24 hour time-travel wormhole is dropped into Earth's orbit, then the Earth would cease to age yet the Sun would be unaffected and continue to age. What happens billions of years later when the sun starts to expand as it becomes a red giant.

If no time passed for Earth, then how could the Earth be destroyed by the future sun yet if an observer is watching from the moon and the Sun isn't going through a similar worm-hole, then it would age, and would expand, and would destroy the Earth. What does an observer on the moon actually see happen to the Earth in the future?

Is this a paradox or is my understanding flawed?

 

[JesseM]

I've been reading about the possibility of time travel by accelerating one end of the wormhole and then returning it to its original position giving you a wormhole with an entrance in the present and exit in the past.

If time travel using wormholes as described above is possible, can anyone explain whether this is a paradox or an error in my thinking.

Hypo: A wormhole is created that is equal in diameter to the Earth and the wormhole exit is accelerated so that it is 24 hours in the past. If such a wormhole were dropped into the Earth's orbit, then every 24 hours the Earth would pass through the wormhole with the effect

The Earth's orbit takes 1 year to complete, not 24 hours. 24 hours is the time it takes the Earth to perform one rotation on its own axis.

being that the Earth never aged because with at the end of every revolution around the sun, the Earth would age 24 hours, fall into the wormhole, and become 24 hours younger again, ad infinitum so that the Earth would cease to age.

Nothing "becomes younger" by traveling through a wormhole, your proper time (time as measured by your own clock) continues to tick forward before, during and after going through the wormhole, it's only the external universe that could appear to get younger from your perspective. In your example, if the wormhole was placed at a set point in Earth's orbit so the Earth passed through it once every year according to its own clock, then the Earth would see other objects in the solar system like the Sun and Mars cease to age (i.e. Earth would see them age normally by one year as it orbited outside the wormhole, but then immediately after traveling through it would see them as 1 year younger than they were right before it entered the wormhole).

If I go to the moon and watch as the 24 hour time-travel wormhole is dropped into Earth's orbit, then the Earth would cease to age yet the Sun would be unaffected and continue to age. What happens billions of years later when the sun starts to expand as it becomes a red giant.

If the Earth keeps doubling back to the same date and position then there's going to be a problem with different versions of the Earth crashing into each other. I suppose we could arrange the wormhole so that the position in the Earth's orbit where it emerges is not exactly the same as the position as it was 1 year ago, but is instead slightly offset. So then you might place the wormhole slightly behind the Earth's position in its orbit in 2006, and you might immediately see another Earth appear out of the wormhole with its calendar date reading 2007 (and whose inhabitants remember just making their first trip through the wormhole), trailing just behind the Earth that's never been through the wormhole and whose calendar date reads 2006...then trailing the 2007 Earth would be a 2008 Earth, trailing the 2008 Earth would be a 2009 Earth, and so on, from your point of view they'd all emerge from the wormhole one after another in 2006, then one year later when from your point of view it was 2007, you'd see them all enter the other mouth of the wormhole in succession and disappear. Of course if this went on indefinitely you'd still have the problem of multiple Earths crashing into each other in 2006, so eventually in 2007 you might move the wormhole out of the way so that the last of the Earths in the series that exited in 2006 (say, an Earth which exited in 2006 with its own date reading 2050, and whose date now reads 2051 as it's on the verge of entering the wormhole again) does not reenter the wormhole but instead continues to orbit normally. In this case, from your point of view, what you've seen is that before 2006 the Earth's orbit contained only one Earth, then from 2006-2007 it contained a whole bunch of Earths that kept looping back to 2006, then in 2007 they all disappeared into the wormhole save one (the Earth whose date now reads 2051) which continues to orbit normally now that the wormhole has been moved aside.

 

[Marc76Law]

Thanks for the correction with respect to rotation versus revolution. I feel stupid making such a basic error.

Nothing "becomes younger" by traveling through a wormhole, your proper time (time as measured by your own clock) continues to tick forward before, during and after going through the wormhole, it's only the external universe that could appear to get younger from your perspective. In your example, if the wormhole was placed at a set point in Earth's orbit so the Earth passed through it once every year according to its own clock, then the Earth would see other objects in the solar system like the Sun and Mars cease to age (i.e. Earth would see them age normally by one year as it orbited outside the wormhole, but then immediately after traveling through it would see them as 1 year younger than they were right before it entered the wormhole).

I understand the part about the Earth not getting younger, just the external universe and follow what you're saying about the external universe appearing to age 1 year and then reverting to one year in the past, but how does the account for a completely external object like the Sun. Would the Earth going through the wormhole each year ever experience the sun expanding into a red giant?

I don't understand the mechanism by which a variety of Earth's come into being and could crash into each other. I've read about the theory of a multiverse with respect to time-travel and if that's what your explaining, it was my understanding that each Earth would be in a parallel universe, not all in one universe.

 

[JesseM]

I understand the part about the Earth not getting younger, just the external universe and follow what you're saying about the external universe appearing to age 1 year and then reverting to one year in the past, but how does the account for a completely external object like the Sun. Would the Earth going through the wormhole each year ever experience the sun expanding into a red giant?

Not if it could keep doubling back to 2006 without multiple Earths crashing into each other; in that case people on Earth would only ever see the Sun as it was in 2006-2007 (according to a calendar in the external universe).

I don't understand the mechanism by which a variety of Earth's come into being and could crash into each other. I've read about the theory of a multiverse with respect to time-travel and if that's what your explaining, it was my understanding that each Earth would be in a parallel universe, not all in one universe.

No, nothing about multiple universes, just lots of Earths at different ages. Instead of Earth, suppose we talk about human travelers--suppose you have just turned 29 on Jan. 1 2006 and make a plan to wait one year until you're 30 on Jan. 1 2007, then step through the wormhole back to Jan. 1 2006, then live out your 30th year in the 2006-2007 period, then step through the wormhole again on Jan. 1 2007 and return to Jan. 1 2006, and live out your 31st year in the 2006-2007 period as well, and not make any further trips back in time. In this case, from my point of view as an external observer, on Jan. 1 2006 I'll see two older versions of you emerge from the wormhole, one aged 30 and one aged 31, who will join the 29 year old you who has yet to travel through the wormhole. The three versions of you will live their lives and age normally throughout 2006, and then on Jan. 1 2007 the youngest version of you (who was 29 in 2006 and is now 30) will disappear into the wormhole (becoming the 30-year-old I saw emerging on Jan. 1 2006), as will the middle version of you (who was 30 when he appeared in 2006 and is now 31, and who will become the 31-year-old I saw emerging from the wormhole in 2006), while the oldest version (who is now 32) is done with his travels and won't go back. So because you doubled back to the same date twice, during the 2006-2007 period there were three of you coexisting; if you had doubled back 50 times on the same dates, then there'd have been 51 versions of you at different ages coexisting in the 2006-2007 period.

The same principle would apply to the Earth, if the Earth kept going through the wormhole many times then there'd be many Earths at different ages coexisting in the same physical orbit during a one-year time span, and the Earth's orbit only has a finite amount of room so you probably couldn't have billions of differently-aged Earths occupying the same orbit during a single year. And if only a smaller number emerge during that period, there must be some reason that the oldest version of the Earth isn't going to travel back through the wormhole in a year, either because the wormhole is moved out of the way at the point it's about to enter, or because it's trying to exit too close to the position another younger Earth is trying to exit and crashes into it or has its path disrupted gravitationally (though the mere fact that this oldest Earth did exit the wormhole when it was one year younger implies that the younger Earth doesn't get too badly damaged/disrupted by the encounter to continue traveling through the wormhole for some more rounds).

 

[Marc76Law]

Now I'm starting to understand you're explanation with the example you gave of different versions of me. However, how does this explanation not violate the law of the conservation of matter.

If matter can neither be created nor destroyed, where does the matter or mass come from to create the multiple Earths or multiple persons.

 

[JesseM]

Now I'm starting to understand you're explanation with the example you gave of different versions of me. However, how does this explanation not violate the law of the conservation of matter.

If matter can neither be created nor destroyed, where does the matter or mass come from to create the multiple Earths or multiple persons.

Well, conservation of matter/energy gets complicated in GR, as I understand it the conservation law still applies "locally" in region of spacetime that's small enough so that spacetime curvature can be neglected (see the http://www.aei.mpg.de/einsteinOnline/en/spotlights/equivalence_principle/index.html for some info on how GR 'locally' reduces to the same laws as in SR), but in larger regions it's more ambiguous how to define "energy"...see this article for more:

Is Energy Conserved in General Relativity?

In the specific case of wormholes it may be that since the spacetime is "asymptotically flat" (meaning the curvature can be treated as approaching zero as you get far from the wormhole) some of the energy conservation problems above might not apply. If you look at post #6 by "pervect" on this thread he says that the mass of the wormhole mouth would actually change when an object enters or exits it in a way that would allow overall mass to be conserved...one problem with this is that if enough objects exit a given mouth the mouth's mass will actually have to become negative to compensate, but traversable wormholes already require the assumption of negative energy/mass to keep them from collapsing, so perhaps this is no more implausible than the idea of traversable wormholes themselves.