Modification of twin "paradox" with a wormhole.

[md65536]

Consider a stationary Earth, with one end of a wormhole fixed nearby, and the other end fixed one light year away. The wormhole is a shortcut of negligible length connecting pairs of events, one at (0, t) and one at (1 LY, t). There are two twins, one on Earth and one in a rocket that travels at a relativistic speed relative to Earth (say 0.6 c or choose a convenient number). Who ages more in these 4 scenarios?:

1. The rocket leaves Earth, travels to and enters the far end of the wormhole, and ends up back at Earth, having been inertial the whole time.

2. The rocket leaves Earth through the near end, exits at the far end, and travels back to Earth, inertial the whole time. (How do these appear different, in terms of Doppler effect? Is there something here that you would say is "equivalent to acceleration" of the basic twin paradox?)

3. Suppose the same as (2), except that when the rocket exits the far end of the wormhole, it's moving in the opposite direction, and turns around (in negligible time) before travelling back to Earth. (How is this different from (2) in terms of measurements and/or what an observer sees?)

4. The rocket leaves Earth, travels to the far end of the wormhole, but it is the Earth that goes through the wormhole, and meets the rocket at the far end. (Is there an "equivalent to acceleration" here?)

 

I have guesses and explanations I can post later, but like before I'm curious about other people's answers and how intuitive relativity is.

[Kino]

What's the metric of your wormhole? If you don't know that then the answer is "not enough information".

[md65536]

8 hours ago, Kino said:

What's the metric of your wormhole? If you don't know that then the answer is "not enough information".

Is it not possible to describe a simple bridge between the two locations, assuming the simplest configuration without additional complications if they're not specified? Eg. no gravitational properties given, so assume no additional gravitational effects? No discontinuity given so assume it's continuous? The proper time needed to traverse the wormhole is negligible because its length is specified as negligible, right? This of course is a mathematical problem and not a physically possible experiment or anything.

What other information is needed? Is it a case where there are eg. gravitational effects that contribute to the answer and can change it by a lot, or is it that there are multiple possible wormholes that equally fit the inadequate description above, but have opposite answers to the question?

[md65536]

10 hours ago, Kino said:

What's the metric of your wormhole? If you don't know that then the answer is "not enough information".

Does ds^2 = 0 make sense as a metric, describing a wormhole with negligible interior length and no other specifications? Or is it the external length (which I doubt because there's no requirement of external connections for general wormholes), something like [math]ds^2 = -c^2 dt^2 + dx^2 = 0 + (1 LY)^2[/math]?

 

I think that enough information is given, with the assumption that we start with the basic twin paradox and nothing more (so assume flat spacetime, no gravity etc.), then add just the wormhole information given (is that even possible while keeping external spacetime flat?). Wormhole traversal events are described; an object enters one location at time t and exits at the other 1 LY away at time t in Earth's frame. All other information can be determined because there is also the 1LY external connection, I think. Any other complications could be added if you want, and you get a different answer, but the same it true for the basic twin paradox setup. Have I oversimplified things to the point of nonsense?

Edited April 20, 2021 by md65536

[Ghideon]

1 hour ago, md65536 said:

Does ds^2 = 0 make sense as a metric, describing a wormhole with negligible interior length and no other specifications? Or is it the external length (which I doubt because there's no requirement of external connections for general wormholes), something like ds2=−c2dt2+dx2=0+(1LY)2 ?

I find your question interesting! I have not done the math to check this (yet) but I get a feeling that the wormhole introduces a scenario that special relativity does not adress. Does the formulas used to resolve the twin "paradox" apply when faster than light travel is possible? Can an observer, stationary relative to the earth, apply SR to calculate the time experienced by the twin travelling through the wormhole?  

My reply is a quick intuitive answer and there may be more to your question than I realise at this time.

Edited April 20, 2021 by Ghideon
spelling

[md65536]

2 hours ago, Ghideon said:

I find your question interesting! I have not done the math to check this (yet) but I get a feeling that the wormhole introduces a scenario that special relativity does not adress. Does the formulas used to resolve the twin "paradox" apply when faster than light travel is possible? Can an observer, stationary relative to the earth, apply SR to calculate the time experienced by the twin travelling through the wormhole?  

My reply is a quick intuitive answer and there may be more to your question than I realise at this time.

I think it can be resolved with SR. Yes, the rules of SR still apply even if there are other aspects involved that it doesn't resolve, and even when there are aspects that modify the rules. Eg. if there's spacetime curvature involved, you don't throw out the results of SR, but neither does SR give you the complete answer. It's highly possible that my question is ill-posed! My intention was to describe the wormhole so that doesn't add aging effects that can't be described in the domain of SR.

SR can deal with faster than light particles, but shouldn't work for accelerating between faster and slower than light. However the wormhole is just a shortcut through spacetime. Basically it involves taking a shorter path between two events, shorter say than a straight path through flat spacetime that a beam of light might travel along, but it doesn't actually involve moving relative to something else at a speed higher than that of light. The situation above can break some physical laws, eg. it allows being in 2 places at once, but I think that on its own it's set up so that causal paradoxes aren't possible. The events here that an object can pass through "simultaneously" are outside of each other's light cones.

I guess "the time experienced by the twin travelling through the wormhole" would depend on the metric, and that would typically involve things outside of SR, but I tried to specify (not clearly?) that the proper time taken to traverse the wormhole is negligible. You should also be able to calculate some things based on being consistent with what some other observer sees. I may have missed something (ill-posed question) or reasoned wrong, but I came up with unambiguous answers that are consistent between observers. Yes, I'm curious about the intuitive answers! But more along the lines of intuitive reasoning, where you can convince yourself that you have an answer that the maths agree with, not just a guess. Also if you only have a guess, are there results from SR that can fill in the missing information?

[swansont]

11 hours ago, md65536 said:

I think it can be resolved with SR. Yes, the rules of SR still apply even if there are other aspects involved that it doesn't resolve, and even when there are aspects that modify the rules. Eg. if there's spacetime curvature involved, you don't throw out the results of SR, but neither does SR give you the complete answer. It's highly possible that my question is ill-posed! My intention was to describe the wormhole so that doesn't add aging effects that can't be described in the domain of SR.

But curvature means that your location has a time dilation effect, which aren't described by SR.

SR assumes a locally flat spacetime, which you don't seem to have here.

[md65536]

5 hours ago, swansont said:

But curvature means that your location has a time dilation effect, which aren't described by SR.

SR assumes a locally flat spacetime, which you don't seem to have here.

Can you have a multiply connected flat spacetime with a bridge between two locations? I didn't specify that, so one could add other curvature if they want, but with a flat spacetime, SR describes time dilation while outside the wormhole.

But traversing the wormhole itself involves a singularity? Because of that, the time dilation factor is indeterminate? So it doesn't matter if it takes a negligible time to traverse the wormhole, because another clock might elapse an indeterminate amount of time in that instant?

I think that SR is not needed to describe the time dilation that occurs while traversing the wormhole, because I've given that information in the specification of the wormhole. I could post what I calculated for one of the scenarios to see specifically where this reasoning might go wrong, but I was hoping someone would have their own answer to discuss first.

20 hours ago, Ghideon said:

My reply is a quick intuitive answer and there may be more to your question than I realise at this time.

What is your quick answer? Maybe we could discuss it and fill it in more, and perhaps end up in agreement or learn about something we're both missing!

[swansont]

10 minutes ago, md65536 said:

Can you have a multiply connected flat spacetime with a bridge between two locations?

Approximately, I would think. (Once you have curvature I think you have to get infinitely far away to have flat spacetime, but you can say it's "flat enough" at some point where the time dilation between two points is negligible.) Like being far from a star and then passing by it, you would go from flat to curved to flat, as a first-order approximation.

I think you would have to know the details of what curvature the wormhole introduces as others have suggested or implied.

[Ghideon]

29 minutes ago, md65536 said:

What is your quick answer? Maybe we could discuss it and fill it in more, and perhaps end up in agreement or learn about something we're both missing!

I meant that the answer I gave was quick, sorry for being unclear. But here is a quick one with an attempt at using the time dilation formula to illustrate my quick answer.

Time dilation formula: 

[math]t= \frac{t'}{ \sqrt{1- \frac{ v^{2} }{c^{2} } } } [/math]

Assume that an observer "A" , stationary relative to the earth, applies the above formula to calculate the time dilation and/or time experienced by the twin "B" travelling through the wormhole. From A's perspective B moves one lightyear in a very short amount of time (time much less than on year) and hence A uses a velocity v>c in the formula. The result contains the square root of a negative number since [math]1- \frac{ v^{2} }{c^{2} } < 0[/math] and hence SR fails in the scenario.

 

Edited April 21, 2021 by Ghideon
clarification

[md65536]

43 minutes ago, swansont said:

Approximately, I would think. (Once you have curvature I think you have to get infinitely far away to have flat spacetime, but you can say it's "flat enough" at some point where the time dilation between two points is negligible.) Like being far from a star and then passing by it, you would go from flat to curved to flat, as a first-order approximation.

I think you would have to know the details of what curvature the wormhole introduces as others have suggested or implied.

Oh, so any realistic wormhole would curve spacetime around the entrances, and with the Earth being nearby there would be gravitational time dilation for the entirety of the rocket's trips through space. I can see how that could change the answers of who ages more. I'd assumed no gravitational effects since none were specified. I effectively described an unrealistic wormhole, but I think it still works on paper.

 

31 minutes ago, Ghideon said:

From A's perspective B moves one lightyear in a very short amount of time (time much less than on year) and hence A uses a velocity v>c in the formula. The result contains the square root of a negative number since 1−v2c2<0  and hence SR fails in the scenario.

I didn't do it that way. You're using SR to try to calculate the time dilation while traversing the wormhole, but the description of the wormhole gives you that information. I'd say that B's velocity is constant relative to A the entire time, but there is a discontinuity in its location. However since location doesn't show up in that formula, it makes no difference. I don't think that B is actually travelling faster than anything by using the wormhole, it is merely travelling a shorter distance.

But I see what you mean, that was your answer! Yes, I agree it doesn't work to treat the travel through the wormhole as a really fast travel through the space between the mouths. It would have to be a path through a different part of spacetime.

 

Edited April 21, 2021 by md65536

[swansont]

3 minutes ago, md65536 said:

Oh, so any realistic wormhole would curve spacetime around the entrances, and with the Earth being nearby there would be gravitational time dilation for the entirety of the rocket's trips through space. I can see how that could change the answers of who ages more. I'd assumed no gravitational effects since none were specified. I effectively described an unrealistic wormhole, but I think it still works on paper.

Doesn't a wormhole have curvature throughout its extent? It would seem odd to me if it didn't (but this isn't in my wheelhouse, so I could be mistaken)

[Ghideon]

3 hours ago, md65536 said:

You're using SR to try to calculate the time dilation while traversing the wormhole, but the description of the wormhole gives you that information. I'd say that B's velocity is constant relative to A the entire time, but there is a discontinuity in its location.

Yes and no, I think. My intuitive approach was to not look at the wormhole and just treat B's travel as a "Black Box" according to A. From the point of view of observer A the equations of special relativity should predict the outcome regardless of how B managed to travel. Or in other words; A should be able to apply the equations of SR to the flat spacetime in A's frame of reference and get a valid result. Or leave SR and apply other formulas* but that, in my opinion, deviates too far from the twin paradox  But I am not sure my approach is a correct way to try to apply SR in this scenario you have created.

(Thanks by the way for an interesting and refreshing discussion!) 

*) probably General Relativity?

Edited April 21, 2021 by Ghideon

[md65536]

2 hours ago, Ghideon said:

My intuitive approach was to not look at the wormhole and just treat B's travel as a "Black Box" according to A. From the point of view of observer A the equations of special relativity should predict the outcome regardless of how B managed to travel. Or in other words; A should be able to apply the equations of SR to the flat spacetime in A's frame of reference and get a valid result. Or leave SR and apply other formulas* but that, in my opinion, deviates too far from the twin paradox  But I am not sure my approach is a correct way to try to apply SR in this scenario you have created.

That makes sense, to treat objects in the same inertial frame the same, regardless of what's in their black box. I think it would be easier to deal with proper time, because if you have the proper time of B between 2 points, then you really don't care what it does between those 2 points, or even if it remains inertial or has constant speed.

I think it doesn't work with coordinate time in this case. According to A, two different objects say B and B' that travel between the same two events, will age the same if they travel any path between the events at the same speed. By necessity, if they do that (same speed ending up at same place, regardless of what path they took to get there through a simply connected flat spacetime ie. no wormhole) then they'll have travelled the same odometer distance relative to A's inertial frame, for the same amount of time. But if B travels through the wormhole and B' travels through space, they can end up at the same location, with the same speed, and yet have travelled different distances. They couldn't travel different distances at the same speed and end up at the same event (same time).

The way I set up the wormhole, I'm assuming that B ages negligible proper time when traversing the wormhole. If B' makes the same trip through space at a speed faster than light relative to A, it should age a negative proper time I think.

5 hours ago, swansont said:

Doesn't a wormhole have curvature throughout its extent? It would seem odd to me if it didn't (but this isn't in my wheelhouse, so I could be mistaken)

I think it wouldn't matter, if it's possible to say "the proper time to traverse the wormhole is negligible", but I don't know if that's valid, eg. if what I described has a singularity. The relative time that passes outside the wormhole during the trip, is already specified. Effectively, if an object goes through the worm hole at Earth time t, proper time 0, they emerge at time t (in Earth's frame, aka. according to a local clock on the far end of the wormhole which is Einstein synchronized with Earth's clock) and proper time 0.

Basically, the shortest shortcut possible. That might be geometrically impossible for some reason I'm unaware of.

Edited April 21, 2021 by md65536

[md65536]

The calculations I get are that

Spoiler

the rocket twin ages less in every case.

The wormhole I described is apparently not realistic. But, the same calculations can be done by removing the worm hole and replacing it with 2 planets on each end, with synchronized clocks, and then calculating a one-way rocket trip. You're then comparing the rocket's proper time and the Earth's coordinate time so it's not the same as the twin paradox, but that's fine, SR lets you calculate that. The wormhole is set up to connect events that are simultaneous in the Earth's frame, and an object traversing it in an Earth-frame's instant is sort of like saying that two clocks at either end and synchronized in the Earth frame, represent the same clock. It doesn't matter who goes through the wormhole or what their inertial frame is, the wormhole represents an instant jump in the Earth frame, generally not in other frames.

I wrote out a bit of time dilation and Doppler analysis for each of the cases, but it ended up being more tedious than I expected. I could post that if anyone's interested.

As usual, if you take out any magic or mystery from SR "paradoxes" and look at it geometrically, it's straightforward.

[Halc]

22 hours ago, md65536 said:

The wormhole is set up to connect events that are simultaneous in the Earth's frame, and an object traversing it in an Earth-frame's instant is sort of like saying that two clocks at either end and synchronized in the Earth frame, represent the same clock. It doesn't matter who goes through the wormhole or what their inertial frame is, the wormhole represents an instant jump in the Earth frame, generally not in other frames.

You seem to be envisioning something like a stargate, a portal connecting the non-parallel worldlines of say two different planets.  If one thinks of it as such, there's no reason why it cannot look just like a window.  One can eat at a table that is half here and half there, and you can talk to the other person light years away as if they were in your presence, and can pass the salt and such.

The characteristics of such a setup depend on said unspecified metric. Imagine a clock at each end of the table.

If the clocks stay in sync despite the relative motion of ends of the wormhole, then one end can be accelerated as in the twin scenario, starting and ending in near proximity.  When the other end of the gate returns, the table is still there and both observers (neither of which have passed anything except the salt through the gate) will have aged the same, resulting in the wormhole/portal becoming a time portal instead of a spatial teleport setup.  In other words, the portal can be changed from connecting events with space-like separation to connecting events with time-like separation. Similarly, the stargate could be set up at constant separation but vastly different gravitational potentials.  If the clocks tick at the same rate, it quickly becomes a time machine.

If the two clocks do not stay in sync, the two parties eating at the table will have difficulty communicating since one will be 'faster' than the other.  The setup could be used to determine the absolute rest frame since the clock that objectively ticks faster must be moving slower. The principle of relativity would be violated.

 

[md65536]

17 hours ago, Halc said:

You seem to be envisioning something like a stargate, a portal connecting the non-parallel worldlines of say two different planets.  If one thinks of it as such, there's no reason why it cannot look just like a window.  One can eat at a table that is half here and half there, and you can talk to the other person light years away as if they were in your presence, and can pass the salt and such.

Yes, that sounds like what I'm describing.

Another way to describe it is with paper. If you draw out gridlines representing time and one spatial dimension on a flat sheet of paper, that can represent flat spacetime, and the rules of SR can be applied. If you bend the paper without stretching it, you're not distorting the relations between events along paths drawn on the paper. The rules of SR still apply. The wormhole represents bending the paper and making 2 points touch, and adding a zero-length path or connection between those 2 points.

17 hours ago, Halc said:

If the clocks stay in sync despite the relative motion of ends of the wormhole, then one end can be accelerated as in the twin scenario, starting and ending in near proximity.  When the other end of the gate returns, the table is still there and both observers (neither of which have passed anything except the salt through the gate) will have aged the same, resulting in the wormhole/portal becoming a time portal instead of a spatial teleport setup.  In other words, the portal can be changed from connecting events with space-like separation to connecting events with time-like separation. Similarly, the stargate could be set up at constant separation but vastly different gravitational potentials.  If the clocks tick at the same rate, it quickly becomes a time machine.

If the two clocks do not stay in sync, the two parties eating at the table will have difficulty communicating since one will be 'faster' than the other.  The setup could be used to determine the absolute rest frame since the clock that objectively ticks faster must be moving slower. The principle of relativity would be violated.

I specified that the wormhole mouths are at rest in the Earth frame. It sounds like if you added the ability to accelerate them, you'd have a choice of what happens, and would have to further specify that?

As specified, a clock from Earth that goes through the wormhole at vanishing speeds at time t, comes out at time t. So the clock would remain synchronized with Earth's clock, in Earth's frame. To be consistent in other frames, the clocks would have to be out of sync in other frames. Let's say there are 2 clocks at either end of the wormhole, sync'd in Earth's frame. For a rocket traveling away from Earth in the direction of the far mouth, the clock on Earth is behind the one at the far mouth. In this frame, a rocket that enters at Earth and exits "right now" at the far mouth, will not enter until a long time passes! For a rocket travelling in the opposite direction, the rocket that exits "now" has entered a long time ago.

None of this is a problem at all. With the wormhole mouths not accelerating or changing their configuration, the rockets can communicate with each other. If a message can be passed in the Earth frame, it can be passed in any frame (light cones are invariant). Also, if a rocket exits and can say either "I have not yet entered the other side" or "I just entered the other side" or "I entered long ago", it can also switch between those statements simply by accelerating to a different inertial frame where another statement becomes true.

It sounds like, by underspecifying the wormhole details, I've made it possible to add details that violate SR, but it's also possible to leave it so that no added details violate SR.

 

So for example, say a rocket leaves Earth at high speed at Earth-time t and goes through the wormhole, exiting at local wormhole time t. But the time on Earth, according to the rocket, might now be t-0.6 years. The rocket turns around in negligible time. Local wormhole time is t+epsilon, but the time on Earth is now say t+0.6 years. It enters the wormhole at t+epsilon and returns to Earth at t+epsilon. Neither twin has spent significant time moving relative to the other, and both have aged negligibly. (This isn't any of the 4 scenarios above, all of which involve the rocket making a long-duration one-way trip.)

Or you could say, the rocket that exits could see Earth at two different times, one across flat spacetime, say .6 years in the past or future but the image delayed by ... a little or a lot by the travel time of light, and one at time t seen nearby through the wormhole.

Edited April 24, 2021 by md65536

[Halc]

Apologies for slow reply, but I've been logged off for several days.

On 4/24/2021 at 1:34 PM, md65536 said:

Another way to describe it is with paper. If you draw out gridlines representing time and one spatial dimension on a flat sheet of paper, that can represent flat spacetime, and the rules of SR can be applied. If you bend the paper without stretching it, you're not distorting the relations between events along paths drawn on the paper. The rules of SR still apply. The wormhole represents bending the paper and making 2 points touch, and adding a zero-length path or connection between those 2 points.

I specified that the wormhole mouths are at rest in the Earth frame.

If the paper touches at only two points, that’s a one-shot teleport from one event to another. There is no gate held open, and thus the concept of the velocity and acceleration of the ‘gates’ does not apply since an event is a frame-independent fact.

So I think you mean rolling the paper so it intersects at a line following the time axis, thus forming a worldline of your stargate. Yes, you don’t need to distort the paper to do that. If you can create such a thing where the two gates are simultaneous in Earth frame, then you’ve created a time machine. All you need is a second set of gates that with the two ends simultaneous with the frame of some receding object. Go through the Earth gate, come back through the other one, and now it’s 1921.

Just pointing out that if you posit this sort of thing, then circular causality results. Didn’t need to accelerate either end of a gate to accomplish it.

It sounds like if you added the ability to accelerate them, you'd have a choice of what happens, and would have to further specify that?

That was the metric spoken of I think.

As specified, a clock from Earth that goes through the wormhole at vanishing speeds at time t, comes out at time t. So the clock would remain synchronized with Earth's clock, in Earth's frame. To be consistent in other frames, the clocks would have to be out of sync in other frames.

If I get you right, then conversation across our dining table with has one person on helium and the other on Xenon, each respectively from the perspective of the other. It would be like trying to talk with your twin on the fast ship. OK, no time machine there unless you build a 2^nd gate with the properties described above.

Why would the gate time be synced with one end (Earth) and not the other? Wormholes don’t seem to have a preferred end like that.

Let's say there are 2 clocks at either end of the wormhole, sync'd in Earth's frame.

I assume they’re relatively stationary then, else they could not be in sync. OK, they could, but only in a frame where the two endpoints have equal and opposite velocity.

For a rocket traveling away from Earth in the direction of the far mouth, the clock on Earth is behind the one at the far mouth. In this frame, a rocket that enters at Earth and exits "right now" at the far mouth, will not enter until a long time passes! For a rocket travelling in the opposite direction, the rocket that exits "now" has entered a long time ago.

Ouch! That kind of blew my frame reference clutch.

Let me see if I get it. The gate ends are in sync in Earth frame, both say noon, 3.464 light hours apart. The ship moves at .866c and gets there in 4 hours (4PM), but subjectively only 2 hours pass on the ship. If he leaves at ‘noon’ (synced with Earth clock at departure), then the destination clock moves from 3:00 to 4:00 during the two hour trip the clock makes moving to the ship, so yes, the Earth clock (noon) is 3 hours behind the destination clock that reads 3:00. When the destination clock passes him at 2PM, the Earth clock simultaneously reads 1PM.

If someone enters the gate at noon, he’ll appear at the destination at noon (according to both local clock and his watch). Not sure how that can be construed as not entering until a long time passes. Ok, he not only steps through the gate, but does so at .866c. In that frame, he exits the gate at noon, and the receding Earth clock currently reads 9AM, which is 3 hours behind. So I read you so far. He’ll not enter that end of the gate until 6 hours from now when it’s noon over there.

None of this is a problem at all.

You are opening a portal between events that are separated in a space-like manner. Nothing violated by that (except locality) until you open a second one synced in a different frame. The main reason it isn’t a big problem yet is because you’re avoiding accelerating the ends, which is why I attempted to explore that part.

It sounds like, by underspecifying the wormhole details, I've made it possible to add details that violate SR, but it's also possible to leave it so that no added details violate SR.

I think SR has problems with wormholes in the first place. There’s just no such concept in Minkowskian spacetime. You have to jump to GR for that, and SR is suddenly inapplicable anywhere and thus cannot be violated.

So for example, say a rocket leaves Earth at high speed at Earth-time t and goes through the wormhole, exiting at local wormhole time t. But the time on Earth, according to the rocket, might now be t-0.6 years. The rocket turns around in negligible time. Local wormhole time is t+epsilon, but the time on Earth is now say t+0.6 years.

The acceleration of the rocket doesn’t have any affect on what time it is back on Earth. You make it sound like a causal relationship when in fact it is just a changed choice of coordinate system, which can be done without actually accelerating.

[md65536]

On 4/28/2021 at 4:22 PM, Halc said:

So I think you mean rolling the paper so it intersects at a line following the time axis, thus forming a worldline of your stargate. Yes, you don’t need to distort the paper to do that. If you can create such a thing where the two gates are simultaneous in Earth frame, then you’ve created a time machine. All you need is a second set of gates that with the two ends simultaneous with the frame of some receding object. Go through the Earth gate, come back through the other one, and now it’s 1921.

Just pointing out that if you posit this sort of thing, then circular causality results. Didn’t need to accelerate either end of a gate to accomplish it.

[...]

You are opening a portal between events that are separated in a space-like manner. Nothing violated by that (except locality) until you open a second one synced in a different frame.

Yes, that's what I should have said, thanks for the correction. I was mixing up having a space and time coordinates drawn as a grid on the paper, and then mistakenly thinking of the paper as a space that exists in time. It should be the two worldlines of the respective ends of the wormhole, joined and then effectively sewed together.

As you hint at, and I argued in another thread, it wouldn't be enough to have two wormholes, with one going backward in time, to allow causal loops. The mouths would also have to be close enough together. There are no closed time-like curves if the curve is space-like. For example, if you go through a wormhole that brings you 100 years back in time, but also puts you 200 light years away, there is nothing at the entrance that you can affect before you leave. Or, if you travel far away "instantly" using one wormhole, and then back 100 year and ending up near Earth with another, but you have to travel 200 light years to get from one wormhole to the other, you also can't affect anything from before you leave.

On 4/28/2021 at 4:22 PM, Halc said:

Why would the gate time be synced with one end (Earth) and not the other? Wormholes don’t seem to have a preferred end like that.

I didn't mean they wouldn't be. The clock goes through at vanishing speeds. I was calling it "the Earth frame" but they share that frame.

On 4/28/2021 at 4:22 PM, Halc said:

Not sure how that can be construed as not entering until a long time passes. Ok, he not only steps through the gate, but does so at .866c. In that frame, he exits the gate at noon, and the receding Earth clock currently reads 9AM, which is 3 hours behind. So I read you so far. He’ll not enter that end of the gate until 6 hours from now when it’s noon over there.

Yes, that sounds like what I was thinking of. When I wrote "long time" I had the Doppler effect on my mind. But really, only relativity of simultaneity (3 hours) and time dilation (gamma=2) contribute to the long time, so let's say "it's a stretch" to call it a long time. The delay of light due to the (length-contracted) distance to Earth as well as the (not length-contracted) increasing distance of the Earth between the rocket's now and 6 hours later when it enters the wormhole at the other end, contribute to the much longer time until the rocket sees the Doppler-shifted image of itself entering the wormhole.

On 4/28/2021 at 4:22 PM, Halc said:

You are opening a portal between events that are separated in a space-like manner. Nothing violated by that (except locality) until you open a second one synced in a different frame. The main reason it isn’t a big problem yet is because you’re avoiding accelerating the ends, which is why I attempted to explore that part.

I wasn't thinking that far, I was only trying to describe the simplest case I could. But I guess you'd have to decide if the wormhole remains set up for the Earth frame, or if it behaves as an object that changes to a new inertial frame. In the paper analogy, eg. if you have 2 joined worldlines and you move them separately across the paper, do they stay sewed together like they originally were, possibly stretching and distorting? Or do they slide across each other as you skew the paper, and end up like they were set up for their new reference frame as much as they were for the old one? I have no thoughts on that, but I think that since this wormhole is not based on anything real or theoretical, you'd have to make it up.

On 4/28/2021 at 4:22 PM, Halc said:

The acceleration of the rocket doesn’t have any affect on what time it is back on Earth. You make it sound like a causal relationship when in fact it is just a changed choice of coordinate system, which can be done without actually accelerating.

Yes, I agree completely. I was only talking about "according to the rocket", meaning the coordinate time of the distant Earth in the rocket's standard (Minkowski?) coordinates.

The whole point is the acceleration doesn't have a causal effect, because the change in coordinate time on Earth when the rocket accelerates also corresponds with a change in relative simultaneity between the mouths of this wormhole. So for example, if two twins leave Earth and travel together "the long way" to the far end of the wormhole, and only one turns around (thereby having the distant Earth's time in their coordinates "jump ahead"), and they enter the wormhole roughly at the same time, they exit at roughly the same time too (travelling in opposite directions here). As in your example, they both age 2 hours while Earth aged 4, despite only one experiencing a change in relative simultaneity in its coordinates.