[Zer0 (OP)]

If the Identical Twins are replaced with Two Individual Atoms.
(Preferrably Hydrogen)

Both closely video recorded.

One is kept on Earth.
Other sent on the Space Trip.

After whatever Earth years (x) the Other arrives back Younger.(x/2)

Say if it was for a year...
First Atom on Earth would have a year old long length video recorded.
(x = 1yr)

But will Second Atom only have (x/2 = 1yr/2 = 6months) of Video length recorded?
No Right.
Coz for Second Atom, time never slowed down or things were never in slow motion.
Instead, from the Second Atom's perspective, it could be the First Atom on Earth whose time speed up.

(1) Right?

If both videos are same length.
If shown to an individual who has no idea about the Experiment.

(2) Would they be able to make out or tell a difference?

(3) Is it the Motion, or Rate of Change of Motion(Acceleration) that makes the real dilating difference?

(4) If there ever was such a way of closely monitoring & observing an Atom in different inertial states, would one really see any observable differences inside the inner workings of the Atom?

Short & Simple.
🍭

[Zer0 (OP)]

So eventually...

Why is Time Dilating then?

If the Earth Twin eats 100 mangoes in 1 Earth Year

The Space Twin would have eaten 100 mangoes in it's 1 Space Year Journey.

If both video recordings are Compared, there would be 100 mangoes in each.

So like the Space Twin's video would be Shorter?
6months?

So then if that SpaceTwin video is played at 0.5x speed(half) then it would resemble Earth Twin's video length?

But then that would show SpaceTwin in Slow Motion Right?

Jeez!
🥴
Mind goes Mangoes understanding all this...wat do u intelligent folks eat?

ps - does anyone feel Utube songs are sloww n play them at 1.25x
Dis has nothin 2 do wit da pond n everythin 2 do wit a DiscoToad!
🤭
(Solly)

[Halc]

If the Identical Twins are replaced with Two Individual Atoms.
(Preferrably Hydrogen)

Both closely video recorded.

An atom is too small to make a video of it. Just saying, be careful of treating quantum size objects as classical things.  I think we can do what you're asking here, but stay aware of the pitfalls.

One is kept on Earth.  Other sent on the Space Trip.
After whatever Earth years (x) the Other arrives back Younger.(x/2)
Say if it was for a year...
First Atom on Earth would have a year old long length video recorded.
(x = 1yr)

But will Second Atom only have (x/2 = 1yr/2 = 6months) of Video length recorded?

The video can be of anything, yes. It will have recorded 6 months if the trip was fast enough for a 2x dilation factor. It matters not a hoot what the video was recording.

Coz for Second Atom, time never slowed down or things were never in slow motion.

Time never slows down for a human either. Any occupant at any speed will not notice anything abnormal. The trip simply consumes 6 months of normal time for the 2nd atom/video/person.

Instead, from the Second Atom's perspective, it could be the First Atom on Earth whose time speed up.

No. From any one inertial frame, it is the moving thing that slows down. So for the outbound traveler, the home guy ages half as fast. But the traveler is never stationary in any one given reference frame, so in no frame does he always age faster.

If both videos are same length.

They're not if the video equipment is moving with the thing it is recording. If the camera on atom 2 stays home and just uses a long lens, then it will show the recorded thing slow way down on the way out and be speedy on the way back, mostly due to Doppler shift. Not sure what an unchanging object looks like slowing down, but that's another problem. An atom doesn't really age unless it's by radioactive decay or something, which is a one-time event, not a process.

If shown to an individual who has no idea about the Experiment.
(2) Would they be able to make out or tell a difference?

No. They'd say each was a long boring video of an atom, which is unchanging in both recordings.

(3) Is it the Motion, or Rate of Change of Motion(Acceleration) that makes the real dilating difference?

Motion relative to an arbitrarily selected inertial frame is what makes the difference. Acceleration breaks the symmetry, but doesn't directly cause any dilation.

(4) If there ever was such a way of closely monitoring & observing an Atom in different inertial states, would one really see any observable differences inside the inner workings of the Atom?

OK, the caesium atom has these states that switch back and forth in the presence of controlled radiation. They use that to make accurate clocks. There would be no difference in that atom in the two cases. Any segment of either video would be indistinguishable, and mandated by the Galilean relativity: Physics works the same relative to any inertial reference frame.

Why is Time Dilating then?

Time isn't dilating. The worldline of path2 between a pair of events simply has a shorter temporal length than the worldline of path1, the less accelerated thing (Earth) begtween the same pair of events. Time itself is unaffected by this.

If the Earth Twin eats 100 mangoes in 1 Earth Year
The Space Twin would have eaten 100 mangoes in it's 1 Space Year Journey.

Space twin's journey is only 6 months, no? At that pace, he eats 50 mangos.

If both video recordings are Compared, there would be 100 mangoes in each.

Not if each person eats a mango each 3.6 days.

So like the Space Twin's video would be Shorter? 6months?

Yes

So then if that SpaceTwin video is played at 0.5x speed(half) then it would resemble Earth Twin's video length?

It would take a year to play at that pace, yes. Watching it, everything would appear in slow motion.

But then that would show SpaceTwin in Slow Motion Right?

Watching anybody's video at half speed shows them in slo-mo.

ps - does anyone feel Utube songs are sloww n play them at 1.25x

A few are, and need to be played fast, yes. It is done to prevent the videos from being removed as fast as it helps fool the bots that look for copyright violations. All such videos are illegal, else they'd have no reason to do that.

[evan_au]

If the Identical Twins are replaced with Two Individual Atoms. (Preferrably Hydrogen)

A hydrogen atom does not "age" in any obvious and consistent way.
- It can absorb a photon, but it will promptly re-emit one or more photons, and go back to the initial state.

However, if you started with two collections of (say) 1 million tritium atoms:
- These do decay at a known rate, in a non-reversible manner
- And so if you compared the number of remaining Tritium atoms when they came back together, you could estimate how much time had elapsed between them.
- You can estimate the number of remaining Tritium atoms by measuring the level of radioactivity (eg with a Geiger Counter or similar).

[Janus]

So eventually...

Why is Time Dilating then?

If the Earth Twin eats 100 mangoes in 1 Earth Year

The Space Twin would have eaten 100 mangoes in it's 1 Space Year Journey.

If both video recordings are Compared, there would be 100 mangoes in each.

So like the Space Twin's video would be Shorter?
6months?

So then if that SpaceTwin video is played at 0.5x speed(half) then it would resemble Earth Twin's video length?

But then that would show SpaceTwin in Slow Motion Right?

Jeez!
🥴
Mind goes Mangoes understanding all this...wat do u intelligent folks eat?

ps - does anyone feel Utube songs are sloww n play them at 1.25x
Dis has nothin 2 do wit da pond n everythin 2 do wit a DiscoToad!
🤭
(Solly)

Assume The space twin travels at 0.8 c both out and back.
The round trip takes 1 Earth year, meaning he got 0.4 ly from earth when he turned back ( as measured from the Earth)
Our earth observer watching him will see him eat mangoes 1/3 as fact as he himself does (a part of this is due to the time delay caused by the increasing distance between them.)*
After 1/2 year, the space twin reverse direction back towards Earth. however since this occurs when he is 0.4 light years from Earth, the Earth twin doesn't see this occur for another 0.4 years or 0.9 years after the space twin left. During this whole time, he sees the Space twin eating mangoes 1/3 as fast. Since he has eaten 90 mangoes by this time, he will have seen the Space twin eat 30 mangoes. Once he sees the space twin reverse direction, he will see him eating mangoes 3 times faster than himself.**
But by the time he actually sees the space twin eat 30 mangoes and reverse direction, the space twin has already traveled most of the way back to him, and arrives back at Earth just 1/10 of a year later (having been gone a total of 1 year) during this time the Earth time eats another 10 Mangoes, while watching the space twin eat 30 mangoes.
Thus he eats a total of 100 while the space twin eats just 60.

If we look at things from the space twin's perspective, we get this:
The space twin recedes at 0.8c from the Earth, during which time he sees the Earth twin eat mangoes 1/3 as fast. After he has eaten 30 mangoes he reverses direction to head back to Earth ( the fact that he has eaten 30 mangoes by the time he turns around is something that he and the Earth twin must agree on.)
during this time he sees the Earth twin eat 10 Mangoes.
Now here comes a major difference between him and the Earth twin. The Earth twin had to wait to see the effects of the Space twin turning around because that event took place 4/10 of a light year away. But for the space twin, his reversing direction happens right there, so he has no delay. He immediately sees The Earth twin start eating mangoes 3 times faster than himself.  He eats the same number of mangoes for the return trip as on the outbound trip, so he eats 30 more mangoes, for a total of 60. During the return leg he sees the Earth twin eat three times as much as himself, or 90 mangoes, plus the 10 he'd seen him eat during the outbound leg gives 100 for a total.  This is the same end result the Earth twin arrived at.
* what he sees is the combination of this increasing lag and time dilation. If he were to factor out the light delay, he would conclude that the space twin was eating mangoes at  rate 6/10 his rate.
** again a combination of a ever decreasing time lag and time dilation, factoring out the light delay has him concluding that the space twin once again is eating mangoes 6/10 as fast during the return leg.

[Zer0 (OP)]

Why is Time Dilating then?

Time isn't dilating. The worldline of path2 between a pair of events simply has a shorter temporal length than the worldline of path1, the less accelerated thing (Earth) begtween the same pair of events. Time itself is unaffected by this.

🙄

If the Earth Twin eats 100 mangoes in 1 Earth Year
The Space Twin would have eaten 100 mangoes in it's 1 Space Year Journey.

Space twin's journey is only 6 months, no? At that pace, he eats 50 mangos.

SpaceTwin's journey was 6 months for EarthTwin.
But from STs perspective it was a whole year Right?

If both video recordings are Compared, there would be 100 mangoes in each.

Not if each person eats a mango each 3.6 days.

hmm.

But then that would show SpaceTwin in Slow Motion Right?

Watching anybody's video at half speed shows them in slo-mo.

🤭

ps - does anyone feel Utube songs are sloww n play them at 1.25x

A few are, and need to be played fast, yes. It is done to prevent the videos from being removed as fast as it helps fool the bots that look for copyright violations. All such videos are illegal, else they'd have no reason to do that.

Ya, I've seen movies in reverse lol.
I was just referring to a silly habit of mine.
The songs aren't illegal.
Just playing a normal song at 1.25x speed on Utube somehow adds to the Disco effects.

[Zer0 (OP)]

Janus
🙂

What if Identical Twins were in Automated AI equipped Identical Space Ships with no windows.

The AI specifically designed with main objective to Not let any Movement/Motion be detected by either Twins.

They Synchronise Clocks n shut the door.

AI commences executes n performs task.
Experiment ends & doors open.

Twins observe their Clocks are out of sync.

So the obvious conclusion is the Twin who's clock shows less time has passed is the One who Travelled, Right?

The Other, who's clock shows more time has passed was Stationary?

[Eternal Student]

Hi.

The AI specifically designed with main objective to Not let any Movement/Motion be detected by either Twins.

    This is going to be a problem.  We don't know of any anti-gravity or anti-acceleration technology.   How is the AI going to prevent one of the twins experiencing an acceleration as they make the turn to come back home?

     I guess the AI supervisor can just knock each of the twins unconscious with a hammer and they they won't notice anything?   But then again, the AI can also interfere with their clocks while they are unconscious and since we're only talking about 6 months of actual difference in age, perhaps they won't notice anything what-so-ever.

   Best Wishes.

[Zer0 (OP)]

Hey Eternal
🙋

No no, let's not knock them.
Let's feed em a big fat dinner n sing lullabies.
😇

But Yes, you are Correct.
Humans would detect motion.
& One of them Has to be Stationary.

To notice a single micro second difference between synced atomic clocks would still need considerable amount of acceleration, Right?

What if they both moved in exact same direction n speed.
Or opposite direction but same speeds.
Could there be a way in which the AI could move them both in sync n stop them & then No Time Difference is noticeable?

ps - time is relative.
👍
Isn't that Incredible!

[Halc]

What if Identical Twins were in Automated AI equipped Identical Space Ships with no windows.

It isn't a function of what is seen outside the window, so no change from other descriptions.

Twins observe their Clocks are out of sync.
So the obvious conclusion is the Twin who's clock shows less time has passed is the One who Travelled, Right?

Right. The travelling twin's worldline diverged significantly from the fairly straight worldline of the first twin.

The Other, who's clock shows more time has passed was Stationary?

There is no absolute 'stationary', so at best, you can say that the home twin was nearby the location in space of both start and end events, and the travelling twin wasn't.

This is going to be a problem.  We don't know of any anti-gravity or anti-acceleration technology.
 How is the AI going to prevent one of the twins experiencing an acceleration as they make the turn to come back home?

Not a problem. Just have both of them accelerate at the same g the whole time, but the home one goes around in circles and the other accelerates out and back. The turning can be done gradual enough to not be detected without gyros and such. A good inertial guidance system would know which is which.

[Eternal Student]

To notice a single micro second difference between synced atomic clocks would still need considerable amount of acceleration, Right?

      In the conventional "twins paradox" situation of special relativity, the difference in elapsed time for the two twins is determined by these two things:
1.   The speed of travel.
2.   The distance travelled.

   The accelerations are assumed to be rapid (usually instantaneous) just for convenience in calculations.  However, there is no requirement for the accelerations to be high.    Provided you are prepared to take long enough journeys then even small accelerations will eventually get the spaceship to a high speed and then a noticeable difference in elapsed time for the two twins will be observed.

What if they both moved in exact same direction n speed.

    Then they follow the same world-line.  They see no differences between themselves and experience the same elapsed time.

Or opposite direction but same speeds.

     
     Then they don't ever get to the same place and so they can't compare time on their clocks easily.  The situation becomes un-interesting. 
     You may need to explain what you meant more clearly.  Did you mean each twin leaves earth (but in opposite directions) and each does an equivalent (but mirror image) trip through space, turns around and comes back to earth?
    If that's what you meant then each twin experiences the same elapsed time, so they remain the same age as each other.  However, they will both notice that more time had elapsed for people who remained on earth.

Could there be a way in which the AI could move them both in sync n stop them & then No Time Difference is noticeable?

   Yes, you've mentioned two possibilities already.   Move the twins together, or move them symmetrically but in different directions.

Best Wishes.

[Zer0 (OP)]

Thank You All for being Supportive & Helpful.
😊
🙏

Just to sum it up.

I stay on earth with a clock.
Other synced clock is sent in space.

While it's moving away from me at incredible speeds, clock seems little reddish in color & ticks slow.

While returning it's bluish & tickin faster.

When i compare my earth clock with the spacey clock...

They both would be in sync back again?
Coz they both are now in same location frame of reference?

Or Not.
Bcoz even thou Spacey clock travelled in positive direction(away) from me n then in negative direction(closer) to me...

Directions were reversed, but speed/momentum was always forward.
No Negative Speed Right!
Hence both clocks even thou being back at the same location, would still be ticking in their own respective frames of reference.

Or i still don't get it?
🤔

& Also, if I'm video recording that Spacey clock for whole one year from earth.

I'd have a rather interesting 1year long video of a clock that ticks slow pace 6months & then covers up for lost ticks by ticking at fast pace for 6months.

Then if a selfie video recorder was sent along with the Spacey clock, Will it record 6month long video?
Would the clock in the video tick at varying speeds? Like fast pace slow pace?

Whatever may be, that 6month video should show a clock ticking for a whole one year.
It has to.
I got video evidence on earth video recorder Right???

ps - Thanks All!
👍

[Eternal Student]

Hi Zero,

   You know the twins paradox is confusing and it is difficult to understand.   You seem to be doing what most humans beings are very good at:   You are looking for symmetry and finding things that seem to be symmetric.    This is a good strategy for most science problems, however in this one it is the apparent symmetry that causes the confusions and makes the whole thing look like a paradox.
     To resolve (break) the paradox we have to look very hard to find the asymmetry (the things that are not symmetric).   Most of us look back at the twins paradox several times, each time convinced that we have spotted something else that must be symmetric and forces the twins to experience the same elapsed journey time.  However, it should always be possible to break that symmetry or find something that was wrong with our reasoning.
    Let's have a quick look at some of the things you have just said:

While it's moving away from me at incredible speeds, clock seems little reddish in color & ticks slow.

    Yes this is true.

While returning it's bluish & tickin faster.

     This is only partially true.   
     The clock on the spaceship is actually still ticking at a slower rate than a clock on earth.   (Sorry).

    However, an observer on earth might "see" the spaceship clock ticking faster just because the spaceship is coming towards them fast.  After each second, the spaceship is much closer to the earth than it was,  so light from the spaceship doesn't take as long to travel to the earth.   Assuming the spaceship is travelling at 0.7c, then it does a good job of chasing after the light that would tell people on earth  "our clock shows  09:04:56 ".   When the spaceship finally releases light that will tell the people on earth "our clock shows 09:04:57 " it is only about 0.3 light-seconds behind the first light that was released and is travelling to earth.
     Consider doing the same thing with ordinary message delivery systems.  Suppose you live in Australia and I live in a little island near the North pole.  There is an ordinary postal service running between the two which uses a variety of links such as boats, aircraft and postal delivery vans to get a letter from you to me.  It takes the postal service about 48 hours to get a letter to me.  Over a week you write one letter every day that just says "Today is Monday", and then the next one you write on the next day and it says "Today is Tuesday"..... etc. etc.   Provided you don't move and I don't move, then your letters arrive at my house with about 24 hours between each one.   It's always old information as far as I'm concerned but at least the time between each letter is 24 hours.   Then something strange happens, I get a letter that says "Today is Wednesday" and also another letter that says "Today is Thursday".  I don't think "wow, a whole day passed for you in 0 time for me".  Instead I check the post marks on the envelopes and realise that you have been travelling from Australia toward the North pole and you have actually managed to chase the postal delivery service and catch it up.  You managed to write the second letter and get it on the same delivery airplane as the other letter.
   The spaceship is doing something similar.  It is chasing after the earlier light signals, it doesn't quite catch them up but it's not far behind them.

    There are many YouTube videos, articles and texts written about the twins paradox.  As I've said, most of us need to look at more than one explanation and return to examine the paradox several times before we are "happy" enough with it.

Best Wishes.

[Janus]

Thank You All for being Supportive & Helpful.
😊
🙏

Just to sum it up.

I stay on earth with a clock.
Other synced clock is sent in space.

While it's moving away from me at incredible speeds, clock seems little reddish in color & ticks slow.

While returning it's bluish & tickin faster.

When i compare my earth clock with the spacey clock...

They both would be in sync back again?
Coz they both are now in same location frame of reference?

They would tick at the same rate, but would no longer be in sync.

Or Not.
Bcoz even thou Spacey clock travelled in positive direction(away) from me n then in negative direction(closer) to me...

Directions were reversed, but speed/momentum was always forward.
No Negative Speed Right!
Hence both clocks even thou being back at the same location, would still be ticking in their own respective frames of reference.

Or i still don't get it?
🤔

that is not what frames of reference are. They are not "attached to object".  Location also doesn't matter.  Two objects can "be in the same reference frame" just as long as they are at rest with respect to each other. ("In the same reference frame is basically shorthand for "at rest with respect to the same reference frame"

& Also, if I'm video recording that Spacey clock for whole one year from earth.

I'd have a rather interesting 1year long video of a clock that ticks slow pace 6months & then covers up for lost ticks by ticking at fast pace for 6months.

Then if a selfie video recorder was sent along with the Spacey clock, Will it record 6month long video?
Would the clock in the video tick at varying speeds? Like fast pace slow pace?

Whatever may be, that 6month video should show a clock ticking for a whole one year.
It has to.
I got video evidence on earth video recorder Right???

ps - Thanks All!
👍

Go back my Mango post, the same argument holds. The Earth observer will not see the space clock ticking fast for 6 mo and slow for 6 mo.  With the space clock traveling at 0.8c, he would see it tick slow for 10.8 mo, and fast for 1.2 month.
Since the respective rates are 1/3 and 3, he would see it tick off 3.6 mo while ticking slow, and then another 3.6 mo while ticking fast, ticking off a total of 7.2 months in the one year the Earth observer is watching. Besides ticking slow for 6 mo and fast for 6 mo, would not "cancel each other out", ticking  1/3 at fast for 6 mo accumulates 2 mo, while ticking fast for 6 mo, accumulates 18 mo, for a total of 20 months.
Now, as I pointed out in the mango example, the spacey clock would see the earth clock tick slow for half the time and fast for the other half.   But the spacey clock only ticks off 3.6 mo during each leg of the trip. So, for during the first leg it sees the Earth clock tick off  3.6/3 =1.2 months and during the second leg it sees it tick off 3.6 x 3 = 10.8 months, for a total of 12 mo during the whole trip.

[Eternal Student]

Hi @Janus
   Your reply looks good.   Just check these numbers:

and fast for 1.8 month.
Since the respective rates are 1/3 and 3, he would see it tick off 3.6 mo while ticking slow, and then another 3.6 mo while ticking fast,

I think that  1.8   should be edited to  1.2,      everything else seems OK   3 x 1.2 = 3.6   etc.

[Janus]

Hi @Janus
   Your reply looks good.   Just check these numbers:

and fast for 1.8 month.
Since the respective rates are 1/3 and 3, he would see it tick off 3.6 mo while ticking slow, and then another 3.6 mo while ticking fast,

I think that  1.8   should be edited to  1.2,      everything else seems OK   3 x 1.2 = 3.6   etc.

Yep, thanks for catching that. I went back and edited it.

[Zer0 (OP)]

Hi Everyone!
😊

Thanks for playing along with words like ' Spacey ' & ' Mangoes ' ha ha.
Helped to understand it better.
👍
When you all select my quotes, things that i have said, n then confirm it by sayin ' Correct, Right n True ' that is reaffirming & quite fun.

I think i get it now.
I knew bout " Time Dilation " & the Twins Paradox before too.
I just didn't understand it before.

Just pondering over it a bit...

A clock the size of the Sun would tick slower in comparison to a clock the size of the Moon.
Right?
(Assuming they both are at Rest)

Coz not just direction & momentum, even gravity affects Time.

When 2 clocks are Synchronised...
Synchronization is making sure they both Tick at the exact same moment.
Right?

But does that truly Sync them both?
Like, those 2 clocks are at a different location in space-time to begin with.

Mmm..like say the Atoms that make up those Clocks, aren't they all out of sync with each other.

The Space-time continuum would have a different x,y,z,t for each & every atom.

Wouldn't one have to Merge both the clocks in such a way that they become 1 Clock.
(An indistinguishable copy of them both)
Sync?

Just one more lil thingy plzzz!

What if 2 Synced clocks at rest with each other, start getting separated as the Space between them starts expanding.

Would they remain in Sync?
🤔

There is no motion, no force, no gravitational field involved.

Cheers!
✌️

[Halc]

A clock the size of the Sun would tick slower in comparison to a clock the size of the Moon.

A clock the mass of the sun (in isolation) would tick slower than a lower mass clock, but that's the same as a small clock nearby the sun. The size of the clock doesn't matter. It being a clock in the presence of mass is what matters.

Coz not just direction & momentum, even gravity affects Time.

Correct. Gravitational potential to be exact, not the strength of the gravitational field.

Synchronization is making sure they both Tick at the exact same moment.

It means several different things. Two clocks in each other's presence are in sync if they display the same time, regardless of their respective tick rates.
Two clocks spatially separated from each other but 1) at the same gravitational potential and 2) with no relative motion can be synced relative to any one arbitrary inertial frame, which means that relative to that one frame, the two clocks read the same time. Relative to the other frames they don't read the same time and are thus not synced, but they still tick at the same rate.

But does that truly Sync them both?

It means they're synced to each other. It isn't meaningful for A to be synced to B and B not synced to A.

say the Atoms that make up those Clocks, aren't they all out of sync with each other.

Synced means they read the same time. It doesn't mean the two clocks are identical or even mechanically similar. You can sync a sand-hourglass with a fuse on a bomb if you like.

The Space-time continuum would have a different x,y,z,t for each & every atom.

Events have x,y.z,t. Atoms are not events. Atoms have a location at a given time (x,y,z) and trace a 1 dimensional line of events through spacetime since they exist for more than just one moment.

What if 2 Synced clocks at rest with each other, start getting separated as the Space between them starts expanding.

Excellent question.
If the clocks are in each other's presence, spatial expansion cannot separate them since you can multiply zero by any factor you like and still get zero.
If they are spatially separated and synced relative to some inertial frame, then expansion does not effect that inertial frame and the clocks will remain equidistant and synced relative to that one frame.

If on the other hand the two clocks both are stationary (zero peculiar velocity) relative to comoving (expanding) space, then they are stationary but increasing in proper separation due to the expansion.  Such clocks, relative to cosmological time, will remain in sync despite the increase in proper separation. More generally if two clocks are both moving at the same peculiar speed but in different directions, they can still be synced to the cosmological time and be running slow due to their motion, but they will remain in sync as they slow over time. They would not by synced relative to any inertial frame.

[Eternal Student]

Hi.

A clock the size of the Sun would tick slower in comparison to a clock the size of the Moon.
Right?

   Yes, provided you're careful.   You're talking about gravitational time dilation.   The size of an object (in metres by metres by metres) is less interesting than it's density   (Mass squashed into a small space).   You're also assuming the clock is on the surface rather then being the whole mass of the object.   If the clock was the entire moon then this has a varying gravitational field as you go under the surface and toward the centre, the gravitational field drops to 0 at the centre.  So most of the inner mechanisms of a moon sized clock would actually be in a very low strength gravitational field.

When 2 clocks are Synchronised...
Synchronization is making sure they both Tick at the exact same moment.
Right?
But does that truly Sync them both?
Like, those 2 clocks are at a different location in space-time to begin with.

   Well these are very good questions.   It's easy enough to synchronize clocks when they are in the same place.  Yes it does mean just making sure they tell the same time ("it's 09:05:01").   
    There is another thread that discusses the problems of synchronisation when two clocks are separated by some distance.   There is a generally accepted procedure for doing this and it assumes the speed of light is the same in every direction (it travels left-to-right  with the same speed as it would travel right-to-left).   The equivalence of the speed of light in any direction cannot be tested or proved, it must be assumed.
    Note that Synchronising clocks just means getting them to show the same initial time   (e.g.  "it's  00:00:00  right now" ) at what should be the same instant.   The clocks are assumed to be accurate and reliable  - they will genuinely record the correct amount of time that elapses (we use perfect clocks not cheap watches from the street-market etc.)
    The very last part of your comment could be much harder to answer,  if we are working with General Relativity then one of the clocks we want to synchronise might be in a region of curved space.  For example, we could put a clock inside the event horizon of a black hole.  There is no way to synchronise clocks across an event horizon as far as I know.  Let's just ignore this and assume we are synchronising clocks in standard flat space.

Mmm..like say the Atoms that make up those Clocks, aren't they all out of sync with each other.

   Maybe.  It doesn't matter too much.  We are treating "the clock"  as an idealised object.   For most of Special and General Relativity we don't really care if an actual real clock can be placed somewhere and what that clock might be made of.   Indeed the simplest clocks we can use are just things that bounce light up and down between two mirros and count each bounce as a tick of the clock.  These are not real objects, you can't buy a light clock at the local shop in the highstreet.      All we want is the concept of time for our models and we can assume that a perfect idealised clock exists with which we can measure the passage of time.

What if 2 Synced clocks at rest with each other, start getting separated as the Space between them starts expanding.

Would they remain in Sync?

    Then we need General Relativity to explain what might happen.  The spirit of your question is that the clocks are always co-moving  (they are co-moving with the co-ordinates of an expanding universe).  They were initially close together and that's why they seemed to be (almost) at rest with respect to each other.  As the universe expands the separation between them increases and the relative velocity between them increases.
    I will also assume that this universe has a metric that is typical of an expanding universe (a FLRW metric).   In this case co-moving objects do experience local time at the same rate (and quite conveniently this is also the rate of flow of the universe-wide co-ordinate time but that is just a bonus).     
    So, yes the clocks will stay synchronised.
    Technically, you stated the clocks were initially at rest and if there was a small distance between them then this would mean one of the clocks could not have been co-moving, it must have had a small peculiar velocity through space.  This is a minor adjustment and more complicated to explain then it's worth.  One clock would tick ever so slightly faster than the other clock to begin with but as time eveolves their rate of flow of time would become more and more closely matched until eventually they are perfectly matched.

Best Wishes.

(Looks like Halc has also replied, sorry for any overlap).

[Halc]

You're talking about gravitational time dilation.   The size of an object (in metres by metres by metres) is less interesting than it's density

Ah thanks. I didn't mention that, and it's important. A clock at the center of a solar mass crammed into the volume of the sun runs slower than one at the center of a solar mass expanded out to Earth orbit (like it will several billion years from now).

You're also assuming the clock is on the surface rather then being the whole mass of the object.   If the clock was the entire moon then this has a varying gravitational field as you go under the surface and toward the centre, the gravitational field drops to 0 at the centre.  So most of the inner mechanisms of a moon sized clock would actually be in a very low strength gravitational field.

Yes, but the lowest gravitational potential is there at the center, so the clock at the center of the moon runs slower than one on the surface. You seem to imply that it's a function of the gradient of potential, and not a direct function of the potential.

(we use perfect clocks not cheap watches from the street-market etc.)

But the street market watches all say 'Rolex' on them.

There is no way to synchronise clocks across an event horizon as far as I know.

Can be done, at least briefly. Just have to pick an appropriate coordinate system. You've said that I'm a stickler for coordinate systems. Syncing a pair of separated clocks is coordinate dependent, so it simply means picking one that works across the EH. Given the curvature at non-local scales, they won't stay in sync, so technically it is no different than saying that there is no way to synchronise clocks across a change in gravitational potential, which means even a pair of clocks on different floors of a building.

What if 2 Synced clocks at rest with each other, start getting separated as the Space between them starts expanding.

They were initially close together and that's why they seemed to be (almost) at rest with respect to each other.  As the universe expands the separation between them increases and the relative velocity between them increases.

Given linear expansion (no gravity or dark energy as specified in our simplified case), the rate of increase in proper separation between any two comoving objects remains constant forever. For it to increase, you need to add energy, and that's why we've excluded dark energy.  Besides the point. The clocks will remain in sync relative to the cosmological frame even with (uniform) gravity and dark energy, and even with nonzero (but identical magnitude) velocity relative to that frame.

Apologies for appearing to be nitpicky.