[Eternal Student]

Hi Halc,

    Gravitational potential    vs.  gravitational field strength   
You're right it's the potential that matters.    I mentioned the field strength because I was sloppy.  The key is that the potential will vary a little as you start to go under the surface.  It's just a needless mess to work out what might actually happen to the local time for the inner mechanism of the clock.

   Synchronising clocks:
    Well I was going to mention how it is usually done.    Put a man half-way between the two clocks and tell him to flash a light. etc.    Someone would have mentioned that it's harder if there is no path that light can take to reach one of the clocks. 
    Anyway, .... ok    change co-ordinates and say they are synchronised... but you know that no one else will call that  a "synchronisation procedure".

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.

   I agree with this, if we assume linear expansion means the scale factor, a(t) = k.t     (arbitrary constant k as usual).  It's taken me about half an hour to check this but I am old and slow.  Anyway, we aren't going to assume this sort of expansion because it isn't reasonable for the general audience.

   Most people have seen Hubble's law:
    Speed   =    H₀  .  D

   Now, in reality  H is dependant on time but it's not a terrible approximation to assume H ≈ H₀  over small ranges of time centred on right now.
    Then we have,

Separating variables we have,   
Hence, 

  So that the scale factor approximates exponential growth.   For this type of expansion, which is a reasonable approximation for the universe around the current time, the rate of change in separation distance of two co-moving objects is given by the Hubble law.  In particular, it will increase as the distance of separation increases.

Apologies for appearing to be nitpicky.

   It's always a pleasure speaking to you.  Thank you for your time, I would otherwise have just done nothing this evening.

Best Wishes.

[Halc]

It's just a needless mess to work out what might actually happen to the local time for the inner mechanism of the clock.

You could compute the speed of a rock dropped into a hole dug to the middle of the moon. I'm sure that's published somewhere. The rate difference between there and the clock at the surface can be directly worked out from that.

Synchronising clocks:
    Well I was going to mention how it is usually done.    Put a man half-way between the two clocks and tell him to flash a light. etc.    Someone would have mentioned that it's harder if there is no path that light can take to reach one of the clocks.

Works fine for our two clocks crossing the EH. Light easily reaches both clocks from the midpoint, regardless of which side of the EH that midpoint is.
Just remember that locally, spacetime is Minkowskian, and spacetime near the EH is no exception to that. It isn't a physical singularity.

Anyway, .... ok    change co-ordinates and say they are synchronised... but you know that no one else will call that  a "synchronisation procedure".

I do. It was a pretty trivial procedure.

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.

... Anyway, we aren't going to assume this sort of expansion because it isn't reasonable for the general audience.

But it is reasonable since expansion is almost linear now. Dark energy has barely started to make headway against gravity.

Most people have seen Hubble's law:
    Speed   =    H₀  .  D

   Now, in reality  H is dependant on time but it's not a terrible approximation to assume H ≈ H₀  over small ranges of time centred on right now.

It is a terrible approximation if you do the integration you do here. You're assuming the constant is flat instead of a much better approximation of H=1/<age-of-universe> which it almost exactly is.

So that the scale factor approximates exponential growth.

While it will one day be exponential like that, the curve is almost exactly linear right now.
I do agree that long term, two objects with no change in proper separation will begin to drift apart. But really long term. Two clocks a light year apart will take millions of years to be measurable separated more than that, whereas it might take but thousands with the exponential expansion that you assume. Hubble's constant is anything but a constant, despite the fact that it is projected to someday approach some constant value.

It's always a pleasure speaking to you.  Thank you for your time, I would otherwise have just done nothing this evening.

Oh good. I always picture myself as the old cynic. I've even been occasionally complimented on it such as "You're my kind of cynic".

[Eternal Student]

Hi again,

About the procedure for Synchronizing clocks  ..... Halc said:

Works fine for our two clocks crossing the EH.

   No, I'm not sure it will work, Halc.   It works OK if the clocks and the man in middle with the flash light can all be at rest in the same inertial frame (free-falling into the Black Hole).     However, the space is only Minkowski flat space locally.   For a small black hole in particular, these local inertial frames become significantly non-inertial with only the smallest step of distance away from the origin.   The clock that is outside the Black Hole event horizon cannot extend their inertial frame to cover all of space from where they are to where the man in middle is and further to where the other clock is located.
    Or to see it another way,  we can take a view from infinity:  If the man with the flashlight is inside the EH then no light signal from the man can reach a clock while it is outside the EH.   The clock on the outside and the man on inside can do their very best to be at rest in an inertial frame just by going into free-fall toward the Black Hole (sadly this won't be exactly the same frame for both of them - but it's the best we can do).  Indeed, the light emitted from the man may eventually reach that clock but it can't happen until the clock has crossed the EH.
     Another minor concern is that the r co-ordinate is time-like inside the EH.  A man in the middle between two clocks isn't really half-way between the clocks,  he is half-time between the two clocks and the clocks aren't really telling the time anymore they are just reporting on the t co-ordinate.   I suppose the procedure is synchronizing lengths -  It's all too much to think about late at night.
- - - - - -
   Anyway, we have drifted off the OP.  Are you OK, Zer0?

[Halc]

   No, I'm not sure it will work, Halc.   It works OK if the clocks and the man in middle with the flash light can all be at rest in the same inertial frame (free-falling into the Black Hole).

Those two statements seem to contradict each other in my book. 

However, the space is only Minkowski flat space locally.   For a small black hole in particular, these local inertial frames become significantly non-inertial with only the smallest step of distance away from the origin.

Which just means that 'local' becomes 'really local'.  The clocks might need to be microns apart or less in order for spacetime between them to be sufficiently approximated by Minkowskian spacetime.

space from where they are to where the man in middle is

I don't think a 'man' will fit between a pair of clocks a micron apart, let alone survive the spacetime curvature that necessitates the clocks being that close. Making clocks small enough to be separated by such a small distance is an engineering problem, not a physics one.

Or to see it another way,  we can take a view from infinity:  If the man with the flashlight is inside the EH

Oops! In this view, the man with the flashlight never gets inside and never turns the thing on. Neither does either of the yet-to-be-synced clocks.

Another minor concern is that the r co-ordinate is time-like inside the EH.  A man in the middle between two clocks isn't really half-way between the clocks

Only in Schwarzschild coordinates. In the man's inertial CS, yes he is halfway between them. In fact, the clocks can be bolted to a stick (with the man belted to the middle) so 2nd order effects don't draw them apart. The stay at a fixed spatial distance so long as the stick doesn't break, at which time spacetime wasn't Minkowskian any more anyway.

[Zer0 (OP)]

Plz continue...
Squeeze a bit more of techy words n sciency jargon innit.
Throw a dash of equations, equal slices of formulaes & sprinkle a few mind crunching numbers onnit.
" On Da Rockz "
🍻
Physics beats beer, anytime.
🤭

Jokes apart...

A second remains a second throughout the Universe?

In the ' Postal Letter ' analogy..the pace of the delivery system remains constant.
The poster changes locations.

Similarly, one second takes exactly one second to process everywhere.
Just the transfer of information gets delayed or speeds up.

In Other words, Time never dilates.
Our perception of it does yup?

Ps - I'm reminded of a Utube vid.
The maker, i felt, was tryin to sensationalise the Synchronization Convention.
Making claims light Could travel in a different speed in a specific universal direction.
But
If A = B, then B must equal A.
So what if it's just a mutually beneficial understanding agreement.
👍

[Halc]

A second remains a second throughout the Universe?

Regardless of what X is, X must be X. It would violate the law of identity if X was not X. So maybe we can consider asking a different way.

In the ' Postal Letter ' analogy..the pace of the delivery system remains constant.
The poster changes locations.

That seems an illustration of Doppler effect, not relativity.

In Other words, Time never dilates.

Of course it dilates, but dilation doesn't mean a second not being a second.

Our perception of it does yup?

It's actually a philosophical topic, and thus depends on your philosophical definition of time.

The relativistic view says that spacetime is one thing, not space within time. Objects don't move in spacetime, but rather occupy worldlines (paths) within it. Time is defined as the proper length of a given worldline, not as anything that 'flows'. You can have two different worldlines that connect a pair of common events, and the proper lengths of those two worldlines are different. That is measurable time dilation, using the relativistic definition.

Then there's the presentist camp (which you seem to find intuitive). It posits that there is a present moment, that this present moment progresses at some rate into what is termed 'the future', and that time is the rate at which this progression takes place. There is no empirical way to measure this progression, but for standardization purposes, it is typically defined that the rate of flow of the present on Earth is '1' and everything else relative to that.
In that scenario, time dilation means travelling away from Earth can make your clock run faster or slower than '1', but an accelerated round trip always has enough 'slower' segments that the clock will accumulate less time than had it stayed on Earth.
I find this interpretation to be quite naive, but being far more intuitive for somebody who doesn't regularly depart from this singular reference frame, it is the more commonly held view except among physicists.

The maker, i felt, was tryin to sensationalise the Synchronization Convention.
Making claims light Could travel in a different speed in a specific universal direction.

An absolutist claim, completely unbacked by empirical evidence, but also cannot be falsified by empirical evidence. It works under any absolute interpretation like Lorentz aether theory, of which presentism is a subset. (Lorentz did not posit presentism, but most of his followers add that).

[Eternal Student]

Hi everyone.

   There's not much wrong with Halc's answer but I'm not sure it's what you're after,  Zer0.  I'll have a go at providing some answers.

A second remains a second throughout the Universe?

   That would be so nice.  I wish we could just say "yes" and move on but we can't.
Let's confine our attention to special relativity.
    A clock that moves with an observer, works like a clock should work.  It records the passage of time for the observer.  Fortunately, every second that the clock says has passed really does seem like a second for the observer.   Everything is fine and perfectly alright with the world provided the clock moves with the observer.
    Special Relativity introduces one interesting effect.  If a clock does not move with an observer but instead it takes it's own path, then it does not match a clock that was moving with the observer.  This is just weird, no two ways about it, it's just weird.   It's not that the second clock was broken - we can put a second observer with the second clock and keep them moving together.  Now each observer will report that the clock that travelled with them recorded the passage of time perfectly normally, every second reported by the clock seemed like a proper second.   However, the clock that didn't travel with them reports the wrong times as far as they are concerned.   There's plenty enough written about this in textbooks, Pop Sci articles and YT videos if you want to learn more.
     One problem we have is trying to work out which clock actually showed the "correct" or "true" time.  The fundamental assertion in special relativity is that all inertial frames of reference are equally good.   There is no way to know which frame of reference or which clock is "better" or more "truthful" than any other.  We have two choices now:
   1.   Abandon the notion that there is one true time.
   2.   Keep trying to find something that can be intrinsically measured or observed and would allow us to identify that one frame of reference is "different" and possibly "preferred" over another frame of reference.

    Special relativity takes option 1.  -  accept that all inertial frames are equally good and accept that the rate of flow of time is not universal.
    We then have a weak answer to your question:   1 second is always 1 second, a local measure of the passage of time, wherever you are in the universe and whatever speed you might have.   Sadly 1 second for you is not experienced as 1 second by everyone else.  There is no sense in assuming that one rate of flow of time is better or more truthful than any other.

In the ' Postal Letter ' analogy..the pace of the delivery system remains constant.
The poster changes locations.

    I can't find any mention of a postal delivery system in this thread.  If it is what I think it is, then yes the pace of the delivery system remains constant.  I hope you appreciate how vague this reply is.

In Other words, Time never dilates.
Our perception of it does yup?

     This is usually a great cause for concern among teachers and texts written to explain Special Relativity.  It is best if you do not think that Time dilation is just an issue with perception.   Two clocks can be made and tested.  When they are kept together (and therefore move together), they stay synchronised.    Now, if we accelerate one of those clocks and send it on a journey around the galaxy and back to meet the other clock at the end then something has changed -  the two clocks are NOT synchronised anymore.  We haven't done anything to mess with the clocks other than sending one on a journey.   A human being wasn't there to see the clocks all the time, they didn't need to be, this wasn't a perception issue, the clocks measure time and they do not require a person to be there watching them.  When the clocks finally come back together, they are NOT synchronised.  This is a physical difference not a perception issue.  You can close one eye and squint a bit while looking at the clocks if you want, it doesn't help, the hands of one clock are not pointing to the same position as the hands on the other clock.
      We assume that both clocks were working as clocks, they recorded the correct passage of time.  We are left with the conclusion that a different amount of time elapsed for the two objects.

Best Wishes.

[Zer0 (OP)]

A specific quantity of Water would boil in a Specified Time Interval & evaporate at 100°c.

The same amount of water Can boil & evaporate at a lower temperature below 100°c.

The same amount of water Can boil faster & evaporate even faster at 100°c.

Keyword - seal level pressure.

So if We observe two boiling pots at same temperature, but one evaporates completely before the other.
For a Layman, that's Time Dilating.

Similarly, what factors in while travelling through SpaceTime?

Is it the movement, or acceleration.

Say, 2 clocks at rest in a common reference frame.
They start moving apart slowly.
They become unsynchronized.
But Why?

What connection does inertial movement have with Time.

Supposedly, clocks stop ticking around the event horizon of a BH.
No doubt, not arguing, agreed!
👍

But again, Why?
What relation does mass have wirh Time?

Ps - i know most of the times responders might be getting irritated by the same & same round & round futile questions.
You Folks are Superb Teachers!!!
Thanks All.
🙏

[Halc]

A specific quantity of Water would boil in a Specified Time Interval & evaporate at 100°c.

At normal pressure, water at 100c will not boil without input of heat from somewhere. Any boiling will cool the water, and if one boils water say by reducing pressure instead of adding heat (I've done this), it will eventually freeze.

So if We observe two boiling pots at same temperature, but one evaporates completely before the other.
For a Layman, that's Time Dilating.

If you're using two identical setups of boiling water as a clock, yes. Being the same temperature has little to do with it since I can put two pots on my stove with a liter each and get both boiling, but one will boil away long before the other because I put it on higher heat. Both pots are the same temperature (100c). This is not time dilation.

Is it the movement, or acceleration.

Under SR, it is movement. But movement is frame dependent. Something moving fast in one frame is stationary in another, so its time is dilated only relative to the abstract frame in which it has rapid movement.
Acceleration has nothing to do with it since I can subject two clocks to identical speed but different acceleration and they will remain in sync.

Say, 2 clocks at rest in a common reference frame.
They start moving apart slowly.
They become unsynchronized.[/quote]Even before they start moving apart, they're not synced in other frames, only ones in which they have no motion along the direction of the line connecting them.
Once moving apart, they can still be in sync in any frame in which they have identical speed.

What connection does inertial movement have with Time.

It's just the way the geometry of the situation works. Pathlengths change if you change the path, and a bent line through spacetime of a moving object traces a path that is different in length along both spatial and temporal axes than a straight line.

Supposedly, clocks stop ticking around the event horizon of a BH.

Only relative to some reference frames. Certainly not relative to the frame of the clock. A person observing his watch as he falls through the event horizon will notice nothing unusual. He can't tell when he passes it. Physics is still the same on both sides.

What relation does mass have wirh Time?

Mass alters the geometry. Hence the phrase 'bending spacetime'.

[Eternal Student]

Hi again,  I hope you are well.

Keyword - seal level pressure.

  Did you really mean "seal"?   That's an animal.  If you meant atmospheric pressure which is highest at sea level, then that's OK.

Similarly, what factors in while travelling through SpaceTime?

Is it the movement, or acceleration.

    At the simplest level, we can use Special Relativity.   It is the velocity of one object relative to another that is important.  The objects do not need to experience an acceleration.
    The only reason why an acceleration might be important (as it is in the "twins paradox") is that we can't easily compare two clocks unless we bring them back to the same place eventually.  We need some acceleration to alter the straight line motion of an object and bring it back to the other object.

Say, 2 clocks at rest in a common reference frame.
They start moving apart slowly.
They become unsynchronized.
But Why?

     Because of stuff in special relativity.  I could say it again if you wanted but there are plenty of other articles and videos about this.  I suppose we could say that "time dilation must happen because otherwise there isn't a way to explain why light always seems to travel at the speed c in a vaccum".
     This is essentially saying that it must happen and avoiding explaining WHY it happens.  The deeper explanation of why it happens is beyond me and the subject of many articles across many fields like Physics and Philosophy that would fill this entire forum.

The other comments you make and questions you ask are similarly complex:
   What connection does inertial movement have with Time?
   What relation does mass have with Time?

- - - - - - - - - -
   It seems that Halc has replied before I finished this post....
  A lot of his comments relate to a  "Triplets paradox"   which isn't a real thing  but just a sensible way of extending the  "twins paradox" that has become a conventional part of Special Relativity.
    Consider triplets this time and not just twins.    We keep one sibling on planet earth as usual.  Meanwhile we send the other two siblings off into space for a round journey.  We send these two in opposite directions and keep everything (like the speed of travel and distance travelled) perfectly symmetric but just mirror-images of the journey as far as the sibling staying on earth would be concerned.
     We see that we have done the twins paradox situation twice.   The sibling on earth disagrees with the elapsed time that the space travellers report.  However, the two space travelers will agree on their elapsed times.  The two space travellers were moved apart and brought back together again but it was done in such a way that their total elapsed times agree.
     Time dilation is complicated and it is possible to expose two clocks  (the space traveling siblings) to different accelerations (opposite in our case) but if we can find an inertial reference frame in which their speeds (a scalar quantity not a vector) were always the same then the clocks remain in sync in that frame throughout the entire motion.   In our example, we can choose the frame of reference for the sibling who stayed on planet earth.

Best Wishes.

[Zer0 (OP)]

I Really Really Appreciate the Time & Effort & Gray cells that You spend inorder to give such detailed crystal clear expectations.
🙏
(Wish i had Teachers like U back in school, well Glad you Exist)
🍭

Spacetime bends and has different pathlenghts.
No wonder it's SpaceTime, One Word.
If one changes, so does the other.
👍

I accept my " Why " is somewhat answered.
& If i still have a problem with it, then I'd be in disagreement with Reality.
✌️

Sometimes, somethings, are just the way they are Right!

(Ok now go work on much more important things)
All thee Best!
🤍