Whose Clock Slows Down in Relativity?

[pulmonata]

So I did a degree some time ago and after watching a programme on TV, I decided to revisit relativity and set myself some questions. One of these was the time difference by spending time on the space station. I worked out that roughly for every 15 mins that pass, the time difference will be 1 min, or after 3 months there will be a difference of just over 6 days. My question is: For Who? Being on Earth, the space station is moving relative to us so if we could observe the clock on the station, it would run 1 min slow after 15 mins. But then being on the space station, the Earth is moving relative to it, so one would assume that the same would apply, that observed from the station, the clock on Earth will run slow. But clearly one clock will be faster than the other when brought to the same location and compared side by side. So whose clock really runs slow? Late night thoughts that will keep me up for even longer!

 

[ghwellsjr]

Your calculations are way off and they appear to be based on Special Relativity where the time dilation is reciprocal. But in this case, the main contribution to the slower clocks on the space station is due to the difference in gravity and the fact that the space station is in "free fall" since it is in orbit. The difference in clock rate between the space station and on the Earth is very, very small and not reciprocal.

 

[pulmonata]

ok so i understand that my conceptual understanding of what is happening is wrong, that just because the object is moving fast relative to a stationary object does not mean that it is moving at the same speed relative to a stationary object on earth. So apply it to an object moving away from earth. With both objects moving away at speed, the phenomenon will be observed on whatever scale and the question still stands: they move fast from us, their clock is slow, we move fast from them, our clock is slow. So whose is slower when compared side by side during and after the event?

 

[GrayGhost]

To simplify this, assume that there is a space station (Deep Space 9) in free space far from gravity source. The USS Voyager, after synchronising their clock with the station, departs and accelerates to some high luminal inertial speed. After some time, the Voyager burns thrusters, reverses course, and returns to Deep Space 9 for a clock comparison over lunch.

The answer to your question is this ...

Although both always view themselves as position zero and all others relative to themselves, the Voyager clock will read less than the Deep Space 9 clock (on return). That is to say, the Voyager experienced a lesser passage of time than did Deep Space 9 over the same interval ... ie departure thru return. In such scenarios, the clock that does not remain "always inertial", is the clock that ages least.

To explain why this is the case is a complex discussion, and requires an indepth understanding of the Lorentz transforms, and likely Minkowski spacetime diagrams as well. So the short answer is this ... for 2 observers who reside at both events (departure & return) marking the spacetime interval, he who remains "always inertial" always ages the mostest.

GrayGhost

 

[ghwellsjr]

The issue of whose clock is slower can easily be determined, as Gray Ghost pointed out, by saying it is the one who accelerated (well he said the one who ages the most is the one who remained inertial--meaning never accelerated), but you should not conclude that acceleration is the direct cause of time dilation, it isn't. It's spending time at different relative speeds that causes the difference in aging when they reunite. See this thread for more discussion on this topic:
https://www.physicsforums.com/showthread.php?t=464460

 

[yuiop]

ok so i understand that my conceptual understanding of what is happening is wrong, that just because the object is moving fast relative to a stationary object does not mean that it is moving at the same speed relative to a stationary object on earth. So apply it to an object moving away from earth. With both objects moving away at speed, the phenomenon will be observed on whatever scale and the question still stands: they move fast from us, their clock is slow, we move fast from them, our clock is slow. So whose is slower when compared side by side during and after the event?

In SR there is no clear definition of which clock is actually "really" ticking slower until they are both brought back to the same location. At that point, the one that aged the least is the one that has taken the longest path through space time. This is very easy to sketch on a graph of space against time.

So I did a degree some time ago and after watching a programme on TV, I decided to revisit relativity and set myself some questions. One of these was the time difference by spending time on the space station. I worked out that roughly for every 15 mins that pass, the time difference will be 1 min, or after 3 months there will be a difference of just over 6 days. My question is: For Who? Being on Earth, the space station is moving relative to us so if we could observe the clock on the station, it would run 1 min slow after 15 mins. But then being on the space station, the Earth is moving relative to it, so one would assume that the same would apply, that observed from the station, the clock on Earth will run slow. But clearly one clock will be faster than the other when brought to the same location and compared side by side. So whose clock really runs slow? Late night thoughts that will keep me up for even longer!

There is a fairly easy way to calculate this to a reasonable degree of accuracy using Schwarzschild coordinates. Technically we should use Kerr coordinates because the Earth is rotating, but Schwarzschild coordinates are simpler and adequate for the low speed of rotation of the Earth. In the gravitational context we take the product of the gravitational time dilation \sqrt{(1-2GM/(rc^2))} and the velocity time dilation \sqrt{(1-v^2/c^2)}. The Earth's rotation velocity and the space station's orbital velocity are taken relative to the background stars in the same way that sidereal orbital periods are calculated relative to the background stars. Using Kepler's laws we can calculate the velocity of a satellite as a function of its orbital radius and the velocity time dilation \sqrt{(1-v^2/c^2)} can then be expressed as \sqrt{(1-GM/(rc^2))}. The time dilation of a satellite at radius r_S relative to clock at infinity is:

\sqrt{1-2GM/(r_Sc^2)}*\sqrt{1-GM/(r_Sc^2)} \qquad (Eq1)

The time dilation of a clock on the surface of the Earth (radius r_E) relative to a clock at infinity, uses the rotation speed of the Earth for the velocity and is given by:

\sqrt{1-2GM/(r_Ec^2)}*\sqrt{1-v^/c^2} \qquad (Eq2)

The speed of the satellite clock relative to the Earth clock is found from (Eq1)/(Eq2). Given that the rotational velocity of the Earth surface is 465 m/s and the radius of the Earth at the Equator is 6384 km and the orbital radius of the space station is 6384+333 = 6717km we can calculate the time dilation of the space station is 0.999705 relative to a clock on the surface of the Earth. This means time does pass slightly slower on the space station, but not as slow as your initial calculations suggest. Interestingly, for a clock on a GPS satellite at an altitude of 20200 km above the Earth, the time dilation relative to a clock on the surface of the Earth is about 1.00045 which is faster than a clock on the surface of the Earth. Note that for satellites which require GR, whether the clock is ticking faster or slower is not simply determined by considering acceleration. Both the space station and the GPS satellite are following geodesics so they that feel no proper acceleration and are locally inertial (while a clock on the surface of the Earth does experience proper acceleration) yet one satellite ticks slower and the other ticks faster than the Earth reference clock.

The calculation for the space station is here and the calculation for the GPS satellite is here.

The plot of time dilation versus radius with the space station radius as a minimum and the GPS satellite as a maximum can be found here.

There are other factors to consider besides the fact the Earth is not exactly Schwarzschild if you are interested in accuracy such as varying density of the Earth and irregular non spherical surface, and elliptical orbits with varying radius that are not always over the equator, but these are fine details that were probably not even factored into early GPS satellite system calculations.

 

[Mike_Fontenot]

[...]
So apply it to an object moving away from earth. With both objects moving away at speed, the phenomenon will be observed on whatever scale and the question still stands: they move fast from us, their clock is slow, we move fast from them, our clock is slow. So whose is slower when compared side by side during and after the event?

https://www.physicsforums.com/showpost.php?p=2934906&postcount=7

 

[GrayGhost]

In SR there is no clear definition of which clock is actually "really" ticking slower until they are both brought back to the same location. At that point, the one that aged the least is the one that has taken the longest path through space time. This is very easy to sketch on a graph of space against time.

You meant the shortest path thru spacetime, yes? The pathlength is the length of the spacetime interval (s), which is nothing more than the proper time experienced by the oberver at both events. Although s is the length of the hypthenuse (slanted worldline of the traveler) on a Minkowski diagram, it is temporally shorter than the stationary observer's vertical time axis (over the same interval).

It's a tricky subject, the relative aging deal. Inherent in SR, is the fact that the rate at which time passes by one per himself is always the very same rate at which time passes by another per herself. That is, the rate of "proper time" is the same for all. And (as I think you meant), he who ages the least is he who travels the shorter path thru the continuum between the 2 events. So, the relative aging is more about the comparison of accrued duration over paths of differing length, as opposed to one clock ticking faster or slower than the other. However since we cannot witness (or measure) space as moving others do, we don't see their path's length thru the continuum. Although the LTs can predict it, it goes unbeknownst visually. So all we have is the relative comparison of clocks, a frame-to-frame comparison deal. So an invariant rate of proper time manifests itself as a relative time rate differential. Therefore, it's valid to say "his clock ticked slower", since that's pretty much all we have by observation. It's just another way of viewing the mechanism. It's all relative :)

GrayGhost

 

[GrayGhost]

https://www.physicsforums.com/showpost.php?p=2934906&postcount=7

During the constant-speed legs of the trip, BOTH twins conclude that the other twin is ageing slower. But when the trip is over, they both agree that the stay-at-home twin is older. How is that possible?

It's possible because, during the turnaround, the traveler will conclude that the home twin quickly ages, with very little ageing of the traveler. The home twin concludes that neither of them ages much during the turnaround. When you add up all these segments of ageing, you get the result that the home twin is older (and both twins exactly agree on that).

Indeed. The math requires it. But trying to convince folks "why" that happens and "what it all means", is the harder part. Brian Cox summed it up pretty good ... we each coexist at all points upon our own worldline, since birth, even though this goes unbeknownst.

GrayGhost

 

[Phrak]

Indeed. The math requires it. But trying to convince folks "why" that happens and "what it all means", is the harder part. Brian Cox summed it up pretty good ... we each coexist at all points upon our own worldline, since birth, even though this goes unbeknownst.

GrayGhost

Comment From the Correct Langauge Police:

/ An abuse of language too often found in discussions of relativity. As used above, "Coexist" is a present tense verb.

To correct the above (, ignoring the dubious implication that somehow we and our world line are two different things): "We Coexisted in the past with our past world line.", "We Coexist now with our world line.", and "We Will Coexist with our future world line, perhaps."
/

 

[GrayGhost]

As I said, and as you just confirmed, it's the harder part.

Anyone who doesn't like the use of the word "coexist" in discussions of relativity, likely doesn't yet fully understand the theory. It's easy to throw the equations around and obtain solutions, but explaining the meaning of the theory ... is not so easy.

Brian Cox is a pretty smart fellow. He begins discussing this matter at about 1 minute into this short video clip. Take it from him, if not from me ...

http://www.youtube.com/watch?v=CGetCESW-dM"

GrayGhost

 

[Phrak]

I am sure that Brian Cox is well meaning but statements like "All moments in time already exists," is just a meaningless string of words; as meaningless as "yesterday I will ride my bike." If he were to invoke a hypothetical meta-time he would make some sense, and perhaps he did this, but editors of that clip had left it out.

 

[GrayGhost]

I am sure that Brian Cox is well meaning but statements like "All moments in time already exists," is just a meaningless string of words; as meaningless as "yesterday I will ride my bike." .

Well, your point is well taken, and its not as though I have not also argued your same points at length. Whether the statement "all moments in time already exists is meaningless", depends upon your view of what TIME really is. Relativity theory has extended the meaning of space and time, and so all words associated with them also extend. I admit, it ain't easy.

Bottom line, you see me a distance off traveling luminally. I exist in your NOW across space as you now perceive it. Since we move relatively, our sense of simultaneity is rotated wrt one another. Therefore, the I that you hold, holds you at some point upon your own worldline in your past, and "you are truly there". Hence, you coexist in both your past and your present, even though you never experience it. Given such, you must conclude that the prior version of yourself (that I now know is there) will eventually arrive at your present moment, and thus your future must coexist as well as your present and past.

I raise this matter only because of the ongoing debate as to what is real versus merely apparent, ie are relativistic distortions real. They are real. I also responded to Mike Fontenot's reference of another thread in relation to this topic, ie how clocks spin wildly during twin B's acceleration. Now most folks simply assume relativistic effects are apparent vs real, a mistaken assumption. Therefore, I decided to explain WHY inertial clocks advance wildly per those who undergo proper acceleration, such that real physical meaning could be atrributed to the relativistic effect. Not easy to do in a post forum format. Briefly though, from the POV of he who undergoes proper acceleration, remote inertial clocks spin wildly because said clock advances or digresses (in a manner different from what an inerial observer would record) along its own worldline due to the relative change in the accelerating observer's POV. The best part, per the theory, it is all real.

GrayGhost

 

[Phrak]

Well, your point is well taken, and its not as though I have not also argued your same points at length. Whether the statement "all moments in time already exists is meaningless", depends upon your view of what TIME really is. Relativity theory has extended the meaning of space and time, and so all words associated with them also extend. I admit, it ain't easy.

OK. Give me a definition of what "time really is" (your verb tense, not mine) where "I rod my bicycle tomorrow," is a statement consistent with the definition of time within the scope of relativity theory.

 

[ghwellsjr]

https://www.physicsforums.com/showpos...06&postcount=7

During the constant-speed legs of the trip, BOTH twins conclude that the other twin is ageing slower. But when the trip is over, they both agree that the stay-at-home twin is older. How is that possible?

It's possible because, during the turnaround, the traveler will conclude that the home twin quickly ages, with very little ageing of the traveler. The home twin concludes that neither of them ages much during the turnaround. When you add up all these segments of ageing, you get the result that the home twin is older (and both twins exactly agree on that).

Indeed. The math requires it. But trying to convince folks "why" that happens and "what it all means", is the harder part. Brian Cox summed it up pretty good ... we each coexist at all points upon our own worldline, since birth, even though this goes unbeknownst.

GrayGhost

Mike claims that his idea is the only correct conclusion that the traveler can come to. But he is switching frames of reference to come up with this idea. If he would analyze the entire scenario from any arbitrary single frame of reference, then he would see that his idea is wrong. Or if he would analyze what the traveler actually observes, then he would see that his idea is wrong.

If you read his paper and his other posts, you will see that he also claims that under other circumstances a traveler will conclude that his twin actually gets younger at a rapid rate! And he persists in claiming that his idea is the only correct one.

So, the math does not require it, if by "it" you mean what Mike is promoting.

 

[GrayGhost]

OK. Give me a definition of what "time really is" (your verb tense, not mine) where "I rod my bicycle tomorrow," is a statement consistent with the definition of time within the scope of relativity theory.

As I said, relativity has extended the meaning of space and time, and so our typical usage of those words, or words associated with those words, no longer apply so easily in all cases.

Let's assume Brian Cox (as well as many other leading physicists) has it right, that one's worldline has always existed in its entirety. That is to say, your entire progression of your own life was laid out before you were born, and will continue to be laid out after you pass. Your entire worldline simply sits there in the spacetime continuum (or Minkowski's 4-space if you prefer). It may then be stated that all events in your life "simply co-exist".

What flies in the face of this, is the fact that we always live in the NOW, which is everchanging, and always seems to pass at some steady rate. Therefore, when relativity is taken at face value, it must be assumed that you exist in all NOWs of your worldline (or timeline if you prefer), even though you "for some yet unknown reason" never realize it. IOWs, there's a version of yourself at all points along your own worldline experiencing his own NOW, concurrently, and none of those versions are aware of this ... except by considering the implications of relativity theory on the grander scale.

In answer to your question, I can only say this ... draw any Minkowski worldline illustration of 2 observers who move luminally relatively, and that, as best as can be done today, explains what time (and space) is. I'm of course simplifying here, because we all know GR would provide a fuller meaning, however ... consider the full implications of that worldline illustration, and what Brian Cox says must be true. BTW, said Minkowski illustration would basically model the situation I stated in my prior post.

So words such as present, future, past, now, before, after, will, did, etc ... all do apply in daily existence, because we only ever experience an everchanging NOW from our own experience. Yet, relativity theory shows that there is more to space and time than casually meets the eye. And I agree, that our existing vocabulary falls short in particular situ, however the theory still stands none-the-less with its extended implications. Bottom line, there is still much work to be done. There are those who will quit once they learn enough to throw the SR or GR equations around, and then there are those who will always ask the next question ... why? And it's a good thing too, because otherwise the Earth would still be flat and we'd still be in the dark ages.

GrayGhost

 

[GrayGhost]

Mike claims that his idea is the only correct conclusion that the traveler can come to.

... If you read his paper and his other posts, you will see that he also claims that under other circumstances a traveler will conclude that his twin actually gets younger at a rapid rate! And he persists in claiming that his idea is the only correct one.

So, the math does not require it, if by "it" you mean what Mike is promoting.

No, I haven't read thru Mike's work, so I did not intend to promote his hypothesis. I'll try to look at his work at first opportunity. So by "it", I was speaking in relation to mainstream relativity, which apparently do not support Mike's hypotheses. Sorry for the confusion there.

GrayGhost

 

[PAllen]

There is a fairly easy way to calculate this to a reasonable degree of accuracy using Schwarzschild coordinates. Technically we should use Kerr coordinates because the Earth is rotating, but Schwarzschild coordinates are simpler and adequate for the low speed of rotation of the Earth.

This has come up before: IF the Earth were rotating fast enough to matter, you *could not* use Kerr coordinates. There is no analog of Birkhoff's theorem for rotating bodies, and it is known that the field around massive rotating body is not closely approximated by a Kerr metric (which only applies to the final state of a rotating black hole). I posted some papers about this in another thread here, where they discuss how to actually approximate field outside massive rotating bodies under some simplifying assumptions.

[EDIT: I found my reference:

http://arxiv.org/abs/gr-qc/0205127

]

 

[Mike_Fontenot]

So by "it", I was speaking in relation to mainstream relativity, which apparently do not support Mike's hypotheses.

My results are COMPLETELY consistent with Special Relativity. In fact, the reference frame I adopt for the accelerating traveler is the ONLY possible reference frame for him (among all possible frames in which he is perpetually stationary) which does NOT contradict the traveler's own elementary measurements and elementary calculations.

At least two other alternative reference frames for an accelerating traveler have been endorsed on this forum: Dolby & Gull's frame (explained by Fredrik), and a frame endorsed by Passionflower (I don't recall the name of that frame's originator). Those two frames, and ANY other frame (besides mine) for an accelerating traveler, ALL answer the following question differently from the answer that my frame gives:

"How long must a formerly accelerated traveler remain unaccelerated, in order to henceforth be considered a full-fledged inertial traveler?"

My answer is that for ANY segment of unaccelerated motion, no matter how short, the traveler is a full-fledged inertial observer during that ENTIRE segment. For example, he can, during that entire segment, legitimately use the standard time-dilation result to conclude that all inertial clocks moving relative to himself are running slow by the factor gamma.

ANY other answer to the above question is inconsistent with the traveler's own elementary measurements and elementary calculations. I consider consistency with the traveler's own elementary measurements and elementary calculations to be an absolutely necessary requirement for any legitimate frame in which the traveler is perpetually stationary.

Mike Fontenot

 

[GrayGhost]

My results are COMPLETELY consistent with Special Relativity. In fact, the reference frame I adopt for the accelerating traveler is the ONLY possible reference frame for him (among all possible frames in which he is perpetually stationary) which does NOT contradict the traveler's own elementary measurements and elementary calculations.

Well, you do sound confident in your assertion. Was your prior hyperlink reference the paper I should be looking at? If so, I'll take a closer look.

"How long must a formerly accelerated traveler remain unaccelerated, in order to henceforth be considered a full-fledged inertial traveler?"

My answer is that for ANY segment of unaccelerated motion, no matter how short, the traveler is a full-fledged inertial observer during that ENTIRE segment. For example, he can, during that entire segment, legitimately use the standard time-dilation result to conclude that all inertial clocks moving relative to himself are running slow by the factor gamma.

Well, that sounds right to me. The hard part, of course, is explaining how to map the heavens during accelerations per he who accelerates.

GrayGhost

 

[PAllen]

My results are COMPLETELY consistent with Special Relativity. In fact, the reference frame I adopt for the accelerating traveler is the ONLY possible reference frame for him (among all possible frames in which he is perpetually stationary) which does NOT contradict the traveler's own elementary measurements and elementary calculations.

At least two other alternative reference frames for an accelerating traveler have been endorsed on this forum: Dolby & Gull's frame (explained by Fredrik), and a frame endorsed by Passionflower (I don't recall the name of that frame's originator). Those two frames, and ANY other frame (besides mine) for an accelerating traveler, ALL answer the following question differently from the answer that my frame gives:

"How long must a formerly accelerated traveler remain unaccelerated, in order to henceforth be considered a full-fledged inertial traveler?"

My answer is that for ANY segment of unaccelerated motion, no matter how short, the traveler is a full-fledged inertial observer during that ENTIRE segment. For example, he can, during that entire segment, legitimately use the standard time-dilation result to conclude that all inertial clocks moving relative to himself are running slow by the factor gamma.

ANY other answer to the above question is inconsistent with the traveler's own elementary measurements and elementary calculations. I consider consistency with the traveler's own elementary measurements and elementary calculations to be an absolutely necessary requirement for any legitimate frame in which the traveler is perpetually stationary.

Mike Fontenot

I believe all of these frames agree on local measurements. They disagree on interpretation of distant events, and, in particular, the two you disagree with say distant events should be interpreted with knowledge of the history of their apparent motion rather than just their current instantaneous motion. The actual core of disagreement was your claim that distant simultaneity has a unique best answer for a given observer. Almost everyone else took the view that your interpretation was perfectly good, but so are many others within broad constraints.

 

[bobc2]

As I said, and as you just confirmed, it's the harder part.

Anyone who doesn't like the use of the word "coexist" in discussions of relativity, likely doesn't yet fully understand the theory. It's easy to throw the equations around and obtain solutions, but explaining the meaning of the theory ... is not so easy.

Brian Cox is a pretty smart fellow. He begins discussing this matter at about 1 minute into this short video clip. Take it from him, if not from me ...

http://www.youtube.com/watch?v=CGetCESW-dM"

GrayGhost

Just finished watching the video with Brian Cox. Thanks for that one. I had not seen it and thought he did an excellent job of presenting the classical special relativity picture of the 4-D universe populated with 4-dimensional objects.

I felt that when he brought in the quantum mechanical picture with the help of his colleague, they did nothing to explain how you get "becoming" phosophy back again without a contradiction in accounting for the 4-D objects. Just because the 4-D space is grainy doesn't mean it couldn't still be 4-dimensional. Just because the laws of physics cannot accurately predict the paths of world lines in the future does not mean that they are not there, any more than the classical SR world lines are there. The world lines can still be there... they are just grainy. I felt that was a very weak aspect of the presentation.

We've had a running discussion over objective philosophy vs. ideal philosophy over in the philosophy forum, and if you don't mind I think I'll reference your post over there.

 

[PAllen]

I believe all of these frames agree on local measurements. They disagree on interpretation of distant events, and, in particular, the two you disagree with say distant events should be interpreted with knowledge of the history of their apparent motion rather than just their current instantaneous motion. The actual core of disagreement was your claim that distant simultaneity has a unique best answer for a given observer. Almost everyone else took the view that your interpretation was perfectly good, but so are many others within broad constraints.

This is in response to Mike Fontenot.

Another issue was arbitrary definition (or lack thereof) of what 'elementary measurements and calculations' are. For example, I gave the description Dolby & Gull's notion of simultaneity in terms of local measurements (which are apperently different from the local measurements you arbitrarily allow):

Imagine attached to any object you can see a clock and a mirror (you, the observer, have a clock too). What you see on the object's clock shows proper time progress for the object. The image of your clock in the object's mirror tells you how to map the event you are now seeing to your own history - back from now halfway to the time you see in the reflection.

 

[christodouloum]

and the question still stands: they move fast from us, their clock is slow, we move fast from them, our clock is slow. So whose is slower when compared side by side during and after the event?

Once you have acceleration we are talking about general relativity (going around the Earth is being accelerated in a centrifugal way, the acceleration is always pointing orthogonally to your orbit). Your question is very well posed within special relativity and the answer through this theory (it applies when things move uniformely, i.e. constant speed and no turning,constant in the full vector description of speed) is that both regard the other guy's clock as running slower but actually there is no objective viewpoint to take on which one is unambiguously running slower. That is why it is called relativity. That is why there is no point, in Einstein's powerful opinion, of talking about The time and The space since these two are a matter of understanding between the two or infinitely many scientists, of subjective viewpoint, of inertial reference frame to put it in many ways. So we talk of spacetime where both would agree on a new kind of interval (not time that passes or distance walked) the spacetime interval. This gives as again a notion of spacetime as a background in an unambiguous universal sense. Try to do the calculation on taking each other's viewpoint you will see that each thinks that it is the other clock that is running slower.

Finally you ask who's clock will be slower when compared side by side. Well that would be the guy that was accelerated. You cannot go away and come back from somewhere without being accelerated can you? That would involve general relativity (in this theory acceleration is the same thing as gravitational field which actually means the spacetime has curved :P ) and the lorentz transformations alone do not suffice to do the job. Why is it the guy that accelerated that gets away with it? Well think of it this way. It is called time dilation. That means from your viewpoint everything that moves has time flowing slower. That means that your own time flows in the fastest rate possible. That still holds in general relativity. So it's the guy that didn't get accelerated that ages the most (the why and how is indeed technical).

 

[GrayGhost]

bobc2,

You're welcome. Brian Cox's videos are usually intriguing, let alone entertaining. Glad you liked it. Feel free to reference it, it's public domain. I too question their notions regarding the production of quantum spacetime and its relation to future events. But then, its difficult to convey abstract concepts in short video clips. Sit down with her for awhile, and we may feel differently. The videos serve only to convey general concepts, and promote interest in the general public for funding. BTW, you should watch all 6 parts of that video, if you haven't already. I merely referenced part 6 of 6, because it related to the discussion at hand, and cut to the chase.

GrayGhost

 

[Phrak]

As far as Brian Cox's and your own arguments to conflagurate past, future and present, they are simply wrong.

You've begun arguing in circles. I doubt Cox authority. You offer proof by contradiction then argue this by invoking Cox as authority. This doesn't cut it.

Cox adds a layer of metaphysics outside of, and in contradiction to relativity theory in a quest to bedazzle an unsophisticated audience. I'm not at all impressed by the psychology behind his actions.

As I am not interested in pursuing it further, we will have to agree to disagree.

You meant the shortest path thru spacetime, yes? The pathlength is the length of the spacetime interval (s), which is nothing more than the proper time experienced by the oberver at both events. Although s is the length of the hypthenuse (slanted worldline of the traveler) on a Minkowski diagram, it is temporally shorter than the stationary observer's vertical time axis (over the same interval).

You have something inverted. Unlike a straight line in spatial dimensions, the interval of a freely falling object in relativity between two spacetime events is not a minimal but a maximal displacement measured in proper time. This result is obtained in applying variational analysis where all neighboring, infinitesimally displaced paths between two events require less proper time.

 

[Mike_Fontenot]

Was your prior hyperlink reference the paper I should be looking at?

Yes. You might want to start with the link to my webpage. Near the end of that webpage, I include the abstract of my paper.

The hard part, of course, is explaining how to map the heavens during accelerations per he who accelerates.

Actually, using my reference frame for the accelerating traveler, it's not hard at all. At any instant in the traveler's life, his reference frame exactly agrees with the inertial frame with which he is momentarily stationary at that instant. And that inertial frame's conclusions about spacetime are simply given by the Lorentz equations.

Here are some previous posts that might be of some help:

https://www.physicsforums.com/showpost.php?p=2923277&postcount=1

https://www.physicsforums.com/showpost.php?p=2957404&postcount=5 .


Mike Fontenot

 

[Mike_Fontenot]

[...] the two you disagree with say distant events should be interpreted with knowledge of the history of their apparent motion rather than just their current instantaneous motion.

For the Dolby & Gull reference frame, it's worse than that: their simultaneity depends on what is going to happen in the FUTURE, not just on what happened in the past. I.e., it is NON-CAUSAL. That is a fatal flaw.

Here's a previous post that elaborates on this issue:

https://www.physicsforums.com/showpost.php?p=2812867&postcount=50 .

The actual core of disagreement was your claim that distant simultaneity has a unique best answer for a given observer. Almost everyone else took the view that your interpretation was perfectly good, but so are many others within broad constraints.

I disagree that the other alternatives are "perfectly good". I maintain that ALL of the possible alternatives (other than mine) have one or more fatal flaws.

Mike Fontenot

 

[Dale]

Hi Mike, I see you are back at your same games as always

My results are COMPLETELY consistent with Special Relativity.

This is correct.

In fact, the reference frame I adopt for the accelerating traveler is the ONLY possible reference frame for him (among all possible frames in which he is perpetually stationary) which does NOT contradict the traveler's own elementary measurements and elementary calculations.

This is not correct, as we have discussed endlessly in other threads which you have never fully answered.

Please stop with this nonsense. When you are dealing with non-inertial reference frames then you can use any coordinate system you like whatsoever and we are not restricted to your preference. All non-inertial coordinate systems are equally valid. You are certainly free to use yours exclusively, but we are also free to use others. The "fatal flaws" that you identify are all just things that you personally dislike about alternatives, but your personal preference does not make them invalid coordinate systems for doing physics.

 

[Mike_Fontenot]

Another issue was arbitrary definition (or lack thereof) of what 'elementary measurements and calculations' are.

Those elementary observations and elementary calculations are given, in detail, in my paper. I'm not willing to reproduce them here. But I have provided some hints in some previous posts:

https://www.physicsforums.com/showpost.php?p=2960120&postcount=41

https://www.physicsforums.com/showpost.php?p=2978931&postcount=75 .

Mike Fontenot

 

[Dale]

I'm not willing to reproduce them here. But I have provided some hints in

And that really is one of the key problems in this whole discussion. It is hard to have a rational discussion when one party uses secret definitions of important terms which he only hints at.

 

[PAllen]

For the Dolby & Gull reference frame, it's worse than that: their simultaneity depends on what is going to happen in the FUTURE, not just on what happened in the past. I.e., it is NON-CAUSAL. That is a fatal flaw.

This is false. At any given point on an observer's worldline, Dolby &Gull only maps spacetime in the prior lightcone of that event. In that sense it is more conservative than Fermi-Normal coordinates. Now, in looking at complete spacetimes, where you have the future 'available' you can construct complete mappings, but the proper point of view for a physical observer is non-causally connected parts of spacetime are simply not-mapped.

 

[bobc2]

As far as Brian Cox's and your own arguments to conflagurate past, future and present, they are simply wrong.

Obviously there are differing philosophies in the realm of metaphysics. However, GrayGhost and Brian Cox are in good company with a number of respected philosophers engaged in thinking about a 4-dimensional objective universe populated by real 4-dimensional objects (Putnam, for example). To say that "...they are simply wrong" brings nothing fruitful to the discussion.

You might have said something like, "I prefer an idealistic philosophy of spacetime and here are a couple of the more forceful arguements for my position, bolstered as well by statements of the philospher, ..."

 

[GrayGhost]

As far as Brian Cox's and your own arguments to conflagurate past, future and present, they are simply wrong.

Easy this to say. Hard to prove. If you think about it long enough, my bet is that you change your opinion on that matter. The only proof we need in support of said contention is a standard Minkowski diagram, and the assumption that what is predicted by the LTs "is real". It's not in conflict with the LTs in any way. It merely addresses the nature of time, and hence a deeper implication of the nature of spacetime.

It's not as though Brian Cox was the first to attain this understanding of relativity. It's been around since Einstein. I have no doubt that Minkowski held the same view. COX merely stated it to layman's in layman terms.

Brian Cox (particle physicist) works on cutting edge projects at the LHC, where relativity must be understood indepth and used routinely. I'd think more than twice before proclaiming his understanding of relativity wrong ... not that anyone's perfect.

As I am not interested in pursuing it further, we will have to agree to disagree.

OK, I'm good with that. Thanks for your opinion.

GrayGhost

 

[Mike_Fontenot]

For the Dolby & Gull reference frame, it's worse than that: their simultaneity depends on what is going to happen in the FUTURE, not just on what happened in the past. I.e., it is NON-CAUSAL. That is a fatal flaw.

This is false.

Interesting that you would think that. About a month ago, I exchanged several emails with Steve Gull. I had asked him if he realized that his simultaneity was non-causal. He replied that "The construction is SHAMELESSLY acausal. Probably should have made a point about that...".

Fredrik (who explained the Dolby & Gull simultaneity in one of his posts) also realized that D&G is non-causal, although he didn't consider that a "show-stopper", like I do.

Mike Fontenot

 

[PAllen]

Interesting that you would think that. About a month ago, I exchanged several emails with Steve Gull. I had asked him if he realized that his simultaneity was non-causal. He replied that "The construction is SHAMELESSLY acausal. Probably should have made a point about that...".

Fredrik (who explained the Dolby & Gull simultaneity in one of his posts) also realized that D&G is non-causal, although he didn't consider that a "show-stopper", like I do.

Mike Fontenot

The difference is using it as operational definition in the real world versus an arbitrary coordinate system. For the former, it is strictly causal. You can only define what event in the past of your timeline was simultaneous to an event you have become aware of. The most you can do about events not yet causally related to you is predict based on your imperfect knowledge; for this purpose, it works as well or poorly as your prediction reliability (including predicting your own future). Note that Dolby & Gull did not invent their notion, they just wrote a paper about it that got attention. In my view it is strictly causal because it only operationally maps your past lightcone.

Given a complete spacetime model, you can build a complete coordinate mapping. In such a mathematical construction, time is just a coordinate, all past and future are determinate, so worrying about causality is not meaningful.

 

[Dale]

Fredrik (who explained the Dolby & Gull simultaneity in one of his posts) also realized that D&G is non-causal, although he didn't consider that a "show-stopper", like I do.

It isn't a "show stopper" in any way other than your personal preference. You are certainly not required to use any other coordinate system than your preferred one, but you are flat out, simply, and without reservation WRONG in your claim that your pet system is the only valid one.

 

[Mike_Fontenot]

Previously, I wrote:

Another issue was arbitrary definition (or lack thereof) of what 'elementary measurements and calculations' are.

Those elementary observations and elementary calculations are given, in detail, in my paper. I'm not willing to reproduce them here. But I have provided some hints in some previous posts:

https://www.physicsforums.com/showpost.php?p=2960120&postcount=41

https://www.physicsforums.com/showpost.php?p=2978931&postcount=75 .

To understand why ALL of the other alternative reference frames for an accelerating traveler (other than mine) contradict the traveler's own elementary measurements and elementary calculations, you have to understand two different things:

1) You first need to understand what those measurements and calculations are, for a PERPETUALLY INERTIAL traveler,

and

2) you need to know how to then apply your understanding of item 1 to the case of a traveler who accelerates.

Anyone who has a good understanding of the fundamentals of special relativity should be able to do item 1. In fact, going though the analysis yourself will be much more beneficial to your understanding than just hearing the solution from me or from anyone else. The required analysis IS elementary, but it is easy to get wrong. Keep at it ... you haven't succeeded until your result agrees with the Lorentz equations. The problem to be solved is described here:

https://www.physicsforums.com/showpost.php?p=2960120&postcount=41 .

If you can't get the solution on your own, read my paper.

Item 2 requires that you consider what happens if and when the accelerating traveler decides to stop accelerating, and to remain unaccelerated for some segment of his life, before deciding to accelerate again. It is necessary to ask, and correctly answer, the following two questions:

"When an accelerating traveler stops accelerating, how long must he wait before he can be legitimately considered to be an inertial observer (and thus be able to legitimately use the Lorentz equations to determine simultaneity at a distance)?"

and

"When a here-to-fore inertial observer begins to accelerate, at what point in his life, before he accelerates, does he lose his status as an inertial observer?".

The correct answer is that for ANY unaccelerated segment of the life of the traveler, no matter how short the segment, the traveler is a full-fledged inertial observer during that ENTIRE segment (and he is thus able, during that entire segment, to legitimately use the Lorentz equations to determine simultaneity at a distance).

You may be able to arrive at that answer yourself, if you spend some quality time thinking about those questions. If you can't, read my paper.

Mike Fontenot

 

[bobc2]

Easy this to say. Hard to prove. If you think about it long enough, my bet is that you change your opinion on that matter. The only proof we need in support of said contention is a standard Minkowski diagram, and the assumption that what is predicted by the LTs "is real". It's not in conflict with the LTs in any way. It merely addresses the nature of time, and hence a deeper implication of the nature of spacetime.

Good post, GrayGhost. Let me take a shot at sketching the Minkowski diagram. I'll use the setup with a blue guy moving to the right at relativistic speed with respect to a black rest system. A red guy is moving to the left in that same system with the same speed (blue and red guys moving in opposite directions with same relativistic speed). In the upper left sketch I have marked off equal distance positions along the respective world lines for the blue guy and the red guy. The fact that we use this symmetric spacetime diagram assures that the calibrations of distances for the blue and red coordinates are the same (otherwise it would be necesary to use the hyperbolic calibration curves).

The blue and red guys each reach the position number 9 at proper times that match each other. The rockets and all objects inside the rockets, including clocks and human bodies, are 4-dimensional objects, so it can be perplexing trying to comprehend who or what is doing the moving. The usual language is that each observer moves along his own world line at the speed of light, c. Einstein's colleague, Hermann Weyl, said something like, "...the observer crawls along his own world line." So, some distance traveled along the 4th dimension would be dX4 = c(dt). To avoid a sidebar on that issue, for now, let's just play like there is some aspect of nature associated with consciousness that moves along the 4th dimension at speed c. It is interesting to ponder the enormous length of a life sized 4-D object along its X4 dimension as compared to their almost negligible X1 length.

So far we have a sketch of the R4 manifold (supressing X1 and X2) with the world lines of the 4-D objects. So, we're having really a purely spatial discussion. The X1' coordinate (blue) and the X'' coordinate (red) is oriented such that the photon world line always bisects the angle between X1 and X4, X1' and X4', ...X1'' and X4''. A photon world line would always bisect the angle between any observer's X1 axis and X4 axis. As a result of this circumstance of the 4-dimensional world, every observer will observe a ratio of 1:1 between displacement along the X4 axis and displacement along the X1 axis for any photon world line. In other words, all observers measure the speed of light as c.

Now, having set up the Minkowski spatial picture, we see in the upper right sketch that one can form a right triangle using X4'', X1', and X4'. Given this purely 4-dimensional relationship, it requires only high school math, Pythagorean theorem, to derive the Lorentz transformations. Here, for example, we derive the transformation formula (rotational) for time, t'' (a clock reading in the red rocket), as a function of the blue clock's time, t'. (see lower right sketch)

The formula indicates that the red guy's clock will lag behind the blue guy's clock as observed by the blue guy. The blue guy's instantaneous 3-D cross-section of the universe, blue's "NOW", intersects the red guy's 4-dimensional rocket at red position 8 when the blue guy is at position number 9. This explains quite clearly the reason for the time dilation effect of special relativity.

But, red's "NOW" (when the red guy is at his position number 9) 3-D cross-section intersects blue's 4-dimensional rocket at blue's position number 8, and sees that the blue guy's clock is lagging behind his own.

Likewise, the lower left sketch illustrates the length contraction effect of special relativity.

 

[GrayGhost]

bobc2,

Thanx. Your post looks fine to me as well. As drafted, I do not see any problem with the illustration. However, what was the general point you were intending to convey in your post?

The use of the Loedel figure does speed things up in proving the point regarding the spacetime interval length s, however it works only because of the particular symmetry used in Loedels whereby the fictitious stationary ict bisects the moving worldlines. IOWs, try applying your formulation process to a standard Minkowski illustration with 1 stationary and 1 moving observer, see what happens. If you do, and are successful, you'll find out how Minkowski formulated it and the meaning of the imaginary i.

EDIT: I was mistaken in my highlighted statement. It the formulation works just the same from a standard Minkowski illustration as from a Loedel figure.

GrayGhost

 

[Dale]

ALL of the other alternative reference frames for an accelerating traveler (other than mine) contradict the traveler's own elementary measurements

No they don't. You have never demonstrated this and it is false. All frames will agree on the result of any measurement.

 

[GrayGhost]

The pathlength is the length of the spacetime interval (s), which is nothing more than the proper time experienced by the oberver at both events. Although s is the length of the hypthenuse (slanted worldline of the traveler) on a Minkowski diagram, it is temporally shorter than the stationary observer's vertical time axis (over the same interval).

You have something inverted. Unlike a straight line in spatial dimensions, the interval of a freely falling object in relativity between two spacetime events is not a minimal but a maximal displacement measured in proper time.

)

I missed this one before ...

Incorrect. I was talking SR. He who resides at both events always measures the least amount of time for the interval. If 2 observers reside at both events, eg in the twins scenario, then the one who travels the shorter path thru the continuum ages less over the interval, because he accrues less proper time.

GrayGhost

 

[bobc2]

bobc2,

Thanx. Your post looks fine to me as well. As drafted, I do not see any problem with the illustration. However, what was the general point you were intending to convey in your post?

I was wanting to place emphasis on the aspect of the pure geometric description of the 4-dimensional universe.

The use of the Loedel figure does speed things up in proving the point regarding the spacetime interval length s, however it works only because of the particular symmetry used in Loedels whereby the fictitious stationary ict bisects the moving worldlines. IOWs, try applying your formulation process to a standard Minkowski illustration with 1 stationary and 1 moving observer, see what happens. If you do, and are successful, you'll find out how Minkowski formulated it and the meaning of the imaginary i.
GrayGhost

Yes, I'm familiar with that, and of course you are correct. But, the ict seems not so significant when you simply swap signs in the derived metric, then arbitrarily substitute the (ict')^2 and (ict'')^2 for -(X4')^2 and -(X4'')^2. I can do the same thing with any simple right triangle on a piece of paper without anyone assigning anything mysterious or unusual about one of the lines (legs) of the triangle.

Einstein begins Appendix II of his book, "Relativity - The Special and the General Theory" by stating, "We can charcterize the Lorentz transformation still more simply if we introduce the imaginary ict in place of t, as time-variable... x4 = ict.

 

[GrayGhost]

the ict seems not so significant when you simply swap signs in the derived metric, then arbitrarily substitute the (ict')^2 and (ict'')^2 for -(X4')^2 and -(X4'')^2. I can do the same thing with any simple right triangle on a piece of paper without anyone assigning anything mysterious or unusual about one of the lines (legs) of the triangle.

Yes, however I see I was mistaken ... on the matter of your spacetime interval formulation not applying for normal Minkowski illustrations with 1 stationary and 1 moving observer. It in fact does apply, of course. Thinking one thing, talked about another. It just doesn't work for Loedel's (geometrically per the diagram) if the 2 moving worldlines are not symmetric about the stationary ict axis.

The best formulation of a spacetime interval, is one which presents the y-axis as well ... so an x,y,ict worldline illustration. The 2-d lightcone (versus a ray's lightpath) helps as well. It explains 2 things ... (1) the meaning of imaginary axes, and (2) why the magnitude of s is always less than the magnitude of ct.

GrayGhost

 

[bobc2]

Yes, however I see I was mistaken ... on the matter of your spacetime interval formulation not applying for normal Minkowski illustrations with 1 stationary and 1 moving observer. It in fact does apply, of course. Thinking one thing, talked about another.

You know, I had no idea what a Loedel diagram was. I made an assumption in my mind about the diagram I figured you had in mind (one used for the not-symmetric case when one traveler has motion with changing directions, etc.) and knew that you were correct in commenting that one would not use the symmetric diagram in that case.

It just doesn't work for Loedel's (geometrically per the diagram) if the 2 moving worldlines are not symmetric about the stationary ict axis.

I'm sure you realize that as long as each observer is moving at constant velocity, you can always find a rest system for which you can then use the symmetric diagram (just find the system for which the red guy and blue guy are moving in opposite directions at the same speed). I can do the twin paradox using a symmetric diagram for the trip out. I add in the hyperbola calibration curves, then use those same calibration curves with a second shifted rest system with increased speed for the guy making the return trip. The calibration curves show clearly the shorter 4-D path for the return trip.

I like the symmetric diagram (now I know to call it the Loedel diagram), because you can illustrate all kinds of seemingly paradoxical situations with it, like the pole-in-the-barn example below.

The best formulation of a spacetime interval, is one which presents the y-axis as well ... so an x,y,ict worldline illustration. The 2-d lightcone (versus a ray's lightpath) helps as well. It explains 2 things ... (1) the meaning of imaginary axes, and (2) why the magnitude of s is always less than the magnitude of ct.GrayGhost[/QUOTE]

Yes, I like that presentation as well.

 

[GrayGhost]

Bobc2,

Wrt your illustration at ...

https://www.physicsforums.com/showpost.php?p=3106843&postcount=39"​

When RED is at 9, BLUE is at 8 in RED's sense of NOW (cosmos wide). Yet, that very same BLUE fellow knows that "the only RED that presently exists" is over yonder with a clock readout of 7 (not 9). So whatever RED was doing before (two units of time prior to 9, mayby scratching his chin) is what is presently happening wrt BLUE, and its real, and RED (who is at 9) knows it per the theory. This suggests to RED (and anyone else) that there exists in-the-future a BLUE who holds RED himself presently at 9, and some future RED that holds said BLUE fellow in the present moment. All are correct, as all LT predictions are assumed real and not illusionary effect.

One conclusion that may be drawn is that an infinite number of REDs exist, each occupying some unique moment upon his worldline, all progressing steadily thru his path w/o knowledge of each other. IOWs, you exist everywhere along your lifecycle at once, and each moment of yourself progresses in unison at equal rate unaware of each other. This is what Brian Cox was taking about, an implication of relativity theory when taken at face value. The worldline sits there in the continuum, in its entirety, always.

OK, let the bullets fly :)

GrayGhost

 

[Phrak]

I missed this one before ...

Incorrect. I was talking SR. He who resides at both events always measures the least amount of time for the interval. If 2 observers reside at both events, eg in the twins scenario, then the one who travels the shorter path thru the continuum ages less over the interval, because he accrues less proper time.

I leave your education in the overtaxed hands of the PF staff.

 

[bobc2]

Bobc2,

Wrt your illustration at ...

https://www.physicsforums.com/showpost.php?p=3106843&postcount=39"​

When RED is at 9, BLUE is at 8 in RED's sense of NOW (cosmos wide). Yet, that very same BLUE fellow knows that "the only RED that presently exists" is over yonder with a clock readout of 7 (not 9). So whatever RED was doing before (two units of time prior to 9, mayby scratching his chin) is what is presently happening wrt BLUE, and its real, and RED (who is at 9) knows it per the theory. This suggests to RED (and anyone else) that there exists in-the-future a BLUE who holds RED himself presently at 9, and some future RED that holds said BLUE fellow in the present moment. All are correct, as all LT predictions are assumed real and not illusionary effect.

One conclusion that may be drawn is that an infinite number of REDs exist, each occupying some unique moment upon his worldline, all progressing steadily thru his path w/o knowledge of each other. IOWs, you exist everywhere along your lifecycle at once, and each moment of yourself progresses in unison at equal rate unaware of each other. This is what Brian Cox was taking about, an implication of relativity theory when taken at face value. The worldline sits there in the continuum, in its entirety, always.

OK, let the bullets fly :)

GrayGhost

Good job, GrayGhost. Some philophers/metaphysicists take from the 4-D universe picture that there must be something involving consciousness doing the moving along the world lines. And it has been suggested that there are at least two different models for this: 1) the one you just described (consciousness all along the world line) or 2) one global 3-D consciousness moving in a way that results in a single 3-D becoming experience for the separate observers (this one is particularly grotesque; it results in zombies along the world lines that are not participating in the single 3-D NOW).

Appologies for getting too far off topic and bringing in too many contrived ideas for a physics forum. I must say that I just have no idea what is really going on with time, consciousness, experience, or reality in general, either from the standpoint of ontology or epistemology. I think nature has much mystery yet to be unraveled (if indeed it is ever possible to unravel it). I do think the objective 4-D universe populated by 4-D objects is valuable as a pedagogical tool in understanding special relativity.

 

[Dale]

If you are going to talk about consciousness, ontology, and epistemology, please start a thread in the philosophy section. It does not belong here.

 

[GrayGhost]

bobc2,

Yes, you did bring in the use of philosophy and used the word metaphysics in your prior post. No apology necessary, as I personally have no issue with that. I myself was not addressing philosophy. I was addressing the theory from a science POV, although when discussing implications of the relativity, it always ventures near that gray area between known physics and the metaphysics. One can argue that to discuss the meaning of the theory, over and above the mere usage of existing formulae, is to engage in metaphysics. Yet, wrt my position (and Brian Cox's, as well as many others), I spoke only in relation to the existing accepted transformations. I was not speaking of metphysical topics in which the math has not been developed.

It should also be pointed out that Newton, Faraday, Maxwell, Einstein, Lorentz, and even Hawking did so. Every idea these great minds ever had was metaphysics, "until" they developed the math for their theory AND had it accepted by the leading scientific community at large. I mean if physics forums such as these existed back in 1905, just imagine what would have been thought of a patent clerk who claimed time slowed down relatively, or asked "what would it be like to ride a beam of light?". On the other hand, for every metaphysical idea, the vast vast majority of them are either incomplete or bogus, so I do understand why the forum here prefers such discussions don't happen. I personally do not believe I crossed that boundary in this thread, however I find that you seem as much interested in philosophy as in the physics (which is fine as well). So yes, the discussion went off target it seems, and the forum appears unhappy. The forum here does not wish to engage into the meaning of theories, but instead, only the usuage of the equations. No problem, I can comply.

One last comment on the matter ... when someone asks "are the contractions real or not", no one in this forum can adequately answer that question to satisfaction, because it would have to be deemed "metaphysics" to explain WHY a proper length and moving contracted length can exist concurrently in nature. IOWs, "look at the equations and solns" is the most likely response you'll get. Another typical response ... "because observers are allowed to disagree, given the 2 postulates true". Yet, no one ever takes the time to ponder how nature must be designed if those things are in fact true. It's easier to say ... "don't ask, just use the formulae". Those kind of responses alone pretty much guarantee few will ever grasp the meaning of it, even though they learn to toss the equations around well enough, and gain superficial understanding of the model. This is unsatisfactory IMO, however I'll comply with the forum's wishes. From here out, any posting I do in the physics forum here will possesses nothing but the equations, and the always typical verbal descriptions, to keep everyone happy.

GrayGhost