The Peripatetic Albert, Round 3

[ostren]

... I think you misunderstood. ...you can't say its the marker that's "really moving" (with respect to the spaceship) because its the spaceship, not the marker, that's undergoing acceleration..

I misunderstood nothing. When geometer said, "If one system is accelerating, even uniformly, with respect to another system it is no longer an inertial system, and you should be able to detect acclerations and thereby determine who is really moving." But acceleration does nothing to differentiate "who is really moving". There's no such determination. There's no preferred frame! If one pilot fires his thrusters and thence feels a G-force, that's no indication what*so*ever as to which of two arbitrary frames "is really moving". They are each moving relative to one another and acceleration cannot qualify that any better.

 

[geometer]

I misunderstood nothing. When geometer said, "If one system is accelerating, even uniformly, with respect to another system it is no longer an inertial system, and you should be able to detect acclerations and thereby determine who is really moving." But acceleration does nothing to differentiate "who is really moving". There's no such determination. There's no preferred frame! If one pilot fires his thrusters and thence feels a G-force, that's no indication what*so*ever as to which of two arbitrary frames "is really moving". They are each moving relative to one another and acceleration cannot qualify that any better.

I beg to differ ostren. You did misunderstand. Russ is right on. My point is that given two inertial coordinate systems, there is nothing you can do to tell which is moving relative to the other. However, as soon as one coordinate system experiences an acceleration, from whatever source, now you can tell who is moving with respect to whom. Accleration does differentiate which of two systems is moving with respect the other.

 

[geometer]

You can locally make acceleration disappear by changing the referential. Only locally. The free-fall observer not noticing the Earth gravitational field has a limited spave around him, otherwise he would notice the spherical shape.

Jumping back a few posts, I'd like to make a comment on this post also. Acceleration is actually a tensor of order 1. One of the main properties of tensors is that if they are non-zero in any coordinate system, they are non-zero in all coordinate systems. So, you can't make acceleration disappear by changing the reference frame.

 

[russ_watters]

ostren, if what you are saying were correct, there would be no possible resolution to Einstein's "twins paradox." You could accelerate a spaceship away from earth, claim Earth is the one accelerating while the spaceship is stationary, have the spaceship turn around and come back to earth, and calculate that the clock on Earth should have less elapsed time than the clock on the spaceship. But then you'd compare the clocks and find that its the clock on the spaceship that has less elapsed time.

And again, nothing in any of this has anything to do with absolute motion. You keep bringing it up, but it isn't what geometer meant and nothing I said implies it exists. Again, answering the question "who is really moving?" does not require an absolute frame of reference.

 

[ostren]

I have carefully read geometer's post #52 as well as Russ Watters' post #54 and I beg to differ with both. To geometer's claim that

..as soon as one coordinate system experiences an acceleration, from whatever source, now you can tell who is moving with respect to whom. Accleration does differentiate which of two systems is moving with respect the other.

I say you are way wrong and I stand by my post #51 unaltered. And your latter sentence above makes no sense: because each is obviously moving with respect to the other -- it's not lop-sided.

And to Russ's claim that

answering the question "who is really moving?" does not require an absolute frame of reference.

I have to ask what in tarnation you mean by the qualifier "really"?? Ah! some motion is "real" and other motion "imaginary".. is that your contention?

As for violating the twin paradox, Russ, you've got it all wrong. The turnaround twin's acceleration doesn't mean that time dilation is his alone because he is the one who is 'really' moving. Ooh, there's that funny word again.

No, the twin paradox can be resolved even assuming that the astronaut twin is STOCK STILL in space the entire time! For example, at my website, Addendum IV.

 

[kawikdx225]

I have carefully read geometer's post #52 as well as Russ Watters' post #54 and I beg to differ with both. To geometer's claim that I say you are way wrong and I stand by my post #51 unaltered. And your latter sentence above makes no sense: because each is obviously moving with respect to the other -- it's not lop-sided.

If you put an accelerometer on the spaceship and another on Earth you will find that it is "lop-sided" as the spaceship records acceleration and the Earth does not.

 

[ostren]

kawikdx225: This is about "who is really moving", a phrase employed by geometer and echoed by russ_watters. This is not about who is really experiencing a G-force.

 

[geometer]

And to Russ's claim that I have to ask what in tarnation you mean by the qualifier "really"?? Ah! some motion is "real" and other motion "imaginary".. is that your contention?

I don't mean to speak for you Russ, but what I meant by "really" refers back to the post that started this thread. In that post, OneEye commented that it was equally valid to say his turning of his car's steering wheel caused the Earth to turn as it was to say it caused his car to turn. He cited the equivalence prinicple from special relativity for this statement.

My comment was that by his turning of the steering wheel, he had introduced an acceleration into the picture, so the frame centered on and moving with his car was no longer an inertial frame and that the special relativity equivalence principle didn't apply any more. Further, since he felt the acceleration, it was possible for him to determine that it was actually he that was moving with respect to a frame stationary on the earth, and it was not the Earth that was moving with respect to a frame centered on his car.

 

[ostren]

Thank you for the compromise, geometer, but I was only partly right, I think. The car's brakes or gas pedal would produce acceleration that can NOT be differentiated from gravitation by means of the local study of light. But any rotational motion, any spinning, that component I do believe is confirmed by light's behavior to be unambiguous.

 

[geometer]

I think we still have a basic disagreement here ostren. Given a similar situation: If I am in my brand new Ferrari (hey, if we're going to pretend...) traveling down the freeway in a perfectly unaccelerated condition there is no way for me to tell whether I am moving with respect to the Earth outside my window or whether the Earth is moving and I am stationary. However, as soon as I step on the gas to pass Granny in her 57 Chevy I have introduced an acceleration into the picture and I can determine who is stationary with respect to whom.

It doesn't matter that I can't tell if the acceleration is due to gravity or to my pushing on the gas pedal; all accelerations produce equivalent effects. But, it does destroy my inertial status and enable me to differentiate the frame moving with me from any other inertial frame.

 

[ostren]

... as soon as I step on the gas .. I have introduced an acceleration into the picture and I can determine who is stationary with respect to whom.
.. it does destroy my inertial status and enable me to differentiate the frame moving with me from any other inertial frame.

Differentiate Ok, but NOT deduce anything of stationary (your word) versus moving. Please use scenarios in deep intergalactic space to make your point, and you'll see that it doesn't play, this "stationary" discernment of which you speak -- or that "really in motion" discernment of which you earlier spoke.

 

[reilly]

geometer says, " However, as soon as I step on the gas to pass Granny in her 57 Chevy I have introduced an acceleration into the picture and I can determine who is stationary with respect to whom."

Nothing could be further from the truth. Einstein would certainly concur. You see, the very definition of acceleration requires that if A is accelerating with respect to B, then B is accelerating with respect to A. Acceleration is, of course, the second time derivative of displacement, which is necessarily relative.

But, the asymmetry that lurks around this issue can be distrurbing. Not so ,with the car. The force that actually accelerates the 57 Chevy is equally and oppositely accelerating the Earth -- no way to tell who is on a special frame.

But, what about , say, a train and a car. In an observer's frame the train is in uniform motion, and the car accelerates. Clearly the car exerts no discernable force on the train. So if the car accelerates with repect to the train, where does the force come from to accelerate the train relative to the car? In fact such a force is frame dependent, much like a Corioulis(sp?) force. It's derived by using the coordinate transformation that brings the car to rest. When applied to the motion of the train, the time derivative of momentum brings in extra terms, some times called fictitious forces, which mimic the force necessary to give the train the proper acceleration in the rest frame of the car. Space invents the necessary force in order to keep the notion of motion strictly relative.

In other words, the very structure of space-time of physics completely precludes anything but pure, relative motion.

Regards,
Reilly Atkinson

 

[geometer]

Differentiate Ok, but NOT deduce anything of stationary (your word) versus moving. Please use scenarios in deep intergalactic space to make your point, and you'll see that it doesn't play, this "stationary" discernment of which you speak -- or that "really in motion" discernment of which you earlier spoke.

OK. Scenario 1. Consider two spaceships in deep space alongside each other, close enough that they can see each other. Initially, they are stationary with respect to each other. We note that "stationary with respect to each other" means that with respect to a third observer, they could be not moving with respect to each other or they could be moving with the same velocity with respect to each other. In this case, observers in the spaceships will not be able to tell if any motion is occurring.

Now, let's assume one of the spaceships experiences an acceleration, say from a programmed rocket firing. Now, an observer in that spaceship could detect that acceleration and from that information deduce that he/she is now in motion with respect to the other space ship.

Scenario 2. Now we assume only one spaceship in deep space. We further assume this spaceship represents an inertial frame. Under these conditions, an observer in that spaceship will not be able to tell if she/he is in motion or is stationary with respect to any other frame of reference. Now, assume that this spaceship experiences an acceleration. Again, the observer in the spaceship will be able to detect the acceleration and can deduce that he/she is in motion with respect to some other inertial frame. The spaceship might be speeding up, slowing down or turning, I'm not sure the observer aboard that ship could tell which, but she/he can tell some kind of motion is occurring.

Note that the key to these scenarios is that an observer in an isolated lab can detect accelerations. I find support for this statement in "Concepts of Modern Physics, Fourth Edition," by Arthur Beiser, where he states "The general theory of relativity, developed by Einstein a decade later, treats problems that involve frames of reference accelerated with respect to one another. An observer in an isolated lab can detect accelerations."

The fact that you can uambiguously determine, in non-inertial conditions, the state of motion of one frame with respect to another does not imply the existence of a preferred frame. The laws of physics are the same in any frame you care to examine. (the Strong Equivalence Principle).

 

[russ_watters]

Differentiate Ok, but NOT deduce anything of stationary (your word) versus moving. Please use scenarios in deep intergalactic space to make your point, and you'll see that it doesn't play, this "stationary" discernment of which you speak -- or that "really in motion" discernment of which you earlier spoke.

If I fire my engine and I start to feel an acceleration, and I check an accelerometer on the marker I just dropped off my ship and see that it is not accelerating, I most certainly can deduce several things:

-There is no gravity field affecting these results (I consider it unreasonable to assume a gravity field coincidentally appeared at the instant I fired my rocket).
-I am accelerating, the marker is not.
-I am moving with respect to the marker, not the other way around.

Certainly, there are a lot of calculations that work fine assuming either to be accelerating (calculating the distance, for example), but not every one makes sense that way.

I have to ask what in tarnation you mean by the qualifier "really"?? Ah! some motion is "real" and other motion "imaginary".. is that your contention?

How many times do I have to say this before you accept it? All motion is relative. The words "real" and "imaginary" have nothing to do with anything.

Heck, I'm not even saying that you can't consider either stationary in your calculations, if you want to be pedantic. But it makes for much more complicated calculations since you now have to add forces that didn't exist before: to consider the marker to be moving and the spaceship stationary, you need add arbitrary forces to both. You need to add a force that cancels the force of the rocket while accelerating the marker: for example, a planet materializing out of nowhere at the exact instant the rocket started firing. Of course, if you do that, you have just added a 3rd reference frame with which to define the rocket as stationary...

No, the twin paradox can be resolved even assuming that the astronaut twin is STOCK STILL in space the entire time! For example, at my website, Addendum IV.

Is your explanation the same as Einstein's?

The force that actually accelerates the 57 Chevy is equally and oppositely accelerating the Earth -- no way to tell who is on a special frame.

Minor nitpick, reilly - the force is equal and opposite, the acceleration is not. f=ma, and the Earth is a lot more "m" than that '57 chevy. Its good to bring us back to that example though: in the rocket example, the force of the engine acts on the rocket alone and a gravitational pull would act on the rocket and marker proportionally. In the car example, you have only one force and it violates f=ma to say that its the Earth that is accelerated due to that force alone.

 

[russ_watters]

Let me try my point another way:

If you have 2 objects and the distance between them is changing, you can reasonably say that either is "moving."

If the rate of that change in disance is changing, you can reasonably say that either is "accelerating"

If a force is measured between the two objects, it is now unreasonable to choose one of them arbitrarily and say it is accelerating and the other is not.

In the car example, a force exists and is measurable that cannot cause the measured acceleration of the earth.

 

[humanino]

If you have 2 objects and the distance between them is changing, you can reasonably say that either is "moving."

Un less the space between them is expanding...

 

[kawikdx225]

But, the asymmetry that lurks around this issue can be distrurbing. Not so ,with the car. The force that actually accelerates the 57 Chevy is equally and oppositely accelerating the Earth -- no way to tell who is on a special frame.

You say there is no way to tell who is on a special frame. If this is true I could do a physics experiment in the accelerating car and on the Earth and the results would be identical. This is not true.

Lets try another thought experiment.
Two cars are sitting at a red light.
Granny in her 57 Chevy and some punk kid in a hotrod.
Each car has a passenger that is tossing a quarter in the air then catching it.
When the light turns green the kid in the hodrod hits the throttle but granny fell asleep.
The passenger in granny's car notices no change in his physics experiment (tossing and catching the quarter) but the passenger in the hotrod must now make a correction to catch his quarter.

 

[ostren]

OK. Scenario 1. Consider two spaceships in deep space alongside each other, close enough that they can see each other. Initially, they are stationary with respect to each other. We note that "stationary with respect to each other" means that with respect to a third observer, they could be not moving with respect to each other or they could be moving with the same velocity with respect to each other. In this case, observers in the spaceships will not be able to tell if any motion is occurring.

Now, let's assume one of the spaceships experiences an acceleration, say from a programmed rocket firing. Now, an observer in that spaceship could detect that acceleration and from that information deduce that he/she is now in motion with respect to the other space ship.

Any such deduction is nonsequitor. Perhaps the deducer doesn't understand relativity. If he is now in motion with respect to the other ship, then the other ship is equally in motion with respect to his own. Per relativity, it is proper to attribute the G-force (of acceleration) to an unusual passing gravitational field. That may sound whacky, but it makes computations easier. You cannot ascribe relative motion to be lop-sided. The motion between the two ships is relative and utterly mutual.

2. Now we assume only one spaceship in deep space. We further assume this spaceship represents an inertial frame. Under these conditions, an observer in that spaceship will not be able to tell if she/he is in motion or is stationary with respect to any other frame of reference. Now, assume that this spaceship experiences an acceleration. Again, the observer in the spaceship will be able to detect the acceleration and can deduce that he/she is in motion with respect to some other inertial frame. The spaceship might be speeding up, slowing down or turning, I'm not sure the observer aboard that ship could tell which, but she/he can tell some kind of motion is occurring.

No he cannot determine unequivocal motion because there is no unequivocal 'background' frame with respect to which he could say he is moving. Sorry, but that's the essence of relativty.

You started all this bickering by using the phrase, "can determine who is really moving", which doesn't fly because there's no viable definition of the word "really" in that phrase.

You saw the post by Reilly Atkinson, and he seconds my opinion. He is obviously more credentialed than I am.

 

[pervect]

Let me try my point another way:

If you have 2 objects and the distance between them is changing, you can reasonably say that either is "moving."

If the rate of that change in disance is changing, you can reasonably say that either is "accelerating"

If a force is measured between the two objects, it is now unreasonable to choose one of them arbitrarily and say it is accelerating and the other is not.

In the car example, a force exists and is measurable that cannot cause the measured acceleration of the earth.

The way I see it, the existence of a force between the objects isn't really the issue - the issue is the application of Newton's laws.. One is perfectly free to adopt any coordinate system one wants - but one should not expect Newton's laws to work in such an arbitrary coordinate system. Measuring a force or forces between the objects isn't the big issue here, IMO, the big issue is applying the formula f=ma.

As others have pointed out, an observer can tell whether or not f=ma "works" for them just by doing local experiments, this can be done without thinking about the forces between the two bodies.

 

[geometer]

Granny in her 57 Chevy and some punk kid in a hotrod.

Are you calling me a punk kid?

 

[geometer]

As others have pointed out, an observer can tell whether or not f=ma "works" for them just by doing local experiments, this can be done without thinking about the forces between the two bodies.

This is my point pervect. Since I can detect acclerations in my frame, I can therefore apply f=ma in my frame and conclude that I am moving. It may be difficult for me to tell if I am speeding up, or slowing down or turning depending on what other data I have available, but I can tell I am moving.

Depending on what data I have about the acceleration it may also be impossible for me to tell if I have suddenly encountered a gravitational field or the acceleration came from another source

 

[ostren]

If I fire my engine and I start to feel an acceleration, and I check an accelerometer on the marker I just dropped off my ship and see that it is not accelerating, I most certainly can deduce several things:

-There is no gravity field affecting these results (I consider it unreasonable to assume a gravity field coincidentally appeared at the instant I fired my rocket).

YOU consider it "unreasonable", yet relativists consider it not only reasonable, but necessary.

-I am accelerating, the marker is not.
-I am moving with respect to the marker, not the other way around.

No, the motion is perfectly mutual. You are drifting back into pre-relativity recidivism.

How many times do I have to say this before you accept it? All motion is relative. The words "real" and "imaginary" have nothing to do with anything.

Right... so you shouldn't go around saying things like in your post #50

"who is really moving" is talking about which of two objects is moving with respect to the other

There's no "which" to it; the motion is mutual!

And you shouldn't go around saying things like in your post #54

answering the question "who is really moving?" does not require an absolute frame of reference.

There's no "really" to it; whaddya mean by that silly word?

Is your explanation the same as Einstein's?

Refering to my website Addendum IV -- yes, it's the same as Einstein's.

 

[ostren]

You say there is no way to tell who is on a special frame. If this is true I could do a physics experiment in the accelerating car and on the Earth and the results would be identical. This is not true.

Lets try another thought experiment.
Two cars are sitting at a red light.
Granny in her 57 Chevy and some punk kid in a hotrod.
Each car has a passenger that is tossing a quarter in the air then catching it.
When the light turns green the kid in the hodrod hits the throttle but granny fell asleep.
The passenger in granny's car notices no change in his physics experiment (tossing and catching the quarter) but the passenger in the hotrod must now make a correction to catch his quarter.

In the hotrod, the quarter is tugged toward the rear. But the driver feels a tug toward the rear as well. And if you carefully examined a ray of light moving transverse to the road, it would appear to the hotrod's occupants to be tugged in that very same direction. So YES, there is asymmetry, but it is perfectly consistent with the presence of a gravitational field. Under relativity, you must ascribe the asymmetry to gravity, rather than claim someone is "really" in motion. I would have to think much too hard to tell you why... perhaps it's because a claim of unequivocal "motion" entails too too many confusing ramifications.

 

[kawikdx225]

In the hotrod, the quarter is tugged toward the rear. But the driver feels a tug toward the rear as well. And if you carefully examined a ray of light moving transverse to the road, it would appear to the hotrod's occupants to be tugged in that very same direction. So YES, there is asymmetry, but it is perfectly consistent with the presence of a gravitational field. Under relativity, you must ascribe the asymmetry to gravity, rather than claim someone is "really" in motion. I would have to think much too hard to tell you why... perhaps it's because a claim of unequivocal "motion" entails too too many confusing ramifications.

hmmmm... brain hurting!
Well if what you say is true then (as Russ said) the twin paradox would not be a paradox and both twins would be the same age after the trip. This disagrees with Einstein. Or am I missing something.

Here's what I think I know.
"I am really moving" is a meaningless statement
"I am really moving with respect to" is valid
"I am really accelerating" is valid

motion is relative
acceleration is absolute

geometer:
lol, no. just tryin to keep it colorful.

 

[ostren]

... Well if what you say is true then (as Russ said) the twin paradox would not be a paradox and both twins would be the same age after the trip. This disagrees with Einstein. Or am I missing something.

Thanks for keeping it colorful!

What you're probably missing about the Twin paradox is that the Lorentz tranform mandates more than just length contraction and time dilation (each of which is direction independent), but also a third element, that of time dissynchronicity, which is direction dependent. Actually there are a handful of adequate resolutions of the Twin Paradox, but of course none of them could be based on the astronaut twin being really*truly the one in motion. Didn't you check out my version yet?

 

[ostren]

... Minor nitpick, reilly - the force is equal and opposite, the acceleration is not. f=ma, and the Earth is a lot more "m" than that '57 chevy. Its good to bring us back to that example though: in the rocket example, the force of the engine acts on the rocket alone and a gravitational pull would act on the rocket and marker proportionally. ..

That's directed to Reilly but I wish to venture this contribution. The gravitational potential gradient that mysteriously arises is precisely of such steepness as to stall your rocket -- ie. counter the engine thrust -- and accelerate the marker just so, as it is thence perceived to travel. It may be helpful to recall that the deflections/trajectories caused by gravity acting on an object do not depend on that object's MASS. Tower of Pisa, y'know?

 

[geometer]

We seem to be really hung up on the word really. What I really meant to really say was that in OneEye's original example he could conclude that he was moving with respect to the Earth and not vice-versa.

Ok Ostren - let's assume we are in a totally isolated laboratory in deep space, initially in an inertial condition. This laboratory has no visual communication with the space outside, and is empty except for you, me and an acclerometer of some kind. We now experience an acceleration (let's assume it's due to a preprogrammed rocket firing just for definiteness). What can we now conclude about our state of motion?

 

[ostren]

*** Nada ***

 

[geometer]

*** Nada ***

Why can I not apply f = ma and conclude that I am moving?

 

[pervect]

We seem to be really hung up on the word really.

Yep. And the word "truly" too.

What I really meant to really say was that in OneEye's original example he could conclude that he was moving with respect to the Earth and not vice-versa.

Really and truly? :-)

It seems to me that this thread has been about 50% arguing over semantics, and the other 50% over philosophy. I think everyone knows what happens next in all of the examples given, it's just a big argument about how to describe it.

 

[ostren]

Why can I not apply f = ma and conclude that I am moving?

I think that it may be because then relativity would not be such an elegant and consummate theory of physics. Even if you apply f=ma, as I said earlier that might only indicate that you are DEcelerating (to a stop?). There is too much ambiguity and yet if you stick to the application of relativity's Equivalence Principle, then nothing is left to be ambiguous. This is my best quick reply although perhaps a bit wanting :-)

 

[HallsofIvy]

Why can I not apply f = ma and conclude that I am moving?

You could apply f= ma and conclude (if f is not 0) that you are accelerating. The point is (and goes back to "Galilean Relativity") that force (which what you "feel") is proportional to acceleration, not velocity so you couldn't use that to determine your velocity- or even whether it is 0 or not.

It was the discovery that eletro-magnetic field DO depend on velocity rather than acceleration that led to the idea that we could use some sort of electro-magnetic experiments (i.e. light) to determine and "absolute" velocity. Relativity developed out of the fact that those experiments still didn't find anything!

 

[kawikdx225]

I don't think anyone was saying that you could determine your absolute velocity by studying your acceleration.
The argument is that if you know for a fact that you are accelerating (by use of an accelerometer) then doesn't that imply you are moving. Since acceleration is by defination a "change in velocity with respect to time" meter/second² ?

In other words, how can you accelerate if you are not moving? Granted you don't know how fast your velocity is or if you are speeding up or slowing down or just turning in a circle but you are moving.

 

[selfAdjoint]

You stomp on the gas and feel that push in your back. Must be moving! You look out the window and the telephone poles are standing still. Whaaaat?

You're driving on an endless belt that your spinning wheels drive backwards. Or, make up your own explanation. Acceleration does not always produce motion.

 

[geometer]

You stomp on the gas and feel that push in your back. Must be moving! You look out the window and the telephone poles are standing still. Whaaaat?

You're driving on an endless belt that your spinning wheels drive backwards. Or, make up your own explanation. Acceleration does not always produce motion.

Wait a minute! In this case you haven't accelerated yourself. You've acclerated the belt so you wouldn't detect the accleration. And, this acceleration did produce motion - it increased the rate at which the belt moves underneath you.

 

[selfAdjoint]

But the belt is running friction free on a planetoid that JUST HAPPENS to be rotating in the opposite direction.

The point, as I indicated, isn't the particular mechanism, but the general principle that the push you feel or don't feel has nothing to do, in and of itself alone, with how you are or are not moving. You have to look at the larger picture, or make reasonable assumptions like the designers of inertial guidance systems.

 

[kawikdx225]

But the belt is running friction free on a planetoid that JUST HAPPENS to be rotating in the opposite direction.

The point, as I indicated, isn't the particular mechanism, but the general principle that the push you feel or don't feel has nothing to do, in and of itself alone, with how you are or are not moving. You have to look at the larger picture, or make reasonable assumptions like the designers of inertial guidance systems.

geometer is right.
In your thought experiment an accelerometer in the car would measure 0 meters/second² therefore you are not accelerating.

Anytime your accelerometer reads 0 m/s²you cannot say"I am in motion" you must say "I am in motion relative to".

Anytime your accelerometer reads anything other than 0 m/s² you can say "I am in motion but I have no idea how fast or in which direction"

 

[ostren]

Kudos to geometer & kawikdx225 for finding selfAdjoint's error. The driver would NOT feel any push in his back in that scenario.

 

[geometer]

Seems like we have kind of run out of steam here, but it's been a great discussion and I think, has sharpened my thinking on this. Thanks!