Triangle GR thought experiment

[kamenjar]

Hi all!
I am still a GR noob and this example has been puzzling me for a while, so I need some help that's hopfully.

Let's imagine that there are stars A and B at 10 LY from Earth (at point C) each and 10 LY from each other.

A signal is sent every year from Earth (C) indicating the beginning of a new Earth year.
Stars A and B would receive such signal at the same time and "synchronize" their clocks.
A twin (Mr. X) is sent on a rocket from A to B carrying the same clock-device and sending the signal out.
Another twin (Mr. Y) is left at A
When X receives a signal, he sets out on a trip "instantly" accelerating to .866 light speed, decelerating to stop at B.
X immediately returns to A in a similar manner.

Contradiction and my limited understanding of GR implies least one of these statements has an error or is an over/under-statement:
(1) After after the trip from A, X arrives at B. During this trip, X should have received 10+10=20 signals from Earth (c).
(2) After after the trip from A, X arrives at B. During this trip, X ages 10 years.
(3) After the return trip A-B-A, X arrives at B. During this trip, X should have received 20+20=40 signals from Earth (c).
(4) During the trip A-B, X observes that its time is running "slower" with respect to C because it received 20 signals while only generating 10.
(5) During the trip A-B, Earth observer at C observes that X's time is "slower" with respect to C because it received 20 signals from X during the period of 40 years.
(6) During the return trip A-B-A, Y observes that X's time is running "slower" with respect to Y because it received 40 signals from Earth and counted 40 years, while only getting 20 signals from X during their trip.
(7) Wikipedia is wrong to state "Time dilation due to relative velocity symmetric between observers"
(8) X's biological clock matches the clock that X carries - clock which is generating signals

Which statement is in error, or is over/under-stated?

 

[JesseM]

Hi all!
I am still a GR noob and this example has been puzzling me for a while, so I need some help that's hopfully.

Let's imagine that there are stars A and B at 10 LY from Earth (at point C) each and 10 LY from each other.

A signal is sent every year from Earth (C) indicating the beginning of a new Earth year.
Stars A and B would receive such signal at the same time and "synchronize" their clocks.
A twin (Mr. X) is sent on a rocket from A to B carrying the same clock-device and sending the signal out.
Another twin (Mr. Y) is left at A
When X receives a signal, he sets out on a trip "instantly" accelerating to .866 light speed, decelerating to stop at B.
X immediately returns to A in a similar manner.

Contradiction and my limited understanding of GR implies least one of these statements has an error or is an over/under-statement:
(1) After after the trip from A, X arrives at B. During this trip, X should have received 10+10=20 signals from Earth (c).

What led you to conclude that? Say in the rest frame of the stars, the first signal arrives at A in 2000, and that's when X sets out towards B at 0.866c in this frame. Since the distance from A to B is 10 light years, it will take X a time of 10/0.866 = 11.547 years to reach B in this frame, so he'll arrive in 2011.547. Since X has received the same number of signals that were received by A and B (during his trip he received each signal slightly before A and B since his distance to C was smaller, but the total number is the same), X has received 11 signals when he arrives at B.

(2) After after the trip from A, X arrives at B. During this trip, X ages 10 years.

If the trip took 11.547 years in the rest frame of A-B-C, and X's aging was slowed by a factor of 2 during the trip, then X aged 11.547/2 = 5.7735 years during the trip.

(3) After the return trip A-B-A, X arrives at B. During this trip, X should have received 20+20=40 signals from Earth (c).

Takes X another 11.547 years in the A-B-C frame to get back, so a total of 23.094 years, meaning X has received 23 signals.

(4) During the trip A-B, X observes that its time is running "slower" with respect to C because it received 20 signals while only generating 10.

X only sends out a signal once every 2 years in the A-B-C frame, but the distance from X to C is different for each one, so it's a bit complicated to figure out how many signals C has received at the time X makes it to B. Also, because of the relativity of simultaneity, the answer to the question "how many signals has C received at the same moment that X reaches B" will depend on your choice of frame, it would be different in the frame where X is at rest during the journey from A to B than it would be in the frame where A-B-C are at rest, since these frames disagree about what event on C's worldline is simultaneous with the event of X arriving at B.

(5) During the trip A-B, Earth observer at C observes that X's time is "slower" with respect to C because it received 20 signals from X during the period of 40 years.
(6) During the return trip A-B-A, Y observes that X's time is running "slower" with respect to Y because it received 40 signals from Earth and counted 40 years, while only getting 20 signals from X during their trip.

Again, your numbers are wrong, it would take some work to figure out the right ones and first you'd have to specify what frame's definition of simultaneity you want to use.

(7) Wikipedia is wrong to state "Time dilation due to relative velocity symmetric between observers"

It's symmetric between inertial observers, it no longer becomes symmetric if X stops moving inertially and accelerates to turn around. If X was just moving in a straight line forever at constant speed, C's view of X's signals would be just like X's view of C's, though the rate of signals would depend on the amount of time before or after the moment of closest approach due to the Doppler effect which increases the perceived frequency of signals when an object is coming towards you and decreases the rate when it's moving away from you (the Doppler effect has to do with the fact that different signals are emitted at different distances and therefore take different times to reach the other observer--time dilation is what's left when you factor out these signal delays due to the finite speed of light)

(8) X's biological clock matches the clock that X carries - clock which is generating signals

Yes, that's true.

 

[ghwellsjr]

First off, this is not at GR problem, it is just an SR problem.

Jesse, you have done a good job of setting the record straight with regard to the timing of events, namely that it will take X just over 23 years to make the round trip but since his clock is running at half speed, he will have sent out half that many signals or about 11. But kamenjar is stating in points 4, 5 and 6 that each observer agrees that X's clock is running at half speed and he is wondering where's the time dilation symmetry?

He did not explicity state anything about a frame of reference but he implied one when he described and analyzed the situation in the one frame of reference in which A, B and C are at rest. And this is the proper way to do it. There will never be any ambiguity or confusion or paradox when an entire scenario is described and analyzed according to a single frame of reference.

There was no need to bring up another frame of reference when discussing his point 4, although what you say is true, especially the point about it being very difficult to transform a scenario from one frame of reference to another. What I would point out is that the observer at C will not see the signals from X until long after the entire trip is over because it takes time for the signals to travel the approximate ten light years from X to C. But kamenjar is right, even though his numbers are wrong, that C will receive far fewer signal from X than C is sending out in the same interval of time, so C knows that X's clock is running slow. And for similar reasons, X knows that his clock is running slower, and Y knows that X's clock is running slower. Everybody agrees and there is no ambiguity or confusion or paradox because it is all being discussed and analyzed in a single frame of reference.

So the answer to kamenjar's question has nothing to do with who is traveling at what speed relative to another person, it has to do with analyzing a situation from one frame of reference and getting one definition of which clock is running slow (X's in this case because he is the only one traveling) compared to analyzing the same situation from another frame of reference where different observers are now traveling. Whoever is traveling in the frame of reference which is being used to analyze the situation is the one(s) whose clock(s) is running slow. The faster they are traveling, the slower their clocks are ticking. It's really very simple.

 

[JesseM]

But kamenjar is stating in points 4, 5 and 6 that each observer agrees that X's clock is running at half speed and he is wondering where's the time dilation symmetry?

He did not explicity state anything about a frame of reference but he implied one when he described and analyzed the situation in the one frame of reference in which A, B and C are at rest. And this is the proper way to do it. There will never be any ambiguity or confusion or paradox when an entire scenario is described and analyzed according to a single frame of reference.

I don't think he was talking about a frame of reference at all, I think he was arguing that there'd be an asymmetry just in terms of how frequently C got signals from X vs. how frequently X got signals from C (each using their own clock to time the frequency of incoming signals from the other), which is a frame-independent issue. This might be true if X turns around, but as long as both X and C are inertial there should be no asymmetry.

 

[ghwellsjr]

Agreed, he wasn't talking about a frame of reference (that's what I said) but he also wasn't making any argument. He was simply making eight statements from his noob point of view and realizing that some of them must be in error but he doesn't know which ones. He's asking us to tell him which ones are in error. It turns out that every one of them except the last one is in error, as you have pointed out.

But his seventh point is his important one because he is making an observation based on the previous six, which although they are in error in the details, they still lead him to the correct observation in number seven, namely that there is no time dilation symmetry because he (inadvertantly) described the scenario from one consistent point of view (or frame of reference).

Suppose he had all the numbers correct in his first six points and still made the observation that there is no time dilation symmetry in spite of what the wikipedia says. How would you have responded to him then? My response is that the time dilation symmetry only comes about when you are using two different frames of reference to describe the same situation.

For example, let's take the same Mr. X and Mr. Y (who are twins according to kamenjar). We will forget about A, B and C. The two of them are traveling away from each other at .866 c. First we analyze the situation from the frame of reference where Mr. Y is stationary and we see that Mr. X's clock is running at half speed. Now we start over again and analyze the same exact situation from the frame of reference where Mr. X is stationary and we see that Mr. Y's clock is running at half speed. And now we have demonstrated the symmetrical time dilation that the wikipedia is talking about.

 

[kamenjar]

You guys simply did an awesome job correcting my calculation mistakes - I got a little carried away in thought .866 means 2x dilation and made a mistake there. Now I get that...

My main issue was with (7) actually, so that's why I came up with the example...

ghwellsjr almost got to the point, C conclides that X's clock is running slow but...

Y's clock is running at half speed. And now we have demonstrated the symmetrical time dilation that the wikipedia is talking about.

Y (I should've used A) is not sending the signlas, so that X's percepion of them is "distorted". We use C in this case to use as a "Clock reference". The assumption is that X's relative movement toward's C is not significant enough to cause huge dilation so that X can logically conclude that C's clock is running slow, and therefore Y's (A's) clock is running slow.

X is counting signals from C. He's making a logical deduction based on that. We all agree that it is 23 signals for the round trip. In a way, X observes 23 Earth rotations around the sun through the window of the ship while it "thinks" that 11 years of X's time have passed (right?) Or does it receive most of the C's signals during that instant acceleration ?

I would assume that most of C's signals are coming during the "inertial" motion. With that in mind (if it is correct) I can not see why even considering relative slight motion towards and away from C would distort X's perception to cause X to logically conclude that C's clock is running slower.

The best way for me would maybe be for someone to try to "explain a picture" of what X sees across the window side window while observing Earth do 1-year revolutions around the Sun and on the "signal ticker" X is reading.

So a bit more detail from edit:
X must get a total of 23 signals for the round trip.
It should (or could it not?) be the case that X receives them in "regular" intervals. When I say regular intervals, I _think_ that it doesn't happen that C's signal intervals are received in less than 12 of Y's months. Even if it is like 10 months, Y should be able to logically compensate.
I guess the main point that I don't get is why is X's perception about C's clock so distorted that X can't conclude that C's clock is running faster.

Maybe it's just a "human thinking thing"... If I could imagine myself traveling near the speed of light, I can understand that the world in front of me is blueshifted (aging faster), and behind me redshifted (ticking slower), so to get the real idea on how fast I am going (or how slowed my time is), I look out the side window and see things "happen faster than usual" and also blueshifted slightly.

 

[Ich]

The best way for me would maybe be for someone to try to "explain a picture" of what X sees across the window side window while observing Earth do 1-year revolutions around the Sun and on the "signal ticker" X is reading.

I think the most important point is that X doesn't see C through the side window, but rather through the front window.
Due to aberration, he sees C approaching him at an angle, even if it is straight to his left or right at that moment.
Naturally, an approaching object's light is blueshifted (=shorter intervals). This effect exactly reverses the ratio that you'd expect from time dilation alone.
As X knows about aberration, he'd correct his observations for it and conclude that C is time dilated.

 

[ghwellsjr]

Let me first answer your specific questions and then I'll address your overall concern:

In a way, X observes 23 Earth rotations around the sun through the window of the ship while it "thinks" that 11 years of X's time have passed (right?)

Correct.

Or does it receive most of the C's signals during that instant acceleration ?

No.

X must get a total of 23 signals for the round trip.
It should (or could it not?) be the case that X receives them in "regular" intervals.

Regular intervals.

Maybe it's just a "human thinking thing"... If I could imagine myself traveling near the speed of light, I can understand that the world in front of me is blueshifted (aging faster), and behind me redshifted (ticking slower), so to get the real idea on how fast I am going (or how slowed my time is), I look out the side window and see things "happen faster than usual" and also blueshifted slightly.

Now I can see where you are coming from and why you presented your thought experiment the way you did.

You should not think that time dilation is merely caused by a doppler shift which is when the frequency of a repetitive signal arrives at a higher rate when the source and destination are getting closer together (aging faster, as you said) and when the frequency of the repetitive signal arrives at a lower rate when the source and destination are getting farther apart (ticking slower, as you said). If that were the case, then your scenario makes sense because you are trying to minimize the doppler shift by having Mr. X travel at approximately right angles to the direction of the signals between him and the earth.

The first thing wrong with your thinking is that the time dilation is related to the direction of motion. It is not. But you are correct that the apparent frequency shift is related to the direction of motion and you might think that it would be best to minimize this to see the real time dilation but that is the more difficult situation in which to understand what is going on because the doppler shift is constantly changing throughout the experiment and the analysis is correspondingly cumbersome.

If you think just about two observers traveling in a straight line that goes between them (your Mr. X and Mr. Y), then the doppler shifts due to their motions becomes fixed values (as long as they are always moving away from each other at a constant relative speed). If there were no time dilation, then I think it might be easy to understand that what each one will observe when receiving the signals from the other person would not be symmetrical. They would both be emitting a repetitive signal at the same rate (and we can assume at the same times) but Mr. X moves away so that by the time he detects the signals from Mr. Y it will take longer than for Mr. Y to detect the signals that were emitted by Mr. X at the same times. So the doppler frequencies without time dilation would not be symmetrical. (If you need more help on this, look up "Doppler effect" in wikipedia.)

But now consider what happens if the moving one, Mr. X, is experiencing time dilation. He will be emitting the signals at a slower rate and so Mr. Y will be receiving them at a slower rate than he would have without time dilation. And Mr. X will be receiving the signals from Mr. Y at a "faster" rate than he would without time dilation because his clock is taking longer to tick. The net effect is that with time dilation of the moving observer, both observers will measure the tick rate (doppler frequency) of the other observer symmetrically, that is, at the same perceived rate. (If you need more help on this, look up "Relativistic Doppler effect" in wikipedia.)

So relativistic doppler is the indirect means by which we can actually measure symmetrical time dilation. It is very difficult to otherwise measure or demonstrate that time dilation is symmetrical. In my analysis of your scenario presented earlier, it was only because of the round trip of Mr. X that I could legitmately make the statement that he was the one who had experienced time dilation and not any of the other observers.

 

[kamenjar]

I think the most important point is that X doesn't see C through the side window, but rather through the front window.
Due to aberration, he sees C approaching him at an angle, even if it is straight to his left or right at that moment.
Naturally, an approaching object's light is blueshifted (=shorter intervals). This effect exactly reverses the ratio that you'd expect from time dilation alone.
As X knows about aberration, he'd correct his observations for it and conclude that C is time dilated.

Great explanation! This really helped me "visualize" dilated travel. I want more! Got any book recommendations? (less formulas more logic and words) :)

By the way, what is the observed redshift of "stationary" objects directly to side of the ship? Blue, red, or neutral?

Also, so near planets appear as "squished balls" depending on the observation angle?

...
So relativistic doppler is the indirect means by which we can actually measure symmetrical time dilation. It is very difficult to otherwise measure or demonstrate that time dilation is symmetrical. In my analysis of your scenario presented earlier, it was only because of the round trip of Mr. X that I could legitmately make the statement that he was the one who had experienced time dilation and not any of the other observers.

Thanks, I think I understand now.

Does that mean that there would be a method like "triangulation" by which a "blinded" C and Y could determine to make fairly accurate conclusions about their relative positions or their own time dilation? Maybe using CMB? It almost sounds like the missing information problem.

 

[ghwellsjr]

Does that mean that there would be a method like "triangulation" by which a "blinded" C and Y could determine to make fairly accurate conclusions about their relative positions or their own time dilation? Maybe using CMB? It almost sounds like the missing information problem.

I'm not sure I understand your questions but at least for drawing conclusions about your own time dilation you can always conclude that as long as you are not accelerating, you will not be able to measure or sense any time dilation, no matter what your past history of acceleration is. How could you? All your clocks and aging processes are affected in the same way so whatever you use to measure time will always look the same to you.

But if you want, you can define a frame of reference by a point in time when you are not accelerating, and then measure your acceleration until you reach a speed with respect to your original frame of reference (or measure it some other way) and say that you are now experiencing time dilation but you won't be able to detect it. And you can say that another observer who is still at rest in the original reference frame is not experiencing time dilation. In fact, that is what I thought you were proposing in your original thought experiment.

I'm not sure what you mean by "blinded" but if you mean the ordinary kind of triangulation that is used in astronomy to measure distant objects, then yes, that is a legitimate way to determine distances up to a certain point.

If you are asking if the CMB (Cosmic Background Radiation) is a way to determine an absolute rest point, the answer is no, because it has a black body spectrum which means that it will have an identical spectrum when viewed from any speed. If you measure the spectrum and then accelerate to any arbitrary high speed and measure it again, it will look identical so there is no way to use that information to conclude when you are at rest with respect to the CMB.

And I never heard of the missing information problem. What's that?

 

[kamenjar]

Per what Ich pointed out I think that this is the point:

For the example when X is moving away from Y, each sees each other's clock to be slower when they compesate for each other's redshifts and perceived distances. Because X is the one whose time is in fact running slower, unless they know that fact, there is no way for them to compensate correctly. So that is the theory of relativity means in this case.

 

[ghwellsjr]

No, the only reason we can say that X is the one whose time is "in fact" running slower, is because he made a round trip and came back to rest where he started in the frame in which we are discussing the situation. While he is traveling with respect to Y (and C), if Y were to send the same signal out that X is sending out, they would both measure exactly the same relativistic doppler frequency. We have to make a distinction between what is perceived and measureable and what we declare based on our frame of reference. We can analyze the entire situation for everbody from start to finish from any arbitrary frame of reference (that's the point of SR) and get the same answers for things that have an answer, like comparing the clocks between X and Y after they reunite.

 

[Passionflower]

If you are asking if the CMB (Cosmic Background Radiation) is a way to determine an absolute rest point, the answer is no, because it has a black body spectrum which means that it will have an identical spectrum when viewed from any speed. If you measure the spectrum and then accelerate to any arbitrary high speed and measure it again, it will look identical so there is no way to use that information to conclude when you are at rest with respect to the CMB.

So are you saying that we cannot determine for instance the velocity of the Earth wrt the CMB?

 

[JesseM]

Per what Ich pointed out I think that this is the point:

For the example when X is moving away from Y, each sees each other's clock to be slower when they compesate for each other's redshifts and perceived distances. Because X is the one whose time is in fact running slower, unless they know that fact, there is no way for them to compensate correctly. So that is the theory of relativity means in this case.

In the theory of relativity there is no frame-independent physical fact about "whose time is running slower", different inertial frames say different things about whose clock runs slower, and all these frames make identical predictions about all local facts like instrument readings, what two clocks read at the moment they meet at the same position, etc. (also the equations of the laws of physics look the same in all inertial frames, so that's another reason there's no physical basis for judging one frame to be more correct than another)

 

[Ich]

I want more! Got any book recommendations? (less formulas more logic and words) :)

Sorry, no recommendations. What I'm talking about is aberration and transverse doppler effect, if you want to look up these things.
Aberration is quite simple and logical: you see the object now exactly where it was when it emitted the light that reaches you now.
For an approaching object:
When it's exactly side by side with you (at some disctance), you see it where it was when it sent the light: still in front of you.

It's true that for X, C is time dilated. But because X sees C approaching, there is additional blueshift, such that X sees C's clock going faster. That reconciles both viewpoints (C and X expect each other to be time dilated) in your thought experiment. It's not a statement about some clock really going slower than the other.

 

[ghwellsjr]

So are you saying that we cannot determine for instance the velocity of the Earth wrt the CMB?

Yes, at least not based on the spectrum of the CMB.

 

[yuiop]

If you are asking if the CMB (Cosmic Background Radiation) is a way to determine an absolute rest point, the answer is no, because it has a black body spectrum which means that it will have an identical spectrum when viewed from any speed. If you measure the spectrum and then accelerate to any arbitrary high speed and measure it again, it will look identical so there is no way to use that information to conclude when you are at rest with respect to the CMB.

So are you saying that we cannot determine for instance the velocity of the Earth wrt the CMB?

I have to agree with Passionflower here. The velocity of the Earth wrt the CMB can and has been measured. You can always tell if you are at rest with or moving relative to the the CMB. What is in question is that if you are at rest with the CMB then how do you know that the CMB rest frame is truly at rest in any absolute sense? I think you are thinking of a very distant observer maybe millions of lightyears away from Earth that appears to be moving at high velocity relative to the Earth and yet that observer can also measure themselves to be at rest with the CMB. Distant observers are at rest with the local CMB and spacetime and yet appear to be moving relative to us because "space itself is expanding" but this is getting into cosmological issues that are perhaps beyond the remit of this thread. If you are interested in finding out more about the CMB and expanding space, then the cosmology forum is perhaps the best place to pursue it.

 

[ghwellsjr]

No, I wasn't addressing the issue of whether the CMB was at rest in an absolute sense.

I said what I said. I didn't say that there may be some other way to determine if you are at rest with the CMB, I said you couldn't use the black body spectra information to determine if you are at rest with the CMB. The reason is that a black body spectrum will transform into an identical spectrum when using the Lorentz transform.

I don't want to research the cosmology forum, can you please just answer the question if the rest with the CMB was determined by just the spectrum or some other information?

 

[Passionflower]

Let's get thing clear about the CMB, surely nobody wants to give an impression that the wool is pulled over people's eyes.

If you are asking if the CMB (Cosmic Background Radiation) is a way to determine an absolute rest point, the answer is no, because it has a black body spectrum which means that it will have an identical spectrum when viewed from any speed. If you measure the spectrum and then accelerate to any arbitrary high speed and measure it again, it will look identical so there is no way to use that information to conclude when you are at rest with respect to the CMB.

So is your answer still no?

Yes, at least not based on the spectrum of the CMB.

Ok, so it seems you agree that one can. Then why bring op the spectrum?

I said what I said. I didn't say that there may be some other way to determine if you are at rest with the CMB, I said you couldn't use the black body spectra information to determine if you are at rest with the CMB.

Ok, but then can you see that you might confuse people?

 

[JesseM]

I said what I said. I didn't say that there may be some other way to determine if you are at rest with the CMB, I said you couldn't use the black body spectra information to determine if you are at rest with the CMB. The reason is that a black body spectrum will transform into an identical spectrum when using the Lorentz transform.

What's your source for the claim that the black body spectrum is unchanged under the Lorentz transform? That sounds a little implausible, even stars are often regarded as having an approximately blackbody spectrum, but obviously stars are blueshifted or redshifted if you move towards or away from them at relativistic speeds (perhaps the redshifted or blueshifted spectrum would still look like a blackbody spectrum, but from a different type of star with the peak of the spectrum at a different frequency, so not an 'identical spectrum'). And the page what would a relativistic interstellar traveller see? from the Physics FAQ on John Baez's site says at the very end:

At this extremely high ultra-relativistic velocity, radiation from the universe would emanate from a single point in the direction of travel, and all radiation, even the cosmic background, would be Doppler shifted out to gamma ray wavelengths or far radio, with next to nothing in between.

 

[Ich]

a black body spectrum will transform into an identical spectrum when using the Lorentz transform.

I think you've misread something. It's true that a black body spectrum transforms into a black body spectrum under the Lorentz transform. But both will have different temperatures.

 

[ghwellsjr]

What's your source for the claim that the black body spectrum is unchanged under the Lorentz transform? That sounds a little implausible, even stars are often regarded as having an approximately blackbody spectrum, but obviously stars are blueshifted or redshifted if you move towards or away from them at relativistic speeds (perhaps the redshifted or blueshifted spectrum would still look like a blackbody spectrum, but from a different type of star with the peak of the spectrum at a different frequency, so not an 'identical spectrum').

I couldn't find the Scientific American article from years ago that (I thought) I remembered reading that a black body spectrum looks the same from a different frame of reference, but I did find other articles that addressed what you guys are saying, especially May 1978, "The Cosmic Background Radiation and the New Aether Drift" by Richard A. Muller. So I stand corrected and I thank you all for setting me straight.

I wasn't trying to pull the wool over anyone else's eyes or confuse anyone else. I was the one who had the wool over my own eyes and I, myself, was confused. I apologize. You can disregard anything I said on this thread relating to CMB. I will study the above mentioned article and try to learn what it is all about. Again, thanks for bringing this to my attention.

 

[Passionflower]

I wasn't trying to pull the wool over anyone else's eyes or confuse anyone else. I was the one who had the wool over my own eyes and I, myself, was confused. I apologize. You can disregard anything I said on this thread relating to CMB. I will study the above mentioned article and try to learn what it is all about. Again, thanks for bringing this to my attention.

And just to make sure you understand, I was not implying you did that, I was simply referring to a possible impression. And mistakes do happen, I know, since I make a lot of them!