[macaw / caveman1917 discussion from an ATM thread]

[caveman1917]

Mod note: posts #1 to #27 in this thread taken from here: http://www.bautforum.com/showthread....ield-equations


ETA: nevermind, i see the thread has evolved by now, i forgot to read the last page before replying

The two of you are trying to solve different problems.

macaw you're solving for the general solution of a stationary body and an orbiting satelite.
pogono is only solving for the specific instant where the two are at the same position, that's why he's setting all his receiver/dt terms to zero but keeps non-zero his emitter/dt terms.

So for him and even though
That's because he is just solving for the instant at which they pass at the same location.

[macaw]

ETA: nevermind, i see the thread has evolved by now, i forgot to read the last page before replying

The two of you are trying to solve different problems.

macaw you're solving for the general solution of a stationary body and an orbiting satelite.
pogono is only solving for the specific instant where the two are at the same position, that's why he's setting all his receiver/dt terms to zero but keeps non-zero his emitter/dt terms.

Nope, is also true for an orbiting satellite, except that the orbit is non-circular, i.e. . Since pogono stipulated , this is not the case.

So for him and even though
That's because he is just solving for the instant at which they pass at the same location.

Err, wrong. As I explained earlier, you can't substitute for in the general solution SELECTIVELY, you must do it throughout. You either do it throughout or you don't do it at all. No half-assed job.
Secondly, pogono didn't "solve" anything, he just failed repeatedly to do a simple substitution in the general solution I gave him.
Thirdly, as explained to him several times (in case you missed that), the stationary observer is IRRELEVANT for the case of the GPS. The starting point was the post containing the false claims pogono made about his "solution" being "used by the GPS engineers". I pointed out over several pages that his claim was false, instead of admitting that, he moved the goal posts inserting the stationary observer and trying to solve a different problem. I merely pointed out that the solution to his new problem was a subset of the general solution I gave him.
Fourthly, if you are intent on solving yet another problem, the ratio of time dilation between a stationary observer and an observer on an elliptic orbit, my general solution produces the answer easily. But it is not the answer that pogono came up with.
Fifthly, and most importantly, whatever the scenario, the time dilation is not additive and it makes no sense adding up proper time intervals with coordinate time intervals as pogono has done repeatedly in this thread.
Sixthly, none of the above has anything to do with pogono's ATM with the exception of his basic misunderstandings about GR, the Schwarzschild solution, observers, time dilation, GPS, etc. When one points out the various mistakes in his ATM, he simply manufactures a scenario that showcases his misconceptions about mainstream physics in general and relativity in specific.

[caveman1917]

Nope, is also true for an orbiting satellite, except that the orbit is non-circular, i.e. .

Which is what i said:

Since pogono stipulated , this is not the case.

No, it is the case. The is not valid over the entire orbit (ie the general case you're solving for), it's valid at one specific instant of time. It's not saying the orbit is circular, it's saying the solution is for the specific instant where .

Err, wrong. As I explained earlier, you can't substitute for in the general solution SELECTIVELY, you must do it throughout. You either do it throughout or you don't do it at all. No half-assed job.

It's not a substitution, again it is not a general solution, but a solution for the specific instant where the observer and the satelite are at the same location.

The problem he is solving for is this:
1. Take the schwarzschild metric
2. Take an observer that is stationary.


3. Take a satelite that is orbiting in a non-circular orbit.


4. Solve for the relative time dilation at the instant where they pass at the same location (ie the moment the satelite passes the observer)



You're only considering the first 3 statements, therefor trying to solve for the relative time dilation in the general case (ie over the entire orbit) and therefor considering the as a substitution of with in the metric, where the is really only a simple (and unnecessary!) constant saying at which r they pass eachother.

Try solving for all 4 given statements, you'll see that the solution was correct. And so was yours, you are just doing different problems.
Not that there is any need to drag the schwarzschild metric into it in the first place, one could just use SR in this case since the manifold approximates minkowski locally anyway.

Secondly, pogono didn't "solve" anything, he just failed repeatedly to do a simple substitution in the general solution I gave him.

He solved his problem, you solved yours. He failed yours and you failed his, but that's just because you're doing different problems.

Thirdly, as explained to him several times (in case you missed that), the stationary observer is IRRELEVANT for the case of the GPS. The starting point was the post containing the false claims pogono made about his "solution" being "used by the GPS engineers". I pointed out over several pages that his claim was false, instead of admitting that, he moved the goal posts inserting the stationary observer and trying to solve a different problem. I merely pointed out that the solution to his new problem was a subset of the general solution I gave him.
Fourthly, if you are intent on solving yet another problem, the ratio of time dilation between a stationary observer and an observer on an elliptic orbit, my general solution produces the answer easily. But it is not the answer that pogono came up with.
Fifthly, and most importantly, whatever the scenario, the time dilation is not additive and it makes no sense adding up proper time intervals with coordinate time intervals as pogono has done repeatedly in this thread.
Sixthly, none of the above has anything to do with pogono's ATM with the exception of his basic misunderstandings about GR, the Schwarzschild solution, observers, time dilation, GPS, etc. When one points out the various mistakes in his ATM, he simply manufactures a scenario that showcases his misconceptions about mainstream physics in general and relativity in specific.

Yes i'm not arguing for his ATM in general, i'm just saying that in that specific bit of it the problem was that the two of you were considering different problems.

[macaw]

It's not a substitution, again it is not a general solution, but a solution for the specific instant where the observer and the satelite are at the same location.


The problem he is solving for is this:
1. Take the schwarzschild metric
2. Take an observer that is stationary.


3. Take a satelite that is orbiting in a non-circular orbit.


4. Solve for the relative time dilation at the instant where they pass at the same location (ie the moment the satelite passes the observer)

The above is NOT what pogono is claiming, you need to read the thread before you manufacture your own problem statement. Here is what pogono is claiming as a problem statement. It is fair to mention that pogono has moved the goal posts several times in the meanwhile.

IF that were the case (it isn't, for reasons explained above), pogono's solution is just as wrong. Do you know what the correct general solution would be in the case of a satellite in a non-circular orbit?

Not that there is any need to drag the schwarzschild metric into it in the first place, one could just use SR in this case since the manifold approximates minkowski locally anyway.

Nope, it cannot. The reason is that the satellite is orbiting a gravitating body, so the solution depends on the Schwarzschild radius. You can't get solutions dependent of in SR because SR has no concept of Schwarzschild radius. I can understand pogono making such a rookie mistake but you?

He solved his problem,

Once again, he didn't "solve" anything, he simply kept fiddling with the problem statement in the hope that he'll make it agree with his "solution".

you solved yours.

Actually this is wrong, I produced a general solution. You can derive the solution for any particular case from my general solution.

[caveman1917]

The above is NOT what pogono is claiming, you need to read the thread before you manufacture your own problem statement. Here is what pogono is claiming as a problem statement. It is fair to mention that pogono has moved the goal posts several times in the meanwhile.

I was working from his problem statement here where he did a step by step. My comment is to be taken about the discussion from that post forth.

IF that were the case (it isn't, for reasons explained above), pogono's solution is just as wrong. Do you know what the correct general solution would be in the case of a satellite in a non-circular orbit?

Of course, just plug the right equations into the schwarzschild metric, it's not that hard.

Nope, it cannot. The reason is that the satellite is orbiting a gravitating body, so the solution depends on the Schwarzschild radius. You can't get solutions dependent of in SR because SR has no concept of Schwarzschild radius. I can understand pogono making such a rookie mistake but you?

I'm afraid the rookie mistake is yours, but i'll grant you that it isn't easy to spot when you just look at the equations.

Firstly by reasoning, since we are talking about the relative time dilation at the moment the satelite passes the observer, you know we can use SR from the perspective of the observer since we are talking about infinitesimal change around a single event in spacetime, ergo it is minkowskian.

Secondly, using the standard schwarzschild metric. I'm sure we both agree that for this (ie at the moment the satelite passes the observer) the relative time dilation is given by


We can rewrite this as


Recall that the velocity of the satelite as seen by the observer is


And recall that relative time dilation in SR is simply


You can see that we're almost there, but there is this extra term next to the radial component of the velocity, which is why it is hard to spot. The trick is in recalling that the schwarzschild metric in its standard form has anisotropic light speed, which is the reason for the appearance of that term. Cancel the anisotropy (which you need to cancel to use SR) and you cancel the term and everything falls in place.

Thirdly, you can use the isotropic form of the schwarzschild metric by substituting . Then if you calculate it in those coordinates you won't get the extra term and it falls out nicely, just recall for the final equation that in those coordinates the coordinate speed of light (while isotropic now) equals

Actually this is wrong, I produced a general solution. You can derive the solution for any particular case from my general solution.

Yes i know, which is the basis for the misunderstanding from at least post 75 onwards. He was doing a particular case of in the sense of that being the radial point for the instant they passed eachother where you had interpreted that as meaning that the satelite was in circular orbit. Both of you were right, you were just talking next to eachother. At least from post 75 onwards, i'm not making a statement about anything before that.

[macaw]

I was working from his problem statement here where he did a step by step. My comment is to be taken about the discussion from that post forth.

Well, you are working off the wrong one, next time, pay attention before you jump in.

I'm afraid the rookie mistake is yours, but i'll grant you that it isn't easy to spot when you just look at the equations.

Firstly by reasoning, since we are talking about the relative time dilation at the moment the satelite passes the observer, you know we can use SR from the perspective of the observer since we are talking about infinitesimal change around a single event in spacetime, ergo it is minkowskian.

Secondly, using the standard schwarzschild metric. I'm sure we both agree that for this (ie at the moment the satelite passes the observer) the relative time dilation is given by


We can rewrite this as


Recall that the velocity of the satelite as seen by the observer is


And recall that relative time dilation in SR is simply

Yes, you posted the same mistake as pogono did, you seem like a good candidate to work together with him on his ATM. Here are the correct calculations:

[LaTeX ERROR: Image too big 1028x40, max 650x600]

Make , and you obtain the correct answer:

[LaTeX ERROR: Image too big 679x40, max 650x600]


If you insist in doing the comparison when you get:



See, the term in ? You can't have that in SR, no matter what. You inadvertently missed it by rolling it into your

[caveman1917]

Well, you are working off the wrong one, next time, pay attention before you jump in.

Someone has to correct your mistakes when you make them.

Yes, you posted the same mistake as pogono did, you seem like a good candidate to work together with him on his ATM.

Post 75, when including the clarifications of the problem statement in later posts, is correct. Actually there is only a single minor mistake in it. Can you find it?

See, the term in ? You can't have that in SR, no matter what. You inadvertently missed it by rolling it into your

You are of course missing the entire point. Have you tried doing it in isotropic coordinates? What is the solution then? Maybe then you might see it.

For everybody else who can just reason about it without needing to calculate everything: if i am an observer stationary somewhere in a gravitational field equipped with a clock and measuring rods and a satelite passes at my exact location, i can use simple SR to calculate the relative time dilation between the satelite at that moment and my clock. It's called the principle of locality, or local lorentz invariance. It's a simple consequence of the fact that the tangent space at every point on a lorentzian manifold is minkowskian.

[macaw]

Someone has to correct your mistakes when you make them.

Please stop injecting your own ATM theories into other people's ATMs. Open your own.

You are of course missing the entire point. Have you tried doing it in isotropic coordinates? What is the solution then? Maybe then you might see it.

You are missing the point, your is equal to . Try as hard as you may, there is no in special relativity. Instead of repeating the OP's fringe claim, open your own ATM.

For everybody else who can just reason about it without needing to calculate everything: if i am an observer stationary somewhere in a gravitational field equipped with a clock and measuring rods and a satelite passes at my exact location, i can use simple SR to calculate the relative time dilation between the satelite at that moment and my clock.

Actually, if there is a gravitating body present, you can't. Because the answer is a function of , the Schwarzschild radius of the gravitating body. There is no in special relativity. If you think otherwise, I suggest that you open your own ATM thread on the subject rather than hijacking this ATM.

Have you tried doing it in isotropic coordinates?

Isotropic coordinates are part of GR, not SR. They are nothing but a convenient coordinate transformation and you cannot transform away gravitational fields. You ought to know that.

[caveman1917]

Please stop injecting your own ATM theories into other people's ATMs. Open your own.

Just because you don't understand something doesn't give you the right of throwing ATM accusations around.

You are missing the point, your is equal to . Try as hard as you may, there is no in special relativity. Instead of repeating the OP's fringe claim, open your own ATM.

Actually, if there is a gravitating body present, you can't. Because the answer is a function of , the Schwarzschild radius of the gravitating body. There is no in special relativity. If you think otherwise, I suggest that you open your own ATM thread on the subject rather than hijacking this ATM.

Neither depends on because they are measurements. is a measurement on my clock, i don't need to appeal to any parameters. If i have a clock and a rod and a satelite passes by me, i get the velocity of the satelite as a measurement without ever having to come up with any .

Isotropic coordinates are part of GR, not SR. They are nothing but a convenient coordinate transformation and you cannot transform away gravitational fields. You ought to know that.

You are once more missing the point. It's not about trying to transform away gravitational fields. The only reason i'm bringing up the isotropic form of the schwarzschild metric is because it makes it easier to see the underlying point.

Did you do the calculations in them or not?

If you are an observer stationary in a gravitational field with a clock and a measuring rod and a satelite passes at your location, can you measure the instantaneous velocity of the satelite? Yes.
Can you then use the simple to get the relative time dilation between your clock and the satelite's clock at that moment? Yes.
Must this be the same result as doing it all from the schwarzschild metric? Yes.
So can we equate the two results? Yes.

If you think otherwise, you are contradicting the basic premise of GR that locally SR applies, and i suggest you open your own ATM thread on that.

[macaw]

Just because you don't understand something doesn't give you the right of throwing ATM accusations around.



Neither depends on because they are measurements. is a measurement on my clock, i don't need to appeal to any parameters.

This is clearly false , on two accounts:

1. From the perspective of the distant observer, the term is very real and you cannot explain it away using SR. His clock shows and not and his answer is directly dependent on that pesky .


2. Remember that the satellite is moving, move a fraction of mm away from the emitter/receiver coincident position and you will measure

[LaTeX ERROR: Image too big 679x40, max 650x600]


MQ_to_Caveman17_ATM_1: Explain, using special relativity, the presence of the factor in the measurement.


MQ_to_Caveman17_ATM_2: Explain, using special relativity, the presence of the factor in the measurement.

If i have a clock and a rod and a satelite passes by me, i get the velocity of the satelite as a measurement without ever having to come up with any .

This is false, see above.

If you are an observer stationary in a gravitational field with a clock and a measuring rod and a satelite passes at your location, can you measure the instantaneous velocity of the satelite? Yes.
Can you then use the simple to get the relative time dilation between your clock and the satelite's clock at that moment? Yes.
Must this be the same result as doing it all from the schwarzschild metric? Yes.
So can we equate the two results? Yes.
If you think otherwise, you are contradicting the basic premise of GR that locally SR applies, and i suggest you open your own ATM thread on that.

Please answer the two questions above. Do you still claim that you can use SR to solve the problem?

[macaw]

We can rewrite this as


Recall that the velocity of the satelite as seen by the observer is

Not in GR, it isn't. See below.

And recall that relative time dilation in SR is simply


You can see that we're almost there, but there is this extra term next to the radial component of the velocity, which is why it is hard to spot. The trick is in recalling that the schwarzschild metric in its standard form has anisotropic light speed, which is the reason for the appearance of that term.

Then your from SR is no longer your from the above formula. No amount of your handwaving will make the two expressions equal because you can't make go away.

Cancel the anisotropy (which you need to cancel to use SR) and you cancel the term and everything falls in place.

The above is contradicted by the simple fact that , in GR, the proper speed of the orbiting satellite is given by:



where:




and not by your naive pythagorean addition formula. That's one problem with your claim. The second problem is that you can't use your handwaving to bring the expression to the form given the fact that you got wrong to begin with.


MQ_to_Caveman17_ATM_3: If you still think otherwise, you would need to do the calculations to prove it, waving hands doesn't count as proof.

[caveman1917]

This is clearly false , on two accounts:

1. From the perspective of the distant observer, the term is very real and you cannot explain it away using SR. His clock shows and not and his answer is directly dependent on that pesky .

Irrelevant. It is not from the perspective of the distant observer, but from the perspective of the stationary observer. It is his measurement of the speed of the passing satelite that can be used in a purely SR manner to calculate the relative time dilation between his clock and the satelite's clock. The observer at infinity doesn't come into it.

2. Remember that the satellite is moving, move a fraction of mm away from the emitter/receiver coincident position

Irrelevant. The measurement is made at the instant when , not when a fraction of a mm away. Changing the problem statement and then attacking the solution as if it were to a different problem statement is a pure straw-man.

MQ_to_Caveman17_ATM_1: Explain, using special relativity, the presence of the factor in the measurement.

MQ_to_Caveman17_ATM_2: Explain, using special relativity, the presence of the factor in the measurement.

There is neither in the measurement, because the measurement is by the stationary observer, not the one at infinity. All he needs is his own rod and clock to measure the speed of the satelite at the moment it passes by him. Since he is doing the measurement at one instant of time at his exact location, he can use SR.

Please answer the two questions above. Do you still claim that you can use SR to solve the problem?

Of course. If you weren't so enamoured with the observer at infinity but considered only the reference frame of the stationary observer you would no doubt immediately see that.

[caveman1917]

caveman17 thinks that your post 75 is correct, so let's debunk his statement. For the purpose of the argument, let's admit that you have just moved the goalposts and you are no longer working on your "solution to the GPS problem" and you are just comparing the clocks on the satellite with the clock of the (totally irrelevant) stationary observer.
So, the correct expression would NOT be what you wrote above but:




You missed the differentiation throughout, let's be generous and chalk this as a typo.

That is indeed the single minor mistake i was talking about earlier.

The more serious problem is that you are differentiating values measured by the emitter wrt the time of the receiver, as in
, something that is physically meaningless.

What is the coordinate tangential speed of the satelite from the frame of the stationary observer?

It is refreshing to see that you found an advocate for your ATM in caveman17.

Again, i'm not advocating his ATM, only the correctness of his post 75.

[macaw]

Irrelevant. It is not from the perspective of the distant observer, but from the perspective of the stationary observer. It is his measurement of the speed of the passing satelite that can be used in a purely SR manner to calculate the relative time dilation between his clock and the satelite's clock. The observer at infinity doesn't come into it.



Irrelevant. The measurement is made at the instant when , not when a fraction of a mm away. Changing the problem statement and then attacking the solution as if it were to a different problem statement is a pure straw-man.



There is neither in the measurement, because the measurement is by the stationary observer, not the one at infinity. All he needs is his own rod and clock to measure the speed of the satelite at the moment it passes by him. Since he is doing the measurement at one instant of time at his exact location, he can use SR.



Of course. If you weren't so enamoured with the observer at infinity but considered only the reference frame of the stationary observer you would no doubt immediately see that.

Actually, quite relevant in pointing out your incorrect ideas that you can use SR in order to solve the problem.

[macaw]

T



What is the coordinate tangential speed of the satelite from the frame of the stationary observer?

Definitely not the mixed coordinates.

Again, i'm not advocating his ATM, only the correctness of his post 75.

But you are.

[caveman1917]

Then your from SR is no longer your from the above formula. No amount of your handwaving will make the two expressions equal because you can't make go away.

The only reason the term is there is because coordinate speed of light is not isotropic in the standard form of the schwarzschild metric, and since the in means the coordinate speed of an object as a fraction of the coordinate speed of light, the radial component of the velocity has the extra term since the coordinate radial speed of light is different from the coordinate tangential speed of light.
If you can't see it that way, for the umpteenth time, use the isotropic form of the schwarzschild metric and the term goes away.

The above is contradicted by the simple fact that , in GR, the proper speed of the orbiting satellite is given by:

Irrelevant. The proper speed has nothing to do with it, v is the coordinate speed in the frame of the stationary observer.

The second problem is that you can't use your handwaving to bring the expression to the form given the fact that you got wrong to begin with.

CM1: Why are you now claiming that the v in is proper speed? That's such a basic mistake it is refreshing to see you state these things.

[caveman1917]

Definitely not the mixed coordinates.

Then what is it? Give the formula for the instantaneous coordinate tangential speed of the satelite from the frame of the stationary observer at the instant when they are at the same location.

It is most obvious that the answer is .
You say it is something else, then show it.

[macaw]

Then what is it? Give the formula for the instantaneous coordinate tangential speed of the satelite from the frame of the stationary observer at the instant when they are at the same location.

It is most obvious that the answer is .
You say it is something else, then show it.

It should be obvious that the correct answer is:

[caveman1917]

I'm quite sure he'll be thrilled.

I'm quite sure your mindreading skills need polishing.

[macaw]

Sorry it can't be done.

The problem is that none of this generalizes. If you move them to even a little difference in r coordinate, you'll have the gravitational component kicking in and it doesn't work anymore. And most importantly, the decomposition itself only works in specific types of spacetimes, there is nothing general about it.

It is refreshing to see that you have come around and you finally adopted my position on the subject.

[caveman1917]

It is refreshing to see that you have come around and you finally adopted my position on the subject.

I've not come around to anything, since that was not the problem statement. The problem statement was about the instantaneous velocity at the instant where they are at the same r, i was merely explaining why that doesn't generalize to when they are at different r, which is a different problem to solve and not the one which was under consideration.

I find it impossible to adopt your position since the gamma function takes coordinate speed and not proper speed, and i stand by everything i have said on the subject as stated. In any case, i'd be happy to explain it to you if you'd like to start a thread about it in S&T since this continued discussion in this thread will only get us in trouble again anyway.

[macaw]

I've not come around to anything, since that was not the problem statement. The problem statement was about the instantaneous velocity at the instant where they are at the same r, i was merely explaining why that doesn't generalize to when they are at different r, which is a different problem to solve and not the one which was under consideration.

Your sentence , outlined in red, is an exact copy of my sentence (point 2. , minus the associated math) from post 133.

I find it impossible to adopt your position since the gamma function takes coordinate speed and not proper speed,

Err, you miss the simple thing that I gave the correct expression for the proper speed, as a counter to the incorrect equations that you and pogono were using for coordinate speed. You surely know that you can easily derive the coordinate speed from the proper speed , that's all. The only thing necessary is the correct expression of the proper speed, so I don't understand why you continue to beat this strawman.

and i stand by everything i have said on the subject as stated. In any case, i'd be happy to explain it to you if you'd like to start a thread about it in S&T since this continued discussion in this thread will only get us in trouble again anyway.

Not interested, thank you.

[caveman1917]

Your sentence , outlined in red, is an exact copy of my sentence from post 133.

Applied to a different problem. If you meant it in the same way i did, namely that the result doesn't generalize, then my response that you linked to was correct, since at that time we were considering wether the solution was correct for the problem, not wether it generalizes to different problems.

Err, you miss the simple thing that I gave the correct expression for the proper speed, as a counter to the incorrect equations that you and pogono were using for coordinate speed. You surely know that you can easily derive the coordinate speed from the proper speed , that's all.
The only thing necessary is the correct expression of the proper speed, so I don't understand why you continue to beat this strawman.

I just found it interesting that when asked for coordinate speed in post 143 you gave proper speed in post 144, even though the simple transformation between the two reproduces the result from post 143. It almost seemed as if you thought giving the wrong speed would obscure things better, that's all.

That is, unless you of course think that given a proper speed of and a relative time dilation between satelite and observer of the coordinate speed from the frame of the observer would be different from ?

Not interested, thank you.

Of course not.

[macaw]

Applied to a different problem.

Nope, the same exact one, you really need to read point 2.

If you meant it in the same way i did, namely that the result doesn't generalize, then my response that you linked to was correct, since at that time we were considering wether the solution was correct for the problem, not wether it generalizes to different problems.

I didn't say that your sentence is incorrect , I said that it copies mine verbatim, so you came around.

I just found it interesting that when asked for coordinate speed in post 143 you gave proper speed in post 144,

I was simply pointing out that the approach that you an pogono had, to add up speeds in euclidian fashion is incorrect and I posted the correct formula of the proper speed, that's all.

even though the simple transformation between the two reproduces the result from post 143. It almost seemed as if you thought giving the wrong speed would obscure things better, that's all.

It couldn't since your naive formula for coordinate speed is incorrect. Speeds do not add in euclidian fashion in curved spacetime.

[caveman1917]

I didn't say that your sentence is incorrect , I said that it copies mine verbatim, so you came around.

I meant my sentence that it was irrelevant. At the time we were only looking at the specific problem of instantaneous velocity when they are at the exact same location. If you were trying to say that it doesn't generalize, sure i agree, but that was not what was under consideration at that time.

I was simply pointing out that the approach that you an pogono had, to add up speeds in euclidian fashion is incorrect and I posted the correct formula of the proper speed, that's all.

It couldn't since your naive formula for coordinate speed is incorrect. Speeds do not add in euclidian fashion in curved spacetime.

It does locally when we're only considering an infinitesimal region around a single event, since locally the manifold is just flat minkowski, which i'm sure you know. Again it doesn't generalize, but the question was wether it was correct for that specific problem statement, which it is.
The problem was that he thought the result generalized beyond a specific local problem statement, but with all conditions as stated the solution was correct, you might want to review the conditions as stated in post 75. A lucky shot perhaps, but correct nevertheless.

[macaw]

We can rewrite this as


Recall that the velocity of the satelite as seen by the observer is


And recall that relative time dilation in SR is simply


You can see that we're almost there, but there is this extra term next to the radial component of the velocity, which is why it is hard to spot. The trick is in recalling that the schwarzschild metric in its standard form has anisotropic light speed, which is the reason for the appearance of that term. Cancel the anisotropy (which you need to cancel to use SR) and you cancel the term and everything falls in place.

There are three problems with what you are saying:

1. The result you got is and you cant's "cancel out" the factor no matter how much you wave your arms.

2. The speed in curved spacetime does not add naively according to the euclidian formula ANYWAYS , you need to derive correctly, from the Euler-Lagrange equations. They would give you the proper speed and you can obtain through a simple multiplication the coordinate speed. I posted the coordinate speed a little earlier:

In GR, the proper speed of the orbiting satellite is given by:



where:




3. Lastly, I am mystified by your insistence in applying the SR hack when correct application of GR produces the general solution, applicable to all cases.

It does locally when we're only considering an infinitesimal region around a single event, since locally the manifold is just flat minkowski, which i'm sure you know. Again it doesn't generalize, but the question was wether it was correct for that specific problem statement, which it is.
The problem was that he thought the result generalized beyond a specific local problem statement, but with all conditions as stated the solution was correct, you might want to review the conditions as stated in post 75. A lucky shot perhaps, but correct nevertheless.

Given the fact that pogono has started with the general problem (GPS transmitter/receiver rotating at different speeds and at different altitudes) only to keep moving the goal-posts every time his claims were shown wrong and given that his problem statement has been more than obscure (some due to severe language problems, some due to inability to formulate the problem statement in a clear way), it was difficult for me to trust his posted problem statements. This, coupled with his fringe treatment of trivial college subjects made for a very difficult interaction.

[caveman1917]

I'll open another thread on it tomorrow, where we can discuss this further.

[caveman1917]

There are three problems with what you are saying:

1. The result you got is and you cant's "cancel out" the factor no matter how much you wave your arms.

Let subscripts r and t denote the radial and tangential components respectively, and let denote the coordinate speed of light (which is normally just c, but since we got an anistropy we need to specify, and let's use c=1)
In SR: (1)

For the standard form of schwarzschild metric, for a stationary observer at radial coordinate R:





=> (2)



=> (3)

(4)

(5)

Putting 1 to 5 together gets us which is the result we needed. The term is only there because the standard form of the schwarzschild metric is anisotropic, it doesn't appear in the isotropic form (which is why doing it that way might be easier to see).

2. The speed in curved spacetime does not add naively according to the euclidian formula

True, but we don't even need to consider any curved spacetime since we're in the local tangent space which is just minkowski. But what you say is indeed one of the reasons why the result doesn't generalize.

3. Lastly, I am mystified by your insistence in applying the SR hack when correct application of GR produces the general solution, applicable to all cases.

It's not a hack, it's a basic feature of the theory.

Given the fact that pogono has started with the general problem (GPS transmitter/receiver rotating at different speeds and at different altitudes) only to keep moving the goal-posts every time his claims were shown wrong and given that his problem statement has been more than obscure (some due to severe language problems, some due to inability to formulate the problem statement in a clear way), it was difficult for me to trust his posted problem statements. This, coupled with his fringe treatment of trivial college subjects made for a very difficult interaction.

I suppose we can then agree that his solution was correct for the specific problem it was intended for, but that it doesn't generalize?

[macaw]

Let subscripts r and t denote the radial and tangential components respectively, and let denote the coordinate speed of light (which is normally just c, but since we got an anistropy we need to specify, and let's use c=1)
In SR: (1)

Err, no. In SR:

There is no such thing as and , there is only . Did u get the above from a book or did you make it up?

[pzkpfw]

... Did u get the above from a book or did you make it up?

Post #28 was polite, you don't need to add this in your reply.