[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

[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.

[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]

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]

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]

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

[caveman1917]

You keep saying there is a gravitating body there, calculate the curvature tensor and prove it.

Besides, is not the schwarzschild radial coordinate, the schwarzschild radial coordinate is . is a constant, it is the constant radial position of the stationary observer and has been defined unambiguously as such many times throughout this thread. Don't pretend you suddenly think it is the schwarzschild radial coordinate, you have been using it as the constant radial position of the observer yourself many times throughout this thread.

Please answer CM1 and CM2

[macaw]

You keep saying there is a gravitating body there, calculate the curvature tensor and prove it.

MQ13: So, why are you using the Schwarzschild solution as a starting point?

Besides, is not the schwarzschild radial coordinate, the schwarzschild radial coordinate is . is a constant, it is the constant radial position of the stationary observer and has been defined unambiguously as such many times throughout this thread. Don't pretend you suddenly think it is the schwarzschild radial coordinate,

It is not my fault that you don't define your variables. Besides, the presence of is the consacrated notation for Schwarzschild radius.

you have been using it as the constant radial position of the observer yourself many times throughout this thread.

Please answer CM1 and CM2

I did.

MQ14: So, what is if not the Schwarzschild radius?
MQ15: How did you derive the new metric?

[pogono]

Let me say a word, as a root cause of above discussion.

It is not my fault that you don't define your variables. Besides, the presence of is the consacrated notation for Schwarzschild radius.

Maybe we should really start from constants and variable definition to define common language. We have still the same misunderstanding, I am afraid.

Let it be:

OBS - Hanging Observer Reference Frame
SAT- Satellite Reference Frame
IO - Infinite Observer, observer Resting in infinity

- let it be constant radial coordinate where Hanging Observer stays in one place.
- Proper Time increment for Hanging Observer

- temporarily radial distance for Satellite (he moves)
- temporarily angle coordinate for Satellite (he moves)
- Proper Time increment for Satellite

- time increment, for IO reference frame
- point in space where OBS stands. Satellite passes this spot.
[tex]t_0[tex] - point in time in IO reference frame when satellite passes point X
- Schwarzschild Radius

Now, velocities:




Satellite is passing the same spot, where HO stands.
Now, could you please express your point of view with common language to make things clear. I also would like to know, how it works. Really.

If you add new reference frame, constant or variable, please define it.

If anyone deny to use common language, we should assume, he is not interested in achieving agreement but making smokescreen.

[caveman1917]

You know what, let's do this slowly step by step and we'll see your objections about each step before we move on to the next one.

First step.

Let and be real constants where
Note that we can define any constant we want, the reason for choosing those will become obvious later on (if not already), right now they are just simple constants.

Now let's use the following metric for Minkowski space:



Do you agree this is just plain old Minkowski?
Any other objections about this step?

[macaw]

You know what, let's do this slowly step by step and we'll see your objections about each step before we move on to the next one.

First step.

Let and be real constants where
Note that we can define any constant we want, the reason for choosing those will become obvious later on (if not already), right now they are just simple constants.

Now let's use the following metric for Minkowski space:



Do you agree this is just plain old Minkowski?
Any other objections about this step?

1. Why did you first claim that you are using the Schwarzschild metric?
2. Why did you next claimed that you are using the metric?
3. How did you arrive to this third and final metric? Derivation, please?
4. What's with the and ? How do they relate to the geometry of the problem? (especially, how does relate to anything in the problem?)
5. If both are constants , why don't you simply write: ?
6. What are the exact meanings of the variables ?

[caveman1917]

So do you agree that is a metric for a Minkowski space?

1. Why did you first claim that you are using the Schwarzschild metric?
2. Why did you next claimed that you are using the metric?

These are not objections to the first step as posted.

3. How did you arrive to this third and final metric? Derivation, please?

There is no derivation, it is a specific case of the general form of the minkowski metric if we do not require an orthonormal basis.

4. What's with the and ?

That is not an intelligible question.

How do they relate to the geometry of the problem?

They do not. They relate to the chosen basis, and as you should know choosing a different basis does not change the geometry.

(especially, how does relate to anything in the problem?)

No more than being the expression for a constant.

5. If both are constants , why don't you simply write: ?

Because this way of putting it will turn out useful in later steps. If that's not good enough: because i choose to.

6. What are the exact meanings of the variables ?

They are respectively the result of an inner product, and variables representing an ordered basis.

[macaw]

So do you agree that is a metric for a Minkowski space?

1. Why did you first claim that you are using the Schwarzschild metric?
2. Why did you next claimed that you are using the metric?

These are not objections to the first step as posted.

I need you to answer before we get any further.

[caveman1917]

I need you to answer before we get any further.

1. Why did you first claim that you are using the Schwarzschild metric?

1. I was choosing a basis in MINKOWSKI space such that it would correspond to what a local stationary observer would have in his local tangent space if he used the standard schwarzschild coordinates and his own proper time as a basis. In other words, i was in minkowski space all the time, not the schwarzschild geometry. I was just carefully choosing a basis.

2. Why did you next claimed that you are using the metric?

2. I never did. I only used it as a simple example to ask you wether you knew that was minkowski space, since i didn't know wether you kept seeing the schwarzschild solution because you didn't realize that the coefficients were constants or you didn't know that with constant coefficients it indeed is minkowski. It turned out to be the latter.

Now could you please just stick to the step by step derivation, the reason i'm redoing it that way is exactly because that way your previous misunderstandings won't get dragged on ad infinitum.

[caveman1917]

Since you have no more objections to the first step, and in the process learned that your crackpot claims in, among many others, post 33 are incorrect, we can move on to the second step.

Second step.

Using the metric given in the first step we can calculate the radial and tangential coordinate speeds of light at some given radial coordinate, which we will set equal to . We shall call those two quantities and respectively.

For the radial speed.



For the tangential speed.



We see that we have light speed anisotropy in Minkowski space, so you've learned yet another fringe claim of yours to be found incorrect.

Any objections about this step?

[macaw]

Since you have no more objections to the first step, and in the process learned that your crackpot claims in, among many others, post 33 are incorrect, we can move on to the second step.

Second step.

Using the metric given in the first step we can calculate the radial and tangential coordinate speeds of light at some given radial coordinate, which we will set equal to . We shall call those two quantities and respectively.

For the radial speed.



For the tangential speed.



We see that we have light speed anisotropy in Minkowski space, so you've learned yet another fringe claim of yours to be found incorrect.

Any objections about this step?

No, you cannot move to the second step since you gave false answers to the questions from step 1.
You inserted the anisotropy by hand, by inserting (also by hand) the gratuitous factor into the metric. This is why I have asked you repeatedly to derive the metric. Metrics are not pulled out of thin air, they are solutions to field equations, yours isn't a solution to anything. Try the same exercise with the Minkowski metric used in mainstream physics:




Let's see the light speed anisotropy. While you are at it, please cite any experiment that detected any light speed anisotropy in accordance with the predictions of your "theory" above.

[caveman1917]

No, you cannot move to the second step since you gave false answers to the questions from step 1.

Those weren't questions from step 1, and they weren't false.

Metrics are not pulled out of thin air, they are solutions to field equations, yours isn't a solution to anything.

Wrong. Metrics are distance functions on sets, if you disagree please state a source that defines a metric as a "solution to field equations", or retract the claim. In this specific case it is a metric on a Minkowski space. If you disagree, i have asked you before, please state which condition on a vector space for it to be a Minkowski space isn't satisfied, or retract your claim that it's not Minkowski space. ETA: I'll help you out on this one, a Minkowski space is a (1) real vector space, with an inner product that is (2) bilinear, (3) symmetric and (4) nondegenerate and has a signature of (5) (n-1,1). So you can just state which condition you think isn't satisfied.

Try the same exercise with the Minkowski metric used in mainstream physics:



Let's see the light speed anisotropy. While you are at it, please cite any experiment that detected any light speed anisotropy in accordance with the predictions of your "theory" above.

If your unit of distance in one direction is smaller than in the other direction, it is perfectly simple that light will pass more units of distance in one direction than the other in the same time. In other words, coordinate light speed anisotropy. I say that if i use a basis where the basis vectors don't all have the same length i will have light speed anisotropy. Your objection that if they have the same length there is no light speed anisotropy is besides the point, and not even an actual objection. The actual objection would be that if they don't have the same length i won't get light speed anisotropy, or if i have light speed isotropy i can have a basis where they have not the same length, please review the modus tollens.

So do you have any actual objections to step 2? Otherwise i'll just move on to step 3.