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Originally Posted by macaw
While mathematically you are free to do any basis change
Good, so you retract the claim in post 33, and analogously your claims that the given metric is "phony", "in error", "certainly NOT minkowski", etc?

, there is no physical justification for the above basis change.
There is never a physical justification for a basis change. As you should know changing bases does not change any of the actual physics, it is always for mathematical reasons that basis changes are performed, for example making calculations easier or some things easier to see.

Besides, justification is in the result. And in the case of the simpler one in post 32, to test your level of knowledge on the subject as it was a complete give-away.

Also, show that someone else (rather than you) elected to stick the Schwarzschild radius into the Minkowski metric.
Huh? Here is someone:
Originally Posted by macaw
Problem is, the above result is in disagreement with the result that mainstream physics obtains by using the correct approach, via the Schwarzschild solution:

You did learn by now that the above, once you set the coefficients to constants by setting , is just minkowski in a non-standard basis, right? Not the schwarzschild geometry you kept seeing in it.
So you might as well ask yourself the same questions, maybe you'd see how silly they are.
Last edited by caveman1917; 2011-Oct-06 at 09:10 PM.

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Originally Posted by caveman1917
Good, so you retract the claim in post 33, and analogously your claims that the given metric is "phony", "in error", "certainly NOT minkowski", etc?
Your metric is just as incorrect as in your sleigh of hand in terms of trying to match the Schwarzschild answer with the one you get from your contrived metric. So, no retraction, you are just as mistaken as you've always been.

There is never a physical justification for a basis change. As you should know changing bases does not change any of the actual physics, it is always for mathematical reasons that basis changes are performed, for example making calculations easier or some things easier to see.

Besides, justification is in the result.
Your "result" is a gross error. I don't see why you insist.

You did learn by now that the above, once you set the coefficients to constants by setting , is just minkowski in a non-standard basis, right? Not the schwarzschild geometry you kept seeing in it.
Funny thing is, I gave the general solution(that includes the particular case you keep getting wrong) earlier on. Of course, it includes the particular solution and it is derived by using GR. Now, you have struggled to match the general solution by using a hack, by trying to match the GR answer with the metric that you cooked up for this particular purpose notwithstanding that you couldn't provide any physical justification for it. Unfortunately for you, your "metric" produces the wrong answer (see below). Rather than being honest and admitting to error, you have tried all kinds of subterfuges to prove that your method is correct. And they all backfired since I have exposed the errors in all your approaches.

So you might as well ask yourself the same questions, maybe you'd see how silly they are.

The correct answer, via the Schwarzschild solution:

MQ10: Do you see the difference?

Even IF you set your error persists:

The correct answer, via the Schwarzschild solution:

MQ11: Do you see the difference?
Last edited by macaw; 2011-Oct-06 at 10:41 PM.

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Originally Posted by macaw
So, no retraction, you are just as mistaken as you've always been.
Since you continue to maintain the by now painfully obvious incorrect claim that is not minkowski, back to basic math and physics you go then.

The only discrepancy is in your understanding, unless you're deliberately putting up strawmen again of course. As anyone can tell you've just quickly changed the time coordinates and hoped nobody would notice. See how you subtly dropped the prime on the t coordinate so that you could pretend it was the schwarzschild t, rather than the as it had been defined.

At least you've finally come around to using in the minkowski metric yourself

In any case, if you can't even admit to your most obvious errors and ATM claims, there's apparently no point in trying to teach you this basic stuff anyway. Good luck in your course on introductory linear algebra.

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Originally Posted by caveman1917
The only discrepancy is in your understanding,
Not really, I simply pointed out the miscalculations in your method. I notice that you are unable to answer using science (outside your standard personal attacks and insults).

At least you've finally come around to using in the minkowski metric yourself
You mean you can't recognize your own hack vs. the Schwarzschild solution? I am using the GR approach, I just contrasted it against the result obtained using strictly your approach.

there's apparently no point in trying to teach you this basic stuff anyway. Good luck in your course on introductory linear algebra.
The amusing thing is that I published a book on advanced numerical techniques used in linear algebra (reduction to Jordan canonical forms, calculation of the Penrose pseudoinverse matrix , eigenvectors and eigenvalues for complex valued matrices, etc). The fact that you are resorting to insults points out to the fact that you are losing the scientific argument.
Last edited by macaw; 2011-Oct-06 at 11:33 PM.

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A funny one to end this with.

My solution:
Originally Posted by caveman1917
Let's drop the "sat" subscripts for consistency of notation

1)

Now let's try a substitution 2)

This gives us after some trivial calculation

3)

Now, you claim 3 is correct but 1 incorrect, therefor the substitution 2 is incorrect, thus your claim A:

But then we also have a claim B from you:
Originally Posted by macaw
Originally Posted by caveman1917
For the standard form of schwarzschild metric, for a stationary observer at radial coordinate R:

True
As anyone can tell about statements A and B (using standard symbolic logic notation):

Now let us introduce another statement C as "the moon is made of cheese". And let us use the well-known principle of explosion.

So it follows that and thus you are claiming that the moon is made of cheese. Got any reference for that?

Note that this could be done with any of your multiple mutually contradictory statements, but using this one also makes it plain clear that you just subtly changed the time coordinates in order to come up with your latest non-existent "objection".
Last edited by caveman1917; 2011-Oct-06 at 11:58 PM. Reason: r -> R

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Originally Posted by macaw
The amusing thing is that I published a book on advanced numerical techniques used in linear algebra (reduction to Jordan canonical forms, calculation of the Penrose pseudoinverse matrix , eigenvectors and eigenvalues for complex valued matrices, etc).
It is indeed amusing that, if that is true, you then still can't recognize or calculate a simple change of basis in a 3d vector space.

The fact that you are resorting to insults points out to the fact that you are losing the scientific argument.
Not really, that's just the old "Quid pro quo" adage.

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Originally Posted by caveman1917
It is indeed amusing that, if that is true, you then still can't recognize or calculate a simple change of basis in a 3d vector space.
Actually, I can. What I am criticizing is the fact that your "choice of base" is unphysical. It is just a failed attempt to get your results to line up with the correct results.

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Originally Posted by caveman1917
A funny one to end this with.

My solution:

Now let's try a substitution 2)
You don't know that, you are using SR, so you DO NOT KNOW anything about gravitational redshift, therefore is off limits for you. If you are going to claim that the above is "just a substitution" I will tell you that you are just using numerology since you have no way of knowing the relationship between and without USING GR.

This gives us after some trivial calculation

3)
Well, it is trivial but it isn't a "calculation", it is yet another sleigh of hand since (see above) you cannot possibly know the GR relationship between and without USING GR.

Now, you claim 3 is correct but 1 incorrect, therefor the substitution 2 is incorrect,
You got things backwards, what I claimed (since post 6) and what I continue to claim is that:

*3 is correct (I obtained it in ONE line of calculation directly off the Schwarzschild metric)

*2 is trivially correct (actually is a trivial consequence of 3, SO YOU and pogono CANNOT USE IT SINCE IT IS DERIVED FROM the Schwarzschild solution, something that is CANNOT be part of YOUR "THEORY". You are using SR, you CAN'T put in by hand).

*1 was "derived" by you through a sleigh of hand that involved an obviously incorrect derivation of . You have just shown that you are also using that is off limits for you.

On the other hand, if one uses the trivial substitution of 2 into my general solution 3 while operating within the scope of GR ALL ALONG, one correctly arrives to 1. The way you arrived to 1 isn't correct. If you look at the posts in all this thread, I have challenged your "derivation" all along.

Is it clearer now for you?

You know, I really don't understand why you keep insisting in this stance after I have given you the correct general answer since post 6. Why do you insist in using an obvious cheat that is riddled with mistakes for solving one trivial subcase when you have already been given the general solution derived from base principles? Just to support pogono's fringe claim that he has a valid derivation?
Last edited by macaw; 2011-Oct-07 at 03:56 AM.

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Originally Posted by macaw
Actually, I can.
Good, then you will have no problem telling me what you wrote down here (let's ignore the typo about instead of )
Originally Posted by macaw
Problem is, the above result is in disagreement with the result that mainstream physics obtains by using the correct approach, via the Schwarzschild solution:

is either the minkowski metric or the schwarzschild solution?

Let's see wether you really can.

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Originally Posted by caveman1917
Good, then you will have no problem telling me what you wrote down here (let's ignore the typo about instead of )

is either the minkowski metric or the schwarzschild solution?

Let's see wether you really can.
The correct, general and rigorously derived solution based on the Schwarzschild solution has been posted since post 6.

The errors and the cheats in your attempt at the solution have been listed repeatedly, the last time in post 98.

Lastly, I suggest that you drop the professorial tone, the errors in your approach clash with the tone. Thank you.

11. Everybody in this thread is confused. I fear I may be joining them shortly. I don't have the time to untangle this mess properly, so I'll only comment on the first page.
Originally Posted by caveman1917
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
This is correct, at the particular instant specified.

Originally Posted by caveman1917
Recall that the velocity of the satelite as seen by the observer is
It isn't. The observer sticking a measuring rod radially would have its coordinate length related to its proper length in accordance to grr. More formally, the frame of a static observer is given by
.
So, if we take dρ = dr/√(1-rs/R) as the observer's measuring rod, the radial velocity component is given by

Which is what you wanted in the previous equation. Waving one's hands at this instead is not helpful.

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

where:

That's not right. Since ∂t and ∂φ are Killing fields, giving the specific energy and angular momentum constants of motion that you have above. Substituting into gives:

[Edit: Corrected important minus sign typo.] This is your expression for v², and it is the radial part (dr/dτ)² only. The satellite four-velocity is

This is the Lorentz gamma measured by the stationary observer, and it's the same same as rescaling (dt/dτsat) by (dt/dτobs). Therefore, we could just worry about dτsat/dτobs for arbitrary motion, as caveman1917 has done; though his equation is only correct at a particular instant, that's all he claimed of it.

Originally Posted by caveman1917
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.
The only step shared with SR is the final dot product of observer and satellite four-velocities. As you say, they're co-located at that instant, those vectors are in the same tangent space, making it valid at that instant. But calling the whole thing SR is rather confusing, because the entire derivation prior to that step is manifestly not SR. It also would've helped immensely to frame the problem in the right way. Coordinate speed is conceptually irrelevant in general, though it has some uses in special cases. For example, the following comment is very strange:
Originally Posted by caveman1917
Irrelevant. The proper speed has nothing to do with it, v is the coordinate speed in the frame of the stationary observer.
Ok, suppose you have an arbitrary static spacetime, , where T is independent of t, a static observer at some location xi, which has some satellite pass by it. Then:

Because of hij, that's very clearly not the coordinate speed. But multiplying through by (dτobs/dτsat)² gives:

In isotropic coordinates, hij = h²δij, and we can say:

The factor (h/T)² is simply the square of the coordinate speed of light, so your statements about isotropic coordinates are not wrong. However, we see from the above that the proper speed is always relevant directly, isotropy or no.

Originally Posted by caveman1917
For the standard form of schwarzschild metric, for a stationary observer at radial coordinate R:

=> (2)
I've no idea what you're doing on the second line, but the first line and the second equation in the third are correct. But that's not what coordinate speed means. It seems that rather than using the Schwarzschild radial and time coordinates, you're mixing and matching the Schwarzschild radial coordinate and the proper time of the observer. What a strange thing to call 'coordinate speed'! If you're going to associate some coordinates with the observer instead of using the Schwarzschild ones, you might as well be building a frame, as done above... this doesn't just rescale Schwarzschild time to proper time, but Schwarzschild radius to proper radial distance, so that you can forgo the entire mess altogether.
Last edited by Kuroneko; 2011-Oct-12 at 11:06 PM. Reason: -h²/r² → +h²/r²

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Originally Posted by Kuroneko
Everybody in this thread is confused. I fear I may be joining them shortly. I don't have the time to untangle this mess properly, so I'll only comment on the first page.

This is correct, at the particular instant specified.
Yes, it is trivially correct. Except that caveman1917 derivation is invalid, that it the whole point of the thread.
The errors are outlined in post 98, the correct derivation is given in post 6. There is absolutely no reason for this thread to have run over 100 posts.

That's not right. Since ∂t and ∂φ are Killing fields, giving the specific energy and angular momentum constants of motion that you have above. Substituting into gives:

This is your expression for v², and it is the radial part (dr/dτ)² only.
Slight correction to your above derivation:

Sorry, I used the notation from Rindler, v is certainly , the proper radial speed, As an aside, I have repeatedly told caveman1917 throughout the thread that working in coordinate speed is meaningless.

The satellite four-velocity is
You can see that I put down the other two components of the four velocity in the second and the third equations of the post you cited. More precisely:

and

I've no idea what you're doing on the second line, but the first line and the second equation in the third are correct. But that's not what coordinate speed means. It seems that rather than using the Schwarzschild radial and time coordinates, you're mixing and matching the Schwarzschild radial coordinate and the proper time of the observer. What a strange thing to call 'coordinate speed'! If you're going to associate some coordinates with the observer instead of using the Schwarzschild ones, you might as well be building a frame, as done above... this doesn't just rescale Schwarzschild time to proper time, but Schwarzschild radius to proper radial distance, so that you can forgo the entire mess altogether.
Absolutely, this is not an SR problem, nor is the messy attempt to solve the problem using contrived Minkowski metrics valid. This is an elementary GR problem that is solved in one line using GR.
Last edited by macaw; 2011-Oct-12 at 04:14 PM.

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Originally Posted by Kuroneko
It seems that rather than using the Schwarzschild radial and time coordinates, you're mixing and matching the Schwarzschild radial coordinate and the proper time of the observer.
That was not my choice, it was the original problem statement in the other thread where this got taken from. The problem defined using a coordinate basis as follows: as time coordinate the proper time of an observer at stationary r, as spatial coordinates the standard schwarzschild coordinates at the same stationary r.

What a strange thing to call 'coordinate speed'!
It is a coordinate speed, don't be misled by the appearance of .
The calculation of the coordinate speeds of light is just as a shortcut for a basis change to an orthonormal basis. When working in minkowski with a general orthogonal basis, division of the velocity in some direction by the coordinate speed of light in that direction gives the velocity in that direction from an orthonormal basis. That's the reason for the use of the coordinate speeds of light.

If you're going to associate some coordinates with the observer instead of using the Schwarzschild ones, you might as well be building a frame, as done above... this doesn't just rescale Schwarzschild time to proper time, but Schwarzschild radius to proper radial distance, so that you can forgo the entire mess altogether.
True, it can be done a lot easier, but the entire point was to show that the mess also works.

You might want to look at post 70 instead, where the argument is given a bit less messy ( and are real constants such that )

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Originally Posted by macaw
The errors and the cheats in your attempt at the solution have been listed repeatedly, the last time in post 98.
It is not in error.

As far as i can tell your objections have been reduced to the following:

1. I can't transform without "using GR".
2. You claim there is something wrong with the expression for
3. You claim there is something "unphysical" about the choice of basis.

For objection 1, i will give you two statements
A. I can scale a vector in a vector space (such as minkowski) by multiplying with a real number a.
B.

meaning that i can write any real number as

Since you claim i need to "use GR" to do this, you must disagree with at least one of those statements. Which one?

For objection 2:
You are given a minkowski space in a certain orthogonal basis such that the metric in those coordinates is expressed as

Calculate the expression for in those coordinates. Show how it differs from my expression.

For objection 3:
First you (correctly) claim that nature doesn't care about anything that starts with "coordinate", but now you claim that certain coordinate systems are "physical" while others are "unphysical". Please provide a reference showing exactly which properties a coordinate system must have to be "physical".

You probably mean that you're not used to working in such a coordinate system, which is true, but coordinates are just coordinates. The entire point of this exercise is doing it in a non-standard basis. That was the original problem statement, remember?

15. Originally Posted by caveman1917
That was not my choice, it was the original problem statement in the other thread where this got taken from. The problem defined using a coordinate basis as follows: as time coordinate the proper time of an observer at stationary r, as spatial coordinates the standard schwarzschild coordinates at the same stationary r.
...
You might want to look at post 70 instead, where the argument is given a bit less messy ( and are real constants such that )
Looks good. I'm going to restate it for a more general stationary spacetime in coordinates for which the metric is diagonal, because frankly I think it's clearer that way, and perhaps it will be worthwhile for others paying attention to this thread. Suppose the metric has some arbitrary coefficients: . Then:

[Edit: Corrected missing square typo. Thanks for corrections, again. *]
So the squared quantity in question is definitely consisting of the coordinate speed components of the satellite scaled by coordinate speed of light in that direction (ψi/T), as you have it. You've made a coordinate transform to t' satisfying dt' = Tdt and hence new T' = 1. This is completely legitimate, since it's just a coordinate transformation that changes nothing essential:

Additionally, a step you've done that I've left out until now is evalutationg the metric at a particular event (time and place of observer meeting the satellite). I've omitted it only to make it clear that your derivation of the instantaneous time dilation factor is completely equivalent: you evaluated at the start, I at the end. Since we're only interested in that instant, either way is fine.

The reason I found it initially rather confusing is to what extent this can be said to 'use SR'. Yes, the tangent space of every event is Minkowski, but that's not what's commonly understood by approximated by Minkowski spacetime or having locally Minkowski coordinates. Local means "on a neighborhood" rather than "at a point". That's definitely not the case here; to do so, we'd (probably) have to match up the acceleration of the observer with the Rindler chart to first order.

But I see now that's that's more an issue of vocabulary rather than anything essential to your derivation. And has apparently been addressed in this thread anyway (nearly lost among the commotion as it was).
Last edited by Kuroneko; 2011-Oct-14 at 09:29 PM. Reason: exponent dτ→dτ²

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Originally Posted by Kuroneko
The reason I found it initially rather confusing is to what extent this can be said to 'use SR'. Yes, the tangent space of every event is Minkowski, but that's not what's commonly understood by approximated by Minkowski spacetime or having locally Minkowski coordinates. Local means "on a neighborhood" rather than "at a point". That's definitely not the case here; to do so, we'd (probably) have to match up the acceleration of the observer with the Rindler chart to first order.
I see the confusion. The emphasis on the statement of "locally approximating minkowski" wasn't meant on the "locally" but on the "minkowski". In the sense of establishing that the correct environment for this is minkowski, if the tangent space was something else, say euclidean, none of this would have worked. It was not meant as a statement about the neighborhood of an event rather than at an event, just establishing the correct space to use.

17. Well, then there's absolutely no problem.

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Now that macaw's been permabanned, what becomes of this thread?

19. Originally Posted by Nereid
Now that macaw's been permabanned, what becomes of this thread?
That's a question you could ask the mods, but here IN the thread it's off-topic meta discussion. (However, the answer is: here is a thread, and here it stays. For now it's not closed, there may yet be more useful discussion to add).

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Originally Posted by Kuroneko
Just correcting a minor typo where the exponent on the first term has dropped:

Perhaps it would be worth it to point out that the einstein summation convention is used in the above.
for 3 dimensions of space, likewise for all repeated indices of i.

21. Hi,
I would like to add something.

As I see we have at least achieve the goal and we know that my statement in original thread was true. It was the foundation of this thread's polemic.

Let me ask then, Kuroneko, caveman1917, am I right saying that:

1. We may consider gravity as anisotropy in infinite small, locally Minkowski (limit where space -> "spot") metric for every Hanging Observer. Am I right?

2. If - instead analyzing satellite, we consider photon - then for every Hanging Observer at every "r-distance" in his "local" spot where photon passes his way we obtain "local" Minkowski in form of:

where:

and where I have exchanged with to stress that it is for any "r"

Am I right?

3. Then we can transform above to form of:

introducing:

- just spartial increment for photon

we obtain:

Am I right?

4. So, we should then blame dz for gravitational force, and expect that it has a lot to do with:

just as I claim. Am I right?

5. And at last step we may just recapitulate field existence with short formula for some imaginary move, transforming dz formula as follows:

Am I right?

I appreciate if you validate my reasoning.
Last edited by pogono; 2011-Oct-18 at 12:05 PM. Reason: tau->dtau - thanks to caveman1917

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Originally Posted by pogono
1. We may consider gravity as anisotropy in infinite small, locally Minkowski (limit where space -> "spot") metric for every Hanging Observer. Am I right?
No that is not correct.
The anisotropy that was encountered in the Minkowski metric associated with the observer is due to the peculiar (non-orthonormal) basis being used, not due to gravity. You can see that because it is a linear transformation of the standard basis, is a constant since is a constant. This is not the same as a transformation by since is a variable, not a constant. Basically, the Minkowski metric cannot have non-constant coefficients. And whenever you're describing gravity you must have non-constant coefficients. It's either one or the other.

2. If - instead analyzing satellite, we consider photon - then for every Hanging Observer at every "r-distance" in his "local" spot where photon passes his way we obtain "local" Minkowski in form of:

where:
You have forgotten the differentiation operator on throughout, but that's not the main issue.
What you have here is the expression for null intervals on the schwarzschild metric, where you have set .
This is not the Minkowski metric, it becomes the minkowski metric when you set the coefficients to constants, for example (this includes the !).

3. Then we can transform above to form of:

introducing:

- just spartial increment for photon

we obtain:

Am I right?
Other than the differentiation operator on that is correct, but you've just rewritten things in different symbols. It's still the exact same expression for a null interval in the schwarzschild metric as you started with.

4. So, we should then blame dz for gravitational force, and expect that it has a lot to do with:
You've put the non-linear coefficients in , so in that sense you could say that you "blame " for the gravitational force. But don't forget that also your depends on because of the , so in your language you should "blame" both. This is just a consequence of the spherical symmetry of the schwarzschild metric though.

just as I claim. Am I right?
What does mean?

5. And at last step we may just recapitulate field existence with short formula for some imaginary move, transforming dz formula as follows:

Am I right?
I'm not sure what you mean by "recapitulate field existence", but if you mean "rewriting the gravitational field descibed by the schwarzschild metric in a different form", then yes, sure. I'm just not sure what the point is you're trying to make.

You seem to be under the impression that you're working in the Minkowski metric, which is not true, because the coefficients are not constants.
A rule of thumb is that if you can perform a linear transformation to the standard form of the Minkowski metric, you are in Minkowski space and you have no gravity. If you cannot, you're not in Minkowski space and you do have gravity (at least as long as your definition of gravity includes non-inertial frames in flat spacetime).

23. Originally Posted by pogono
Hi,
I would like to add something. ...
No, not when it's (despite your claims) ATM outside of the ATM forum.

24. Thank you caveman1917.
In few words I will explain what I try to achieve by transforming Schwarzschild metric such way.

But at first, I would like to ask mods, to notice, that I am not going to support my ATM idea. It is just Schwarzschild metric transformation. Even if I have started my life as ATM supporter, it does not mean, that everything that comes out from my mouth has to be ATM.

I learn physics (I have just finished to learn Rindler's transformation), and here I would like to ask if I understand it the right way. Even if I make some mistakes it is still mainstream (macaw also made mistakes here and he was not suspended for making mistakes).

Originally Posted by caveman1917
You've put the non-linear coefficients in , so in that sense you could say that you "blame " for the gravitational force. But don't forget that also your depends on because of the , so in your language you should "blame" both.
Hanging observer's proper time may be expressed by dt. By easy transformation we obtain:

Since we know:
Originally Posted by pogono
introducing:

- just spartial increment for photon

we obtain:

it means:

But we know, that for photon, dt=dx, what drives to:

Originally Posted by caveman1917
Originally Posted by pogono
I'm not sure what you mean by "recapitulate field existence", but if you mean "rewriting the gravitational field descibed by the schwarzschild metric in a different form", then yes, sure. I'm just not sure what the point is you're trying to make.
As we may see by easy substitution we obtain:

Is this ok?

25. Originally Posted by pogono
But we know, that for photon, dt=dx, what drives to:
We don't know that. In the Schwarzschild spacetime with the coordinates as you've defined them, it's definitely false.

I don't know what you're trying to do. Photon orbits are easy to derive in Schwarzschild coordinates without any hanging observers; what they observe can be found afterward. Do you know how to calculate Christoffel symbols (aka connection coefficients)? If so, I can show you how to find photon orbits fairly easily. There's also an even easier method, but Killing vector fields are conceptually more complicated. In the meantime, just try calculating the connection coefficients in Schwarzschild coordinates with θ = π/2 (dθ = 0) as we've been doing for simplicity.

26. Thank you Kuroneko for your response.

Originally Posted by Kuroneko
Do you know how to calculate Christoffel symbols (aka connection coefficients)? If so, I can show you how to find photon orbits fairly easily. There's also an even easier method, but Killing vector fields are conceptually more complicated. In the meantime, just try calculating the connection coefficients in Schwarzschild coordinates with θ = π/2 (dθ = 0) as we've been doing for simplicity.
Thank you, I already know that. It is available in my textbook :-) But it is not the point.
P.S. Kiiling vectors as I suppose are root vectors in linear algebra, so I suppose I would be able to understand it.

Originally Posted by Kuroneko
I don't know what you're trying to do. Photon orbits are easy to derive in Schwarzschild coordinates without any hanging observers; what they observe can be found afterward.
Explaining my motivation:
I want to use Hanging Observer, because otherwords he is measuring, static observer.
I am trying to dig tunnel to Quantum Mechanics, so I try to transform Schwarzschild metric to form appropriate for measuring observer, because measuring observer is all we have in QM.

27. Originally Posted by pogono
Thank you, I already know that. It is available in my textbook :-) But it is not the point.
P.S. Kiiling vectors as I suppose are root vectors in linear algebra, so I suppose I would be able to understand it.
Out of curiosity, what textbook are you using?

P.S. A Killing vector field is one for which the Lie derivative of the metric vanishes. They generate an isometry and, by Noether's theorem, a conserved current. This is important because for the Schwarzschild coordinates, the metric components are independent of t and φ. So ∂t and ∂φ are automatically Killing vector fields, which means that for any orbit with four-velocity u, e = -∂t·u = (1-2M/r)(dt/dτ) is a constant of motion (specific energy), and so is h = ∂φ·u = r²(dφ/dτ) (specific angular momentum). You can substitute this into gμνuμuν = 0 for photons to get the (radial) effective potential directly, without messing about with any Christoffel symbols. Also works for timelike geodesics, into gμνuμuν = -1.

Originally Posted by pogono
Explaining my motivation:
I want to use Hanging Observer, because otherwords he is measuring, static observer.
I am trying to dig tunnel to Quantum Mechanics, so I try to transform Schwarzschild metric to form appropriate for measuring observer, because measuring observer is all we have in QM.
If you're doing quantum mechanics in Schwarzschild spacetime, the most natural arena are freefalling frames, because they're unaccelerated. It's actually quite interesting--for observers freefalling from rest at infinity in Schwarzschild spacetime, slices of constant time are completely Euclidean, and dynamically it looks like space being sucked into the singularity at the local escape velocity.

If you're trying to quantize the Schwarzschild solution itself, then your whole approach is mistaken, You should study the Lagrangian for the metric itself (the Einstein-Hilbert action) first, to understand the issues in quantizing GTR. Quantization of classical systems does not and cannot happen by taking a single classical state by itself, outside any dynamical context. And in this case, all of Schwarzschild spacetime would be a single classical state of the metric!

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Originally Posted by Kuroneko
It's actually quite interesting--for observers freefalling from rest at infinity in Schwarzschild spacetime, slices of constant time are completely Euclidean, and dynamically it looks like space being sucked into the singularity at the local escape velocity.
Interestingly enough, that dynamic picture even works for general Kerr-Newman black holes, though slices of constant time are not Euclidean anymore because there is a twist vector at each point of space in addition to the velocity vector.

29. Originally Posted by Kuroneko
Out of curiosity, what textbook are you using?
GR by Bernard F. Schutz and
polish textbook "Very Special Relativity" by Andrzej Dragan
and few ebooks
I appreciate if you advice something else.

Originally Posted by Kuroneko
P.S. A Killing vector field is one for which the Lie derivative of the metric vanishes.
Ok, I have to recalculate it by my own, to understand it. For now it looks suspiciously simply ;-)
Thank you.

Originally Posted by Kuroneko
If you're doing quantum mechanics in Schwarzschild spacetime, the most natural arena are freefalling frames, because they're unaccelerated.
It was not exactly my point. In free-falling frame gravity indeed vanishes...

I will explain my point this way.
Newton said in 1'st law: "if you do not accelerate, you are inertial system" what makes your v=constans what makes your proper time flow constans

I was trying to explain "static space" by collection of measuring observers, that have to accelerate to keep its proper time constans (that is why I recall Rindler's transformation). But it seems to work also for orbiting bodies.

So, other words I am trying to paraphrase Newton, and say: "if you keep your proper time flow constans, you are 'locally inertial system', even if you accelerate" and applicate above to QM.

It is quite complicated even for me :-)
I will explain it in private message.

Originally Posted by Kuroneko
It's actually quite interesting--for observers freefalling from rest at infinity in Schwarzschild spacetime, slices of constant time are completely Euclidean, and dynamically it looks like space being sucked into the singularity at the local escape velocity.
Originally Posted by caveman1917
Interestingly enough, that dynamic picture even works for general Kerr-Newman black holes, though slices of constant time are not Euclidean anymore because there is a twist vector at each point of space in addition to the velocity vector.
Thank you. Sounds enough interesting to take a closer look...

30. Originally Posted by pogono
Ok, I have to recalculate it by my own, to understand it. For now it looks suspiciously simply ;-)
What you should be getting after substituting into gμνuμuν is that every orbit in Schwarzschild spacetime has a specific effective potential like so:

where -1 for timelike geodesics and 0 for lightlike ones. An equivalent formulation is Schutz p. 276. I wrote it in this form so it can be easily compared with the Newtonian one. For lightlike geodesics, only b² = h²/e² is important.

Originally Posted by pogono
I will explain my point this way.
Newton said in 1'st law: "if you do not accelerate, you are inertial system" what makes your v=constans what makes your proper time flow constans
That's exactly what a geodesic is, though: curves of constant velocity. More precisely, a parallel-transporting the 4-velocity of a geodesic along the geodesic does not change it, . Remember that to compare vectors at different locations in spacetime (which includes same place according to some slicing but different time!), you need to be careful about how you're transporting the vector in order to compare them. I can make no sense of , though. Do you mean dτ/dt=const with t Schwarzschild time?

Originally Posted by pogono
So, other words I am trying to paraphrase Newton, and say: "if you keep your proper time flow constans, you are 'locally inertial system', even if you accelerate" and applicate above to QM.
There's nothing intrinsic about Schwarzschild time, though, and if you parallel-transport the 4-velocity of a hanging observer along that observer's worldline, you don't get their 4-velocity there. The proper spacetime interpretation of Newton's "inertial motion = constant velocity" is an affine geodesic, and that's exactly what GTR has, since it makes several assumptions about the connection.