Gullstrand–Painlevé coordinates

[Mordred]

A reference frame does not require a convention of simultaneaty. Simultaneaty is typically lost in curved spacetime as mentioned several times by Markus. The infalling clock falls along a specific curved spacetime path in PG coordinates. That is provided by the PG line element.

 

[externo]

1 hour ago, Mordred said:

A reference frame does not require a convention of simultaneaty. Simultaneaty is typically lost in curved spacetime as mentioned several times by Markus. The infalling clock falls along a specific curved spacetime path in PG coordinates. That is provided by the PG line element.

The line of simultaneity is no less important than the world line. The world line marks proper time and the line of simultaneity marks proper length. So if we can give the world lines as a definition of a frame of reference, we can also give the lines of simultaneity as a definition. The line of simultaneity is absolute in the same way as the world line. This is something that is denied by the conventionalists, who claim that simultaneity does not exist. We can see that they do not know what they are talking about.

Quote

Simultaneaty is typically lost in curved spacetime

It is not lost, it is the line that marks proper length.

The surface of the Flamm paraboloid is the slice of simultaneity of stationary objects in a gravitational field.

Edited July 30 by externo

[Mordred]

The conventionalistics you mentioned are stating differently than your assertions.  So at this point you need to provide the mathematical detail showing otherwise. As Markus has already requested.

 You already concluded that simultaneaty is lost in accordance to the conventional methodologies.

It's up to you to show why that shouldn't be the case mathematically at this point.

Edited July 30 by Mordred

[Markus Hanke]

17 hours ago, externo said:

Here is the detail of Gemini 1.5 pro :

At this point in time, AIs are not considered a valid source of scientific information, because they make too many mistakes. I suggest you stick to proper textbook sources.

But regardless, we’re not dealing in local approximations at a single event here, since the frame of the falling observer is spatially removed from the Schwarzschild observer, so you need to use a global transformation that accounts for curvature.

So once again - the global map between these charts is not a Lorentz transformation, not even approximately.

17 hours ago, externo said:

The clocks of Lemaître's observers share the same global simultaneity

Again - there is no notion of global simultaneity in curved spacetimes.

17 hours ago, externo said:

Same spacetime, but not same physics.

This statement is meaningless. The causal structure of spacetime is not coordinate dependent.

17 hours ago, externo said:

the speed of light in one direction is no longer the same as the other

This has already been addressed - proper vs coordinate speed.

17 hours ago, externo said:

You give here for definition of a reference frame the same thing I gave above.

No, what you gave wasn’t the same.

17 hours ago, externo said:

A reference frame requires a convention of simultaneity.

No it does not.

17 hours ago, externo said:

This definition is wrong.

The definition is from the text I quoted, which is an authoritative source. I didn’t just make this up. You might not agree with the accepted definition, but that’s your own issue.

The point is that, according to this definition, the GP observer constitutes a valid frame - which is what every single textbook on GR says. Working in the GP frame (sometimes called “raindrop coordinates”) is a standard exercise, you know.

17 hours ago, externo said:

Now it turns out that this change of coordinate from (dt,dr) does not constitute a boost, unlike the (dT,dρ) of Lemaître. This is why I said that the Painlevé coordinates were not a reference frame according to relativity.

…and then, a few lines later:

17 hours ago, externo said:

In the Painlevé coordinates, the cone tilts with the faller which is thus in inertia. Only the GP coordinates are conform to the equivalence principle. These coordinates are thus the physical reference frame.

You seem to contradicting yourself now.

15 hours ago, externo said:

The line of simultaneity is no less important than the world line.

No such thing exists. There are only hypersurfaces of simultaneity, as used for example in the ADM formalism - but these are specific to a given observer, and are not generally shared by other observers in that curved spacetime. So once again, there’s no global notion of simultaneity that all observers agree on.

Edited July 31 by Markus Hanke

[externo]

11 hours ago, Markus Hanke said:

At this point in time, AIs are not considered a valid source of scientific information, because they make too many mistakes. I suggest you stick to proper textbook sources.

But regardless, we’re not dealing in local approximations at a single event here, since the frame of the falling observer is spatially removed from the Schwarzschild observer, so you need to use a global transformation that accounts for curvature.

So once again - the global map between these charts is not a Lorentz transformation, not even approximately.

If you are unable to verify whether the calculations are correct, you are not a valid source of information yourself... I said that going from (dt,dr) to (dT,ρ) corresponded to boost. I'm giving you this as an information only. I'm not asking for your opinion in this matter.

11 hours ago, Markus Hanke said:

Again - there is no notion of global simultaneity in curved spacetimes.

Simultaneity is what allows to measure the proper length of an object. Objects have a proper length even in curved spacetime, so there is simultaneity whether we are in curved spacetime or not.

11 hours ago, Markus Hanke said:

This statement is meaningless. The causal structure of spacetime is not coordinate dependent.

What is the causal structure of spacetime?

11 hours ago, Markus Hanke said:

This has already been addressed - proper vs coordinate speed.

The isotropy of the speed of light depends on the synchronization convention. The proper speed of light depends on the choice of synchronization. Only Einstein's synchronization gives an isotropic proper speed of light. You are confusing the change in speed caused by the closure of the light cone, which is a coordinate speed, with the speed caused by the choice of synchronization, which is a proper speed. The proper speed of light is isotropic or not depending on the choice of the synchronization convention, this is what the conventionalists say.

11 hours ago, Markus Hanke said:

The definition is from the text I quoted, which is an authoritative source. I didn’t just make this up. You might not agree with the accepted definition, but that’s your own issue.

The point is that, according to this definition, the GP observer constitutes a valid frame - which is what every single textbook on GR says. Working in the GP frame (sometimes called “raindrop coordinates”) is a standard exercise, you know.

But the simultaneity dr +v dT used in the Painlevé coordinates does not correspond to the simultaneity of the faller's frame of reference according to Einstein's synchronisation. The simultaneity of the faller's frame of reference is dρ. For the cone to tilt, we must act as if dr +v dT were the simultaneity of the faller, so we must create a frame of reference in which dr +v dT corresponds to the simultaneity, so this is what I am saying : (dT, dr +v dT) does not form a valid frame of reference according to relativity. This is very clear.

Moreover, according to conventionalists, simultaneity does not exist and therefore one cannot define hypersurfaces of simultaneity simply with worldlines. One must also choose a convention of simultaneity.

11 hours ago, Markus Hanke said:

 

…and then, a few lines later:

You seem to contradicting yourself now.

No such thing exists. There are only hypersurfaces of simultaneity, as used for example in the ADM formalism - but these are specific to a given observer, and are not generally shared by other observers in that curved spacetime. So once again, there’s no global notion of simultaneity that all observers agree on.

This shows that the theory is self-contradictory. It's not my fault if the Lemaître frame of reference is an proper accelerated frame of reference. You can verify for yourself using Schwarzschild coordinates that the falling object cannot be in free fall, as he has to resynchronized his own clocks all the time and is subject to time dilation.

11 hours ago, Markus Hanke said:

No such thing exists. There are only hypersurfaces of simultaneity, as used for example in the ADM formalism - but these are specific to a given observer, and are not generally shared by other observers in that curved spacetime. So once again, there’s no global notion of simultaneity that all observers agree on.

The hypersurface of simultaneity is just a convention so we can very well define a single hypersurface for everyone (Selleri's synchronization)
Note that I am only repeating what the conventionalists say, if you do not agree with them and think that simultaneity exists, let me know.

 

Edited July 31 by externo

[Markus Hanke]

14 hours ago, externo said:

If you are unable to verify whether the calculations are correct

The calculation is correct, but it’s a local approximation, and thus not applicable across distant frames in a curved spacetime, which is what we’re dealing with here. The global transformation is not a Lorentz transformation.

14 hours ago, externo said:

I'm not asking for your opinion in this matter.

So why did you open up a thread on a discussion forum if you’re not interested in what others have to say? What is your intent for being here?

14 hours ago, externo said:

Simultaneity is what allows to measure the proper length of an object.

Proper length is defined in an object’s rest frame, it has nothing to do with simultaneity of remote observers in a curved spacetime. Even within that frame only, it doesn’t depend on simultaneity.

14 hours ago, externo said:

You are confusing the change in speed

As said before, the locally measured speed of light in all frames is always c, whereas distant coordinate speeds can vary depending on observer.

14 hours ago, externo said:

Einstein's synchronisation

We’re not in Minkowski spacetime here.

14 hours ago, externo said:

Moreover, according to conventionalists, simultaneity does not exist

No one said this. It is always possible to foliate classical spacetimes into a family of space-like hypersurfaces of simultaneity, which is the basis for the ADM formalism of GR. 

The thing with this is that each physical observer will foliate spacetime differently, according to his own clocks and rulers. There are in fact infinitely many such foliation schemes for any given spacetime. That’s because there’s no single notion of simultaneity that is globally agreed on by all observers. So what we are saying is not that no notions of simultaneity exist, but rather that such notions are not globally shared by distant observers separated across a curved spacetime. In essence, each observer has his own natural hypersurface of simultaneity, which isn’t shared by other observers.

I suggest the section on the ADM formalism in Misner/Thorne/Wheeler for further study.

14 hours ago, externo said:

It's not my fault if the Lemaître frame of reference is an proper accelerated frame of reference.

Natural Lemaître observers are “raindrop” observers - they are in free fall starting from rest. Therefore \(a^{\mu}=0\) at all times.

14 hours ago, externo said:

You can verify for yourself using Schwarzschild coordinates

Free fall is defined as the vanishing of proper acceleration at all points - meaning the resulting world line is a solution to the geodesic equation. This is coordinate-independent, and clearly holds for raindrop observers, irrespective of coordinate basis.

As it so happens, I have in fact gone through the procedure of formally solving the geodesic equations for free fall in Schwarzschild spacetime, back when I first learned GR, so I know first hand that time-like free fall geodesics and null geodesics do in fact exist in this spacetime. Unsurprisingly so.

14 hours ago, externo said:

Selleri's synchronization

We’re not in Minkowski spacetime here - for the I-don’t-know-how-many-th time.

Quote

What is the causal structure of spacetime?

It is how distant events in a spacetime are causally related, ie how they can be connected by smooth, everywhere differentiable curves, and the tangent spaces on those curves (ie time-like, null, space-like). This structure is independent of coordinate choices.

—-

To return to the original topic of this thread: your claim was that GP coordinates do not constitute a valid frame of reference. 

This thread is in the mainstream sections, so I’ve used the agreed-upon textbook definition of what constitutes a reference frame to show that the original assertion doesn’t hold. If you wish to suggest an alternative definition, the “Personal Theories” section would be the appropriate place.

Your question has been answered, even if you don’t agree with the answer.

 

Edited August 1 by Markus Hanke

[externo]

8 hours ago, Markus Hanke said:

As said before, the locally measured speed of light in all frames is always c, whereas distant coordinate speeds can vary depending on observer.

It is impossible to measure the speed of light. The speed on a round trip is always measured at c, which means that the narrowing of the cone, which gives the impression of a reduction in the speed of light on a round trip is a coordinate change in speed, but you cannot know whether the speed of light is isotropic or not. So you cannot know whether the light cone is actually tilting or not.

8 hours ago, Markus Hanke said:

In essence, each observer has his own natural hypersurface of simultaneity, which isn’t shared by other observers.

What you are saying here is not what the conventionalists say. According to Reichenbach there is no natural hypersurface of simultaneity. Einstein's convention can be replaced by any other convention. Natural simultaneity does not exist.
It seems that Einstein was a conventionalist and that the consensus today is rather in this direction:
https://philsci-archive.pitt.edu/674/2/epsilon_sim.pdf

8 hours ago, Markus Hanke said:

Natural Lemaître observers are “raindrop” observers - they are in free fall starting from rest. Therefore aμ=0 at all times.

Natural Painlevé observers are “raindrop” observers, not Lemaître's. 

When transitioning from Schwarzschild to Lemaître coordinates, the light cone does not tilt; instead, the faller's hypersurface of simultaneity shifts and adjusts. This necessitates a manual resynchronization of the faller's clocks.

Perhaps you are unaware that when clocks accelerate together without the light cone tilting, as in special relativity, they maintain synchronization from a stationary perspective but desynchronize from their own perspective. Essentially, the one-way speed of light fluctuates relative to the accelerating clocks, necessitating resynchronization to restore isotropic light speed measurement. These clocks undergo proper acceleration.

Consequently, if the light cone remains fixed, the faller must inevitably experience proper acceleration and recalibrate their clocks. This is an irrefutable fact.

On the other hand, if the cone tilts as in the Painlevé coordinates, the faller is in inertia, because the cone does not change orientation relative to him during its fall.

8 hours ago, Markus Hanke said:

Free fall is defined as the vanishing of proper acceleration at all points

Thus Lemaître's faller is not in free fall because it undergoes proper acceleration, unless we assume that the simultaneity hypersurface dr of the Schwarzschild coordinates is not the "natural" simultaneity hypersurface, that is to say that the light cone that it describes is artificial, and that the real light cone tilts according to the  dr +v dT "natural" simultaneity.

8 hours ago, Markus Hanke said:

To return to the original topic of this thread: your claim was that GP coordinates do not constitute a valid frame of reference. 

This thread is in the mainstream sections, so I’ve used the agreed-upon textbook definition of what constitutes a reference frame to show that the original assertion doesn’t hold. If you wish to suggest an alternative definition, the “Personal Theories” section would be the appropriate place.

Your question has been answered, even if you don’t agree with the answer.

Reference frames also define curved hypersurfaces of simultaneity. The hypersurfaces of simultaneity are not the same for Painlevé coordinates as for Lemaître coordinates, although these two coordinate systems share the same time vector field, so one of the two coordinate systems does not contain the right hypersurfaces of simultaneity.

I want to point out that I agreed to follow the definition you gave but it makes things more complicated and it is wrong. A reference frame requires both time and space coordinates and the mathematical definition you give has no physical interest. This is a physics forum, so the definition you give is out of place here.

Edited August 1 by externo

[Mordred]

Do you honestly mean that you do not see the 4  coordinates in the equation for the reference frame supplied by Markus ?

Specifically this term

\[x^\mu\]

Edited August 1 by Mordred

[Mordred]

  Reading over this thread I really have come to the conclusion that although you may or may not understand SR via Minkowskii (remains to be seen) its become evident that you really do not understand GR and curved spacetimes regardless if its the Schwarzschild metric or PG coordinates. Markus has had to repeat the same statements over and over again in particularly that there is no global simultaneity. This is the primary reason the infalling clock (proper-time follows the null geodesic worldline of the metric.) that is true in both the Schwarzschild metric and the PG coordinates. You already disagreed with the mathematics I posted earlier showing that the in-fall signals is symmetric with the outgoing signals simply because you feel its wrong but have yet to actually post any mathematics showing GRs treatments to be incorrect.

What you termed the conventionalist view.

The other reason I feel you do not understand GR is the following statement

On 7/31/2024 at 9:22 AM, externo said:

 

What is the causal structure of spacetime?

 

 I'm seriously hoping the nature of this question is due to English not being your primary language but anyone who knows relativity should be able to answer that question.  As Markus has repeatedly stated The Schwarzschild metric  is not Minkowskii.

 

Edited August 1 by Mordred

[Mordred]

Ok apparently a different approach is needed.

@externo I would like you to consider the following we all agree that particles follow geodesics just as we all agree that the massive particles follow a different geodesic than a massless particle. So here is what you need to figure out 

What are the affine parameters of the null geodesic with regards to the Schwarzschild metric and PG coordinates and why are affine parameters needed in the massless particle case as they are not needed in the massive particle case.

You can find the answers to the above in pretty much any GR textbook though I recommend Wald,  Mathius Blau or Sean Carrol

recall in the null geodesic case \[ds^2=0\]

here is Sean Carrols lecture notes

https://arxiv.org/abs/gr-qc/9712019#

here is Mathius Blau's

http://www.blau.itp.unibe.ch/newlecturesGR.pdf

I would also highly suggest you look at the Levi-Civita affine connections and the Kronecker Delta with regards to Riemannian manifold with the above

Edited August 2 by Mordred

[externo]

7 hours ago, Mordred said:

Ok apparently a different approach is needed.

@externo I would like you to consider the following we all agree that particles follow geodesics just as we all agree that the massive particles follow a different geodesic than a massless particle. So here is what you need to figure out 

What are the affine parameters of the null geodesic with regards to the Schwarzschild metric and PG coordinates and why are affine parameters needed in the massless particle case as they are not needed in the massive particle case.

You can find the answers to the above in pretty much any GR textbook though I recommend Wald,  Mathius Blau or Sean Carrol

recall in the null geodesic case

ds2=0

 

here is Sean Carrols lecture notes

https://arxiv.org/abs/gr-qc/9712019#

here is Mathius Blau's

http://www.blau.itp.unibe.ch/newlecturesGR.pdf

I would also highly suggest you look at the Levi-Civita affine connections and the Kronecker Delta with regards to Riemannian manifold with the above

While I admit I'm not proficient in all the mathematics of general relativity, it's clear that you're hiding behind arguments of authority. Spacetime can be sliced into arbitrary coordinate systems. All these coordinate systems are supposed to yield the same physics. But this only holds true if we define certain slices as physical and others as not. For instance, the Schwarzschild slicing doesn't allow one to cross the horizon, thus it's not physical. Some might say, "Painlevé coordinates are better adapted," but this is meaningless; it's an empty phrase. On the other hand, if I say that Painlevé coordinates are "physical and Schwarzschild coordinates are not", that has meaning. Schwarzschild coordinates offer a slicing into spatial sheets that isn't physical; they are fictitious slices of space. Again, the faller in the Schwarzschild coordinates is not in free fall because his clocks get out of sync during his fall from his point of view. This is a fact. And this shows that this coordinate system is not physical.

Edited August 2 by externo

[studiot]

49 minutes ago, externo said:

While I admit I'm not proficient in all the mathematics of general relativity, it's clear that you're hiding behind arguments of authority. Spacetime can be sliced into arbitrary coordinate systems.

I'm not hiding behind anything  -  I've been away  - , but this is the wole point of 'foiliations' which you seem to misunderstand.

 

Yes indeed a foliation is a way of selecting a subspace ( sub manifold)  of a given one (manifold), exactly as you have said and slicing is a good word.

BUT

53 minutes ago, externo said:

All these coordinate systems are supposed to yield the same physics.

This is not necessarily the case.

 

A simple example relates to the laws of  Mechanics, which are different in spaces of different dimension.

And your 'slices' are another way of invoking the reduction of dimensionality map I referred to earlier.

 

Whether presented as tensors or otherwise the differences in Mechanics between dimensionality 1, 2, 3  and 4 have been taught to Engineers, Geologists and other disciplines for a very long time.

[Mordred]

1 hour ago, externo said:

While I admit I'm not proficient in all the mathematics of general relativity, it's clear that you're hiding behind arguments of authority. Spacetime can be sliced into arbitrary coordinate systems. All these coordinate systems are supposed to yield the same physics. But this only holds true if we define certain slices as physical and others as not. For instance, the Schwarzschild slicing doesn't allow one to cross the horizon, thus it's not physical. Some might say, "Painlevé coordinates are better adapted," but this is meaningless; it's an empty phrase. On the other hand, if I say that Painlevé coordinates are "physical and Schwarzschild coordinates are not", that has meaning. Schwarzschild coordinates offer a slicing into spatial sheets that isn't physical; they are fictitious slices of space. Again, the faller in the Schwarzschild coordinates is not in free fall because his clocks get out of sync during his fall from his point of view. This is a fact. And this shows that this coordinate system is not physical.

Hiding behind authority meaning recognizing and using GR correctly.

 The problem here is you keep repeating claims that run counter to GR itself yet have made zero effort to show mathematically where GR is wrong using mathematics and not mere assertions.

However as you stated your not proficient in the mathematics of GR so quite frankly stating others who are proficient in the mathematics is incorrect isn't a very useful tactic.

A good example is stating the equation Markus posted was wrong when the truth is you didn't understand the equation.

Mathematics aside do yourself a favor take a beach ball and place a ruler on the surface of a beach ball without deforming the ball

The point of contact of the ruler to the ball surface is the only portion (localized region) where the Ricci curvature can be approximated as zero.

That is only region where the Minkowskii metric will work.

You cannot globally apply Minkowsii to a curved manifold. Precisely as Markus has been stating.

PS the point of contact (tangent to the surface) is where the affine connections reside and you are using covectors /contravectors under GR.

Now think of your ruler as the basis tangent vector for your lightcones give you a hint why the lightcones are changing angles ?

Now move that tangent vector around the circumference of the ball ( that's what is being described by the Levi-Civita affine connection) as well the geodesic equation in curved spacetime.

The kronecker delta only applies for the Euclidean portions (localized hyperslices ) with regards to Minkowsii

(The RULER is the particles momentum) the surface of the ball is the Worldline.

 

Edited August 2 by Mordred

[externo]

1 hour ago, Mordred said:

Hiding behind authority meaning recognizing and using GR correctly.

 The problem here is you keep repeating claims that run counter to GR itself yet have made zero effort to show mathematically where GR is wrong using mathematics and not mere assertions.

Ok, I've finally come to understand that my repeated assertion that the falling observer in Schwarzschild and Lemaître coordinates needs to resynchronize his clocks was incorrect.

Let's approach the problem from the other side.

So, the falling observer does not need to resynchronize his clocks for the speed of light to remain isotropic. This means that the light cone actually tilts as one moves through a gravitational field. Only the coordinate system that accounts for this tilting in the necessary proportions is a physical coordinate system.

The key point to understand is that if we accelerate but do not need to resynchronize our clocks, it  means that the light cone tilts. In special relativity, it's the opposite: when we accelerate, the light cone does not tilt, and it is necessary for the moving observer to resynchronize their clocks to maintain the isotropy of the speed of light.

Edited August 2 by externo

[Mordred]

Now your getting it recall \[ x^\mu\] is the particle. X is always the coordinate axis for the particles momentum. In the Minkowskii case the changes to x is only the rescaling for length contraction.

In curved spacetime however the x axis is changing orientation  as it's the tangent vector on the hypershere.

That should be a huge help to making the mathematics of GR easier to understand +1 

[externo]

19 minutes ago, Mordred said:

Now your getting it recall

xμ

is the particle. X is always the coordinate axis for the particles momentum. In the Minkowskii case the changes to x is only the rescaling for length contraction.

 

In curved spacetime however the x axis is changing orientation  as it's the tangent vector on the hypershere.

That should be a huge help to making the mathematics of GR easier to understand +1 

Painlevé metric is ds² = dT² - (dr +√(Rs/r)dT)²

So the "x" axis, i.e. dr +√(Rs/r)dT, changes orientation and goes down in the time coordinate dt. 

Note that (dt,dr) are orthogonal and (dT, dr +√(Rs/r)dT) also. Thus dr +√(Rs/r)dT does not evolve at constant t
I believe the literature says that the surface of the paraboloid is at constant t, but the equations show otherwise. The spatial paraboloid is embedded into time coordinate. The dimension w indicated as the embedding in the literature is actually t.

Edited August 2 by externo

[Mordred]

T being the PG time it might be best to agree on a specific article at this point as nomenclature between treatments can lead to confusion.

We have to look directly at the covectors and vectors at this stage.

Covectors are needed in curved spacetimes whereas in Euclidean space vectors are sufficient.

Edited August 2 by Mordred

[externo]

40 minutes ago, Mordred said:

T being the PG time it might be best to agree on a specific article at this point as nomenclature between treatments can lead to confusion.

We have to look directly at the covectors and vectors at this stage.

Covectors are needed in curved spacetimes whereas in Euclidean space vectors are sufficient.

https://forum-sceptique.com/download/file.php?id=3005

(dt,dr ) are orthogonal

(dT,dR) = (dt,dr +√(Rs/r)dT) are orthogonal

In the figure you see how the instantaneous vectors are oriented.

 

[Mordred]

Yes but look back on the T relations I provided back on page 1 the links to that site never seem to direct me to where it should be going and the site being in French makes figuring out why I'm getting redirected isn't helping.

Been kind of busy atm hopefully this eve or this weekend  I will have time to assist on the Time components specific to PG as it uses its own time for proper time T and replaces \(\tau\)

Edited August 2 by Mordred

[Mordred]

Ok with PG coordinates there is some very different treatments from the Schwarzschild Metric the first is the use of the raindrops. Each raindrop has its own clock falling in from infinity. The other feature is that for the lifting operation (singularity treatment) the time coordinate is tied to the scale factor a in a similar fashion as a commoving observer for the FLRW metric (but don't confuse the two). The light-cones are also defined differently 

in This case the tilting has a different cause. That cause is due to the above two conditions which lead to a different light-cone definition. This is where a reference you have been using would be helpful If you don't have a reliable reference either Markus or myself can likely find one for you. However it may be best to use a reference that you find easier to understand and relate to. 

Recall earlier I linked the following

https://en.m.wikipedia.org/wiki/Lemaître_coordinates

this formula is not included in that link but arises from the raindrop coordinates  using the Symmetries mentioned in the above link

\[dT=\frac{\partial T}{\partial{t}}dt+\frac{\partial{T}}{\partial{r}}dr\]

 

 

I'm going to drop this reference as it certainly will lead to unnecessary confusion particularly since you already stated afterward that you are not strong in GR mathematics. That and were not really concerned with Kruskal diagrams at this stage

https://link.springer.com/article/10.1140/epjc/s10052-023-11370-9

Edited August 2 by Mordred

[Mordred]

@Markus Hanke I've been trying to find a decent method to explain geodesic congruence in regards to PG and Lemaitre frames without getting too technical. Have you perchance come across any decent treatments. Blau and Wald may be a bit too intense

[Mordred]

@externo I would like you to consider the following with regards to the raindrops or Lemaitre frames. Take any number of raindrops. Each raindrop has its own geodesic.

now examine the deviations between each geodesic. In the freefall state the equations of motion are the first order velocity terms (free fall). The geodesic deviations due to the curvature terms (easiest example to understand towards a common center of mass in the Newtonian limit) is the tidal force what we see as the second order acceleration term. Both links I posted earlier (the two lecture notes) include the relevant mathematics.

This is what you will need to understand geodesic congruence in The PG coordinates. You will also need to look into the maximally symmetric spacetimes (local) vs global and how to transform one to the other. All those details in those lecture notes. 

Edited August 3 by Mordred

[Markus Hanke]

3 hours ago, Mordred said:

Have you perchance come across any decent treatments.

Sorry, I don’t immediately know of any good treatments on this particular subject. I’m rather busy in real life for the time being, but if I come across something, I’ll post it here.

18 hours ago, externo said:

Ok, I've finally come to understand that my repeated assertion that the falling observer in Schwarzschild and Lemaître coordinates needs to resynchronize his clocks was incorrect.

This deserves a +1 from me  Learning new things and improving our understanding is what these forums are fundamentally about.

[externo]

In the classical representation of GP coordinates the space is flat and the faller passes the horizon. But dr + vdT does not seem colinear to dr and on the horizon it makes an angle of 90°. So I do not see how the faller can pass the horizon if he descends steeply

It should also be noted that if the GP coordinates are physical both for space (dr + vdT ) and time (dT), they and they alone can tell us about the geometry of space-time and the possibility of crossing the horizon. The other coordinate systems are only abstract coordinates.

Edited August 3 by externo

[Mordred]

As @studiot mentioned were dealing with foliations with the line element its not as straightforward as one might think looking at the equations You have to look at what geodesic congruences are being applied. This typically requires using covariant derivatives provided in the article below for PG coordinates 

\[ds^2=dT^2-(dr^2+\sqrt{\frac{2M}{r}}dT^2-r^d\Omega^2\]

the Euclidean space is the surface of constant T. One of the better articles covering this detail and showing the light-cone shift is here 

 

https://www.sissa.it/app/phdsection/OnlineResources/104/Adv.GR-Lect.Notes.pdf

The above is the reason I was seeking decent coverage of geodesic congruence however there is far too many prerequisite steps such as Leibniz product rule. Knowing one forms and 2 forms etc if you aren't familiar with their usage and reasons for usage under GR. As they are needed in this case

In curved spacetime the only way to remain Lorentz invariance is to use a minimum of a vector and a convector. If you never worked with convectors then that is a preliminary step you will need.

For example with the  above the Kroneckeer delta affine connections using Leibniz gives the maximally symmetric spacetime (Minkowskii, De-Sitter/anti Desitter) spacetime region. (local) the global metric above the affine connections follow the Levi-Civita affine connections. This is used in regards to aspects such as the killing vectors with regards to the Ricci scale term the maximally symmetric spacetimes the Ricci scalar is constant.

That is some of the preliminary details with regards to geodesics and also a huge part of understanding the Christoffel term of the geodesic equation for null geodesis.

You already mentioned your not too familiar with GR how much calculus have you taken ? The latter parts can be found in Calculus textbooks they tend to have the best coverage of the Kronecker delta and Levi-Civita.

The lesson you need to learn is as follows

the acceleration of the object is determined entirely by the connection coefficients of the geometry and has nothing to do with the properties of the object

Edited August 3 by Mordred