# Interpreting Gullstrand-Painlevé Coordinates

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Hi,
I have been playing around with some figures in a spreadsheet linked to the Schwarzschild metric and the equivalent solution in terms of http://en.wikipedia.org/wiki/Gullstrand%E2%80%93Painlev%C3%A9_coordinates" [Broken]. This spreadsheet is only considering the specific case of a free-falling observer {C} such that the relativistic effects of velocity and gravity are both proportional to the coordinate radius [r]. However, there are some aspects that I am not sure I understand.

As a very broad generalisation, it appears that the Gullstrand-Painlevé solution to the normal Schwarzschild metric essentially replaces [dt] with the perception of time in the free-falling frame [dtc]. The objective of this solution is to avoid the ‘coordinate singularity’ at [r=Rs], which occurs with the Schwarzschild metric. While this approach does seem to lead to a more consistent solution in the sense that the free-falling time and velocity remain continuous through the event horizon at [r=Rs], it not clear to me how some of the more fundamental issues are resolved.

If we assume that a series of stationary shell {B} observers are positioned along the free-falling radial path, they would be able to resolve their relative time with respect to some remote distant observer {A} in flat spacetime. Time in each {B} frame would slow with respect to {A} due to gravity only. However, as {C} passes each {B} observer on-route towards the black hole horizon at [Rs]; {B} would observe an additional slowing of time in the {C} frame due to its relative and increasing velocity. As such, it should be possible to prove that the effects of time dilation, as described, are real.

So my first question is that while the free-falling observer may appear to sail through the event horizon, ignoring tidal effects at this point, how do you relate the time in {C} back to {A}?

My second question relates to the velocity of the free-falling observer and the speed of light after crossing the event horizon. A quadratic solution of Gullstrand-Painlevé equation with respect to [dtc] arrives at the classical free-fall velocity:

[1] $$v = -c \sqrt { \frac {Rs}{r} }$$

Using the same approach, but setting the variable [dtau=0], gives up the following velocity of a photon in the {C} frame:

[2] $$v = -c \sqrt { \frac {Rs}{r} } \pm c$$

While this appears to preserve the velocity of light [c] with respect to the free-falling observer {C} is there a suggestion that this process somehow exceeds the normal speed of light?

As a footnote to the last question, the following equation also appears to suggest that time taken to fall from the event horizon [Rs] to the central singularity [r=0] must involve a velocity [v] in excess of [c]?

[3] $$dtc = \frac {2Rs}{3c}$$

Therefore, I would appreciate any further insights, clarifications or corrections to any of the assumptions above. Thanks

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Gold Member
Thanks for the feedback.
the 'raindrop' speed exceeds c when it passes r=2M. I don't think it can communicate back across the horizon, but I'm not sure about this.
Yes, this appears to be the implication of equation [1] in post #1. However, equation [2] then suggests that the speed of the photon is plus/minus [c], e.g. 0 and 2c, as it passes the horizon. The implication is that the photon will suffer a similar fate as the raindrop.

I also understand the event horizon to be one-way only. While the physics inside the event horizon may only be conjecture, it seems strange to have a relativistic process that appears to talk about exceeding [c], although the entire concept of space and time inside [Rs] might be questionable. However, I was really interested what people thought about the first question, i.e.

How do you relate the time in {C} back to {A} outside the event horizon?

Mentz114
How do you relate the time in {C} back to {A} outside the event horizon?

I don't think it's possible. The Schwarzschild coordinates used outside the horizon don't extend past the horizon. Which is why the distant observer will never see the faller reach the horizon.

It seems that outside the horizon and inside ( {A} and {C}) are disconnected realities.

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I don't think it's possible. The Schwarzschild coordinates used outside the horizon don't extend past the horizon. Which is why the distant observer will never see the faller reach the horizon. It seems that outside the horizon and inside ( {A} and {C}) are disconnected realities.
OK, that might be a fair enough position to take, but there is still the question as to how anything got inside the event horizon in the first place. For it seems that outside the horizon there is the issue of the relative time in {C} slowing to stop with respect to {A}. It is claimed that the Gullstrand-Painlevé coordinates avoid the ‘coordinate singularity’ at [r=Rs], but I am still unsure how this approach avoids the apparently ‘real’ effects of time dilation. For example, if the distant observer {A} and the free-falling observer {C} were twins:

How would each twin be aging as {C} approaches the event horizon?
How old would each twin be when {C} passes through the event horizon?

Mentz114
I would guess

1. The {A} twin will be infinitely old ( if you see what I mean) when he sees the faller go through the horizon.

2. The faller will be less than infinitely old. I'll have a go at integrating the world line of the faller when I have time.

Have you seen these papers ?

arXiv:gr-qc/0001069v4 18 Oct 2000

and the 'River Model'

arXiv:gr-qc/0411060v2 31 Aug 2006

I've read that while an infalling observer sees nothing strange very locally to them, 'strange things ' happen as seen by other observers. Specifically, the Fermi-Normal tetrad of the infalling observer rotates such that the horizon becomes its light cone at the point of crossing; everything outside is 'past' everything inside is future. This may not be quite precise, but I recall reading something generally to that effect.

Mentz114
I've read that while an infalling observer sees nothing strange very locally to them, 'strange things ' happen as seen by other observers. Specifically, the Fermi-Normal tetrad of the infalling observer rotates such that the horizon becomes its light cone at the point of crossing; everything outside is 'past' everything inside is future. This may not be quite precise, but I recall reading something generally to that effect.
(my bold)

This sounds right because the horizon is a null surface. If the faller sent out a light beam at r=2m it would be stuck on the horizon, presumably.

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Have you seen these papers ?
http://arxiv.org/PS_cache/gr-qc/pdf/0001/0001069v4.pdf" [Broken]
http://arxiv.org/PS_cache/gr-qc/pdf/0411/0411060v2.pdf" [Broken]
Thanks for pointing to both of these papers. I have only had time for a quick look, but will try to work through some of the details over the weekend. The first one looks more relevant to me, but the second one seems to make reference to the point raised by PAllen in post #7
I would guess:
1. The {A} twin will be infinitely old ( if you see what I mean) when he sees the faller go through the horizon.
2. The faller will be less than infinitely old. I'll have a go at integrating the world line of the faller when I have time.
The spreadsheet I have produced to get a better ‘feel’ for the results being predicted by both the Schwarzschild metric and the Gullstrand-Painlevé version simply uses numeric integration to get an approximation. It is based on a black hole with a mass of 10,000 suns and steps the radius [r] by 0.01*Rs towards the centre. Clearly, twins {A} and {C} end up having very different perceptions of time, but as it works out they both share the same perception of space. The expansive curvature of space due gravity is exactly cancelled out by the contracting effect of velocity in the free fall case.

As a side issue, it is interesting to consider how {C} might actually determine its velocity on-route. While {C} can look at his own watch, it seems more problematic for {C} to determine the distance covered towards the black hole. While we could position a series of stationary {B} observers along the radial route, the integral sum of the separation between the {B} observers would suggest the distance to the event horizon is infinite if the geodesic of curved space is followed - “curiouser and curiouser said Alice”.

Anyway, my example starts at r=10Rs and suggests that {A} would ‘see’ {C} approach within r=1.01Rs after 30.26 seconds, although time is now rapidly dilating towards infinity at this point. In contrast, {C} actually crosses the event horizon after 20.16 seconds and ‘arrives’ at the singularity after another 0.66 seconds. However, one of my main questions was

How do you relate the time in {C} back to {A}?

In essence I agree with both points above. However, the consequence of this conclusion seems to be that {C} ends up occupying a universe which no longer exists in terms of {A}. For {A} eternity has come and gone, while {C} has only aged by 20.16 seconds. Unfortunately, it appears that the universe according to {C} only has another 0.66 seconds of existence. I would be interested in hearing if this conclusion is contested by the theory of relativity. Thanks

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Mentz114
mysearch said:
The first one looks more relevant to me, but the second one seems to make reference to the point raised by PAllen in post #7

The first one gets all the Schwarzschild radial geodesics as a one-parameter family of curves. I can do this with a much shorter calculation, which I can send you if you want a look.

In essence I agree with both points above. However, the consequence of this conclusion seems to be that {C} ends up occupying a universe which no longer exists in terms of {A}. For {A} eternity has come and gone, while {C} has only aged by 20.16 seconds. Unfortunately, it appears that the universe according to {C} only has another 0.66 seconds of existence. I would be interested in hearing if this conclusion is contested by the theory of relativity.

This view seems to be the accepted one. I'm sure someone will put me right if it's not. There's no doubt that {C}'s worldline will terminate at r=0, but the universe will probably still exist.

The time of ~20 secs seems a bit short to fall from infinity, but it will be a finite number I'm sure.

Thre are other charts, like Kruskal–Szekeres which give maximal coverage of the spacetime which lead to similar conclusions.

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The first one gets all the Schwarzschild radial geodesics as a one-parameter family of curves. I can do this with a much shorter calculation, which I can send you if you want a look.
Thanks for the offer; that might be useful. I plan to sit down and try to work through the first paper this weekend. This paper also touches on Kruskal diagrams in its appendix to which you made reference in your last post. I still plan to read the second paper, but need to do more work on differential geometry first; especially Christoffel symbols that seem to be used through the second paper.
This view seems to be the accepted one. I'm sure someone will put me right if it's not. There's no doubt that {C}'s worldline will terminate at r=0, but the universe will probably still exist.
Well that is quite an interesting point. I do not want to strain the forum guidelines by introducing too much speculation, but it seems legitimate to discuss the implications of relativity. Depending on your model of the cosmology, the universe may exist forever in entropy decay or collapse back into a singularity. While second option does not align to current thinking, it conceptually raises an interesting point. If time in {A}, i.e. in distant flat spacetime, effectively ends up running infinity faster with respect to {C}, then could the universe ‘hypothetically’ have time to collapse back on itself before {C} reaches the event horizon. However, this seems to raise some sort of spacetime paradox to me. Of course, an alternative speculation might be that {C} never reaches the horizon; although this option also appears to raise its own set of issues. However, as you say, somebody may be able to correct what is nothing more than speculation.

The time of ~20 secs seems a bit short to fall from infinity, but it will be a finite number I'm sure.

The times quoted in post #9 were not based on infinity, but a starting point of r=10Rs. However, the initial velocity is synchronized to the required free-fall velocity that would have had to conceptually start at infinity. I would also be happy to share my Excel spreadsheet, if it was thought to be of any use to anybody.

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So my first question is that while the free-falling observer may appear to sail through the event horizon, ignoring tidal effects at this point, how do you relate the time in {C} back to {A}?
From the Wikipedia article in the section on "Rain coordinates" you see that they are related by:
$$dt_r=dt-\beta\gamma^2dr$$

So all you have to do is to integrate to obtain the coordinate transformation. In units where c=1 and 2M=1 believe that the result is:

$$t_r=t-2\sqrt{r}+ln\left( \frac{\sqrt{r}+1}{\sqrt{r}-1} \right)$$

http://www.physics.umd.edu/grt/taj/776b/hw1soln.pdf

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While this appears to preserve the velocity of light [c] with respect to the free-falling observer {C} is there a suggestion that this process somehow exceeds the normal speed of light?
Sure, but it is just a coordinate speed, which is not constrained to c even in flat spacetime.

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How would each twin be aging as {C} approaches the event horizon?
How old would each twin be when {C} passes through the event horizon?
This depends entirely on your choice of simultaneity convention.

But for convenience, let's do the following. Let's consider a nearby shell observer {B} who is close enough to the point where {C} crosses the event horizon that there is no significant curvature and let's not worry about {A}. Then we can use the Rindler coordinate transform to convert from {C} to {B}.

In {C}'s local Minkowski frame {B} is travelling pretty fast so he is aging slowly, and {B} is accelerating, so his rate of aging is decreasing, but not infinitely. So, as {C} crosses the horizon {B}'s age is finite and so is {C}'s.

In {B}'s local Rindler frame {C} is travelling pretty fast so he is aging slowly, and {C} is free-falling towards the horizon deeper and deeper into the gravity well. His rate of aging is decreasing infinitely. So, as {C} crosses the horizon {B}'s age is infinite, but {C}'s is finite.

Gold Member
Dalespam,
I really appreciate you taking the trouble to respond on several levels. Let me outline my current perspective on some of the points you have raised:
From the Wikipedia article in the section on "Rain coordinates" you see that they are related by:
$$dt_r=dt-\beta\gamma^2dr$$

http://www.physics.umd.edu/grt/taj/776b/hw1soln.pdf" [Broken]

Thanks for the link to the article above, which I have only quickly skimmed at this point. I have worked through the derivation of the Painleve-Gullstrand solution using the Taylor and Wheeler book on ‘Exploring Black Holes’ see inset on page B-13. They effectively convert [dt] to [dtc] in a two-step process via the Lorentz transform dividing {B} from {C}. I think I follow the rationale of this process, but somebody might prove me wrong on this matter as well:surprised
Sure, but it is just a coordinate speed, which is not constrained to c even in flat spacetime.
This is an interesting point. The velocity of the free-falling observer {C} is given by:

[1] $$v = -c \sqrt { \frac {Rs}{r} }$$

This value can also be derived from classical physics by equating kinetic and potential energy equations. However, the speed of light [c] for this frame can be also determined from either the Schwarzschild metric or the GP solution and appears to have the form

[2] $$v = -c \sqrt { \frac {Rs}{r} } \pm c$$

While this maintains the relative speed of [c] within the {C} frame, it seems to suggest a real and continuous increase in velocity through the horizon towards the central singularity. As indicated in post #1/equation [3], the fall time from the horizon to the central singularity also seems to suggest an aggregate velocity that exceeds [c]. While the whole issues of distance and time inside the horizon seems to be a matter of debate, my spreadsheet seems to remain consistent to [v] increasing without limit. Therefore, how long it takes to get through the horizon seems to be the most interesting point to resolve.
This depends entirely on your choice of simultaneity convention. But for convenience, let's do the following. Let's consider a nearby shell observer {B} who is close enough to the point where {C} crosses the event horizon that there is no significant curvature and let's not worry about {A}. Then we can use the Rindler coordinate transform to convert from {C} to {B}. In {C}'s local Minkowski frame {B} is travelling pretty fast so he is aging slowly, and {B} is accelerating, so his rate of aging is decreasing, but not infinitely. So, as {C} crosses the horizon {B}'s age is finite and so is {C}'s. In {B}'s local Rindler frame {C} is travelling pretty fast so he is aging slowly, and {C} is free-falling towards the horizon deeper and deeper into the gravity well. His rate of aging is decreasing infinitely. So, as {C} crosses the horizon {B}'s age is infinite, but {C}'s is finite.
I have attached a diagram showing the 3 frames of reference being used, i.e. {A}, {B} and {C}. The equations show the relative measure of time [t] and space between all 3 frames, but {A} is considered as the unity reference out in flat spacetime. I would appreciate knowing if anybody disagrees with the basic relationships being shown. To me, there seems to be some advantage to focusing on the effects of time dilation because it is easier to correlate these effects on biological siblings who conceptually populate all frames of reference and who started out being the same age. The suggestion of the equation [3] is that time in {C} still slows to a stop at the horizon with respect to {B}, as it does for {A}, albeit at a slower rate.

[3] $$t_C = t_B \sqrt {\left( 1 - \frac{Rs}{r} \right) } = t_A \left( 1 - \frac{Rs}{r} \right)$$

So while I ‘think’ I understand the issue of coordinate singularity that the GP coordinates seeks to resolve, it seems to me that it does so without attempting to correlate the time in {C} with {A} or {B}.

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While this maintains the relative speed of [c] within the {C} frame, it seems to suggest a real and continuous increase in velocity through the horizon towards the central singularity.
I have an almost pathological distaste for the term "real". However, if you are willing to accept coordinate velocities as "real" then you are completely correct that this is a continuous increase in coordinate velocity through the horizon to coordinate speeds >c. As you point out, however, the speed of the object never exceeds the speed of light even if the coordinate speed does exceed c.

In other words, the coordinate speed has no absolute meaning, and is not limited to <c. What does have absolute meaning is if the worldline is timelike, lightlike, or spacelike, and it is always timelike.

So while I ‘think’ I understand the issue of coordinate singularity that the GP coordinates seeks to resolve, it seems to me that it does so without attempting to correlate the time in {C} with {A} or {B}.
If the transform I gave in post #12 is correct then you can use that to correlate times in {C} with times in {A} if you can't avoid the compulsion.

Gold Member
I have an almost pathological distaste for the term "real".
Sorry to hear about this, I will try to refrain from using the word in my response.
If the transform I gave in post #12 is correct then you can use that to correlate times in {C} with times in {A} if you can't avoid the compulsion.
Clearly, you seem to feel that correlating the times in {A} and {C} is either fruitless or nonsensical. I appreciate that this may all be obvious to you, but I would really like to known why, as it seems to me that relativity is all about understanding the comparative measure of time and space in different frames. Anyway, the transform you gave in #12 is reproduced below and seems to align to the paper you referenced, i.e. ‘http://www.physics.umd.edu/grt/taj/776b/hw1soln.pdf" [Broken]on page 2.

$$t_r=t-2\sqrt{r}+ln\left( \frac{\sqrt{r}+1}{\sqrt{r}-1} \right)$$

To be honest, I am still trying to work through this equation and have become a little confused as it is expressed in geometric units and my spreadsheet is using standard SI units. Equally, other references, which appear to be stating the same equation, seem to have a different form that lead to different results, e.g.

http://casa.colorado.edu/~ajsh/phys5770_08/bh.pdf" [Broken] – see page 13: equation (28)

http://arxiv.org/PS_cache/gr-qc/pdf/0001/0001069v4.pdf" [Broken] – page 1: equation (1.3)

If anybody has this equation expanded in SI units, i.e. M, G and c, I would really appreciate it if you could post it in this thread. However, given that my spreadsheet is effectively doing a numeric integration by summation, I believe I can use your other equation to sum the incremental values:

$$dt_r=dt-\beta\gamma^2dr$$

I have summarised my results in the attached graphs, which I believe show the incremental and summed value for the specific example cited in post #9 and #11. Whether they are right is another matter, but they seem to align to basic relativistic effects cited in the diagram attached to post #15, and were the basis of why I was having trouble reconciling time in {C} to {A}. As far as I can see, twin {C} will pass through the event horizon 20.16 seconds after reaching r=10Rs, based on my example, relative to his own frame. However, it seems to raise some fairly fundamental questions about what has happens in frame {A}. At the end of the day, I am simply trying to get a better understanding of the implications of relativity in this extreme example. I would appreciate both your patience and insights. Thanks

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Clearly, you seem to feel that correlating the times in {A} and {C} is either fruitless or nonsensical. I appreciate that this may all be obvious to you, but I would really like to known why, as it seems to me that relativity is all about understanding the comparative measure of time and space in different frames.
Sorry, I probably shouldn't have phrased it that way. I certainly don't want to discourage your study in any way.

The thing is that in GR the coordinates that you are allowed to use are almost completely arbitrary. You can take the worldline of {A} and the worldline of {C} and you can use either the Schwarzschild coordinates or the GP coordinates to map points on {A} to points on {C} and vice versa. However, you could just as well take any arbitrary smooth 1-to-1 mapping between {A} and {C} that you like and construct a coordinate system that uses that as its simultaneity convention. In the end, all you learn is that the simultaneity conventions are just conventions.

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The thing is that in GR the coordinates that you are allowed to use are almost completely arbitrary. You can take the worldline of {A} and the worldline of {C} and you can use either the Schwarzschild coordinates or the GP coordinates to map points on {A} to points on {C} and vice versa. However, you could just as well take any arbitrary smooth 1-to-1 mapping between {A} and {C} that you like and construct a coordinate system that uses that as its simultaneity convention. In the end, all you learn is that the simultaneity conventions are just conventions.
Thank you. I will reflect further of what you are saying. However, I can’t help feeling that if the effects of time dilation are ‘tangible’ then the full implications of the relative rate of time in {A} and {C} have not been addressed.

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Time dilation is coordinate dependent, so it is in some measure simply an arbitrary human convention. However, there are related things that you can discuss which are not coordinate dependent, such as the Doppler shift of signals sent from one to the other. If you want to get "tangible" effects similar to time dilation then I would recommend starting there.