Another spacetime visualization

[Tomas Vencl]

TL;DR

curved spacetine 2 dimensional diagram

Ispired by PeterDonis remark about "river model" in some thread a time ago I made next visualization picture.
The graph desctibes, how the flat Minkowski spacetime is changed in presence of mass (black hole). It do not need much explanation, almost everything is described at the picture. To me it seems to be very ilustrating, so maybe, it can help someone. Please, any notes and remarks are welcomed.

 

[PeterDonis]

@Tomas Venci, what coordinates are you using to make this graph?

There is one obvious error: the vertical blue line at the horizon is marked, at the top, as "simultaneity line for event when free-falling observer crosses the horizon". That is not correct. That vertical blue line is null, not spacelike, so it cannot be a simultaneity line for any observer.

I'm also not sure about the directions of the ingoing null lines as they get close to the horizon. AFAIK there is no coordinate chart in which those become horizontal at the horizon, which is why I ask what coordinates you are using to make the graph.

I'm also not sure about the directions of the simultaneity lines for the free-falling observer as they get close to the horizon. They don't look right for any of the coordinate charts I'm familiar with.

Finally, I'm not sure what you mean by a "reverse accelerating observer" or how you are obtaining the worldlines for that observer. This observer's worldline appears to also be horizontal at the horizon, but that is the same direction the ingoing null lines are in, and no observer's worldline can be null, it must be timelike.

 

[Tomas Vencl]

Thank you, with coordinates you point exactly to the problem.
This is not any of the coordinates I know, r coordinate is clear, the Y coordinate is the point and I must admit I do not exactly know. Generally I thing that graph is rather heuristic picture than exact graph. It may show the relations between different observers and for example situation, that freefalling observer sees locally flat spacetime (and I wonted to keep it locally exactly Minkowski). The closest coordinate for Y is, I think, function of proper time of which I call reverse accelerated observer, which is observer who feels the same proper acceleration as static hovering observer, but opposite direction. I was also thinking about Y as relabelled free-fall coordinate (free fall propper time * some function to keep locally light cones unchanged). I understand that it looks strange (at least) to do something and do not know exactly what, but the picture makes some sense to me. Of course it may be completely wrong.

About the "the vertical blue line at the horizon is marked, at the top, as simultaneity line for event when free-falling observer crosses the horizon" The line is not one of those vertical blue worldlines for static observers , it is one of the lines which are marked as blue "simultaneity" for static observer. But near horizon, they are also almost vertical.
All other points are conected to the first one-what exact coordinates are there. It is similar to free-fall, but somehow transformed. If it is completely nonsense, I am sorry.

 

[Tomas Vencl]

The light cones looks exactly as those at fig.12. here :
https://www.rpi.edu/dept/phys/Courses/Astronomy/CurvedSpacetimeAJP.pdf

 

[Tomas Vencl]

As I wrote I tried to visualize the river model being interested what it shows. So the basic constuction of the graph was simple. I rotated finite elements of small Minkowski diagrams by angle which is done by free fall velocity dr/dtau=sqrt(rg/r) and connected those elements together to obtain smooth red lines. So they should be free falling worldlines (but on the Y scale is not proper time because proper time is along worldline). The other lines are also simple for example null geodesics must make angle 45 degree to free falling worldline (to keep locally flat Minkowski).
And the result was like it is making sense for me.

 

[PeterDonis]

This is not any of the coordinates I know

Then how did you make the diagram, and why do you think it means anything?

 

[PeterDonis]

The light cones looks exactly as those at fig.12. here :
https://www.rpi.edu/dept/phys/Courses/Astronomy/CurvedSpacetimeAJP.pdf

Yes, but that figure does not have all the other things that yours does. In particular, it does not mark "Schwarzschild lines of simultaneity"--in particular, it does not mark the vertical horizon line as one.

 

[PeterDonis]

I rotated finite elements of small Minkowski diagrams by angle which is done by free fall velocity dr/dtau=sqrt(rg/r) and connected those elements together to obtain smooth red lines. So they should be free falling worldlines (but on the Y scale is not proper time because proper time is along worldline). The other lines are also simple for example null geodesics must make angle 45 degree to free falling worldline (to keep locally flat Minkowski).

If you had limited yourself to doing this, you would basically be doing the same thing as Fig. 12 in the paper you linked to.

But what you say here does not explain how you got the blue "Schwarzschild lines of simultaneity", or the yellow "reverse accelerated observer" lines.

 

[Tomas Vencl]

If you had limited yourself to doing this, you would basically be doing the same thing as Fig. 12 in the paper you linked to.

But what you say here does not explain how you got the blue "Schwarzschild lines of simultaneity", or the yellow "reverse accelerated observer" lines.

The blue lines : They are observers which are moving in original Minkowski systems (before I rotated them) with velocity that after rotation they become static . Simultaneity lines for them I made the same way as in special relativity (locally in those small Minkowski systems). And also after rotation I connected them .
The yellow lines I made exactly the same way, but the velocity has the opposite direction.

If you look at some small limited part of the diagram, it is nothing more than standard special relativity Minkowski diagram (only rotated) with red "static" reference frame and boosted blue (or yellow) reference frame.

 

[PeterDonis]

The blue lines : They are observers which are moving in original Minkowski systems (before I rotated them) with velocity that after rotation they become static .

This doesn't make any sense. "Static" has a well-defined meaning in GR, but it does not appear you are aware of it.

The well-defined meaning is that a static observer is maintaining a fixed altitude above the black hole's horizon. Is that what you intended for your "static" observers?

Simultaneity lines for them I made the same way as in special relativity (locally in those small Minkowski systems). And also after rotation I connected them .

This doesn't make any sense either. The simultaneity lines for static observers in the sense I gave above do not "rotate" as your "local Minkowski systems" rotate; those "local Minkowski systems" are the local inertial frames for freely falling observers, not static observers.

The yellow lines I made exactly the same way, but the velocity has the opposite direction.

What velocity? And how do you know this even makes sense physically?

 

[Tomas Vencl]

This doesn't make any sense. "Static" has a well-defined meaning in GR, but it does not appear you are aware of it.

The well-defined meaning is that a static observer is maintaining a fixed altitude above the black hole's horizon. Is that what you intended for your "static" observers?

I hope yes. The blue vertical worldlines does not change r coordinate, so I think it is static ?

This doesn't make any sense either. The simultaneity lines for static observers in the sense I gave above do not "rotate" as your "local Minkowski systems" rotate; those "local Minkowski systems" are the local inertial frames for freely falling observers, not static observers.

I described only method how I got the lines. But there is another way to obtain blue simultaneity lines which makes the same result . If I suppose, that green light signals are correct (as they seems the same as in the article), then I can use Einstein synchronization convention. I send the signal from one static observer to second which reflect it. When I receive reflected signal, the event at half of my measured propper time period (between sending and receiving) I can declare as simultaneous with event of reflecting signal (second observer). I know, this is only one of the possibilities how to define simultaneity at gravity and I also know that if they compare clocks they measure different clocks rate. So this "synchronization" is only for first moment.
And those blue simultaneity lines at the diagram behave exactly like this. They are connecting the events which are simultaneous by this definition.

The yellow observer is not important, I used it for different purpose and we can forget it. I am interested about the blue one (static), red one (free falling from infinity) and green (light signals).

All this was my attempt to visualize river model of gravity and look if the result is equal to some known gravity visualizations. I thought that yes, but now it seems that I was wrong.

 

[PeterDonis]

The blue vertical worldlines does not change r coordinate, so I think it is static ?

Yes.

If I suppose, that green light signals are correct (as they seems the same as in the article), then I can use Einstein synchronization convention.

And doing that should make it obvious to you that the vertical line that marks the horizon, which you have called a blue simultaneity line, cannot be a simultaneity line--because it is the worldline of the outgoing light signal.

If you dig into what that implies, you will realize that it is telling you that there are no static observers at or inside the horizon. So there can't be any simultaneity lines for static observers in that region either.

 

[PeterDonis]

I described only method how I got the lines.

The method you described first still doesn't make sense to me. But the alternative method you described, of using Einstein synchronization, is sufficient to illustrate the issue at the horizon, as I explained in my previous post just now.

All this was my attempt to visualize river model of gravity and look if the result is equal to some known gravity visualizations. I thought that yes, but now it seems that I was wrong.

The key aspect of the river model with regard to "visualization" is that space itself is treated as moving inward. A static visualization can't really capture that aspect.

 

[Tomas Vencl]

And doing that should make it obvious to you that the vertical line that marks the horizon, which you have called a blue simultaneity line, cannot be a simultaneity line--because it is the worldline of the outgoing light signal.

If you dig into what that implies, you will realize that it is telling you that there are no static observers at or inside the horizon. So there can't be any simultaneity lines for static observers in that region either.

OK, now I see what you mean. This misunderstanding is caused by my attempt graphically show the situation when proper time of free falling observer is finite while crossing the horizon, the distant Schwarzschild observers time for this event is infinite. I understood this line as some limit. Maybe I used wrong expression or so, you see, that this line is glued to graph manually. If I say that this line is coming not from horizon, but some very near point, than it is not from the beginning exactly vertical, and than, of course is not leading to infinite Schwarzschild time, but to some very distant future of Schwarzschild time, is it ok ?

 

[Tomas Vencl]

The method you described first still doesn't make sense to me. But the alternative method you described, of using Einstein synchronization, is sufficient to illustrate the issue at the horizon, as I explained in my previous post just now.

I believe that the situation at the horizon is clear now.

The key aspect of the river model with regard to "visualization" is that space itself is treated as moving inward. A static visualization can't really capture that aspect.

I thought, that this aspect is reflected at the picture by rotation of local inercial systems (which are then moving inward).

 

[Ibix]

In a flowing model, surely the location of an inertial observer should be constant, and features that are time-independent in Schwarzschild coordinates (such as the event horizon) should come up to meet them?

 

[PeterDonis]

If I say that this line is coming not from horizon, but some very near point, than it is not from the beginning exactly vertical, and than, of course is not leading to infinite Schwarzschild time, but to some very distant future of Schwarzschild time, is it ok ?

Yes, that would be OK, since outside the horizon the Schwarzschild lines of simultaneity do behave basically as you have drawn them, at least as far as I can follow what the paper you referenced is doing.

I thought, that this aspect is reflected at the picture by rotation of local inercial systems (which are then moving inward).

The "rotation of inertial systems" shows how observers free-falling into the hole move. That's not the same as saying that "space" moves along with them. You can't really capture "space moving inward" in a static visualization, because the static background of your visualization is what is naturally interpreted as "space" (or "spacetime", with "space" being the horizontal grid lines).

Btw, Andrew Hamilton has some good diagrams of different coordinate charts used on Schwarzschild spacetime, as well as transformations between them:

https://jila.colorado.edu/~ajsh/bh/schwp.html

Note that none of these are the same as the diagrams in the paper you referenced, which is taking a different approach--what it is using is not a coordinate chart on spacetime. That is why some aspects of the diagrams in the paper have properties that no actual coordinate chart diagram would have.

 

[PeterDonis]

In a flowing model, surely the location of an inertial observer should be constant

You can't make a global coordinate chart this way on Schwarzschild spacetime. The best you can do is a Fermi normal coordinate chart covering a narrow "world tube" around one inertial observer's worldline.

 

[Tomas Vencl]

The "rotation of inertial systems" shows how observers free-falling into the hole move. That's not the same as saying that "space" moves along with them.
You can't really capture "space moving inward" in a static visualization, because the static background of your visualization is what is naturally interpreted as "space" (or "spacetime", with "space" being the horizontal grid lines).

Thank you for your answers and your time.This is probably the point. I am not interpreting space as horizontal grid lines and time as vertical. This is another reason why I do not know the coordinate system of the graph. I am interpreting space as red lines marked "simultaneity". I agree it is not natural and it is not completely clear to me how to understand it. I am looking at the diagram the same way how we are looking at standard special relativity Minkowski diagrams. There also space (and time) is given by simultaneity line (and worldline) and each observer has its own space and time direction. (Yes, here is r direction, but r is only given by measuring of circumference.)

Btw, Andrew Hamilton has some good diagrams of different coordinate charts used on Schwarzschild spacetime, as well as transformations between them:
https://jila.colorado.edu/~ajsh/bh/schwp.html

Yes, I know them.

Note that none of these are the same as the diagrams in the paper you referenced, which is taking a different approach--what it is using is not a coordinate chart on spacetime. That is why some aspects of the diagrams in the paper have properties that no actual coordinate chart diagram would have.

This is probably also reason why I did not recognize the exact coordinates of my graph. If you can say anything more about this topic (meaning the "absence" coordinates at the picture in the paper) it would be very helpful. I am still confused about fact that I know how I made my picture but do not know exact coordinates which sounds stupid. Thank you again.

 

[PeterDonis]

I am not interpreting space as horizontal grid lines and time as vertical.

Then how are you interpreting them? And how do you think they are being interpreted in the paper you referenced?

This is another reason why I do not know the coordinate system of the graph.

There isn't one--at least, not a coordinate system on spacetime. What the paper you referenced is doing is constructing a Euclidean model (rather than a Lorentzian model), and using that as a visualization aid. It is not using any coordinate system on spacetime itself.

I am looking at the diagram the same way how we are looking at standard special relativity Minkowski diagrams.

That's not what you describe in your very next sentence--at least, not globally.

There also space (and time) is given by simultaneity line (and worldline) and each observer has its own space and time direction.

This works locally--in a small patch of spacetime, or in a narrow "world tube" around a single worldline. But if you want to try to fit all this together into a single global description, you have to use a single global coordinate chart, and in a curved spacetime there are no global inertial frames.

If you can say anything more about this topic (meaning the "absence" coordinates at the picture in the paper)

As noted above, the paper's diagrams are not drawn in spacetime itself. They are drawn on a Euclidean manifold which has a certain mathematical relationship to spacetime, but it is not spacetime, and not all of the properties of curves, light cones, etc. in the diagrams will be the same as properties of curves, light cones, etc. in spacetime itself.

 

[Tomas Vencl]

Then how are you interpreting them? And how do you think they are being interpreted in the paper you referenced?

I think that picture in the paper is interpeted as that x coordinate is r and y coordinate also is not time any of known observer. As can I understand you you are saying something similar ? :

There isn't one--at least, not a coordinate system on spacetime. What the paper you referenced is doing is constructing a Euclidean model (rather than a Lorentzian model), and using that as a visualization aid. It is not using any coordinate system on spacetime itself.

As noted above, the paper's diagrams are not drawn in spacetime itself. They are drawn on a Euclidean manifold which has a certain mathematical relationship to spacetime, but it is not spacetime, and not all of the properties of curves, light cones, etc. in the diagrams will be the same as properties of curves, light cones, etc. in spacetime itself.

I probably do not understand correctly because, next seems to me be in conflict what you say above.

But if you want to try to fit all this together into a single global description, you have to use a single global coordinate chart, .

I think, I must now take a time and think about what you writed. I am not leaving the discusion, but I can't defend my solution , mainly bacause I am not at your level. If you or anyone wants anything to add, I am still interested. Thank you.

 

[vanhees71]

Hm, I'm a bit puzzled by the above discussion, because I also understand the AJP paper by Jonsson such that it's depicting a 1+1-D Lorentzian manifold in the same sense that a Minkowski diagram depicts the Lorentzian affine space (i.e., Minkowski space) of special relativity.

The difficulty already in the latter less complicated case that we are trained since kindergarten to read diagrams like this as Euclidean planes rather than Lorentzian/Minkowskian planes, but that's what we indeed have to do to read them correctly. You must forget about the lengths in the sense of Euclidean space but must substitute it by the "lengths" of Minkowski space. E.g., when drawing time-space axes for two inertial frames you have to specify the units on each of these axes by drawing the time and spacelike hyperbolae $(c t)^2-x^2=\pm 1$ (Minkowskian "unit length") rather than circles as you'd have to draw in the case of usual Euclidean geometry.

Concerning Johnsson's diagrams I understand them such that he draws some curved surface and puts at each point a local Minkowskian basis (2-bein). However, as @PeterDonis 's rightfully cautioned, he doesn't really draw the surfaces of a Lorentzian manifold, because that's impossible if you want to draw them as curved surfaces embedded in 3D Euclidean space. So instead of drawing coordinate lines of a Minkowski orthogonal set of coordinates of the surface he just draws them as coordinate lines of a Euclidean orthogonal set of coordinates of a surface embedded in 3D Euclidean space and claims that he would somehow visualize the Lorentzian spacetime.

In my opinion that's highly misleading, because all that have the depicted surface in Euclidean space has in common with the Lorentzian spacetime manifold is that the temporal and spatial lengths along the coordinate lines are the same (which he explains himself at the beginning of Sect. 3 of the paper), but as far as I understand, you cannot take the shape of the drawn Euclidean surface as a whole as a true visualization of the Lorentzian surface. I thus think that such attempts to depict 1+1-D spacetime models by Euclidean surfaces does more harm than it helps.

 

[PeterDonis]

I think that picture in the paper is interpeted as that x coordinate is r and y coordinate also is not time any of known observer.

Yes.

I probably do not understand correctly because, next seems to me be in conflict what you say above

No, there's no conflict, because I was talking about two different things.

The diagrams in the paper you referenced are not spacetime diagrams in any coordinate chart. That means they are not global descriptions of spacetime. They are just visualization tools, which attempt to represent certain particular aspects of spacetime but do not represent other aspects.

 

[Tomas Vencl]

Thank you both very much.
Let me summarize discussion up to date.
-picture is not real spacetime diagram, it can be some visualization.
-the method which I obtained it is not checked, but the result (null geodesics, freefalling worldline and simultaneity) generally looks as in the other visualization in paper which I posted. The Schwarzschild simultaneity lines looks generally how they should.
-the description of blue line at the top is not correct, there is no Schwarzschild simultaneity line at the horizon. It could be seen as a limit near horizon.
-The yellow observer is not considered.
-such attempt can be highly misleading because it is not global descriptions of spacetime. It is visualization tool, which can represent some aspects of spacetime but do not represent other aspects.