Light ray paths around a black hole (work in progress)

[Cham]

This topic probably belongs to the Physics & Astronomy thread, but it's about an addon project, so...

I'm very excited about the results I'm getting right now. This may probably ends in an addon about pulsars and black holes. Using Mathematica, I'm solving the EXACT equations for the light deflection around a pulsar (the Mathematica code will be easily adapted for a black hole, later). Here's a link to a PDF document showing some light rays (take note that this is only a preliminary "work in progress" document) :

http://nho.ohn.free.fr/celestia/Cham/Divers/test.pdf

See the last picture, at the end of the document. The black circle represents the pulsar's surface. The curve which is making a full loop inside the pulsar is just there to show what may occur around a black hole. This is reallly fun !

[t00fri]

This topic probably belongs to the Physics & Astronomy thread, but it's about an addon project, so...

I'm very excited about the results I'm getting right now. This may probably ends in an addon about pulsars and black holes. Using Mathematica, I'm solving the EXACT equations for the light deflection around a pulsar (the Mathematica code will be easily adapted for a black hole, later). Here's a link to a PDF document showing some light rays (take note that this is only a preliminary "work in progress" document) :

http://nho.ohn.free.fr/celestia/Cham/Divers/test.pdf

See the last picture, at the end of the document. The black circle represents the pulsar's surface. The curve which is making a full loop inside the pulsar is just there to show what may occur around a black hole. This is reallly fun !

Interesting! I'll get back to this as soon as I'll have a little spare time...

Bye Fridger

[Cham]

Here's a GIF image of various light paths around the center. The circle is the puslar surface (10 km radius). I'm also showing some paths going "inside" the pulsar, which is the same as for a black hole of the same mass.



I may convert the data to built a CMOD version of those paths, so they could be shown within Celestia.

[t00fri]

Cham,

in case noone is interested in these exciting development aspects in the forum, how about pulling such matters out of here into CelestialMatters?

Personally, I think Celestia would profit a lot from more basic development work related to exciting NEW astrophysics phenomena. Instead, we had plenty of "fast" straightforward coding recently...

Who knows, perhaps we manage to open the CM doors for Christmas? ....


Cheers,
Fridger

[Cham]

Here's another picture I just made, few minutes ago. The smallest circle is the event horizon (point of no return) of an 1.4 M_Sol black hole. The medium circle is the BH's "photosphere" and the large circle is a 10 km radius pulsar (usefull to give a scale). The photosphere represents the only circular orbits for light. Those orbits are unstable, and we can clearly see why on the picture below. The source of light is located far away at the left of the picture. All those curves will probably be converted to a CMOD model, when I'll be satisfied with the results.



I also tried to make the same with a Kerr black hole, but the maths are MUCH more involved. It may be a project for the future, since I'm also interested to show the effect of "frame dragging", produced by the rotation of the black hole. However, I'm not sure it may be relevent for a spinning pulsar.

[ajtribick]

This is really interesting work here... I'm interested in this kind of thing for rendering purposes, but unfortunately my coding skills are no way near good enough to implement this kind of thing in, say, POV-ray.

I'd like to see results of a Kerr black hole - I've been wondering about whether it would be significant for a pulsar.

[Cham]

This is really interesting work here... I'm interested in this kind of thing for rendering purposes, but unfortunately my coding skills are no way near good enough to implement this kind of thing in, say, POV-ray.

I'd like to see results of a Kerr black hole - I've been wondering about whether it would be significant for a pulsar.

My coding skills are very limited too (actually, I have none !). I only know Mathematica, which isn't made for rendering, but only for maths solving.

The Kerr black hole is MUCH more complicated to describe mathematically. And I need to perform a coordinates transformation to use some "special" cartesian coordinates (since the light source is far away, where space is euclidian). Doing the transformation to a Kerr metric, and translating all this into a Mathematica code may be out of my skills range.

Also, a real pulsar may have some multipolar field caracteristics (quadrupole moment, etc...), and Kerr metric describes the FINAL STATE of a spinning BH, not a pulsar. However, I may have a solution : Schwarzschild metric with a linear correction to describes the rotation (frame dragging). So I think there's a way. Of course, I'm starting all this with the simplest problem : Schwarzschild field only (no rotation).

[Cham]

What is extraordinary with BH optics, is if there isn't any light falling on it, you will not see the black hole. But if you throw some light directly to the BH (using a flash light, say), the BH will return some of the light directly to you, and you'll see a ring of light all around a black disk. BHs aren't so black, actually, if you throw light on them ! This effect doesn't occur at all on a pulsar, since its photosphere doesn't exist (or is located inside the pulsar, which is irrelevant). The picture above shows this clearly.

[Cham]

Here's a closer sight of what is happening close to the photosphere (middle circle) :

[Cham]

And here we go ... it's in Celestia now (click on the small image for a larger one) :



The small white sphere is the event horizon (not texturised yet, to illustrate the "membrane paradigm" of black holes). The red circle is the "photosphere". I'll add a translucent sphere there.

The curves are exact solutions of the geodesic equation applied to the Schwarzschild metric (in "isotropic" cartesian coordinates).

[Cham]

Here's a test addon for you to try :

http://nho.ohn.free.fr/celestia/Cham/Di ... iation.zip (660 KB zip file)

The black hole is temporarily set in our solar system. Just enter "Event Horizon" in Celestia, and you'll get to a place with many "null geodesics" (light paths) and the "photosphere". Please, tell me what you think of this, crittics and suggestions.

[Cham]

Since I don't get any response from the physics community about my magnetic field interpretation problem, I'm taking a pause from it. I'm now developping a more systematic approach to the light deviation around a black hole, especially since my Mathematica code is now able to export an entire CMOD file, properly formatted. Here's a new prototype. The curves are more precise. The one associated to the critical impact parameter is even doing several complete turns around the "photosphere" ! (the shorter curve shown below, going back to the left). I may add a nice fade out transition to all the curves which are going back to infinity, but I'm not sure this is a good idea for the representation of the light deviation :



Closer shot of the center (the black hole model isn't placed in there yet) :


The curve's color is temporary.

I'll also include the effects of the rotation of the black hole (frame dragging), using the Kerr metric, or maybe just its linear approximation, since the effect is not that strong for a pulsar. We can show that the approximation is justified if :

Code: Select all

F  ==  2 G S/(c^3 r^2)  <<  1   for all "r" > R_star

which is equivalent to

Code: Select all

RotationPeriod >> 0.0000347 sec,

which is very well satisfied by pulsars. Actually, F = 0.001 (approx) for a fast typical pulsar. F < 0.001 for slower pulsars.

[Cham]

This is most probably the final version for the Schwarzschild black hole :





Next, I'll include the frame dragging effect from the rotation of the black hole. Since the spherical symmetry will be broken, there will be MUCH more interesting curves. This may even become a mess, in 3D.

[Cham]

The Mathematica code with rotation effects (frame dragging) is working well. There are LOTS of options to select (sense of rotation of the central body relative to the incoming light rays, rotation period, initial conditions of the light rays, etc).

Here's an amazing example (probably caused by an error somewhere, I'll have to check this). The central body is rotating anticlock wise, rotation period is 0.033 sec (crab's pulsar value), mass is 1.4 solar. The green circle has a 10 km radius (a pulsar's size, for reference), the red circle is the photosphere, and the small black circle is the event horizon.

The initial conditions used for the light rays are exactly the same as previously shown (without the frame dragging effects). Notice how the light rays are falling, and apparently "pushed back" because of the frame dragging effect :



It reminds me a bit the "Penrose process" about energy extraction from a rotating black hole, but I don't think this is related since nothing seems to be falling in the event horizon. Those curves may be wrong, and I'll have to investigate further.

[Cham]

I've found the source of this "repulsion" effect shown on the previous picture : my spin parameter isn't small enough, so the numerical solution isn't accurate close to the event horizon. More preciselly :

Spin parameter = G*S/c^3 r^2 should be much smaller than 1 for my geodesics equation to be accurate. This is the case at the pulsar radius, for M = 1.4 sol mass, rotation period P = 0.033 sec and R = 10 km :

Code: Select all

Spin parameter = G*(2/5 M R^2)*(2 pi / P) ?· (c^3 R^2)

= 4 pi G M / (5 c^3 P) = 0.0005254    <<  1

However, when evaluated at the event horizon, we get :

Spin parameter = 0.0491.

While it's smaller than 1, it isn't smaller enough for the approximation used to hold close to the horizon. I could use a longer rotation period, but then the effect of frame dragging wont be much noticeable on the trajectories. I will have to use the full Kerr geometry or else restrict myself to pulsars without drawing the light paths under their surface (as it should, anyway!).

[Cham]

Here's the correct rendering. The frame dragging effect is weak here and was calculated using a simple linear correction to the geodesics equation. This linear term is accurate for "small" angular momentum, i.e. "large" rotation periods only. I used M = 1.4 Solar mass, Rotation Period = 0.05 sec.

Three pictures for comparison purposes, with exactly the same initial conditions on light rays.

No rotation at all :


Anti-clockwise rotation :


Clockwise rotation :


The interpretation of these pictures is crystal clear : in the anti-clockwise case, the frame dragging "slow down" light on its path and gives gravity the time to pull on the ray, so it falls inward. In the clockwise case, it's the contrary : the frame dragging is giving the opportunity to light to escape.

[Cham]

Here's what happens when we push the approximation too far out of its validity domain :



The effect is interesting anyway. It may be interpreted as an extreme frame dragging effect. The rotation period is 0.001 sec anti-clockwise.

[Cham]

Here's a variation for the Schwarzschild black hole (no rotation, so no frame dragging effect), as seen in Celestia. A ponctual blue light source is placed at left. The black hole is at right. The red sphere is its "photosphere".



Since I have the strong feeling that I'm talking all alone here, I'll stop to publish any results. Obviously, this subject interess nobody.

This topic is now closed.

[Joe]

Cham

I must say that I read every post from you in this thread, trying VERY hard to follow you up on those advanced maths and physics. As a computer programmer, I wish that I could contribute somehow to this wonderful add-on you have been worked on rather than just admiring your published screen-shoots. It would be nice if you would release a try version of this add-on.

[jll]

Hi Cham

I think that it would be a shame to stop this thread, even if you have the right and that I understands your discouragement not to have answers

I learned a lot of from things in this thread.
I thought of testing math?©matica, but this commercial product is rather expensive.

Go on testing these new orientations and to make us benefit from the result.

JLL

PS : If you look at stats, your articles are read many times (much more than mine )