Hi Jorrie,

As always, very interesting. I'm assuming the white in the center represents areas without orbits. I would expect no orbits inside the event horizon, but there appears to be regions outside of the event horizon that don't have orbits. I think that's because you set r = 2.5r_e . Why did you choose 2.5, did you have a specific reason or just an arbitrary point?

You Wrote "However, 3-d is required when a particle orbits arbitrarily around two or more massive objects, which is the real objective of the study. More about that later."

Is the example above an orbit about two or more massive objects, or is the fact that it appears three dimensional actually an optical illusion an it is in fact just a two dimensional orbit. Or is something else going on. I'm confused by that part.

Lastly, if the object were closer at periapsis (say r = 1.5r_e) would relativistic effects distort the orbits? Make them seem bent or something (if viewed in the manner you provided above)? I'm not sure if that question makes sense, so if it doesn't, just let me know.

Best Regards,

Roger

Hi Roger, you wrote:

"I would expect no orbits inside the event horizon, but there appears to be regions outside of the event horizon that don't have orbits. I think that's because you set r = 2.5r_e . Why did you choose 2.5, did you have a specific reason or just an arbitrary point?"

A periapsis of 2.5r_e with the velocity shown is arbitrary, but provides a highly relativistic, yet stable orbit with a periapsis shift of close to 360 degrees. This also means that the particle never comes closer than 2.5r_e from the hole, hence the white area around the event horizon.

The orbit shown is of a single particle around a single, isolated black hole, so the orbit cannot deviate from the original plane it was launched into. The 3D appearance is indeed just an optical illusion.

"Lastly, if the object were closer at periapsis (say r = 1.5r_e) would relativistic effects distort the orbits?"

At r=1.5r_e, the local circular orbital velocity reaches c and only light can orbit there in a quasi-stable fashion. Any free-falling object (even light) venturing closer than 1.5r_e to the hole will spiral into the hole, irrespective of it's angular momentum. That's the ultimate 'distorting effect' that black holes have on relativistic orbits. Relativistic orbits are however already a 'distortion' of Newtonian orbits.

Jorrie

Hi Jorrie,

Thanks for answering my questions. Regarding your answer to my third question:

You Wrote: "At r=1.5r_e, the local circular orbital velocity reaches c and only light can orbit there in a quasi-stable fashion. Any free-falling object (even light) venturing closer than 1.5r_e to the hole will spiral into the hole, irrespective of it's angular momentum"

So are you saying that stable orbits can only exist outside of 2.5r_e? Or is there a threshold between stable and unstable orbits somewhere closer in distance? In other words, if I'm understanding you correctly, there seems to be a region outside of the event horizon where stable orbits can't exist. Is that correct? Why is that? It seems like there should be stable orbits all the way up to the event horizon. Is it caused by a relativistic effect?

Hi again Roger. You asked:

"So are you saying that stable orbits can only exist outside of 2.5r_e? Or is there a threshold between stable and unstable orbits somewhere closer in distance?"

"Stable orbits" must be qualified somewhat. Circular relativistic orbits are only possible when r > 3r_e. That's twice the radius of light's only possible orbit, which is also not stable. With 1.5r_e ≤ r ≤ 3r_e, the slightest perturbation to a circular orbit will let the particle either spiral into the hole or go into an elliptical orbit. Elliptical orbits of material particles must stay outside periapsis radius r_p > 2r_e, otherwise they are also dragged into the hole. Parabolic or hyperbolic (open) orbits can venture as close as r > 1.5r_e before being dragged in.

This has to do with the peculiarities of the energy requirements of relativistic orbits i.t.o. escape velocity and speed of light issues. At r < 3r_e, circular orbits need more total energy (kinetic + potential) than at r = 3r_e, so a single slight loss in energy make them spiral in. Newton's theory does not predict this.

In the range 1.0r_e ≤ r ≤ 1.5r_e, only particles obtaining some external positive (outward) radial velocity energy component can prevent being dragged into the hole, irrespective of how much angular momentum they have. If you want to put a rocket in orbit there, its engine must provide continuous outward thrust, otherwise it will spiral in. This is because ballistic orbital velocity there exceeds the speed of light.

Jorrie

That's fascinating. Are the equations very complicated or can you post them here? Or is there a reference to your website you can provide.

I can see light being in a circular orbit or directly falling into an event horizon, but I can't see light in an elliptical orbit. I say that because the speed of an object in an elliptical orbit changes at different points in that elliptical orbit, but since the speed of light is constant, I wonder if, rather than speeding up or slowing down, it gets red shifted and blue shifted (by the corresponding energy change). Is that possible or are elliptical orbits forbidden for light or am I missing the point when we are talking about light in orbit?

Sorry about all the questions, but its your fault for putting up an interesting post.

Hi Roger. You wrote: "I can see light being in a circular orbit or directly falling into an event horizon, but I can't see light in an elliptical orbit."

Correct. Light can be in a circular or hyperbolic orbit, but never elliptical or parabolic. The issue about the energy of light in the vicinity of a black hole is quite tricky. In Schwarzschild coordinates (that of a distant observer), the speed of light varies with distance and direction relative to a black hole. It is given by:

c_s² = c²( g_tt²cos²θ + g_tt sin²θ),

where g_tt = 1-2GM/(rc²) and θ is the coordinate angle between the instantaneous photon path and the radial (r) towards the black hole of mass M. The second term inside the brackets is due to gravitational time dilation and the first term due to both gravitational time dilation and spatial curvature. The closer to the hole, the slower the coordinate speed of light and it is observable as the Shapiro time delay of light. The energy changes are observable as redshift, so it is in effect both speed and redshift that determine the coordinate energy of a photon near a black hole.

However, when a local observer measures the speed of light, the value is its normal self (c), because that observer's rate of time and length of meter rods are all changed in step by the gravitational field. And obviously, for a local observer there is no redshift of local photons...

The equations for all this are slightly involved. I have posted some on this Blog in the past, here and here. (They are all in Relativity 4 Engineers, of course. Check chapters 5,6,7 and 8).

Hope this clears some of the issues.

Jorrie

Thanks Jorrie. I will read the information you have provided...and reread...and reread and....well you get the picture. Eventually I'll understand it or have it memorized, either way I'll seem smarter.

Hi Jorrie,

this is really great,

is there a similarity to the orbits of an object with a periapsis shift of only 4°?

Are there any possible orbits that have much more than the 364° of periapsis shift?

I assume that the acceleration is high so there should be some energy radiated?, this included?

Can the orbiting object be stable against tidal forces at this small distance to the black hole?

RHABE

Hi RHABE. Your questions:

"Is there a similarity to the orbits of an object with a periapsis shift of only 4°?"

No, because an orbit with only 4° periapsis shift must be much farther from the black hole and will be practically elliptical, not the double-loop sort of orbit.

"Are there any possible orbits that have much more than the 364° of periapsis shift?"

Yep, much more. I have simulated up to some 1000° shifts, meaning the orbits makes a number of full circles at near periapsis radius and then ventures off to a fairly distant apoapsis, before coming back to repeat the process.

"I assume that the acceleration is high so there should be some energy radiated?, this included?"

Nope, in the case of a particle orbiting a black hole, there is no gravitational radiation. It only becomes significant in binary systems, where the orbiter has a significant fraction of the primary mass.

"Can the orbiting object be stable against tidal forces at this small distance to the black hole?"

Yep, if it's a particle! Interestingly, even a spacecraft can be OK there if the black hole is massive enough. At r=2.5r_e, the more massive the black hole, the farther from the hole you are. Tidal forces depend on the spacecraft's size as a fraction of the cube of the distance from the black hole.

Jorrie

Hi Jorrie,

thank you for the answers, I should have known some of these by thinking about.

But what about synchrotron radiation?:

"Nope, in the case of a particle orbiting a black hole, there is no gravitational radiation. It only becomes significant in binary systems, where the orbiter has a significant fraction of the primary mass."

Is there a difference between gravitational and B x I forces?

RHABE

Hi RHABE.

If the orbiting particle is charged and the black hole has a magnetic field around it, then there will be synchrotron radiation, but this is not included in the gravitational forces around a Schwarzschild hole (which is uncharged and not spinning by definition).

I think the hole itself cannot have a magnetic field, but gas and dust falling into it may have. In spinning black holes, that magnetic field causes polar jets of plasma to spurt out from the ergo-sphere around the hole.

"Is there a difference between gravitational and B x I forces?"

I would think so, because they are two different phenomena. The gravitational force is caused by concentrated matter, not "concentrated charge", so to speak. A black hole can have a charge, but the effect on its total energy is insignificant when compared to the effect of its mass.

Jorrie