Why is the photon sphere spherical, but the accretion disk not?

Question

As far as I understand, photons are massless, but are still subject to gravity, must follow the distortions of spacetime (including framedragging etc.), period.

Now the only answer on this site explains the disk shape of the accretion disk with the conservation of angular momentum. But, as far as I understand, photons are subject to this law too, and should be framedragged by the black hole, and following this line of argument, should be ordered into a disk shape or "photon disk".

Now photon spheres are not always spherical, there are different forms, depending on the conditions of the black hole (rotation etc.), but they can hardly be disk shaped.

reason the orbits lie roughly in the same plane: Conservation of angular momentum.

Why do all the planets of the solar system orbit in roughly the same 2D plane?

Because the total angular momentum is conserved, this angular momentum lost by the object that is getting closer must be transported to the rest of the disk that gains the angular momentum. This tends to flatten the disk and align the direction of rotation of all the parts because flattened uniformly rotating disks maximize the angular momentum vs energy ratio. And that's what the dynamics prefers because the kinetic energy is being lost (converted to heat etc.) while the total angular momentum is being increased.

Accretion disk physics - Stellar formation

At this point, one could say, that photons cannot slow down, they cannot transfer their angular momenta. But, photons can and sometimes do transfer their momenta to other objects (solar sail, radiation pressure, mirror etc.), in this case they could transfer their momenta to the black hole's gravitational field itself, thereby "flattening" the photon sphere. And, photons are subject to frame dragging (they must follow the distortions of spacetime), based on the rotation of the black hole.

So there are a few things that come to mind:

1. the accretion disk gets its shape from angular momentum conservation

2. angular momentum conservation holds for photons too

In other words, why is the photon sphere not disk shaped, because of angular momentum conservation, just like an accretion disk? Again, why does this mechanism not "flatten" the photon sphere?

Question:

1. Why is the photon sphere spherical, but the accretion disk not?

Answer 1

the accretion disk gets its shape from angular momentum conservation

... and from interactions between the elements.

There are multiple different orbit collections that are possible that have the same total angular momentum. But if the orbits are with interacting particles, then some will be unstable. You can't have orbits that intersect after a large number of possible intersections have taken place. Collisions will maintain total angular momentum and remove kinetic energy, but they will disturb any intersecting orbits in the process. Over time, particles are removed from those orbits.

At high densities/long lifetimes, only non-intersecting orbits are stable. That reduces to concentric circular orbits all aligned with the angular momentum vector.

Photons don't collide with other photons, so the same mechanism does not appear.

Globular clusters and Oort clouds haven't reduced to a disk because the interactions are too rare compared with their age.

Answer 2

The photon sphere and the accretion disks around a black hole have different components:

• the photon sphere is made of photons which have no mass, whose speed is $ c $. The shape is a sphere since the photons are coming from all directions.

• the accretion disk is made of material particles whose orbiting linear speed is very high but significantly lower than $ c $. The shape is a disk (similar to asteroid rings around planets) and its light is caused by the friction of the particles.

Thus, when you apply the general relativity to a photon or a material particle, the results are different.

To complement this answer, the dimensions of the photon sphere and accretion disks are also different: by using Schwarzschild metrics, with $ R_s=\frac{2GM}{c^2} $, the photon sphere has a radius of $ {3\over 2}R_s $ while the accretion disks may have several radii, whereas the minimum radius is $ r_{ISCO}=3 R_s $ which corresponds to the Innermost Stable Circular Orbit. For a static observer at infinity, the linear speed of the material particles on the ISCO is $ \frac{c}{\sqrt{6}}\simeq\ 0.408\ c $, which is the highest possible speed of a particle in an accretion disk.

Hoping to have answered your question,

Best regards.