Schwarzschild & Kerr black holes geometries

[JeffOCA]

Hello

Let's consider a Schwarzschild BH.

A photon (or a massive particle) crossing the event horizon cannot be static : the r (radial) coordinate becomes a temporal coordinate. Therefore, the photon falls towards the central singularity (r=0). There is no way for him to escape : the horizon is a static limit

Now, the photon sphere is the spherical region of space where the orbital speed is equal to c. Only photons can orbit on the photon sphere but these orbits are very unstable.
If a photon crosses this sphere, can it remain static (without moving or falling towards the central singularity) ? Is going towards the singularity the only movement it can describe ?

Now, let's consider a Kerr BH. There are two horizons (inner and outer). By definition, the ergosphere is the region of space between the static limit and the outer horizon. A photon in the ergosphere cannot be static (it has crossed the static limit) and falls towards the singularity.
But, what happens at the crossing of the outer horizon and, after, when crossing of the inner horizon ? What kind of trajectories are possible ?

Thanks,
Jeff

[George Jones]

If a photon crosses this sphere, can it remain static (without moving or falling towards the central singularity) ? Is going towards the singularity the only movement it can describe ?

If a photon crosses the photon sphere while going from greater to lesser r, it will hit the singularity. A photon can cross the the photon sphere while going from lesser to greater r. For example, a laser hovering between the photon sphere and the event horizon must be pointed "above horizontal" in order for its light to ecape. The closer the laser is to the event horizon, the more above horizontal the direction must be for escape.
Now, let's consider a Kerr BH. There are two horizons (inner and outer). By definition, the ergosphere is the region of space between the static limit and the outer horizon. A photon in the ergosphere cannot be static (it has crossed the static limit) and falls towards the singularity.

Not necessarily. Again, this depends on the direction of the photon.
But, what happens at the crossing of the outer horizon and, after, when crossing of the inner horizon ? What kind of trajectories are possible ?

I'll take a look at my references tomorrow.

[George Jones]

But, what happens at the crossing of the outer horizon and, after, when crossing of the inner horizon ? What kind of trajectories are possible ?

Technical reference: S. Chandrasekhar's The Mathematical Theory of Black holes, Chapter 7 The Geodesics in the Kerr Space-Time, section 63 the null geodesics,.

Non-Technical reference: Kaufmann's The Cosmic Frontiers of General Relativity, Chapter 12 The Geometry of the Kerr Solution.

Each of these books contains a wealth of good information; maybe you can get them from a library, possible through inter-library loans.