Photon sphere around black hole

[lancelot]

I understand that the photon sphere, the circular orbit of a photon, would be at R = 3/2 Schwarzchild radius. This is the geodesic path at that location.

 

But this seems odd. It implies (A) if moving tangentially at greater than R, the photon will pass by with mere deflection. Fair enough.

But, (B) it equally implies that if moving tangentially at less than R, the photon would be in a less than circular orbit, which I can only imagine as a spiral towards the BH. But this would imply that the photon sphere is effectively a Schwarzchild horizon for the photon!

 

Where am I going wrong in this interpretation?

[Janus]

I understand that the photon sphere, the circular orbit of a photon, would be at R = 3/2 Schwarzchild radius. This is the geodesic path at that location.

 

But this seems odd. It implies (A) if moving tangentially at greater than R, the photon will pass by with mere deflection. Fair enough.

But, (B) it equally implies that if moving tangentially at less than R, the photon would be in a less than circular orbit, which I can only imagine as a spiral towards the BH. But this would imply that the photon sphere is effectively a Schwarzchild horizon for the photon!

 

Where am I going wrong in this interpretation?

 

The key here is the word "tangentially". A photon moving at a "tangent" inside the photon sphere follows a geodesic which intersects the event horizon. But a photon traveling at some other angle relative to the tangent may not. An extreme example would be a photon traveling directly away from the BH. Its geodesic carries it out and away from the BH. It is only within the event horizon itself that all geodesics lead back to the BH.

[lancelot]

Thank you Janus. I see my error. A particle could pass on a geodesic into the photon sphere and out the other side if at the right angle of approach. As you point out, only within the event horizon do all geodesics lead back into the BH.