Logarithmic spiral photon orbits

[jpo]

Hello,

It has been said if a photon crosses the event horizon of a black hole it will have a spiral orbit; The spiral is Archimedes $r = r_{i}+r_{0}\theta^2$
There is one such example in "Introduction to tensor calculus, relativity and cosmology" by D F Lawden on page 165

Is it possible that the photon orbit is a logarithmic spiral? Does the spiral shape depend on the metric of the gravitational field?

 

[Bill_K]

jpo, Is he talking about a Schwarzschild black hole or something more general? I'm afraid if he's talking about the photon orbits in Schwarzschild I'd have to disagree with him. At any rate, it's safe to say you won't find an orbit which is a logarithmic spiral.

 

[jpo]

the gravitational field he gives is determined by the metric
$ds^2 = r^2 (dr^2 + d\theta^2) + r(dz^2 - dt^2)$
in quazi-cylindrical coordinates

then he claims a photon emitted at $z=0$ will track the above Archimedes spiral
Also, $\dot{r} = \dot{z} = 0$ initially