Exploring Extreme Redshifting of Photons in Black Hole Orbits

[miscellanea]

I had a few questions about black holes and photons that get stuck in an orbit around them. With some Googling, I was able to get the answers I so badly needed, but, as it is with these sorts of things, I only got curioser and curioser.

So, let's review what I wanted to find out in the first place.

The question that kept me awake last night (fortunately my girlfriend wasn't around, else I'd have been distracted from nookie -- and yes, unfortunately that has happened before with such issues) was a rather simple one. Black holes can bend light, but can a photon get stuck in an orbit around a black hole?

The answer is yes. Thanks to this page: http://www.physics.nus.edu.sg/~phyteoe/kerr/

Coming from a computer science and maths background, it occurred to me that a photon has no mass, so it's not actually "falling" into a black hole while furiously traveling in another direction, therefore getting locked into orbit. Rather, the photon drives straight, it's the road that's developed a bend. But is it possible for a region of space to exist that turns back on itself? Topologically it doesn't make much sense.

But that's not why I'm posting. The way I understand it, light gets blueshifted when it approaches a black hole and redshifted when it moves away from a black hole. But what exactly happens to a photon that's locked into orbit around a black hole?

There can be two explanations. First, the shift is due to the direction in which the photon is moving. Towards a massive object means blueshift and away from a massive object means redshift. In that case once a photon is in orbit, nothing much happens to it. Just keeps on squiggling around the black hole. In the second case, moving through a gravitational field causes redshifting. Period. The "blueshift" is just how an observer falling towards a black hole would see stuff behind him. But that makes me curioser once again and I ask you: how much redshifting can a poor photon take and what happens to it when it can't take no more? It flatlines?

So which is the case? And what actually happens to the poor photon that's been circling a black hole for ages?

 

[pervect]

I had a few questions about black holes and photons that get stuck in an orbit around them.

Photons can indeed orbit a black hole at the photon sphere, as your research has found. This occurs at 1.5x the Schwarzschild radius.

The frequency of the photon stays the same - as long as it doesn't run into anything (like a speck of dust), it can continue to orbit the black hole forever, unchanged.

This is not particularly likely, as the orbit is not stable. It's like trying to balance a needle on its point - you can do it for a while, and in theory if you put a needle exactly on its point it would stay that way forever. In practice, though, the needle will fall over in a fairly short amount of time, due to the instability of the configuration. Similarly, unless the orbit of the photon is exactly aligned, it will tend to drift one way or the other, and will fairly quickly leave the circular orbit.

 

[Oxymoron]

But that's not why I'm posting. The way I understand it, light gets blueshifted when it approaches a black hole and redshifted when it moves away from a black hole.

A photon falling radially into a black hole is blueshifted relative to a far away observer behind the photon, and a photon traveling radially outward toward a far away observer is redshifted. You have to be careful to include what a photon is redshifted/blueshifted relative to when writing this.

Is it at all possible for a photon in a perfect orbit around a black hole to reach a far away observer?

What if the observer was right at the orbit? The observer would see the photon and then see the same photon momentarily later, and then again, and again... So to an observer at the photon orbit there would essentially be a ring of very bright light around them.

But surely this ignores gravitational shifting? An observer sitting in the perfect photon orbit releases a photon tangentially. The photon travels in a straight line, but due to the curvature of spacetime the path of the photon curls around the black hole and continues to orbit forever, assuming the observer moves out of way everytime the photon completes one orbit.

Now, as the observer releases the photon for the first time does he measure any amount of blueshift from the photon? To the observer the photon does not travel into an increasing gravitational field. It moves along a gravitational 'equipotential' line. Similarly, as the photon completes one orbit, if the observer looks behind them, do they see any shift of the approaching photon?

I don't really know this, I am simply posing more questions. I would think, that as the observer releases the photon he sees no shifting, why should he? The photon is not moving out of any sort of gravitational field! Nor should he see any shifting of the photon approaching from behind about to complete an orbit.

...moving through a gravitational field causes redshifting. Period.

Not necessarily. The redshift of a photon is a relative term. Observers measure redshift and the amount of redshift (if at all) depends on where the observer is relative to the motion and position of the photon. An observer moving with a photon through a grav field sees no redshift at all.

how much redshifting can a poor photon take and what happens to it when it can't take no more? It flatlines?

An infinite amount of redshift. Or so I assume. A photon falling inwards directly away from a far away observer toward a black hole redshifts more and more until finally (when the photon reaches the event horizon) the redshift of the photon is infinite. But infinite gravitational redshift is a direct consequence of time dilation at the Schwarzschild coordinate singularity.