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April 1, 2008

A Puzzle About Blueshifts

Posted by John Baez

Greg Egan posed the following problem about rotating black holes, saying “I thought I’d send you this question on the chance that you might have an instant, effortless answer to it. If not, please feel free to ignore it!”

I didn’t know the answer, but it seemed too interesting to just ignore. So, I asked if I could post it here, and he said yes. Maybe someone out there can help?

guest post by Greg Egan

If we happen to know the Kerr metric, it’s easy to show that the blueshift factor on the equatorial plane of the Kerr geometry is exactly the same as for the Schwarzschild metric. In other words, the angular momentum of a rotating hole has (on the equatorial plane) absolutely no effect on g ttg_{tt}. Now, while this doesn’t surprise me in the least, and seems very intuitively reasonable, I haven’t been able to find any simple way to prove it — not without deriving the entire Kerr solution!

Along with g ttg_{tt}, there are two other things that I know of that are unchanged by the angular momentum of a black hole when you’re in the equatorial plane. Working in an orthonormal tetrad whose time vector is parallel to the timelike Killing vector, the tidal squeezing in the direction that points equatorially around the hole is just M/r 3M/r^3. And (working in the same frame) if you compute the covariant derivative of the radial vector with respect to time, the magnitude of that covariant derivative is unaffected by the hole’s angular momentum. (In the Schwarzschild case the derivative just measures the acceleration of the chosen frame; in the Kerr case, it’s a mixture of acceleration and frame dragging. But the magnitude of the derivative is independent of the angular momentum.)

One of these three facts can be taken as a choice of gauge, a physical definition of the rr coordinate on the equatorial plane. So I don’t expect to be able to prove one statement in isolation; you need to define rr, in order for the other statements to have any physical content. But short of pulling the Kerr metric out of a hat, I haven’t been able to show why any one of these three follows from any other. The fact about the blueshift seems so “obvious” that it’s very hard to believe that you really need to know the complete Kerr metric just to prove it, but maybe my intuition about that is simply wrong.

One fun thing I did discover was that if you take these three facts as given (and already know the Schwarzschild geometry), you can derive the whole Riemann tensor for the Kerr geometry on the equatorial plane, with very little work.

Posted at April 1, 2008 7:43 AM UTC

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Re: A Puzzle About Blueshifts

I suspect that the answer to this lies in making use of the fact that the Kerr geometry is Petrov type D, that is, there are two doubly-degenerate principal null congruences.

The calculations aren’t trivial, so I’m not yet sure everything works out, but my hunch is that this is enough to pin down the geometry on the equatorial plane, without resorting to knowledge of the full Kerr solution.

Posted by: Greg Egan on April 5, 2008 5:55 AM | Permalink | Reply to this

Re: A Puzzle About Blueshifts

Knowing the Petrov type does add some information, but not enough.

Posted by: Greg Egan on April 5, 2008 1:19 PM | Permalink | Reply to this

Re: A Puzzle About Blueshifts

Maybe there would have more of a response to this post if it had been posted on a different day.

Posted by: David Corfield on April 6, 2008 4:51 PM | Permalink | Reply to this

Re: A Puzzle About Blueshifts

Ha-ha! It was just a joke: black holes don’t really exist, and neither does Greg Egan.

I actually emailed this question to Jerzy Lewandowski a day before April 1st, but he read it on that day and said “Could you please send me this question some other day?” I resent it the next day, but he still hasn’t answered.

I’d hoped he could help, since he and other people in loop quantum gravity have worked on intrinsic, local geometrical ways to locate the horizon of a rotating black hole. For example:

I still think some real experts on black holes could help Greg with his puzzle. I’m reluctant to bother Ashtekar, since he’s so busy, but maybe I will — he does know his black holes.

Posted by: John Baez on April 6, 2008 10:47 PM | Permalink | Reply to this

Re: A Puzzle About Blueshifts

Dear John,

indeed, I “have worked on intrinsic, local geometrical ways to locate the horizon of a rotating black hole”.

However Greg’s question is interesting and I really
like it:

- it has nothing to do with black holes

because the equality holds at points away from
the horizons, and black hole may or may not be inside

- it is not about the horizons

because the Schwarzschild horizon blue shift factor
corresponds to the Kerr ergosphere blue shift factor
(at the equatorial)

- and more over its nature is global rather
than local

because the \partial / \partial t vector field is
defined by the global properties of spacetime

In conclusion I am just a wrong expert, sorry…

Posted by: Jurek on April 10, 2008 12:04 AM | Permalink | Reply to this

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