Black hole ringdown shows hair?

[Jos Bergervoet]

After the merger of two black holes there is a "ringdown" in which the
new black hole radiates off some energy.

Presumably this ringdown amplitude decays exponentially, so it never
becomes exactly zero. We can also easily imagine (although not easily
test experimentally) the reverse: some incoming gravitational radiation
which is nicely focussed on an existing balck hole makes it "ring up"
instead of ring down..

But if a black hole can have vibration amplitudes, for lots of spherical
harmonics as it seems, doesn't that violate the no-hair theorem?

[Phillip Helbig (undress to reply)]

Post by Jos Bergervoet
After the merger of two black holes there is a "ringdown" in which the
new black hole radiates off some energy.
Presumably this ringdown amplitude decays exponentially, so it never
becomes exactly zero. We can also easily imagine (although not easily
test experimentally) the reverse: some incoming gravitational radiation
which is nicely focussed on an existing balck hole makes it "ring up"
instead of ring down..
But if a black hole can have vibration amplitudes, for lots of spherical
harmonics as it seems, doesn't that violate the no-hair theorem?

No, because the no-hair conjecture (it hasn't actually been proved)
applies to static black holes.

[Phillip Helbig (undress to reply)]

Post by Phillip Helbig (undress to reply)

Post by Jos Bergervoet
After the merger of two black holes there is a "ringdown" in which the
new black hole radiates off some energy.
Presumably this ringdown amplitude decays exponentially, so it never
becomes exactly zero. We can also easily imagine (although not easily
test experimentally) the reverse: some incoming gravitational radiation
which is nicely focussed on an existing balck hole makes it "ring up"
instead of ring down..
But if a black hole can have vibration amplitudes, for lots of spherical
harmonics as it seems, doesn't that violate the no-hair theorem?

No, because the no-hair conjecture (it hasn't actually been proved)
applies to static black holes.

As Jonathan Thornburg pointed out in a followup, "static" should be
"stationary". (Obviously it does apply to rotating black holes, since
angular momentum is one of the three characteristics a hairless black
hole is allowed to have, along with mass and charge, and obviously
something rotating is not static---but it is stationary.)

[Jonathan Thornburg [remove -animal to reply]]

Post by Jos Bergervoet
After the merger of two black holes there is a "ringdown" in which the
new black hole radiates off some energy.
Presumably this ringdown amplitude decays exponentially, so it never
becomes exactly zero. We can also easily imagine (although not easily
test experimentally) the reverse: some incoming gravitational radiation
which is nicely focussed on an existing balck hole makes it "ring up"
instead of ring down..
But if a black hole can have vibration amplitudes, for lots of spherical
harmonics as it seems, doesn't that violate the no-hair theorem?

No, because the no-hair theorem applies to *stationary* spacetimes,
i.e., ones which are (in a suitable sense) time-independent. In practice,
this means that the no-hair theorem is a statement about the t -> infinity
limit of a black hole, and doesn't say anything about the finite-time
behavior of dynamic systems.

A few further clarifications:

Roughly speaking, in general relativity a spacetime is defined to be
"static" if & only if it's time-reversible, or "stationary" if and only
if it's time-independent. (In both cases the precise definition needs
to essentially say that there exists a suitable time coordinate such
that the property holds.)

The distinction between static & stationary is very important: a rotating
black hole spacetime is not static (because time-reversal would reverse
the BH spin, giving a different spacetime), but such a spacetime may be
stationary (time-independent, given a suitable time coordinate). As a
more prosaic example [nicely presented in Ray d'Inverno's fine textbook
"Introducing Einstein's Relativity" (Oxford U.P., 1992)] a pipe with water
flowing in it may be stationary (if the water flow is time-independent),
but it is not static (time-reversal would reverse the direction of water
flow). If the water is not moving, then the system is static as well as
stationary.

Returning to relativity, a Schwarzschild (non-rotating) black hole is
both static & stationary; a Kerr (rotating) black hole is stationary but
not static.

The no-hair (a.k.a black hole uniqueness) theorem states that (roughly,
again I'm eliding a great many mathematical details) in general relativity,
a 4-dimensional spacetime (time + 3 spatial dimensions) which
* has a regular event horizon (in practice, this means that the spacetime
contains a black hole)
* is stationary,
* is electrovac (= the only "matter" fields in the stress-energy tensor
are electromagnetic as per Maxwell's equations), and
* is asymptotically flat (= there is a "far away" region where the
gravitational field is weak and spacetime is very close to flat)
must be Kerr-Newman (= Kerr if there is no electrical charge present).

The Wikipedia page
https://en.wikipedia.org/wiki/No-hair_theorem
has a nice discussion of this.

Note that the various restrictions are essential, e.g.,
* in other numbers of dimensions (different from 4) the theorem doesn't
hold, e.g. in 5 dimensions there can be "black rings"
* if other matter fields are allowed besides electromagnetism, then
there may be "hairy black holes". See
https://link.springer.com/article/10.12942/lrr-2012-7
Cru\'{s}ciel, Costa, & Heusler
"Stationary Black Holes: Uniqueness and Beyond"
Living Reviews in Relativity volume 15, Article 7 (2012)
for a detailed and technical review.