Rotational energy of rotating black holes.

[halfeye]

It seems sort of counter-intuitive to me, but it seems that the slower a black hole of a given mass rotates, the more rotational energy it has.

We know that in a rotating black hole, the singularity is a ring. As that ring gains more energy, it widens. The wider it gets, the more time it takes to revolve.

I'm sort of inclined to think that the speed of the rotation would be c, but if it's slower, the above would still seem to be true.

[Kato]

It seems sort of counter-intuitive to me, but it seems that the slower a black hole of a given mass rotates, the more rotational energy it has.

We know that in a rotating black hole, the singularity is a ring. As that ring gains more energy, it widens. The wider it gets, the more time it takes to revolve.

I'm sort of inclined to think that the speed of the rotation would be c, but if it's slower, the above would still seem to be true.

Okay, I'll start at the end. Nothing with mass can move at the speed of light. So, no, black holes don't rotate at c. My best guess is a newly born black hole rotates at roughly the speed of the star before, which can be pretty fast on human scale but not c.

Second paragraph.. I'll assume you mean a disc? Unless I'm missing something? And even considering that, I feel like most black holes don't rotate nearly fast enough to vary much from their spherical shape. But I will admit to being unsure about that.

With this out of the way we are basically discussing any disc of variable shape that rotates around its center. In which case it's true, iirc, because to preserve momentum while increasing energy you have to widen the radius. But I'm not sure without writing things down right now.. I'm sure if NichG or someone else who's less rusty shows up they can correct me.

[Lord Torath]

When you say you think the rotational speed should be "c", where exactly are you measuring? Rotational speed is given in terms of revolutions per unit of time, while c has units distance per unit of time. To get from rotations per time to distance per time, you need to multiply by pi and a diameter. Which diameter are you proposing using?

[halfeye]

Okay, I'll start at the end. Nothing with mass can move at the speed of light. So, no, black holes don't rotate at c. My best guess is a newly born black hole rotates at roughly the speed of the star before, which can be pretty fast on human scale but not c.

Second paragraph.. I'll assume you mean a disc? Unless I'm missing something? And even considering that, I feel like most black holes don't rotate nearly fast enough to vary much from their spherical shape. But I will admit to being unsure about that.

With this out of the way we are basically discussing any disc of variable shape that rotates around its center. In which case it's true, iirc, because to preserve momentum while increasing energy you have to widen the radius. But I'm not sure without writing things down right now.. I'm sure if NichG or someone else who's less rusty shows up they can correct me.

I'm talking about the rotation of the singularity deep within the event horizon, not the rotation of the event horizon. So no, I'm not talking about the event horizon being a disc, I'm talking about the singularities of Kerr holes (spin but no electric charge), where the singularity is said to be proved to be a ring.

[NichG]

Okay, I'll start at the end. Nothing with mass can move at the speed of light. So, no, black holes don't rotate at c. My best guess is a newly born black hole rotates at roughly the speed of the star before, which can be pretty fast on human scale but not c.

Second paragraph.. I'll assume you mean a disc? Unless I'm missing something? And even considering that, I feel like most black holes don't rotate nearly fast enough to vary much from their spherical shape. But I will admit to being unsure about that.

With this out of the way we are basically discussing any disc of variable shape that rotates around its center. In which case it's true, iirc, because to preserve momentum while increasing energy you have to widen the radius. But I'm not sure without writing things down right now.. I'm sure if NichG or someone else who's less rusty shows up they can correct me.

I'm pretty rusty too, but lets see what we can do... I think the tricky thing is to determine what should be held constant.

Spoiler: Bunch of classical stuff that probably doesn't apply...

Show

For a classical spinning disc (spinning around its center, in-plane) of mass m and radius r, the moment of inertia is I=1/2 mr^2. So it's angular momentum is L = I omega ~ omega r^2, and it's angular kinetic energy is 1/2 I omega^2 ~ omega^2 r^2. So if we just take that object and spin it up, the angular momentum and angular kinetic energy both increase.

Now lets say this disc is being held together by some cohesive force such that it's radius adjusts to keep something about it constant. If this is set up in such a way that the object's angular momentum remains constant when it's spun up, that means that r ~ sqrt(L/omega), so angular kinetic energy ~ L omega - so in that case, it still gains rotational kinetic energy when you spin it up, but with the very peculiar behavior that spinning it up causes it to shrink. Physically, this makes more sense when you consider something which is falling inwards and spins up because it has no way to get rid of its angular momentum, rather than spinning 'causing' it to shrink.

The case where the angular kinetic energy is held constant is kind of pointless with respect to the original poster's question. But something more relevant to gravitational systems is if we consider something like an orbiting particle whose orbit remains circular. Classically, this happens when centripetal acceleration and gravitational acceleration balance, so you have 1/r^2 ~ r omega^2, or omega ~ 1/r^(3/2), or r = 1/omega^(2/3). In this case, rotational kinetic energy ~ omega^2 r^2 ~ omega^(2/3). So rotational kinetic energy still increases as you spin the system up (and also, the orbit gets closer in).

Finally, it might be useful to think of a system of a particle on a nonlinearly-stiff spring - something like F ~ r^3. In this case, unlike the other two we've considered, spinning the system up should cause the particle to move further out rather than closer in. For circular orbits here, r^3 ~ r omega^2, and omega ~ r. Now the rotational kinetic energy is omega^2 r^2 ~ omega^4, so again it still increases when you spin up the system.

Relativistically, I'm less sure. The relevant equations here correspond to the Kerr metric, and those admit ring-shaped solutions. The non-spinning case has to reduce to the Schwarzchild metric, which doesn't. So between the two are some topological transitions, and I'm honestly not sure how to account for those in the energy budget (I guess it just happens smoothly as you increase the angular momentum?).

Edit: Looking at the Wikipedia entry for Kerr metrics, it seems that the radius of the ring singularity scales linearly with the angular momentum, which if interpreted classically does make it look that omega ~ 1/r ~ 1/L, which does seem like would be pretty strange - if nothing else, that seems like it should imply that the speed of a point on the surface of the singularity is a particular constant that depends only on the mass of the black hole, which doesn't seem right. I'm a bit dubious of just using the classical angular momentum here though, and at the kinds of scales these equations are referring to, the mass also varies significantly with omega (and in an annoyingly complex way). Hmm...

[halfeye]

What most interests me is that it seems as if losing energy would speed up the rotation until the limit was reached, when it would (instantaneously?) stop.

[NichG]

What most interests me is that it seems as if losing energy would speed up the rotation until the limit was reached, when it would (instantaneously?) stop.

There are chirps like that in some systems (like a slightly inelastic ball bouncing on a surface - the time between impacts goes to zero in finite time).

I'm still a bit concerned about the mass - velocity relation, and whether it stays classical for all feasible black hole masses.

[georgie_leech]

There are chirps like that in some systems (like a slightly inelastic ball bouncing on a surface - the time between impacts goes to zero in finite time).

I'm still a bit concerned about the mass - velocity relation, and whether it stays classical for all feasible black hole masses.

I seem to recall that being related to how a spinning Black Hole has a smaller Event Horizon, which in turn implies a maximum speed limit (because you can't have a naked singularity). So since more massive Black Holes have correspondingly bigger Event Horizons, they can have greater maximum rotation. That is, since compressing an object makes it rotate faster to conserve angular momentum, and shrinking to a singularity means that would break and go to infinity on the way to becoming the ring-shaped singularity of a rotating Black Hole, it's the mass that tells us how much room the event horizon has to shrink.

But I could be way off. I am nowhere near educated enough to unpack the math that tells us this.

[factotum]

My best guess is a newly born black hole rotates at roughly the speed of the star before

No, it would rotate faster--as a general rule, when you make something smaller (and a black hole is definitely smaller than its precursor star!) it will spin faster due to conservation of angular momentum. This is why pulsars are as fast as they are--the neutron star or white dwarf that is generating the radio signal spin extremely fast compared to a typical star, anything up to several hundred rotations per *second*.

[Rockphed]

No, it would rotate faster--as a general rule, when you make something smaller (and a black hole is definitely smaller than its precursor star!) it will spin faster due to conservation of angular momentum. This is why pulsars are as fast as they are--the neutron star or white dwarf that is generating the radio signal spin extremely fast compared to a typical star, anything up to several hundred rotations per *second*.

Pulsars that are spinning at several hundred rotations per second have picked up additional angular momentum at some point, often by eating an orbiting companion.

Edit: which is not to contradict your main point. The sun spins about once every 8 days. A standard pulsar starts out spinning a couple times a second and slowly slows down (unless it starts to collect mass and angular momentum somehow).

[NichG]

I seem to recall that being related to how a spinning Black Hole has a smaller Event Horizon, which in turn implies a maximum speed limit (because you can't have a naked singularity). So since more massive Black Holes have correspondingly bigger Event Horizons, they can have greater maximum rotation. That is, since compressing an object makes it rotate faster to conserve angular momentum, and shrinking to a singularity means that would break and go to infinity on the way to becoming the ring-shaped singularity of a rotating Black Hole, it's the mass that tells us how much room the event horizon has to shrink.

But I could be way off. I am nowhere near educated enough to unpack the math that tells us this.

The thing in particular that I'm referring to is that, since the radius of the ring singularity (not the event horizon radius, note) is linear in the angular momentum of the black hole (specifically r = L/Mc, then (classically) you would have L = M omega r^2 = rMc, resulting that r ~ c/omega.

But then if we calculate the linear velocity of a point on the rim, it's constant. Specifically, v = r*omega = c. So that would imply that all black holes, regardless of their angular momentum, are rotating at exactly the speed of light at the singularity. Which essentially means that we made a mistake by assuming we could use the classical angular momentum here.

[Bohandas]

I seem to recall that being related to how a spinning Black Hole has a smaller Event Horizon, which in turn implies a maximum speed limit (because you can't have a naked singularity).

IIRC that's never actually been proven and doesn't actually follow from the theoretical framework as it is currently understood.