Support for PBS is provided by World of Warships.
If there’s one thing cooler than a black
hole it’s a rotating black hole. Why? Because
we can use them as futuristic power generators,
galactic-scale bombs, and portals to other universes.
Two months after Einstein presented his complete
general theory of relativity in 1915, a young
German physicist named Karl Schwarzschild
was the first to fully solve its equations
for a realistic situation. Schwarzschild’s
eponymous metric describes the warping of
space and time around a spherically symmetric
mass. And if that mass is sufficiently compressed,
the metric predicts an event horizon - a spherical
surface where the fabric of space cascades
downwards at the speed of light, and where
the flow of time halts from the perspective
of the outside universe. It predicts the
inescapable region of space that we now call
the black hole.
Our observations of the universe have since
told us that black holes are very real. We’ve
seen the gravitational waves caused by their
mergers, we’ve witnessed the havoc they
wreak on their surroundings in distant quasars
and in our own galaxy, and we’ve even taken
an image of a black hole with the event horizon
telescope. But none of these real black
holes are particularly well described by Schwarzschild’s
solution. That’s because the Schwarzschild
black hole has no rotation. But all real,
astrophysical black holes are spinning.
According to the no-hair theorem, black holes
can have three and only three properties:
mass, electric charge, and spin.
Every black hole must have mass. Compacting
a lot of mass into a tiny region is what makes
them black holes in the first place. Essentially
no black holes have electric charge - if somehow
one does acquire charge it would quickly lose
it because it would repel like charges and
attract opposite charges. But essentially
all real black holes will be rotating. The
spin of a black hole comes from the combined
angular momentum of everything that went into
forming it. That includes the rotation of
the star’s core that collapsed into the
black hole in the first place, and the rotation
of any latter object sucked through the vent
horizon. Even non-rotating objects will affect
black hole spin if they fall in at some angle.
Some black holes might not have MUCH spin
- because angular momentum can cancel out
if objects fall in with different spins or
in different directions. But some rotation
is always expected.
Despite the importance of spin in black holes,
it took nearly half a century before Einstein’s
equations were solved for the rotating case.
That was by Roy Kerr in 1963, yielding the
Kerr metric and describing the Kerr black
hole, which has mass and rotation but no charge.
So why did it take so long? Well, general
relativity is hard, and the spherical symmetry
in the Schwarzschild solution eliminated a
lot of complexity. Without that simplification
the algebra was diabolical. But the final
Kerr metric doesn’t look so bad. I guess.
It can be used to calculate the path of any
body moving near or even within a Kerr black
hole - or indeed any rotating mass.
Today we’re going to look at what the Kerr
solution can tell us about the spacetime outside
a rotating black hole. We’ll save the even
weirder details of the Kerr black hole’s
interior for another episode. For a preview
check out our episode on time machines. Yup,
the math says you can visit your past within
a Kerr black hole. For the bit about visiting
other universes you’ll just have to wait.
Let’s start by talking about what is actually
rotating in a Kerr black hole. It’s tempting
to just say that some physical thing deep
beneath the event horizon is rotating. But
if nothing can escape the event horizon, how
can that internal rotation influence the outside?
In fact, how can any effect of gravity extend
from beneath the event horizon? In fact it
sort of doesn’t actually. Both the gravitational
field and its rotation can be thought of as
properties of the spacetime itself.
Black holes are self-sustaining holes in the
fabric of spacetime. Space at the event horizon
cascades downwards, dragging more space behind
it, sort of like how water drags itself near
the edge of a waterfall. In a Kerr black hole,
space above the event horizon is dragged around
in a circle - so less waterfall and more whirlpool.
Water spiraling down a drain in a flat sink
doesn’t know about the hole - it only knows
about the motion of the water around it. In
fact it’s possible to construct a black
hole in general relativity rotating or otherwise
- without any mass. Warp spacetime so it looks
like the exterior of a black hole, and that
warping will persist. So what is rotating?
Spacetime is rotating.
The flow of space around a rotating black
hole is known as frame-dragging. We see it
around any rotating mass. In frame dragging,
any “freefall” trajectory - the path taken
by an object moving freely in the gravitational
field - is dragged in the direction of the
object’s spin. Gravity Probe B measured
the Earth’s frame dragging and it was exactly
as Einstein’s theory predicted - incredibly
weak in Earth’s case. But in the case of
a Kerr black hole, this circular dragging of spacetime
changes everything.
Let’s approach our Kerr black hole. Contrary
to the common misconception, as long as you don’t
get too close to the event horizon it’s
possible to orbit a black hole in a perfectly
stable way. For a non-rotating black hole
you can execute a stable circular orbit as
close as 3 times the radius of the event horizon
- or 3 Schwarzschild radii. Any closer and
no stable orbits exist - unless you’re firing
your rockets like crazy, you must spiral either
inwards or outwards. But for a rotating black
hole, frame dragging gives a you little extra
kick, and so stable orbits exist much closer
to the event horizon. For a black hole rotating
as fast as possible, stable orbits exist all
the way down to the event horizon. As long
as you’re traveling in the same direction
as the black hole spin. If you’re orbiting
in the opposite direction, there are no stable
orbits within 9 Schwarzschild radii.
The name for the size of these innermost stable
circular orbits is ... innermost stable circular
orbit. Or ISCO. For a black hole that is currently
feeding - perhaps devouring a companion star
or, in the case of quasars, a bunch of its
host galaxy’s gas - the ISCO is expected
to eventually be detectable as a dark circle
in the middle of the otherwise insanely bright
accretion disk formed by infalling matter.
And that’s because any gas that gets that close
will quickly be swallowed by the black hole. To date ISCOs have not been directly detected - though there
is tentative evidence in gravitational lensing
studies of quasars.
So yeah, you can orbit “safely” pretty
close to the Kerr black hole’s event horizon.
Just above the event horizon is a particularly
bizarre region called the ergosphere. There,
frame dragging carries space around the black
hole at faster than the speed of light. That
means everything - even light - must move
in the direction of the black hole’s spin.
The situation here is actually similar to
the state below the event horizon where space
moves downwards faster than light. In the
math, that faster-than-light flow of space
is represented in a particularly weird way
- space and time switch places. In particular,
the radial direction becomes time-like, so
downward motion becomes as inevitably one-directional
as time. Well, in the ergosphere the angular
coordinate becomes time-like - it’s as difficult
to resist orbiting the black hole than it
is to travel backwards in time - which is
to say it’s impossible. That same switch
also allows us to extract energy from the
ergosphere, as we’ll see.
The ergosphere extends all the way down to
the event horizon.
Now the event horizon in the rotating case is
not spherical - it’s squished at the poles,
like the rotating Earth. More spin equals
more squished. The ergosphere has a similar
shape - but also dips at the poles to touch
the event horizon - it’s sort of pumpkin shaped.
Now I know you would love to drop below
the event horizon right now - but you’re
going to have to wait. There’s still plenty
to do in the ergosphere - like building a
hyper-advanced black hole engine.
The great Roger Penrose figured this out in
the early 70s. It goes like this: a massive
object is dropped into the ergosphere of a
kerr black hole on a carefully tuned trajectory.
If the object is split into two pieces at
exactly the right point, one half will go
plummeting through the event horizon while
the other is ejected from the ergosphere and
escapes. In fact it escapes with more kinetic
energy than it had coming in - up to 20% of
the energy than was bound up in the mass of
the half that was lost.
This energy is extracted from the rotational
energy in the ergosphere, slowing the black
hole’s spin. To get a little more technical
- it works because the weird space-time flip
in the Kerr metric of the ergosphere allows
one half of the object to acquire negative
energy, which is transferred to the black
hole, while the other half gains the difference
in energy as kinetic energy.
So there’s your black hole engine: maneuver
rocks into a Kerr black hole, blow them apart
at the right instant, and then catch the kinetic
energy of the pieces that get ejected.
Sounds a little messy actually, but you can do this with light. Light that is directed through
the ergosphere in the direction of rotation
will also extract rotational energy and emerge
amplified in a process called superradiance.
This way the rotational energy can be extracted
with, in principle, 100% efficiency. Oh, and
you can also build a black hole bomb this
way - by surrounding the Kerr black hole with
mirrors. Then you just shine a flashlight at it and its photons pass through the ergosphere
again and again, becoming exponentially amplified
until ... boom.
The last process we’ll talk about is actually
important in the real universe. It’s the
Blandford-Znajek process. In this case you
have a magnetic field produced by the flow
of material around the black hole in an accretion
disk. The flow of space in the ergosphere
spins up the magnetic field into a gigantic
particle accelerator. Charged particles are
accelerated along those magnetic field and
can radiate intense light. Ultimately, the
energy of that light is extracted from the
rotational energy of the black hole. It’s
hypothesized that some jets observed from
accreting black holes may be powered by this
process.
Jets produced by fast-rotating black holes
are also a contender for another astrophysical
phenomenon - gamma ray bursts. When a truly
gigantic star collapses at the end of its
life, and if its core was rotating fast enough,
that core will produce a Kerr black hole that
can suck more infalling material into an accreting
vortex and spit it back out in powerful jets,
again powered by the black hole’s rotation.
If we happen to be along the paths of one
of these jets, relativistic effects massively
magnify its brightness. We see these as gamma
ray bursts from over 13 BIllion light years away.
Rotating black holes are very real and powerful
players in the energetics of our universe
- but they’re also very worrying to physicists,
because they threaten several physics-breaking
phenomena - time travel, universe-hopping,
and naked singularities. And we'll encounter all of these
when we drop below the event horizon into the deeper
weirdness of the Kerr spacetime.
Thank you to World of Warships for supporting
PBS. World o f Warships is a naval warfare-themed
massively multiplayer online game, where players
can battle others at random or play cooperative
battle types, battling against bots or in
a player-versus-environment battle mode. Gameplay
features real-world ship physics, that are
based on historical ships from the early to
mid 20th century such as the recently released
USS Charleston. Naval battles are always dynamic
because its not just about firing, it’s
about maneuvers, movement and understanding
the physics of how a ship can move. Because
in reality, the maneuverability of a ship
is influenced by hull shape, ship mass, power
plant output, and many other factors. And
World of Warships captures the feel of complex
motion dynamics without sacrificing gameplay.
For more information and the battleship USS
Charleston, use the link in the description
and use code BATTLESTATIONS2020.
