Normal maps are useless inside black holes. At the event horizon - the ultimate point
of no return as you approach a black hole - time and space themselves change their character.
We need new coordinate systems to trace paths into the black hole interior. But the maps
we draw using those coordinates reveal something unexpected - they don’t simply end inside
the black hole, but continue beyond. In these maps, black holes become wormholes, and new
universes lie on the other side.
Cartographers grid up the surface of the Earth in lines of longitude and latitude so that
every point on the planet can be clearly defined with two numbers. Everywhere but at the north
and south poles, that is. There, all lines of longitude merge, and all directions become
south. We call these points coordinate singularities. A singularity is where a variable in the equation
becomes infinite - and a single swivel at a pole carries you through the longitudes
at an infinite rate. The coordinate singularity of the poles can be banished by shifting the
spherical coordinate system used to grid the earth, or by changing the coordinate system
entirely - for example, you can expand the distance between lines of longitude as you
approach the pole so that those lines don’t converge at all. Unroll the resulting cylinder
and you have the Mercator projection - a perfectly useful map for plotting your path, as long
as you remember that Greenland isn’t really larger than south america.
To map the universe we need 3 dimensions of space instead of two, plus the dimension of
time. And maps of the universe in this 4-dimensional spacetime also have coordinate singularities
- for example around the black hole.
Our first map of the spacetime of a black hole was the Schwarzschild metric -
a relatively simple bit of algebra derived by Karl Schwarzschild just a couple of months
after Einstein published his general theory of relativity. It allows us to calculate the
path of an object moving in the insane gravitational field approaching a black hole. It even works
inside the black hole - beneath the inescapable event horizon. Although it works in both these
regions, the Schwarzschild metric canNOT be used to plot a trajectory that actually crosses
the event horizon. That’s because at the event horizon, time appears to freeze from
the point of view of a distant observer. And the Schwarzschild metric is defined in terms
of that observer’s units of space and time. So if they try to trace a path across the
horizon in terms of their own clock, the moment of crossing never happens. It’s like Achilles
chasing the tortoise in Zeno’s paradox - Achilles covers half the remaining distance at each
step, and so never closes the gap.
Of course Achilles WOULD actually catch the tortoise, and a plummeting cartographer would
fall through the event horizon. The event horizon is just a coordinate singularity like
the earth’s poles, and to make a smooth map through it we need a Mercator projection
of a black hole.
In the Mercator projection, the separation of lines of longitude are multiplied by a
factor that depends on their latitude - and that multiplication factor becomes infinite
at the poles, to cancel out the converging lines of longitude. For black holes we instead
fuse time with a something called a tortoise coordinate, after Zeno’s paradox. It’s
a measure of distance that becomes infinitesimally compact approaching the horizon. That compactification
cancels out the infinite stretching of time so that gridlines pass smoothly across the
event horizon.
The first such scheme was Eddington-Finkelstein coordinates, and they revealed that the singularity
of the event horizon was an illusion. Kruskal–Szekeres coordinates improved our map by enforcing
that the trajectory of light always be at a 45 degree angle. In the resulting Kruskal–Szekeres
diagram, the event horizon becomes is also a 45 degree line, even though it actually
has a constant physical size.
Because nothing can travel faster than light, this makes very clear what parts of the universe
are accessible. Close to the event horizon, even a light-speed path has only a narrow
window of escape. Once inside the event horizon, no such window remains.
These days Penrose coordinates are even more popular for intergalactic travelers. On Penrose
diagrams, space and time also bunch up at infinite distance so tha t the entire universe
fits on the one diagram. Well, the whole universe? Not quite.
In the Mercator projection, we know that lines of longitude and latitude don’t just end
at the edge of the page - they loop.
General relativity uses null geodesics - the paths taken by light rays - to grid spacetime,
and we also assume that those lines don’t just end. There’s no abrupt edge to spacetime
flapping in the wind. The only place geodesics end is at true singularities, like at the
center of the black hole. On our Penrose diagram, we see that light rays can either travel away
from the black hole to infinite distance, or they can travel towards the center of the
black hole and be lost. That’s all fine. We also see that light rays can come in from
far away towards the black hole - no problem there. But what about light rays going in
the other direction? These don’t have a sensible point of origin.
We say this Penrose diagram is geodesically incomplete because there are light rays with
undefined origins. This is equivalent to saying that we have not explored the full range of
the Penrose coordinates within the Schwarzschild description of the black hole. If we trace
those coordinates to their full extent, we get what we call a maximally extended Schwarzschild
solution - and it reveals strange new regions on the Penrose diagram.
If we trace our light ray backwards from our universe we encounter a region that looks
just like the black hole - but with time reversed. This is the white hole, and we’ve this before
- but perhaps we’ll get a little more insight into them today. Today we’re going to explore
an even stranger region. This c orner. The region defined by tracing right-moving light
rays backwards from within the black hole. In our Penrose or Kruskal–Szekeres coordinates
this region LOOKS like our universe. In fact it looks like a mirror-reflected version of
our universe, at least in terms of the coordinates of space and time.
Questions abound: is this parallel universe real? Can we get there? Well before I go on,
I should say - the map I just drew is for the case of an eternal Schwarzschild black
hole. One whose coordinates do not change over time, implying that it always existed.
We’ll see later how things change in the case of a black hole born of the collapse
of a star.
For now, let’s see if we can travel to the parallel universe of the eternal black hole.
The only way to pass between these universes is to travel faster than light. You can see
that by the fact that the only paths shallower than 45 degrees can pass between the universes.
But imagine you could travel at infinite speed - then you could take these horizontal which
would dip beneath the event horizons and emerge in the mirror universe. You’ve just traversed
an Einstein-Rosen bridge - a wormhole. We’ll come back to the detailed physics of wormholes
another time - today we’re interested in what that journey can tell us about the parallel
universe on the other side.
Let’s say you drop into a black hole to try to get to the other side. Within the black
hole, space and time have switched roles. These lines represent steps towards the central
singularity - it’s the old radial direction, but now flows only in one direction - to your
crushing demise. These lines are the old time dimension, but now traversable in both directions.
Once inside the black hole what do you see? Light can reach you from the universe behind
- those are photons that overtake you heading towards the central singularity. Light can
also reach you from below - that’s light from anything that fell in before you. It’s
trying to escape and will ultimately fail, dragged down by the cascading fabric of space.
But for now you overtake that light and get a glimpse of the black hole’s past. You
never actually see the singularity - that is manifest as an inevitable crushing future,
in which the space around you becomes infinitely curved.
So what do you do? You can turn around and try go back the way you came - and if you
can travel faster than light you’ll emerge from the same event horizon that swallowed
you. Or you can plunge faster than light towards the light coming from below - that means going
this way. Against intuition, traveling faster than light in that direction doesn’t get
you to the singularity more quickly - instead you’re ejected through the parallel horizon
into the parallel universe. Within the black hole, you see an event horizon both behind
AND ahead of you. But only superluminal speeds will get you to either.
Assuming you could reach the parallel universe, what would you see? This is where opinion
is divided. Some think that the parallel universe AND the white hole are just coordinate reflections
of the regular universe and black hole - that they don’t have an independent existence.
Just as with the Mercator projection, traveling off the edge of the Schwarzschild spacetime
brings you back somewhere else in the same spacetime. Exactly where depends on how that
reflection works. Perhaps you emerge from the past “white hole” traveling forward
in time, or from the future black hole but traveling backwards in time. Which would just
look like falling into the black hole to someone who themselves are moving forwards in time.
Confusing. And it’s okay that this doesn’t make much sense - faster than light travel
always leads to silly paradoxes because it’s impossible.
Not only is faster than light travel impossible, but eternal black holes don’t exist either.
The parallel universe and white hole are needed in the map of the eternal Schwarzschild black
hole in order for geodesics to have somewhere to come from. But real black holes form from
collapsing stars - there’s no white hole in their past. And within those black holes,
any outgoing light ray can be traced back to the surface of the collapsing star and
to its interior.
Even though the parallel universe of the Schwazschild black hole isn’t likely to be real, there
are intriguing possibilities. That Einstein-Rosen bridge can potentially be made to lead to
different parts of THIS universe, and could be traversed if it could be pried open. That’s
a huge if, but it would allow instant travel between distant locations. And in the case
of rotating black holes, the traversable wormhole and even the parallel universe are not so
easy to dismiss as is the Schwarzschild black hole. In fact we’ll soon follow a sublightspeed
path through a Kerr black holes into parallel regions of spacetime.
Hey Everyone. Welcome to the new Space Time studio aka my apartment because New York is
in lockdown, but as you can see we are doing everything we can to keep bringing you Space
Time every week. Okay today we’re doing comments for the last two episodes which are
on rotating black holes and quantum darwinism. Let’s see what you had to say.
Eddy Mich always thought entanglement could only occur between two particles. What you're
thinking about is the principle of monogamy of entanglement, which states that a given
quantum state can only be maximally entangled with one other quantum state. The key word
here is "maximal". As an example, the spin of a newly created electron-positron pair
are perfectly correlated as being in opposite directions. Measure the spin of one and you
know with perfect certainty the spin of the other. That's maximal entanglement for the
spin state. If the electron then interacts with, say, a photon so that the photon and
electron spin states are entangled, then that will either completely break the original
entanglement or at least introduce some level of undertainty in the correlation. So entanglement
can be completely tranfered, or it can be shared among many particles with reduced strength
to the correlation between any two particles.
SaintCergue gives a nice analogy to describe this idea: the macroscopic "world" is a filter
function which selects quantum states immune to entanglement diffusion. So that's almost
it, but actually ALL entanglement gets diffused. The "filter function" instead causes quantum
states to become correlated with macroscopic observables. The entanglement is dispersed
through the environment, but in a coherent way that enables us to infer the intitial
quantum state.
John Cramer wants to know how big an explosion you can get from a black hole bomb, and kindly
requests a whole video. Well for the video I'm going to directly you to Kurzgesagt, who
have some more details and better animations. But how big is the bomb? Well the limit is
as much rotational energy as the black hole contains. for a maximally rotating black hole
that's 40% of its total mass - all released as energy. To put that in context, think of
any ridiculously energetic process - now multiply that by 10 to the power of some stupidly large
number. Impressive, right?
Cliff86 points out that the way quantum states become increasingly entangled with their environment
seems analogous to the second law of thermodynamics - aka that entropy must increase over time.
Nice observation - but it's not just analogous - entropy and entanglement seem to be fundamentally
connected. I've been meaning to get around to a whole episode on quantum entropy - aka
von neumann entropy, and how it relates to entopy. We'll get there.
Yuval Nehemia points out that Wojciech Zurek looks like the physicist version of Bob Ross.
That was the first thing I thought. Happy little entanglements. Stay safe everyone.