[MUSIC PLAYING]
In "The Legend of Zelda: Majora's Mask,"
Link needs to save Termina from an impending collision
with that planet's malevolent moon.
But what if I told you that the so-called moon is really
a black hole?
[THEME MUSIC]
"Majora's Mask" is probably the darkest game
in the "Zelda" series.
For the unfamiliar, here's a plot recap
to bring you up to speed.
Link arrives in Termina, a province
on a planet whose moon is on a three-day collision
course with the surface.
Now, the planet doesn't seem to have a name, so I'll just refer
to the whole place as Termina.
The approaching moon causes all kinds of havoc.
It has a crazy face, completely with teeth.
And in true "Zelda" fashion, it can only be stopped by Link.
So clearly "Majora's Mask" has some fantastical and
unrealistic elements.
But that doesn't mean that it can't
serve as a springboard to talk about some really
interesting physics.
In fact, the basic physics of "Majora's Mask"
has been analyzed before, both by a thread on Reddit
and by MatPat on an episode of "Game Theorists."
But both of them overlooked a detail
that punches a major hole in their conclusions--
and by hole, I mean black hole.
Today I'm going to argue that the moon in "Majora's Mask"
is not a moon at all, but instead
of a rocky shell surrounding a miniature black hole.
Now before I justify this bold claim,
let me clarify that I'm not dissing on either that Reddit
poster or on MatPat.
Both analyses are extremely well done,
and I'm actually going to rely on MatPat's measurements
in this episode.
As we'll see, the oversight in question is subtle.
I even made it myself at first.
The reason I'm doing this episode at all
is that dissecting that oversight
gives us a chance to explore some often unappreciated
features of how gravity works.
And it gives us a chance to talk about miniature black holes,
which I'm super pumped to have discovered
lurking in a "Zelda" game.
So let's get started.
The root of the problem with both these other analyses
is that they assume Termina's moon has
the same density as our moon, and that's impossible.
For reasons I'll explain shortly,
anything with the density of Earth's moon
would be ripped apart if it got within 12,000
kilometers of Earth's surface.
The same would be true on Termina,
no matter what non-absurd radius and mass we
assign to that planet.
So getting this close with only that density
is simply out of the question.
OK, so how dense is Termina's moon?
Fortunately for us, the game offers an indirect way
to estimate its value.
The key evidence appears in the scenes
just before the moon hits, when loose rocks start flying upward
off the surface of the planet.
You might think this implies that the gravitational pull
on the rocks due to the moon exceeds the gravitational pull
from the planet itself, but that's not correct.
What has to beat the planet's gravity, in this case,
to make the rocks levitate is something
called the moon's tidal force.
This requires some explanation.
Rocks on the surface of Termina are
being pulled towards the moon.
But the planet is also being pulled towards the moon,
chasing after those rocks.
Since the rocks are closer to the moon, where the moon's
gravity is stronger, they do accelerate towards the moon
slightly more quickly than the rest of the planet does.
Now, on its own, this differential acceleration
tends to separate rocks from the planet,
even on the side of the planet opposite the moon.
I mean, think about it.
Rocks on the far side of the planet
accelerate toward the moon even less
quickly than the center of the planet does,
so they would also have a tendency
to leave the planet's surface.
From Termina's point of view, then, the moon's tidal force
manifests itself as an outward push off
the planet's surface, at least along the planet-moon line,
kind of like antigravity.
Ordinarily, though, that outward push
is tiny, so tiny that a planet's own gravity is more than
enough to hold things in place.
For instance, the tidal force from Earth's moon on you
right now it's about 10 million times smaller
than Earth's pull on you.
That's why you don't levitate during a full moon.
Of course, if you bring the moon closer, on Earth or on Termina,
the tidal force from the moon will increase.
The question is, will it be enough to lift rocks?
On Earth, absolutely not.
But what about on Termina?
I've worked out the algebra using that MatPat's measurement
of a mere 30-ish meters for the radius of Termina's moon.
And I put some lower limits on the radius of the planet,
based on the fact that we don't see its horizon
curving in the distance.
You can click over here to see the details,
but here's the bottom line.
If Termina's moon really were only as dense as Earth's moon,
its tidal force when hovering just above the planet's surface
would be 200,000 times smaller than the moon's
current measly tidal force on you.
At worst, it would raise the ocean levels on Termina
by 1/100 of a millimeter.
There's no way it would cause earthquakes,
and it certainly wouldn't cause rocks to levitate.
But in the game, the rocks do levitate.
And that requires Termina's moon to be--
wait for it-- between a billion and 100 trillion times denser
than Earth's moon, depending on whether you think
the tidal force is just whipping the rocks upward
with hurricane-force winds or actually lifting them directly.
This high density would help explain why Termina's moon is
even still in one piece.
After all, the planet also exerts a tidal force
on the moon, which would rip Earth's moon apart
if it got too close to us.
So a trillion times the density of Earth's
moon-- that's ridiculous.
It's beyond insane.
It's the equivalent of cramming 20 million African elephants
into a thimble.
The only solid macroscopic objects
in nature that are that dense are neutron stars.
But here's the problem.
Those objects could only get that dense
in the first place because of their immense self-gravity.
Even if you imagine some esoteric, unnatural way
to synthesize a neutron star, theoretical estimates
say that you'd still need at least 1/10 of the sun's mass,
give or take, to make one.
Termina's moon, even at these densities,
only weighs about half as much of Earth's moon.
Granted, that's a quadrillion times more
massive than MatPat's estimate, but still millions of times
less than what you'd need to make a pure neutron star.
So if it's not a neutron star, how
can Majora's moon be so dense?
Well, maybe it's neutron star material, but only
in the outer layers.
If you could somehow put a much denser mass at the center,
the material around it might have
enough of a gravitational scaffolding, so
to speak, to achieve neutron star
densities with less mass than would normally be required.
So to serve as a gravitational seed,
we need something incredibly small and incredibly dense,
even denser than a neutron star.
OK, what about condensing some of the necessary mass
into a black hole?
That black hole would end up having a radius
less than a millimeter.
Leave some empty space around it for good measure,
and this might work.
How could you ever make such a tiny black hole?
Well, Hawking radiation, for one.
You could start with a more massive and larger black hole
and just let it evaporate slowly.
It would lose mass and size during the process.
And about a gazillion years later,
poof-- moon-ish mass, submillimeter-size black hole.
Piece of cake.
So there's my argument.
The moon in "Majora's Mask" is basically
some super-dense crust with teeth
and with a mini black hole in the middle.
Once you factor in tidal forces, I
don't see how you avoid this conclusion.
Granted, there are some loose ends.
For example, gravity would be billions of times
stronger on Termina's moon than on Earth.
So how does Link get inside and survive the trip?
Look, dude, I'm just a man, OK?
I don't have all the answers.
I just wanted to add a new twist to an old conversation
about "Majora's Mask."
Maybe I also made a booboo.
If I did, I'm sure you Space-Timers will let me know.
You know what to do.
Dig into the comments or to Reddit and discuss.
I'll report any interesting questions
you come up with or conclusions you
find on the next episode of "Space Time."
Last week we talked about why the cosmic microwave background
once made all of space orange.
We're going to talk about that, but first,
a quick announcement.
On April 11 and 12, I will be hosting
part of NASA's International Space Apps
Challenge, a global, multi-city two-day hackathon where
people, including students, can get together
to collaborate on coming up with innovative uses for publicly
available data.
I will be hosting at the main stage in New York City,
along with several astronauts.
For more information and to register to participate,
go to spaceappschallenges.org.
And if you can't participate in person,
you can do so virtually, and it will also be live-streamed.
Now, back to why space used to be orange.
KnowledgePlaylists asks, isn't the color space Cosmic Latte?
You're referring to a recent episode
of "It's OK To Be Smart" where Joe Hanson explained
that the average color of all visible stars in the universe
comes out to an off-white called Cosmic Latte.
The CMB is something else.
Kieron George and Sergio Garza were
discussing whether water could have
existed when space went dark.
No, but not because of temperature or pressure issues.
The problem is that there was no oxygen.
Oxygen gets fused in the cores of stars.
And as far as I know, the first stars
didn't start forming until a couple hundred million years
or so after the sky went dark.
KWeightthree said that I'm just too handsy on camera.
What?
Tharks asks whether given enough time,
the CMB will eventually redshift into the FM radio band.
As far as I know, unless I'm missing something obvious, yes.
Gareth Dean pointed out that the early universe
contains not just hydrogen but also helium,
and asks whether CMB analysis takes this into account.
In more detailed treatments, absolutely.
Look up Recombination on Wikipedia.
Now what imprint that leaves on the CMB,
I don't know off the top of my head,
or even whether it's detectable.
But it can't leave a zero signature.
SD Marlow asked how the CMB can look the same
from all directions if we're not sitting
at the center of the universe.
Well, there is no center to the universe.
The CMB was emitted from all points of the universe
simultaneously.
With each passing moment of time,
any observer sitting anywhere will
see photons that were emitted from progressively
more distant locations.
However, that's only if you're moving along
with the overall cosmic expansion of space.
The Earth has superimposed an individual motion
through the galaxy.
And due to that individual motion,
there is in fact a Doppler shift of the CMB.
It looks bluer in the direction that we're moving and slightly
redder from behind.
All those maps that you see, the false-color maps of the CMB,
subtract this way, which is why you don't see it.
Finally, Paul Ansel emailed me a calculation
working out how much energy it would take to break
Earth apart atom by atom.
Yes, Paul, I got your email.
No, I haven't had a chance to verify the calculations yet.
But once I do, if they check out,
I'll report the answers here on "Space Time."
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