The singularity, the point of infinite density at the core of a black hole
but also so much more.
In mathematics, singularities come in wild and wonderful varieties.
The black hole itself contains more than one.
*intro*
Isaac Newton's universal law of gravitation
was an incredible insight when he figured it out
in the late sixteen hundreds.
In fact, we still use it to fly spacecraft around the solar system today.
However, it has its problems.
Lets look at the math:
Newton's equation gives you the gravitational force exerted between two masses.
m1 and m2.
They're the distance (R) apart.
Straight forward enough, that R squared in the denominator, spells trouble.
It means the force gets larger the closer the masses are to each other.
That makes sense
but what about when R when it gets really close to zero?
Then the result of the equation; the force, becomes extremely large
and is infinite when R becomes equal to zero.
That doesn't really make a lot of sense;
infinite force means infinite acceleration which means
well, physics breaks.
According to Newton's law, in order to fill that infinite gravitational acceleration
you need to get zero distance from an object's centre of mass.
That means all of that object's mass would need to be concentrated at that centre.
A single point of zero size, which means infinite density
and that of course would make it a black hole.
We often use the word "singularity" to describe hypothetically
infinitely dense core of a black hole.
But in math, the meaning of this word is much more general.
You know what? Instead of me trying to explain mathematical singularities,
how about we get a real mathematician to do this properly?
Guys, meet Kelsey Houston-Edwards of the new PBS show infinite series.
Hey Kelsey
Hey matt. Thanks for having me on!
Kelsey, the math for black holes, goes to infinity for different properties
and in different locations.
What does this mathematical weirdness tell us?
Well, mathematicians use the word Singularity pretty broadly
It's really just any point that causes problems
Commonly these problematic points are where quantities become bigger and bigger,
approaching infinity, as they do near a black hole.
Some singularities come about from your choice of reference frame or a coordinate system
An example of a frame dependent singularity that might be familiar to SpaceTime viewers
Is the event horizon of the black hole
ask you to explain here on earth the
north and south pole are examples of
coordinate singularities it's possible
to pass their time zones infinitely
quickly but only because of your choice
of spherical coordinates for the make
sense but the gravitational singularity
of the center of a black hole is a
so-called real singularity right i mean
curvature and the density are infinite
from any friend preference right and
there's no way to avoid a horrible
crushing guests just by switching
coordinate systems but the reality of
the black hole singularity may give
reason to doubt the theory that predict
such a thing
in fact it's happened many times before
from models of the movement of water to
human population growth mathematics
predict physical singularity and we've
been forced to reject the corresponding
theory
so you're saying Einstein is Rome less
for me actually Einstein himself agreed
on this point guide you should check out
culture show infinite series where she
goes into much more depth on the nature
the singularities it's about show by the
way so it's sometimes about real stuff
mathematicians are lucky being limited
by reality is so boring
so does the fact that it includes a
singularity mean there's something
fundamentally wrong with Newton's law of
gravitation well we already know the law
isn't really so Universal when the
gravitational field is too strong saying
their star or a black hole news log
gives the wrong answers and we need I'm
Stein's general theory of relativity
which is the far more complete theory of
gravity so it is general relativity
reserve news pesky singularity and no in
fact it gives us even more singularities
to understand this we need to look at
something called the swath shield metric
that's what you get when you solve the
delightfully complicated Einstein field
equations for the simple case of the
spherically symmetric mass in an
otherwise empty universe we going to
simplify it to only allow movement
directly towards or away from our
massive object in that case it looks
like this okay
that sure is some ass hey this is
space-time we can deal
actually it's really easy to see the
singularities in this equation but let
me just walk you through what it tells
us the swash altmetric allows us to
compare two points or events in
space-time around a mess of object from
the perspective of different observers
for example a short space time passed of
some object so it's worldline might move
an object a distance Delta R over short
time step delta T that motion is towards
or away from the massive object which is
a distance R away
that Delta is squared thing is the
space-time interval and it's a strange
and interesting quantity every inertial
so I accelerating observer will agree on
the same space-time interval for every
pair of events and for every worldline
we talk about this in a lot more details
from our relatively playlist today we're
going to keep it simple as long as our
objects worldline doesn't require
faster-than-light motion then the square
root of the space-time interval is equal
to the amount of time that the object
itself feels over that interval we call
that the objects proper time
oh and our subscript s is a measure of
the mass of the massive object in fact
it's 2 times the gravitational constant
times the mass there would have been
some speed of light through the equation
but we set them equal to 1 because we're
that cool
now the first thing to notice is that
the singularity is still present in a
swath shield metric are the distance to
the center of mass remained in the
denominator just as it was in use law
when you use the trash or metric to
calculate the curvature at r equals 0
that curvature is instant this gives us
the same instant gravitational pool as
Newtonian singularity and just as with
the Newtonian case its gravitational
singularity can only exist if infinite
densities are possible but unlike
Newton's law of gravity the swath shield
metric actually tells us whether or not
that infinite density is expected to see
how we need to look at the second
singularity in this equation a
singularity that Newton's law does not
contain see when distance to the center
of mass is exactly equal to this RS
theme
then RS / r is equal to one at that
point the entire equation starts
behaving very badly it's as much a
mathematical singularity as the one in
the center of the black hole if you
haven't guessed this bad behavior
corresponds to the event horizon and rs
is the SWAT shield radiance imagine an
object sitting at the event horizon but
not moving so Delta R would be 0 but
this bracket is 0 also because 1-1 the
entire space-time interval for a
non-moving point at the event horizon is
zero but remember to sublights being
worldlines the space-time interval kills
us the rate of slow of proper time so
that mean time doesn't pass for an
object hovering at the event horizon not
quite time certainly doesn't pass at the
event horizon no clock ticks can ever
happen there but the prohibition against
objects experiencing time at the event
horizon is actually a prohibition
against objects spending time at the
event horizon no temporal theme nothing
that normally experiences the passage of
time can have a space-time interval of 0
at the event horizon the only way to get
a nonzero space-time interval is to have
a non zero Delta R an object at the
event horizon has to change his distance
from the black hole to keep its clock
ticking that means falling below the
event horizon and what's inside in with
special movement continues to be the
only way to fuel the ticking of an
object proper time clock and come back
to get bit of or some witness in a
future episode
there is one thing that can have the
space-time interval 0 light actually
anything capable of traveling at light
speeds can only have a space-time
interval 0 from its perspective a search
on exist in a single instant and so it
can hang out at the event horizon which
also only exists that one instantly
stretched out instant reactive crossing
the event horizon is where the
singularity really start to behave badly
at the moment of crossing the
denominator here in this workshop metric
is 0 and the whole equation close up to
infinity but what is actually insert
here is nothing physical
it's the fact that even and outgoing
light ray takes infinite time to move
any distance so using boring old time
and distance delta T and daughter are
doesn't let us trace the worldline
smoothly across the event horizon that
horizon is a coordinate singularity just
like guilty talked about what that means
we can fix it
there are ways to construct a US based
on taxes so this singularity just
evaporates for example Eddington single
Stein tortoise coordinates to
compactified I with the stretching of
space time to cancel out the infinities
that's a bit much for right now but
google away my friends anyway the upshot
is that it's really a breeze to drop
through the event horizon physically and
mathematically of course once inside the
event horizon we still have that central
singularity to deal with
unfortunately that one can't be done
away with by simple change in
coordinates
that point of infinite density really
exist actually Einstein's theory and the
squash or solution that is derived from
it suggests it must exist the apparent
inevitability of this singularity may be
evidence that general relativity is
incomplete but you better understand why
the central infinity is unavoidable in
Einstein's theory we have to go back to
that coordinate shift at the event
horizon there's the causal roles of
space and time switch places and the
central singularity becomes not so much
a location in space but an inevitable
future actually to really get this we're
going to need another entire episode
standby to explore what happens when you
switch the causal roles of time vs.
space to space time cheers to kill
Houston edwards for helping us
understand mathematical singularities be
sure to check out the PBS infinite
series episode dealing with earthly
singularities right here and as always
we take you to our patreon supporters to
really make the space-time a lot easier
to do today a special shout-out to
Henry's and students who is supporting
us at the Big Bang level Henry join us
for our patreon google hangout where he
pretty much obliterated the entire
Whitehall hypothesis by correctly
pointing out that if they existed many
great numbers we would see them thanks
Henry for dropping the knowledge and the
dollars it's a huge help
last week we inaugurated the space-time
journal club by looking at held why
tells paper on an apparently positive
vacuum test of the e/m drives the
discussion continued with extreme
enthusiasm in the comments so we'll
definitely be doing more journal club
episodes so now let's see what you have
to say Joshua Hillary asks why the am dr
hasn't been tested more given it isn't
such a complicated experiment
well that may be true but it's still an
issue of resources
it costs a lot to do proper careful
experimental research of any type a
scientist who decided to look into this
is to devote grant money lads days
personnel and most critically a lot of
their own time and energy all of these
us gas in the world of research to warn
doing this sort of effort a given
project has to be promising the a.m
drive is not promising you shouldn't
mistake media and Internet height for
actual potential most physicists just
aren't excited about this because a it
shouldn't work and be there's no
convincing evidence that it really does
work there is the best defense of hints
or come back to that in a moment but the
amount of work being done on it
currently is equal to or greater than
what is warranted by its promise
meanwhile there are many extremely
promising new technologies to work on
including propulsion tech and we don't
even have the resources to give these
they're doing attention
terence console point out that even if
the m dr produces only a tiny thrusts
it's still interesting and i totally
agree if it really does produce
propellants trust then it's enormously
exciting that would be true even if it's
not ultimately useful for spaceflight
which I'm not saying it wouldn't be but
these sorts of huh that's odd moment are
exactly what first opened new fields of
study however it's important to remember
that 999 out of a thousand heart that's
odd moments and due to some unaccounted
factor that is totally within our
current understanding of nature we
investigate anyway because we don't want
to miss that one in a thousand but we
don't pee our pants every time now the
average person doesn't seem most of the
weird unexplainable results that
ultimately proved to be nothing
interesting scientists do
and so if they aren't getting super
excited about the e/m drive it's not
because they're closed minded it's
because they know it for what it
probably is
perhaps it's worth someone somewhere
investigating but it's not worth wet
trousers a few people suggested that the
e/m drive has been shown to work again
and again that's just not the case some
tests to find a trust or the amount of
trust isn't consistent between testers
some tests find a thrust in the wrong
direction or no trust at all
a vacuum test out of tu dresden in
Germany observed trust that found the
trust was the same even if they stood
the am dr vertically a chinese paper
from 2010 that claimed a positive test
was retracted because the authors found
that the thrust vanished when the power
system is placed inside the e/m drive
cavity
this suggested the positive result could
have been due to noise due to the
unshielded power system frankly all this
is about what i'd expect when you
generate temperature differentials and
large magnetic fields around a very
sensitive position measuring device
things are going to move around
Stephen Department asks what I just put
one of these devices in space to see if
it works well see my previous comment
about devoting resources but actually we
may have done this or at least China may
be doing it the state-owned china
academy of space technology announced in
december that it's testing the device in
orbit
they also announced positive ground
tests following the retraction of that
2010 paper but nothing new has been
published as of the making of this video
until then there's no way to assist this
but honestly orbital test don't
eliminate most of the issues of vacuum
chamber ground tips
unless i am dr actually manages to push
a satellite around that would be
convincing a few scattered he reminds us
the first rule of space-time journal
club is we talk over our thoughts and
remain open to all possible ideas and
contributions from others before forming
any solid conclusions
well said Matthew and remember you are
not your job you're not how much money
you have in the bank.
You're not the car you
drive, you are not the content of your
wallet
you are not your car keys... you are star
stuff