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