Kind: captions
Language: en
Imagine you're an astronaut out drifting
in deep space. When you throw a rock as
hard as you can, what's going to happen
to that rock? Well, you would think that
it would continue with constant velocity
in a straight line. That's just Newton's
first law. But what actually happens is
it eventually slows down and
stops. So why does this happen? Where
did all the rocks energy
go? At the turn of the 20th century, the
problem of energy conservation baffled
some of the greatest minds, including
Albert Einstein. Einstein came up with a
possible
solution. But then a littleknown unpaid
mathematician named Emmy Nutter proved
he was wrong. And in doing so, she
created a whole new paradigm for
physics. One that underlies all of
particle physics and explains why
anything is conserved.
It all started in 1915 at the University
of Gingan where Einstein was giving six
lectures on his new theory of gravity,
what would become the general theory of
relativity. The lectures were
wellreceived, but Einstein hadn't yet
settled on the final form of the field
equations. One problem he was facing was
how to show that total energy was
conserved in his new theory. This is the
whole beginning of this story, right?
Classically they thought they had this
understanding of what the energy of a
gravitational field was. All of a sudden
with these new equations they go where
is it? You know is it in the curvature?
You know is it in the stress energy
tensure? Where's the term that we're
looking for? Einstein suggested that the
principle of conservation of energy long
established as a bedrock of physics
might hold the key to working out the
correct field equations. In the
audience, legendary mathematician David
Hilbert was intrigued. So he started to
look for the energy conservation
equations in Einstein's new theory. But
the best he could find was a set of
equations known as the Bianke
identities. They showed that energy was
conserved but only in a completely empty
universe. So for one like ours filled
with stuff, they seemed
useless. Hilbert was
stumped. Fortunately, he knew just the
person for the job. his new assistant
Emmy Nutter.
From an early age, Nutter had dreamed of
following in the footsteps of her
father, a mathematics professor at the
University of Erlingan. She got special
permission to attend lectures at the
university, but they refused to admit
her as an official student. The Erlingan
academic senate held that the admission
of women would overthrow all academic
order. So, in 1903, she spent a semester
at Goodinkan instead. There she learned
about a new way to approach geometry
using
symmetry. Symmetry is one of those ideas
that's easy to recognize but harder to
describe. If I align a mirror on this
triangle like so, then it looks the same
as without the mirror. And that's
because this is an axis of symmetry.
Reflections about this axis leave the
triangle unchanged. And the same thing
happens if I put the mirror like this or
if I orient the mirror like this. So
this triangle has three axis of
symmetry. Now mathematicians generalize
the idea of symmetry further to be any
action you can take that leaves an
object unchanged. So something else I
could do is I could rotate this triangle
by 120° or by 240° or by
360°. Together, these six actions
capture all the symmetries of the
equilateral triangle. But you can also
have more abstract symmetries. For
example, with a mathematical function.
If I shift this function up or down by
some constant amount, call it a, then
all of its y values will change. But if
I differentiate that function, I get the
slope and that remains unchanged
regardless of whatever constant is
added. So you can add any constant to
this function and its derivative always
stays the same. So there is a kind of
translation symmetry and unlike the
symmetries of the triangle this is a
continuous symmetry meaning you can
shift it by any amount you
like. Over the next 12 years Nutter
became a leading expert on symmetry. She
became only the second woman in Germany
to earn a PhD in mathematics and she
used this expertise to help Hilbert and
Einstein with their problem of energy
conservation. The issue had bothered
Einstein so much that he proposed a new
conservation
equation. It said that if we add
together the energy of matter and the
energy of the gravitational field then
that total remains constant. Its change
over time and space is zero.
But when Nutter saw this, she was
convinced Einstein had made a
fundamental mistake because this
equation disregards the foundational
principle that general relativity is
built
on. 10 years earlier in 1905, Einstein
had introduced his special theory of
relativity and it was built on the idea
that the laws of physics were
independent of your frame of reference.
But so far, Einstein had only applied
this principle to inertial frames of
reference. Those are frames that move at
a constant speed. He began wondering
what would it take to generalize that to
consider more general states of motion.
After all, trains on the platform. He
loved trains. Trains would speed up or
slow down. People, you know, moving
around the world don't only move at a at
a single constant speed forever. In
1907, he wrote, "Is it conceivable that
the principle of relativity also applies
to systems that are accelerated relative
to each other?" This made him wonder
perhaps his principle could also be
applied to accelerating and rotating
frames, frames that move in any way in
general. That's the general in general
relativity. So Einstein got to work on
this largely intellectual
pursuit. But then as he was daydreaming
in the patent office, he had what he
called the happiest thought of his life.
He imagined the window cleaner at the
top of the opposite building falling
off.
And Einstein realized that while the man
was falling, he wouldn't feel his own
weight. He would be weightless and
anything he dropped on his way down
would remain stationary relative to him.
It would be just as if he was floating
in outer space. It's ironic. We would
have said, "Oh, he's being pulled down
to the ground because the gravity of the
earth is is exerting a force." But
Einstein said while that person is in
motion, what we call freef fall motion,
they would actually feel no gravity at
all. There must be some equivalence
between accelerated motion and the
action of gravity. So Einstein arrived
at what he called the equivalence
principle. If you were stuck in a rocket
in outer space accelerating at 9.8 m/s
squared, then it would be the exact same
as if you were standing on the surface
of Earth.
And this was huge because it meant that
if Einstein could figure out how to
understand accelerating frames, then he
didn't just get a more general theory,
he would also have a new theory of
gravity. But to achieve this, Einstein
needed to make sure the laws of gravity
had the same form in every frame of
reference. This is the idea of general
coariance, and it's one of the core
tenants of general relativity. To
satisfy it, Einstein knew he had to use
special mathematical objects called
tensors. A simple kind of tensor is a
vector. You can write a vector as a set
of components multiplied by their basis
vectors. For example, this vector can be
written as 3xhat + 2 yhat. But I can
also write this using a different
coordinate system. And with these new
basis vectors, the original vector is
now written as 2 a + 1 b. So the
components, the numbers in this list
changed, but the vector didn't. It
stayed the same. And that's because when
the basis vectors change, the components
adjust in a complimentary way to keep
the vector the same. The vector itself
is independent of which coordinate
system you use. And the same is true for
tensors. Only now instead of having just
two components, a general tensor can
have any number of them in the form of a
matrix. And just like with vectors, you
can change a tensor from one coordinate
system to another and the tensor stays
the same. So that's why Einstein had to
use them to build his new
theory. And that was exactly the problem
that Ner found because when she looked
at Einstein's proposed energy
conservation equation, it contained a
pseudo tensor. And as the name implies,
that isn't quite a tensor. when you try
to transform it from one frame of
reference to another, it doesn't remain
the same quantity in different frames.
The gravitational energy you might
observe in one frame completely
disappears in another. And Einstein, you
know, he had some strange thoughts about
this. I mean, people were trying to
stamp conservation of energy into
relativity by um bending the rules of
mathematics.
So Nutter knew that Einstein's proposed
solution couldn't be the answer and that
made her think what if general coariance
and energy conservation are simply
incompatible and if that's the case then
why general coariance says that laws of
physics must stay the same when you
change reference frames. So that is a
kind of symmetry exactly what Nutter had
spent her career studying. So she
started thinking about the symmetries of
the universe. Beginning with the
simplest possible case, an empty static
universe. Imagine you're an astronaut in
this universe. Since it's empty, there
is nothing special about any particular
point. I mean, it doesn't matter if
you're over here or over there. The
universe is completely symmetric under
translations in space. So suppose you
throw a ball. Well, it'll travel at a
given speed, and after a short amount of
time, it will have traveled some
distance, but since the laws of physics
are the same here as just before, we can
shift the whole universe, and we're back
to the situation we started with. And we
can keep doing this over and over. And
this shows us that the object will
continue with that same speed
indefinitely.
So what we've discovered is that the
principle of conservation of momentum is
a direct result of the fact that there's
a translation symmetry in the universe.
That an experiment done in one spot will
give identical results to that same
experiment done somewhere else. You
could move everything from one place to
another and the physics won't change.
Similarly, the laws of physics don't
depend on whether you perform an
experiment like this or rotate
everything by 90°.
This universe is symmetric under
rotations. So imagine we take a metal
rod and spin it. If we let it rotate for
a minute, then it will have moved
through a small angle. But we can rotate
the whole universe back by that same
angle. And now we're at the starting
position again. And we can keep doing
this so that each instant looks exactly
the same as the one before, which means
the object will keep rotating this way
indefinitely. So the law of conservation
of angular momentum comes from the
rotational symmetry of the
universe. Now another important symmetry
of this universe is time symmetry. The
laws of physics don't change over time.
If you do an experiment today or
tomorrow, you will get the same result.
So what does this symmetry lead to?
Well, to understand this, we're going to
dig into some math and a different way
of doing mechanics using the principle
of least action. Previously on
Veritassium, we learned that everything
always follows the path that minimizes a
quantity known as the action. This is
equivalent to the integral of the
lrangeian L over time. In the simplest
case, that's just the kinetic minus
potential energy. Oiler and Lrangee
found that the principle of least action
is obeyed so long as this set of
differential equations is satisfied. So
Ner used action to see how physics was
affected by different
symmetries. So suppose we do an
experiment where the result is the same
now as some tiny time interval epsilon
later. Then how does this affect the
action? Well, the time is going to
change from just t to t + epsilon. And
as a result the lranchin is also going
to change. So the new lrunchen will be l
prime which is equal to the old lrunchen
plus how much the lrunchion changes over
time. That's just dl by dt multiplied by
how long that change lasts. So
multiplied by
epsilon. But now also remember that the
result is going to be the exact same now
as a little while later. which means
that whatever this term is the dl over
dt doesn't affect the equations of
motion and it's from this symmetry in
the action that we're going to be able
to find the conserved quantity. So let's
take dlddt and rewrite it using the
chain rule. That gives us the partial
derivative of l with respect to x * dx /
dt plus the partial derivative of l with
respect to v * dv / dt. But we can sub
in the partial derivative of L with
respect to x with this term from the
oiler lranch equation. And we can
simplify this further by writing dx over
dt as v. And that gives us this
expression. And now notice what we've
got right here. We've got the time
derivative of some function d / dv times
another function plus that first
function times the time derivative of
the second function. So we can use the
reverse of the product rule to simplify
this to the time derivative of dl over
dv * v. Then as a final step we can
bring dl / dt to the right. So what we
found is that if you take the time
derivative of this quantity it's equal
to zero which means that whatever this
is has to be a constant. So what is it?
Well, remember that in the simplest case
the lrunchen is just equal to the
kinetic minus potential energy which we
can write as 12 mv ^2 minus v. So if we
take the partial derivative of the
lranchinion with respect to v we're just
going to get d /
dt m * v multiplied by v. So this is
going to become mv ^ 2 and then we can
sub in the lrunchion. So this becomes
minus 12 mv ^ 2 minus v but also minus
here. So this becomes plus v and all of
that's equal to zero which we can
simplify to just 12 mv
^2 + v is equal to zero. But wait a
second because this is just a total
energy. So what we've discovered is that
time translation symmetry is equivalent
to saying that energy is
conserved. The principle of conservation
of energy is a direct consequence of
time translation
symmetry. In a theorem, Nutter proved
that all of these examples are no
coincidence. For centuries, people had
no idea where conservation laws came
from. But now Nutter had discovered the
origin of all of them. She proved that
anytime you have a continuous symmetry,
you get a corresponding conservation
law. Translational symmetry gives you
conservation of momentum. Rotational
symmetry gives you conservation of
angular momentum. And time translation
symmetry gives you conservation of
energy. But these are all symmetries of
a static empty
universe. The universe we live in is
very different.
In the 1920s, astronomers measured the
velocities of distant galaxies, and they
realized all of them are moving away
from us. The farther away they are, the
faster they're moving. The implication
was clear. In the distant past,
everything must have been much closer
together. In the 1990s, precise
measurements of supernova revealed that
not only was the universe expanding, but
that expansion was speeding up.
This means over large time scales, our
universe is not symmetric in time. It
was very different 13 billion years ago
and it'll be different billions of years
from
now. Since we don't have time
symmetry, that also means energy as we
usually think of it isn't conserved.
There's no reason for energy to be
conserved anymore cuz you don't have
that symmetry. Think about a photon of
visible light emitted 380,000 years
after the Big Bang. It travels through
the universe unimpeded to arrive at our
telescopes not as visible light but as a
microwave. It has lost 99.9% of its
energy. Where did the energy
go? Doesn't go anywhere. Energy is not
conserved. And this is exactly what's
happening to the rock as well. It starts
off with energy but as it travels
through the expanding universe it slows
down and
stops. The energy doesn't really go
anywhere. It just
disappears. It ends up coming to rest
with regards to the other particles in
the universe. This doesn't violate any
laws of physics because energy and
momentum aren't conserved if there is no
time or spatial symmetry. So once you
you know that symmetries give you
conservation laws and so once you those
symmetries are gone, you don't have to
worry about those conservation laws
anymore. then you can start dropping
these concepts of trying to force
something that you uh want to say is
fundamental into the theory and you just
deal with what the theory gives you. But
if energy isn't conserved in our
universe, then why does it usually seem
like it
is? That's because when you're looking
at the short time scales that we're used
to, time translation symmetry pretty
much holds. An experiment done today
will give the same results as the same
experiment done tomorrow. So that means
for all intents and purposes energy is
conserved. But over large time scales on
the order of millions of years, well
then the expansion of the universe can't
be neglected and the symmetry is broken.
So only when you look at time scales
that big do you notice that energy isn't
conserved. Nutter's first theorem
explains why a rock or a photon loses
energy, but it didn't fully solve the
problem of energy conservation in
general relativity. See, so far nutter
had only dealt with an empty universe
where you could shift the whole universe
and the laws of physics would stay the
same. But this doesn't work in general
relativity where the curvature can
change from one point to another. Now if
you shift the whole universe, rotate it
or let it evolve in time, things don't
stay exactly the same. So you no longer
have these global symmetries. But Nutter
realized there are still other
symmetries left.
See, no matter how you're moving, the
laws of physics always look the same.
That's general coariance. And it is a
kind of symmetry that holds everywhere.
It means that in any small region, we
can always change our frame of
reference. We can transform the points
of space around as much as we like. And
since these transformations aren't
global but local, these are called local
symmetries. In a second theorem, Nutter
proved that for these local symmetries,
you no longer get proper conservation
laws like we're used to in classical
physics. Instead, you get something that
only works locally, a continuity
equation. One example of a continuity
equation describes the flow of water
through a pipe. This first term tells
you how the amount of water changes in a
section of the pipe. And the second
tells you the difference between how
much water is flowing out and how much
is flowing in.
In this case, the first term is positive
because the water level in this pipe
section is increasing and the second
term will be negative because less water
is flowing out of the section than in.
Together, the two terms cancel to give
zero, which guarantees that no water is
created or destroyed. If the total
amount of water changes in a section,
there must either be excess water
flowing in or out. In the case of
general relativity, Nutter found a
similar continuity equation, but with an
important difference. Imagine that now
our pipes are little patches of spaceime
and the water is energy flowing from one
patch to another. In any individual
section, the continuity equation looks
exactly the same as before. So that in
any small region of spacetime, energy is
conserved. But when we link these
sections together, we need to take into
account the curvature of spacetime. And
this changes the equation. Now it's as
if there are little cracks appearing
between different sections of pipe,
between the local patches of spaceime.
And through those cracks, energy can
leak out. In special activity, the pipe
is imperturbable because the pipe is
fixed. And in general activity, you
know, we have to account for the energy
that goes into other kinds of change
over time. And that gets correspondingly
more tricky. Now that we have this new
equation, we can see how it works by
expanding it as a sum of different
terms. This first term is analogous to
the continuity equation from before. The
one which conserves energy within a
local patch of spaceime. But now we have
all these extra terms. These describe
the curvature of spaceime. So as energy
decreases in the first term, these
curvature terms increase. The energy
that you lose from the system you're
tracking, we now start attributing it to
things like the gravitational field,
which has changed because the whole
universe is stretched. We have to
account for the energy that we attribute
to the action of the gravitational field
as well because space and time
themselves aren't sitting still. And all
of this can be described by the
continuity equation Nutter had found.
But when she looked at it, she realized
something. It was exactly equivalent to
the Bianke identities, the half solution
Hilbert had found. He had dismissed it
because it only gave you proper energy
conservation in an empty universe. But
now Nutter proved that it was the best
you could do in general
relativity. With one paper, she had
uncovered the source of all conservation
laws and she had solved the problem in
general relativity that eluded Hilbert
and Einstein. She was so amazing. I
mean, I would go out on a limb and I
would say these two theorems are
probably the most important theorems for
physics of the 20th century. In the
following years, the University of
Gingan took steps to make Nutter's
position more official, allowing her to
do what she loved most, to teach. They
made her a professor, and she even got a
small salary starting in
1923. But all of that changed on the
30th of January 1933 when Hitler became
chancellor of Germany. The Nazis banned
Jewish people from working at
universities and almost immediately one
of her former students told the
authorities of her Jewish heritage and
she was
suspended. Despite this dismissal, she
continued teaching in the kitchen of her
home. Then one day, one of her old
students knocked on her
door. Clothed in the brown shirt of the
Nazi stormtroopers, Nutter let him in.
He had come to learn math and Nutter was
happy to teach him. I love what this
sort of shows about Nutter. You know,
she truly deeply cared about math and
she wouldn't discriminate whether
someone was wearing a Nazi shirt or not.
She taught all. But staying in Germany
became untenable. Fortunately, with the
help of other academics, she managed to
obtain a teaching position at Brin Ma, a
woman's college in America, where she
would teach until her
death. In an obituary for the New York
Times, Einstein wrote that Frowline
Nutter was the most significant creative
mathematical genius thus far produced
since the higher education of women
began.
The reason that Nurther's theorem is so
important is that everybody just changed
their state of mind. All of a sudden the
physicists were thinking about physics
in terms of these symmetries.
Physicists started applying these ideas
to the quantum world too. Realizing that
charged particles like electrons also
have symmetries. Electrons have a phase
which you can think of as an arrow
pointing in some direction. But you can
offset this phase by any arbitrary
amount so long as you do it
simultaneously for all electrons. And
that doesn't change anything physically.
So there's another symmetry. So what
does this offset or gauge symmetry lead
to? Well, it leads to the conservation
of electric
charge. In the 1960s and '7s, Nutter's
insights led directly to the discovery
of new fundamental particles like quarks
and the Higs Bzon. It taught us where
the forces of nature come from, and it
even helped to explain the origin of all
mass in the universe. Not's two
theorems, although little known, are
what has gotten us the closest we've
ever come to a theory of everything. But
all of this and much more will be
covered in a second video. So, make sure
you're subscribed to get notified when
that video comes
out. When Emmy Nutter set out to study
mathematics, she was following in her
father's footsteps. But from there, she
forged her own path. And before long she
was coming up with a whole theory that
reshaped our understanding of the
universe. That is the great thing about
learning. You start by following
instructions, building on what others
have done before you. But then at some
point you start asking your own
questions. You start experimenting and
making discoveries. I've loved watching
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