The Biggest Misconception in Physics

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 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 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 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 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

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

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

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 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

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.

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