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So now we come to the famous equation and

of course we're referring to E equals mc squared.

You may remember that Einstein's paper on The Special Theory of Relativity in 1905,

was published in June.

A few months after that, he published a short note in September,

essentially saying, remember that paper I published a few months ago?

Well I've been thinking more about some of the things contained in it,

some of the implications, and discovered a very interesting relationship.

And, that relationship essentially was, E equals mc squared.

Now, unfortunately to really understand this, to derive it in a sense,

even in a hand waving sense.

Would require us to spend a couple of weeks on concepts of energy and

momentum and a few things like that.

So, we don't have time to do that.

But want to just review a few key things about it and

bring out a few of the key concepts as well.

And we're going to start with the concept of kinetic energy.

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All way back in the 1700s, early 1700s, scientists, they weren't

really scientists at that time, they were called natural philosophers.

Were investigating things with how objects move, and collisions and things like that.

And had identified a certain quantity that seemed to be very important.

And over time, in the 1700s and the 1800s, this quantity came to be known as kinetic

energy and had the form of one half mv squared.

Where m was a mass, so you could imagine a particle,

a tennis ball, something like that, mass.

And then v is velocity, so we're not talking relativity here,

we're just talking everyday type of velocities.

And also, along the way as this developed,

especially in the middle of the 1800s, mid 19th century,

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the idea of conservation of energy came about.

That there were different forms of energy and

you could transform one form of energy into another form of energy.

And so they developed those concepts a little bit more as well.

And one key thing about this was that it really focused on

things like kinetic energy.

And what happened when Einstein came along in 1905, and came up with this idea that

energy, of course here's the famous equation, E = mc squared.

So we see just compared to this equation for kinetic energy, mass is involved and

then there's a velocity squared involved in each case.

This is just the regular velocity of an object,

this is obviously velocity of light, so there's similarity between them there.

But really, what Einstein discovered was not quite this equation.

This is a special form of equation he really discovered.

What he discovered was, and we're going to come back to see how it relates to

conservation of energy here in a minute.

He discovered this, that energy is gamma mc squared.

Okay gamma mc squared,

in other words here's our familiar Lorentz factor coming into play here.

And if we play around with this a minute, let's just remember

back when we were talking about the Michelson Morley experiment.

We used something called the binomial expansion.

So we're going to bring it out of our toolbox one more time here.

And remember it's this, that if you have something 1+x to the n,

some point x to the n power.

If x here is much less than 1, then we can write

this as 1+nx, approximately with that.

So we're going to exploit that here because of course gamma,

what we're going to have here is gamma, in our one form,

square root of 1- v2 over c2 there.

But let's write that in slightly different form as we've done before.

We haven't done this recently, but before we could write it like this.

We have 1 minus v squared over c squared to the minus one half power.

That's just gamma, so I forgot the mc squared here, so

we'll put that on the top, mc squared over that.

So that's this times mc squared, so

again this is just gamma right here times mc squared.

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And note that this has the form of this,

because especially if v is much less than c.

So if our velocity is much less than the speed of light,

what do we mean but by much less?

Well even up to speeds like one tenth the speed of light,

v squared over c squared is still going to be very small value.

So we'll be able to expand this out and

put it into this form here using our binomial expansion.

So if we do that, what happens here is, here's the exponent,

the minus one half is equivalent to the n there.

And I've got a minus here instead of a plus but it works the same way.

So essentially I'm going to have, for this part right here,

this becomes one minus and then the minus one half.

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Now remember this is an approximation for when v is much less than c.

So up to about one tenth the speed of light, someplace in that region.

So still pretty fast compared to our normal everyday experience.

And so note we've got a minus times a negative one half, so

that becomes a plus one half.

And so we'll write this out one more time here,

so we got 1 plus one-half v squared over c squared times mc squared.

Now we're going to bring it back over here in a minute here just to

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Now I've got one half v squared over c squared times mc squared.

Well, the c squared's here cancel and we're left with plus one half

mv squared, that is kinetic energy.

So that falls right out of this general formula for, in the low velocity limit.

Well what is this telling us?

Back before Einstein came along, people would look at kinetic

energy and other forms of energy, and we talked about the conservation of energy.

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What Einstein is saying with this formula is that there's another form of energy,

right here too.

Is not only the kinetic energy that is important, but

this mc squared factor as well.

And note that if we just have v equals 0 here, then energy just is mc squared,

we get our famous form of equation E=mc squared here.

But what this is really saying, going back to conservation of energy.

Is that before Einstein came along with this equation, then the idea was,

well sure, you have a conservation of kinetic energy.

It could go into other forms of energy as well, but

no one really paid attention to the masses involved.

And what Einstein is saying, you know what, you've

got the masses involved maybe in something and you've got some energy involved.

You can actually convert in between them,

and still have the whole thing be conserved.

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So that's what the implication of this equation is.

That in principle you can take some of the mass energy here in a sense.

Sometimes called the rest mass, or

rest energy because the v is zero you're just left the mc squared part here.

You could potentially convert some of this into kinetic energy or

you can go the other way as well.

As long as you if you convert some of this, it turns into this,

if you have some of that, it turn in that.

As long as energy itself, the total is conserved, does not change.

And, again we don't have time unfortunately to be go into

all the details of this.

But essentially what you get out of this, what it leads to eventually

are things like nuclear fission and nuclear fusion, fission and fusion.

So the idea here, fission and fusion,

the idea is, you can turn some just ordinary mass,

it has energy locked up inside of it.

It's really a form of energy, if somehow you can liberate that energy,

you can turn that mass directly into things like kinetic energy.

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And it turns out, in the atomic realm,

really the nucleus of atoms, where the protons and neutrons are.

That if you turn, say,

if you have certain types of uranium, the uranium nucleus can split apart.

And it turns out that if you add up the components that are left, say,

to make it simple.

It splits in to two pieces versus the original uranium atom, the new atoms,

the smaller constituents there have less mass than the uranium atom did.

And therefore where does that missing mass go, it turns into energy.

And that's really the idea of nuclear fission.

That you can split apart certain atoms, certain nuclei.

And that releases the mass energy, in a sense, that's stored inside there.

Nuclear fusion goes the other way.

It turns out that for lighter elements, say Hydrogen, the lightest element.

If you fuse two hydrogen nuclei together to create a helium nucleus,

there are a few other things involved there, as well.

But if you put a helium nucleus out of it, then and you look at it.

The helium nucleus actually has less mass than the two

hydrogen nuclei that you use to put it together with.

So again, where did the missing mass go, it turns into energy.

And that's the whole idea of nuclear fusion, you're fusing

nuclei together in a sense, and this is how the sun works, for example.

So we're all here because of the sun in a sense, and so

we're here because of nuclear fusion.

Nuclear fission was developed, the idea that the concept of it,

over a number of years, 1920s, 1930s.

Ended up development of the atomic bomb or atomic energy in general,

in other words, when you're liberating energy like this.

And if you think about it because c squared is such a big number,

it takes just a little bit of mass, if you can liberate the energy inside there.

You get a lot of kinetic energy and heat energy and

other forms of energy out of that.

In fact just to give you a glimpse of it,

we all sort of have general idea of the destructive power of an atomic bomb.

It doesn't take relatively a small bomb to level a city.

And things like hydrogen bombs which are based on nuclear fusion

are actually have even more power than that.

But In terms of more peaceful uses of it, for example.

If somehow you could liberate all the energy contained in a mass of

just three kilograms, so that's not much mass there.

If you could do that, if you could turn that all into energy,

you could power a city with 100,000 inhabitants for 100 years.

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A city of that size, 100,000 inhabitants needs

an electrical generating station of about 100 megawatts, a 100 million watts.

Therefore just 3 kilograms of matter, again if you could liberate all the energy

inside there, would power a city for up to 100 years.

Now nuclear fission,

nuclear fusion you're just delivering tiny amounts of the mass energy available.

But you're getting a lot of energy out of it, even in those cases.

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Now, you might ask, also going back to Einstein,

what was his role in, especially the atomic bomb.

Because you may know that he wrote a famous letter to President Roosevelt,

Franklin Roosevelt, in the United States.

Alerting the President of the fact that scientist had recently discovered nuclear

fission actually.

So this was in the early 1930s, right around 1930.

And the possibility of either a controlled nuclear reaction or

even an uncontrolled nuclear reaction using nuclear fission.

That would release incredible amounts of energy in a bomb was feasible.

So beyond that though,

Einstein didn't really play much role in the development of it.

He was asked to write the letter by some

of the other scientists involved who were very concerned about this.

And because his name had cache as it were, he could

get the attention of the powers that be, they approached him to write the letter.

But beyond that letter, he really played no role at all in the development of

the atomic bomb and the Manhattan project, during World War II in the United States.

So that's Einstein's role in that.

Certainly in a sense it all does go back to the E=mc squared equation.

The idea that there's an incredible amount of energy stored in regular,

ordinary matter, if we can liberate it in some sense there.

One other point to make here about E=mc squared,

gamma mc squared and the like,

is that we've talked before about invariance.

And the fact that really a better name perhaps for the special theory

of relativity, would be not the theory of relativity but the theory of invariance.

Because one of the key invariant quantities is the speed of light,

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or actually technically it's the speed of massless particles.

But we'll just say the speed of light, invariant.

No matter how you're moving with respect to somebody else, in respect to

a light beam, you will always measure the speed of light to have the value c.

So we talk about invariance.

And turns out that in 1918 so just a few years after the miracle year of 1905.

And it's juts a couple after Einstein introduces general view of relativity.

A mathematician Emmy Noether,

whom Einstein consider one of the greatest mathematicians ever.

She published a very famous theorem that essentially

tied invariance into this idea of conservation of energy.

And she was able to show that other quantities, if you think about just if

we move from here to there, that doesn't change the laws of physics.

That's called translational invariance, so

we can talk about translational invarience.

That if I move from here to there, and I do the same experiment,

if I do an experiment here or do an experiment there.

I shouldn't get anything different in terms of my answers,

assuming everything else is equal there.

In other words,

where you do the experiment in the universe shouldn't matter.

Again I'm assuming the other conditions are equivalent, so

that's translation invariance.

And out of that, Noether showed that

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the concept of conservation of momentum came out of that.

Again, momentum is beyond our course.

But just the idea of translational invariance is connected up with this

idea of conservation of some quantity, in this case conservation of momentum.

Another idea is rotational invariance, as I turn from side to side here,

or point in that direction versus that direction, maybe versus that direction.

I should get the same results.

So that I have rotational invariance

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of the laws of physics in some very general sense.

And that leads to the concept of conservation of angular momentum.

In other words Noether showed that these were equivalent concepts.

If you have rotational invariance you get conservation of angular momentum.

And then finally, related back to our conservation of energy,

she showed, if you have invariance through time.

So time like invariance in a sense,

that is connected directly with the idea of conservation of energy.

So if the laws of physics are the same through time, and of course time and

space time are the key concepts underlying the spatial theory of relativity,

and later the general theory of relativity.

Noether showed that the concept of time and

moving through time, whether I do an experiment now or ten minutes from now.

Or two years from now, again, other things being equal,

that leads directly to energy being conserved.

So in a really deep sense, this all hangs together.

If you really start pushing it down into the depths of some of these concepts,

you'll see that we're not just talking about sort of time dilation and

things like that.

But we're talking about the foundations of the universe itself, and

how it's put together and how it works.

So those are just a few words about again, the famous equation,

the most famous equation.

E equals mc squared, again the real form of it coming out

of relativity is E equals gamma mc squared.

And we show that out of that, you can get the regular form of kinetic energy.

But you really then learn about this concept or

see this concept that mass, itself has energy stored in it.

That mass, matter, and energy are equivalent in some sense and

can be turned from one form into another.

And then, in cases of nuclear fission and nuclear fusion,

release huge amounts of energy potentially.