E=mc2 has got to be the most famous equation in the world. While studying physics. I used
it countless times and saw it derived in lecture slides. But I recently realised that I don't
think I've ever read Einstein's original paper where he derives it for the first time, and
I'm guessing that you might not have read it either. But given how ubiquitous the equation
is, doesn't it seem important, or at least interesting to see its first occurrence in
the flesh, to read it for ourselves? Well, here it is. Here's Einstein's paper from 1905,
where he first mentions the relationship between energy and mass. The paper is only three pages
long, and it's not too difficult to understand. So let's take a look at it. One reason why
you might not have read the original paper before is because it was written in German.
But math is a universal language, right? So when I first took a look at it, I was scanning
to see where E=mc2 appeared. And I couldn't find it. To actually spot it, we need to switch
to the English translation because Einstein never actually wrote E=mc2 in this paper.
Instead, he said it in a sentence. If a body gives off the energy L in the form of radiation,
its mass diminishes by L over c squared. Rearranging that sentence, the change in energy L is the
change in mass times the speed of light squared, we often use E for energy, giving the more
familiar form E=mc2. The equation's message is that mass and energy are interchangeable.
The title of this paper is, in English, "Does the inertia of a body depend upon its energy
content?" we can read inertia as mass here. It was the fourth paper that Einstein published
in 1905, a year known as his miracle year, since the four papers revolutionised science's
understanding of the fundamental concepts of space, time, mass and energy. The other
papers that Einstein published in 1905 were on the photoelectric effect, Brownian motion
and special relativity. They were all published in the same journal, the annals of physics.
This one about the mass energy equivalence wasn't supposed to be a main attraction, it
was an addition to another paper, just more of a quick technical note, which is why it's
so short. The paper starts off by saying that the results of the previous investigation
led to a very interesting conclusion, which is here to be deduced. Those results he's
referring to are from his recent paper called On The Electrodynamics of Moving Bodies, which
presented his theory of special relativity. He summarises a key idea from it, that the
laws of physics must be the same in all non accelerating reference frames. The first equation
here is one from the Special Relativity paper, so he doesn't bother deriving it again, but
it is the change in energy of a wave as seen by a moving observer. It is the relativistic
Doppler effect that he's just discovered in his previous paper and is excited to explore
what else it can do. L is the energy of the light wave, c is the speed of light, and v
is the relative speed between the reference frames. phi is the angle between the source
and the receiver. The Doppler effect is the change in frequency of a wave for an observer
who is moving relative to the wave source. If the source is moving in the direction of
the waves, then those will appear to be more bunched up and have higher frequency. A common
example of the Doppler shift is when a car siren approaches you, and you hear a frequency
that is higher than that of the emitted frequency, and the frequency is lower as it drives away.
The relativistic Doppler shift takes into account the time dilation effects of special
relativity since frequency and energy are related by E=hf, the energy is also changed
by the same factor. Now that we know how light's energy changes when viewing it from a moving
reference frame, Einstein then comes up with a thought experiment for us. Let there be
a stationary body in the system XYZ, and let its energy refer to the system XYZ, be E0.
Let the energy of the body relative to another system moving with velocity v, be H0, that
moving system might be a rocket, let this body send out in a direction making an angle
phi with the x axis, plane waves of light of energy half L measured relatively to XYZ
and simultaneously an equal quantity of light in the opposite direction. Meanwhile, the
body remains at rest with respect to the system XYZ, according to the conservation of energy
and the principle of relativity. As we hit these two equations where E1 and H1 denote
the energy of the body after the emission of light as measured relative to each of the
two systems. These equations just say that the energy of the body after emitting light,
plus the energy of the light pulses in each direction, add to give you what you started
with. That's the conservation of energy. The same is true for the moving frame of the rocket,
but the light pulses are multiplied by the relativistic Doppler effect formula. Since
their energy will appear different to a moving observer. E0 and H0, both describe the energy
of the same body. In E0, it's from a frame at rest, and H0, it's from the rocket, the
difference between E0 and H0 will be the kinetic energy of the body. Remember that according
to the rocket, it's the light emitting body that seems to be moving at high speed, the
rocket observer thinks that they are standing still, H0 minus E0 is the kinetic energy before
the emission of light. And H1 minus E1 is the kinetic energy after the emission of light.
The difference between the two kinetic energies before and after light emission is this quantity
here, which represents the energy of the light emitted. Einstein says the kinetic energy
of the body with respect to the moving frame diminishes as a result of the emission of
light by an amount that is independent of the properties of the body. Moreover, the
difference in kinetic energy, like the kinetic energy of the electron depends on the velocity.
At this step, Einstein then makes a Taylor series expansion of this right hand side.
That means approximating this function as a simple polynomial with respect to v over
c, and centred around zero. And approximation like this might be helpful because once we
decompose a function into its Taylor series, we sometimes see terms that are vanishing
in the limit, and can simplify the expression if we are only interested in finding its limiting
behaviour. For us, the term v over c is so small that we can neglect magnitudes of fourth
and higher orders to get the change in kinetic energy is a half times L over c squared times
v squared. Einstein mentioned that this right hand quantity depended on velocity, just like
the kinetic energy of an electron, which was known to be a half m v squared. So why not
see if it fits that form? Well, it does if L over c squared is equal to the mass. So
that's how Einstein concludes If a body gives off the energy L in the form of radiation,
its mass diminishes by L over c squared. The fact that the energy withdrawn from the body
becomes energy of radiation evidently makes no difference. So that we are led to the more
general conclusion that the mass of a body is a measure of its energy content is the
energy changes by L. The mass changes in the same sense by L over the speed of light squared.
Finishing off Einstein say something visionary. It is not impossible that with bodies whose
energy content is variable to a high degree. For example, with radium salts, the theory
may be successfully put to the test. If the theory corresponds with the facts, radiation
conveys inertia between the emitting and absorbing bodies. Einstein was just starting to wrap
his head around the idea that a mass emitting something massless in the form of light actually
loses mass, and that mass can be transferred in the form of radiation between emitting
and absorbing the bodies. At first, he didn't quite believe it, and wrote his friend about
the whole thing saying the principle of relativity in relation to Maxwell's equations, demands
that mass is a direct measurement of the energy of a body that light carries mass. A noticeable
decrease in the mass must then occur in the case of radium. The thought is funny and infectious.
But whether God is laughing and has played a trick on me, I do not know. This relation
between mass and energy has been tied to Einstein's name and the idea of genius ever since. The
reason that you might not have encountered the original before could be a general lack
of emphasis on the history of science alongside teaching, but it might also be because it
was a bad proof. There are claims that Einstein's original proof contained circular reasoning,
and there are better ways to derive the same thing which are more rigorous and more complete.
The correctness of Einstein's first proof is actually a controversy all of its own.
Here's a paper published last year summarising some of the criticisms, including that Einstein's
proof contains circular reasoning, and that it is only valid to the lowest order approximation
for kinetic energy. Discussion boards online about this topic are surprisingly heated,
bringing in additional factors like the distribution of credit and fame, amplified by just how
famous this equation has become. What's important to realise, though, is that the equation itself
is correct. Disputes over the rigour of the proof are not the same as disputes over the
result. Einstein himself perhaps unsatisfied with his original proof went on to derive
over a dozen versions of the same result in later papers, even more proofs of E=mc2 have
been written by others. If you don't like Einstein's original proof, there are so many
other creative ones to choose from. In a physics class, it might be more common to derive the
energy momentum relation, which is an extension of E=mc2 For bodies with some momentum. Without
the first term massless photons would have no energy. For a body at rest, the first term
disappears and we get E=mc2. And if you're looking for mathematical rigour, why don't
take a look at Terence Tao's version of the proof, which follows Einstein steps but in
a more mathematical language then points out a couple of places where Einstein could have
simplified his proof. Another reason why the original proof isn't plastered everywhere
might be because of another heated controversy surrounding it, which has been tearing apart
physics for 100 years. That is the fight over who deserves credit for relativity, and for
E=mc2.
I only just made it up for air after reading all the criticisms of Einstein's proof. Before
I fell into the cavern of reading about the other people who contributed to the idea,
and how they likely deserve more credit than they got. Einstein wasn't the first to link
energy and mass. Even Newton himself pondered it back in 1717. When he wrote are not the
gross bodies and light convertible into one another, and may not bodies receive much of
their activity from the particles of light which enter their composition? Several scientists
working on electromagnetic fields knew that if a charged body moved through an electromagnetic
field, its mass would be affected. Poincare in 1900, derived an expression for the conservation
of momentum in electromagnetic systems that used E=mc2 in calculation although never explicitly
stated, Einstein seems to have been the first to establish a mass energy equivalence that
was independent of electromagnetic fields, and throughout his life, relentlessly devoted
time and effort to establishing E=mc2 as a universal law applicable to all forms of matter
and energy. So in this sense, it does seem justified to associate the idea with Einstein,
that association would come as a curse when the public started to link the equation with
nuclear weapons. After the bombings of Hiroshima and Nagasaki in 1945, 40 years after Einstein's
paper, E=mc2 was featured in the Smith report, the official US government release about the
development of the atomic bomb. E=mc2 can be used to quantify the amount of energy released
or consumed by nuclear reactions, which convert mass into energy. If you measure the mass
of all the constituents and the mass of all the products, and multiply the difference
by c squared, you get how much energy is released or consumed, usually in the form of light
and heat. With c squared being such a large number, you can see that an extraordinarily
large amount of energy can be released from just a small amount of mass millions of times
more so than in the combustion of chemical explosives. Whilst E=mc2 is useful for quantifying
the amount of energy released, it was not strictly necessary for the development of
the atomic bomb, especially since the fission process was known in 1938. It surprised me
to learn that the neutron wasn't discovered until 1932, 27 years after Einstein's paper
and a much more crucial discovery to the development of the weapon. Einstein himself co signed
a letter to the US president in 1939, urging funding into research for Atomic Energy, warning
that an atomic bomb was theoretically possible. He would later come to regret that letter.
Newton please forgive me, he would write, not referring directly to the bomb, but for
standing on Newton's shoulders, for taking his work and questioning its foundations,
flipping it, building upon it and driving progress to both the good and bad consequences.
The mention of war in the story reminds us how often scientific discoveries are tied
to both amazing and sometimes terrible consequences. Much of the technology which has improved
our lives has also contributed to the destruction of our climate. If you'd like to be involved
in efforts to use science and technology to make a positive impact on our climate, then
consider this video sponsor Wren. Wren is a website where you can calculate your carbon
footprint, discover ways to reduce it, and offset it by funding a diverse mix of carbon
reduction projects like tree planting, mineral weathering and rain forest protection. One
of their innovative technology projects is centred around biochar. Dead wood from Californian
forests, which would normally be burned to ash and release carbon dioxide into the atmosphere
is instead turned into biochar, which locks the carbon for 1000s of years. The biochar
is then mixed with compost and sold to local farmers to use as a natural fertiliser. Offset
your own carbon footprint on Wren. The first 100 People who sign up will have an extra
10 trees planted in their name. Thanks for watching this video and thanks to my Patreon
supporters for making it possible. A special shout out to today's Patreon cat of the day.
Mi-Jin.