Deriving Einstein's most famous equation: Why does energy = mass x speed of light squared?

E=mc^2 is perhaps the most famous equation in all physics, but very few people actually know what the equation means, or where it comes from. In this video I would like to show one method for deriving this equation, as well as provide some insight into what the equation actually means. Along the way we will also touch upon some of the most fascinating features of Einstein’s theory of special relativity, including time dilation, the reason why nothing can travel faster than the speed of light, and the relationship between energy and momentum for massless particles.

References:
Relativity - Albert Einstein
The Meaning of Relativity - Albert Einstein
Why does E=mc^2 - Cox and Forshaw
One, Two, Three, Infinity - Gamow
How to Teach Relativity to your dog - Orzel

[InfuZedShaDoWz]

It's just absolutely insane to me that the human brain is complex enough that it can work stuff like this out. It's like looking into the game files of life.

[WelshyPilotGuy]

When I studied physics I found it hard to accept equations that I had to memorise. It was as though my mind rejected anything that wasn’t proved to me. Learning how to derive an equation was my way of instilling it into my brain for ever and at that point I was able to leap forward. Fantastic video - Thank you

[johnny]

These are the most intuitive and ground up videos I’ve ever seen. No glossing over concepts. Not missing a detail. Absolutely brilliant. Thank you for teaching physics the way it should be taught.

[everettmcinnis]

There are many people on this planet that are capable of doing all of the calculations in this video, but there are very few people that are capable of explaining this topic with the lucidity that you have. Thank you deeply.

[nmgreg]

As a 61 yr old Mechanical Engineer, this channel has me addicted. The teaching style works. Please, keep future (what ever that means, relativistically ) videos coming.

[acpwnd]

I'm so glad your channel exists. I've felt like the way physics is taught is wrong; we're just forced to memorize a bunch of equations. But your channel truly reveals the underlying beauty. Thank you. Don't stop making these videos, I bet they'll inspire many a future physicist.

[Jacob.Peyser]

I watched this video a bit over a year ago out of pure curiosity. But back then I hadn’t even taken physics or learned calculus yet. Now I’ve done both, and coming back to rewatch this video with the new knowledge/intuition of what the heck you’re talking about half the time has made me realize how profound and awesome relativity is. Thank you so much!

[alextaws]

Thanks!!!
I cannot tell you the amount and intensity of pure happiness of understanding and insight your videos give me!
A huge part of that is using the actual math behind the concept (a rarity in educational videos) but didactically so well that I can get even concepts of which I would have never thought that I could! (a difference to, e.g., pbs spacetime)

[jamesdavison]

When I was in high school, I read George Gamow's book "1 . . . 2 . . . 3 . . . infinity!" I read the "gamma" equation and plotted points, and read Heinlein's books about time compression, and tried to get my mind around Einstein's understanding of the universe. I even took courses in differential equations as an engineer and became familiar with the mathematical tools you used. But forty years later, seeing all the dots connected in your video was a personal discovery that I will remember long after tonight's hangover has worn off! Thanks!

[blackstar]

“Which should be familiar to anyone particle physics “.
Thanks for not making me feel stupid 😀

[theshibby]

The fact that my highest education is a trade and I can still understand this, is a tribute to your teaching style.

Thank you

[jaredkiehlmeier]

I’ve been watching videos on relativity for years and this video has brought a lot of conceptual ideas together very concisely . 10/10

[beardydave]

This is one of my favourite derivations in Physics. I remember doing the algebra at Uni and having to redo it so many times to get it right. The maths was brutal, but so satisfying :)

[bangscutter]

It's truly amazing how seemingly unrelated concepts just "pop out" of the math and build on top of one another!

For example, Maxwell's equations deals with classical electricity and magnetism waves, yet the speed of light pops out of it naturally.

Then, the constancy of the speed of light is used to derive special relativity.

Next, special relativity which is about massless light, when applied to an object with mass, the rest mass energy pops out naturally!

After that, considering the momentum of a massless particle and equating with the photo energy from the photoelectric effect, we get the de Broglie wavelength!

The mass particle duality consequence of this, then leads on to Schrodinger's quantum physics!

The self-consistency of the math and physics is truly astounding! It's always been there all these while, just waiting for a brilliant mind to uncover it.

[nirinarabeson]

I'm speechless, this is the most comprehensive video I've seen about physics in general. You made me write down the equations and try to get the same results as your video. Subscribed

[kenhaley]

What a great explanation!

One point I would add for all viewers who get confused by Special Relativity and the common paradoxes. E.g., if moving clocks run slower in my reference frame, then wouldn't my clock run faster from the reference frame of the moving object? (and the answer is no; see 'twin paradox'). And the point is this: You cannot fully understand Special Relativity, unless you understand all three components: Time dilation (discussed here), Lorenz length contraction, and the relativity of simultaneity. Obviously, a deep dive into these topics is beyond the scope of this video, but I think it's worth mentioning, as part of your audience might be new to Special Relativity concepts.

[lakshaygupta]

Stop spoiling us with all the terrific content

[randomstranger]

I’m 15 and I think that if they tought us about the equations we have to ‘memorise’, most people would love physics as much as I do. Great video!

EDIT: Now doing a Physics degree

[davidmthorley]

As a complete dunce when it comes to maths and physics I found your video surprisingly understandable. Very clear, well presented, and using examples that helped me visualise the concepts. Great work.

[ruyewang]

In this video the math can be easily followed, and the physics underneath the math is clearly explained. This is the best derivation and explanation of E=mc^2 (and special relativity) among all popular science talks I have seen, most of which are merely some kind of handwaving in nature. I also enjoyed other videos by you such as the one about Schrodinger equation. I look forward to many more of such videos in the future on more advanced topics, such as general relativity. Although general relativity is based on much more advanced math, but I trust you can still do a better job explaining the theory than any other similar talks I have seen. Please keep producing such high quality videos! Thank you!