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
You can help support this channel via the Physics Explained Patreon account: / physicsexplained
You can follow me on instagram: / physics_explained_ig
You can follow me on Twitter: / physicsexplain1

[@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

[@xxxalphaeverythingxxx8489]

Holy shit this is so relatable to me. I have this feeling of missing out or of not actually knowing something unless i can derive it myself in my mind

[@toughenupfluffy7294]

I'm the same way. I look up word roots for the very same reason, understanding of context allows for deeper learning.

[@User-jr7vf]

@@xxxalphaeverythingxxx8489 same. This is annoying at times but it is the way my brain works and I've come to accept it 😂

[@BD-np6bv]

People who simply memorized equations won't learn anything. It's people who question equations that will grasp the theories.

[@xxxalphaeverythingxxx8489]

@@BD-np6bv Couldn't agree more.

[@nmgreg11]

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.

[@PhysicsExplainedVideos]

Thanks! Plenty more videos on the way

[@hoekbrwr]

As I am even a little bit older now I do understand why I needed to learn all those rules of math in school! That's more than 50 years ago. As I ended up being a practical electric engineer and not done a theoretical job I catch just a tiny bit of it. I really get that marvelous feeling again of understanding all those math 'tricks' as I had to learn 50 years ago. Keep making these excellent videos!

[@daveanderson718]

@@PhysicsExplainedVideos You must be doing something right here. I see I am not the only "old" dude here who once was taught this stuff in a much more complex way.....and subsequently never used it my profession.....only to appreciate what you have done in 30 mins on KZbin. Rather than "whine about the loss", you inspire me to watch it again and your other videos as well. Bravo to you Sir!

[@tim40gabby25]

Jeesh.. anyone here under 60? Old UK duffer here, enjoying the ride :)

[@alirezanabavian771]

@@tim40gabby25 it takes a great deal of experience like decades of engineering to truly appreciate and dive into these equations and walk with Einstein discussing physics though for only a short period of time....yes Albert ..now I see where you're coming from....

[@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.

[@PhysicsExplainedVideos]

I know what you mean!

[@HR-yd5ib]

Only one in A Million brains though! 🤔

[@chrismadaj8751]

That is what I have gotten out from this we are so Superior with our brains but only a few of us can comprehend with mathematics. To come up with these equations But not necessarily saying they're smarter than the Average person. Common sense is a whole another thing.so cool though 😎

[@iancampbell4394]

Not life, my friend. Life is a fundamentally biological concept. This is like looking into the game files of the universe itself.

[@ibrahimkhatib6191]

@Chris Madaj The measure of smartness depends on the smartness frame of reference!

[@johnny14980]

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.

[@jasoncassidy492]

You are easily pleased, John, I find the explanations to be so obfuscated and trite as to be of no use.

[@nerdsunscripted624]

@@jasoncassidy492And you sir put a lot of effort to make your comment more obfuscated than it needed to be. If you think this is hard then maybe the oversimplified mildly incorrect explanations of physics are more for you

[@myzo247]

😊😊😊😊😊😊😊😊😊

[@hfarthingt]

I’m now an expert

[@jasoncassidy492]

​@@nerdsunscripted624 Nothing hard about it, time does not exist and energy has no relation to mass, therefore E= mc^2 is nonsense. F = ma makes perfect sense since a force applied to a mass can cause it to accelerate. However, E = mc^2 suggests energy can be converted to mass and vice-versa. We have no idea what energy is and we define it loosely as the ability to do work. When Einstein created that equation, he was talking about electromagnetic energy being converted to mass. He believed that and he was wrong. Can it also be applied to thermal, electrical, mechanical, and gravitational energy? Don't think so. Einstein lived in a time when these phenomena were just starting to be investigated. I fear he jumped to irrational conclusions.

[@acpwnd2020]

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.

[@PhysicsExplainedVideos]

Thank you so much!

[@Hampo03]

I’m just in the 2nd of 3 years in high school (in Sweden) so it’s not that advanced but my physics teacher is amazing at teaching both. He shows how somethings works and then how to get the equation for it, often using previously learned equations

[@am3rgam3r]

It very much depends on who your professor is. Physics is one of the most amazing and Fulfilling sciences and subjects in school today. ( my opinion )

[@BruceNitroxpro]

Totally and completely true. The simplicity of expression is a wonderment.

[@BboyKeny]

@@am3rgam3r I read the old books by the people of that made the theorems. I find their instructions to be often the best.

[@jamesdavison6290]

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!

[@PhysicsExplainedVideos]

Very glad to hear it, thanks for the comment and feedback

[@daveanderson718]

@@PhysicsExplainedVideos Obviously you are doing something right. Don't stop.

[@alirezanabavian771]

I feel the same way exactly after having been an electronics engineer for 40 years I have gone through the same revelation as you have....more power to him...the best I have ever known...

[@riazhassan6570]

Yes. As an interested layman, I found this helped to take me round difficult corners better than some treatises I read in the past

[@colinnewton7291]

@@alirezanabavian771 Ditto

[@alextaws6657]

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)

[@atticuswalker8970]

I got one for you. E=ymc² where y is the Lorentz factor. Time is 1 dimentional. And spacetime stretches to accommodate the necessary interactions with the Higgs field to maintain mass.

[@musicmakelightning]

Beautiful work. Utterly clear. This series of lectures is masterfully executed. There is much to learn and retain with the minutes spent on this series.

[@blackstar2008]

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

[@toddmarshall7573]

Now that you don't feel stupid, consider this: 9:40 "we'll see the light take a zig zag path": I go off the rails right here. Imagine the train going around me in a circle at 300m/s and 1km away. I'm watching a tennis player riding in a glass boxcar hitting the ball against the wall repeatedly (i.e. your tic tok). I am standing on a rotating disk looking through a telescope such that my view is perpetually perpendicular to the train's velocity. I don't see any zig zag. I don't see any dilation. There is no warping of space or time. Further, if there is a glass box car on my disk with a tennis player doing the same thing as the disk rotates, and I have a split mirror allowing me to observe both simultaneously, their "tic toks" are in perfect sync. Now speed up the train to any velocity you like...say 300e6 m/s. And speed up my disk rotation accordingly. I still don't see any zig zag...yet I am stationary and the train is moving at the speed of light. Further, the tennis players continue to hit the ball (as observed by me) with no zig zag and in perfect "tic tok" sync. And they both have the same mass. The tennis player on the train doesn't enlarge to infinite mass...and if you like, extend the radius of the circle such that he doesn't get squashed by the centrifugal force...and I don't get dizzy with the spinning. There is no magic at the speed of light...or anything to suggest I can't exceed it. Further, a train running the opposite direction but same speed on parallel tracks would appear to approach at twice the speed of light...because it is...and we likely should expect a Doppler effect, just as we observe with sound. 25:88 Then you go on through all kinds of math and end up comparing energy. And you have this spurious mc^2 term that you begin to explain away. But let's start that other train in the opposite direction on parallel tracks. Both trains carry a mass of 1kg. They are both traveling at a velocity of v^2. The KE of those masses is 1/2 mv^2 each. Together it is mv^2. Now, when we get to v = c, just as the masses are about to pass each other in opposite directions, push one mass into the path of the other...taking virtually zero energy to do so (e.g. you moved the mass 5 m perpendicular to the v ...or you move both masses 2.5m ...one toward me and one away from me...while they each travel 300e6 meters in the one second it takes me to do the shoving). And bammo! The masses collide and their velocity adds instantly to zero. And their KE goes instantly to zero...and turns into heat as the masses vaporize. By the time they hit, I'm 150e6 meters away and just see the flash. How much heat? Well, it's 1/2mv^2 plus 1/2mv^2 or mv^2. And if v is c, then it's E=mc^2. Now where is the magic? Where in any of this does relativity come into play? And do you really want me to believe that one tennis player is aging more quickly than the other? Or that the tennis player goes to infinite mass? And you really want me to believe there are massless particles? Why? I'll bet Tesla would get it. I'll bet he wouldn't have to make up photons either.

[@enricolucarelli816]

@@toddmarshall7573 sorry, but all of your thought experiment is wrong from the very beginning, because special relativity only applies to INERTIAL FRAMES, and your frame of reference is rotating, thus anything but inertial. In a rotating frame of reference you experience accelerations, thus you would have to consider space-time warping according to the laws of general relativity.

[@toddmarshall7573]

@@enricolucarelli816 something's peculiar. I can find your reply on my phone but not on my computer.

[@toddmarshall7573]

@@enricolucarelli816 Todd Marshall 38 seconds ago @Enrico Lucarelli something's peculiar. I can find your reply on my phone but not on my computer. ...... and now "having replied on my phone" it shows up on my computer...so I can respond. " sorry, but all of your thought experiment is wrong from the very beginning, because special relativity only applies to INERTIAL FRAMES, and your frame of reference is rotating, thus anything but inertial." And I agree with you. The INERTIAL FRAME is physical...a body in motion tends to stay in motion...and its direction is constant unless acted on by another force...that's inertia. That being said and understood by both of us, and trains A and B moving in the same INERTIAL LINE, which of the following assertions are correct and which are false: (1) The speed of light is constant and the same for all observers. (2) The wavelength(s) of light is (are) constant and the same for all observers. (3) The speed of sound (in its ether of air) is constant and the same for all observers. (4) The wavelength(s) of sound (in its ether of air) is (are) constant and the same for all observers. (5) The length of a ruler is constant and the same for all observers. (6) The "appearance" of a ruler is constant and the same when viewed from all angles. (7) Light from a flash light on a train moving at the speed of light is the same for all observers (8) Two trains moving in the same inertial plane. Train (A) is moving at 80 miles per hour. Train (B) is moving at 80mph plus 150e6 meters per second. On train(B) the wavelength of light from train (A) appears to be twice as long as it really is. (9) In example (8), but train B moving at 90mph plus 300e6 meters per second, the light from train (A) is never observed on train (B) (10) Train (B) cannot move at 90mph plus 300e6 because it's mass grows to infinity at that speed (11) A ruler on Train (B) is a different length than on Train (A) at any speed greater than 80mph. (12) A ruler on Train (B), as observed on (B) is of zero length when moving at 300e6 meters per second (13) A ruler on Train (B), as observed on (B) is of infinite length when moving at 300e6 meters per second (14) A flashlight on Train (B), as observed anywhere has the same mass as on Train (A) My answers: 1-T; 2-F; 3-T; 4-F; 5-T; 6-F; 7-T; 8-T; 9-T; 10-F; 11-F; 12-F; -13-F; 14-T; "In a rotating frame of reference you experience accelerations, thus you would have to consider space-time warping according to the laws of general relativity." More correctly, in a "rotating frame of reference" I experience a force which causes the straight line of inertia to change direction...and that Force is equal to the Mass times Acceleration. That being said and understood by both of us, it suggests "no" change in any of the "rulers" taking measurement...be they space or be they time. And perceptions don't change that.

[@lakshaygupta9061]

Stop spoiling us with all the terrific content

[@jaredkiehlmeier6460]

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

[@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

[@beardydave926]

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 :)

[@iamprox3690]

🤓😒🎃

[@iamprox3690]

l l i’ll lollll l i’ll l l l l l l l l l l l l l l p p p p p p p p p p p p p p p p l p p p p p p p p p p p p p p p p p p p p p i’ll p p p p p p p p p p p p p p p p p p p p p l l p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p l p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p i’ll 😢

[@steveolmore6091]

Thanks for explanation, helps.

[@lightningbolt4419]

@@iamprox3690 agreed. This guy is bullshitting.

[@jasoncassidy492]

Dave...another David...the physicist David Bohm... once stated that an equation with no reality to back it is garbage. Time has no existence and the use of it as the basis of a theory is beyond garbage. The irony is that so many people blindly accept Einstein's theories simply because of his appeal as an authority figure.

[@narfwhals7843]

You really just wanted to say "the ticktock of the moving clock" as often as possible, didn't you? Great video :)

[@PhysicsExplainedVideos]

Hahahaha, I was thinking that as I was saying it

[@Thebigbangisdeadgetoverit]

​@@PhysicsExplainedVideos Keep repeating the lie until becomes the truth. E=MC^2 was originally attributed to Olinto De Pretto, you and your cult of Einstein had destroyed physics. en.wikipedia.org/wiki/Olinto_De_Pretto "Physics explained" my .....

[@bixmcgoo5355]

@@Thebigbangisdeadgetoverit man i wish I was able to be so confident while also being so wrong

[@ayo9344]

@@bixmcgoo5355 Idk it does say on the page that Pretto's formula E = mv^2 (v is the equivalent to c by Pretto) precedes Einstein's E = mc^2 by 2 years. Did you see that part? I just think it's interesting, and tbh just not sure what to think about it.

[@bixmcgoo5355]

@@ayo9344 oh I didn't say that the guy didn't come up with an equivalent theory (which is honestly great, though it's worth noting that the ether stuff he was trying to prove was horseshit). I guess the issue is the previous responder saying that Pretto was the one to which E=MC^2 was originally attributed, which is simply not true. They came up with two essentially equivalent formulae, and the proper formula was attributed to the one who discovered it. It's very likely that Pretto's work wasn't taken very seriously further down the line because, again, his entire line of questioning was in reference to a fundamentally wrong idea that is "the ether"

[@jacobpeyser2169]

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!

[@hareecionelson5875]

Same. First time I saw this I was a year 4 med student, did not understand the chain rule. But, after graduating and working, I reapplied for physics and did more math, now I can follow along and it's even better

[@jasoncassidy492]

Jacob...time to put on the thinking cap and re-evaluate. It took me years after studying electrical engineering to understand that time does not exist. I confirmed that with one of my physics profs years later. He replied immediately that humans invented time to keep tract of change. Physicist David Bohm has claimed the same thing, that humans invented time. If we invented time, which we did, basing it on the rotational period of the Earth, how does it dilate? Anyone can convince most people of anything using math, the trick is to explain what the math means. When you look closely at the math in this video, it's all based on a distortion of the human mind and its inability to keep tract of relative motion. It seems astounding to me that Einstein could have missed that mental issue in his analysis.

[@jacobpeyser2169]

@@jasoncassidy492 Whether or not time is a strictly invented or a discovered phenomenon, it along with all the other fundamental ideas in physics describe our observations of the universe with unparalleled accuracy. In my current opinion, if a theory like special relativity could so precisely describe reality, then surely its implications must be fundamental to reality itself. Now, I never took an official class in special relativity. Hell! In under 2 months, I'll be starting my freshman year at university. So I have a long way to go with my physics education, and my views have much room to change. However, I hold this particular view very tightly. With regard to your opinion, It seems as though you don't get what time dilation is. To argue that time cannot dilate or be relative because time is based on the rotational period of the Earth is almost nonsensical. The whole point of Einstein's special theory of relativity is that there is no universal clock. Every inertial frame has its own unique metric of time, thereby making changes in time relative between observers. Keep in mind that this principle of relativity is mathematically deduced by the constant nature of the speed of light (as laid out in the video) and this characteristic of light was justified experimentally after Einstein's insights. I think the confusion stems from the terminology of time "dilation". Time doesn't physically dilate; it is simply relative. That's it. Einstein was not perfect, but I bet he was a hell of a lot wiser and more intelligent than 99.9% of the average Joe (like you and me) and even the current physics community. Wow... that was some rant over a comment from so long back.

[@jasoncassidy492]

@@jacobpeyser2169 Jacob....what else could time dilation mean? The basis of Einstein's relativity theory is that both time and length change as a mass approaches the speed of light. He got that idea from Lorentz who had hypothesized earlier that time and distance must change when transforming between two axes moving relative to each other, as velocities approach the speed of light. Louis Essen, who invented the atomic clock, pointed out the error in both views. Both Einstein and Lorentz changed reality to make their equations work. Einstein arbitrarily changed the definition of time, and that is not allowed in science without proof. He supplied no proof, his conclusions are based strictly on thought experiments. He also ignored the fact that we humans defined time based on the Earth's rotation.

[@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.

[@gamer19591]

It's always such a bliss seeing you explain such complicated topics with ease. It helps me tie together the theoretical parts from my lectures with some perspective. Thank you for these magnificent videos

[@PhysicsExplainedVideos]

Great to hear!

[@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.

[@PhysicsExplainedVideos]

I totally agree!

[@tim40gabby25]

@@xiaoxiao-kg5np given that the set of um.. Kabbalists does not include Maxwell, then your "all" assertion is false, allowing for your house of cards gently to collapse. Wrong channel old boy, kindly meant :)

[@xiaoxiao-kg5np]

@@tim40gabby25 Maxwell was not in my set of kabbalists that "discovered" quackery. You tried to add him in. I also did not include Newton, although he was a bit of a believer in alchemy and such, as were many others of his day. So please keep the kabbalists in their own camp of numerology and pseudoscientific quackery. My house of made of solid rock, standing on rock.

[@ziqfriq]

@@xiaoxiao-kg5np You should probably check out Gabor Fekete, and get on his email spam list.

[@frankh4069]

@@xiaoxiao-kg5np add Heisenberg to the nope-list

[@77bronc14]

Outstanding video...this is a similar model and description my physics professor used to describe the equation 47 years ago. Great job and brings back some good memories

[@djenning90]

This was a gorgeous and enlightening presentation. Brilliantly done! Very accessible and understandable to me.

[@PhysicsExplainedVideos]

Glad it was helpful!

[@randomstranger2579]

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

[@PhysicsExplainedVideos]

Thanks!

[@jaimeduncan6167]

I am glad you love science, and I can see that you still have a lot of fait in humanity. I believe you are wrong, most people will be asking if it's going to be in the test, and eager to just learn how to calculate, get the grade and forget about it. If you stop to look at it, you will notice that it's not as bad as it seems. I for example, have little interest on following the last fashion, aldo, in a daily basis, it affects us. Even so, if one says "more" people instead of "most" people one will be correct (I am guessing here). Keep your curiosity and your hope, it's a privilege of young age and gives people the power to transform the world, even if it's a little bit, for the better.

[@arniedamaniac6206]

@@jaimeduncan6167 I’m 14 too and I agree with you

[@lillearnero2329]

@@jaimeduncan6167 This is the guy who posted the original comment on another account as I was rewatching the vid and happened to see it. I believe you are correct as I am 16 now and have just finished my GCSEs. A shocking number of people decided to burn their books and notes as soon as we finished which proves your point, I do not really understand the logic behind it but it is just how things are. Fortunately their are still those that learn for curiosity and passion, however few they may be, and so I keep hope that the future can be bright. Thanks for replying and good luck.

[@jaimeduncan6167]

@@frankdimeglio8216 you have no idea what you are talking about. Like no idea, there is plethora of experiments that show that you are wrong. Even more, most of the stuff we see and use in a daily basis will not work if you were correct. I am not saying that the current scientific understanding, that predicts stuff with 10 digits of precision is perfect, we know with certainty that the current models are incomplete but dude you are way off. You really need to study a lot more and be humble. Once you do that your creativity may push you in the right direction.

[@pipertripp]

master class. This was loads of fun. The derivations were very clear and easy to follow. In fact, the bit about time dilation could be taught to kids in geometry call. "See, this stuff you're learning is important and can help lead to mind blowing discoveries."

[@PhysicsExplainedVideos]

Glad you enjoyed it!

[@ruyewang8403]

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!

[@SiLVERSERG]

Beautifully orchestrated video! Excited to watch more of your content!!

[@SeverusVergiliusMaro]

Oh, this is gonna be goooood

[@appleslover]

Say hello to daddy aang

[@obst3085]

Absolutely the same. Every time it's fascinating with how little assumptions and clever actions, such elementary principles can be unveilled (although their primary derivation are works of art by some of the most talented scientists out there *and* their true meaning are beyond grasp for most things still)

[@DLockholm]

I was about to say this same shit lmao

[@magnomaxx2010]

This equation, not was Einstein's creation, but of one italian cientist before.

[@alfredobeltran611]

This is, by far, the best physics channel on KZbin. Very well done. Thanks for posting this videos

[@voy_]

thank you for taking the time to explain even the most basic parts of the math (the part at 18:05 where you bother to explain what "dx" is stands out). it's been a long time since I've been in a math class and these little reminders help me follow along more easily

[@everettmcinnis5858]

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.

[@avihayasraf2404]

"Now we've arrived to this ghastly expression" sounds like my professor. (though you explain way better)

[@neil2444]

No other word in the english language quite describes a bad expression quite like "ghastly", am I right?

[@as007de]

Just finished binging all your videos yesterday! Ready for new content :D Best channel I discovered in a while!

[@PhysicsExplainedVideos]

Glad to hear it, thanks for the feedback

[@JackyKou]

Great video! This channel is my favourite, and is what I rely on if I *really* want to understand some physics mathematically. By the way, at 20:32-21:11, I believe the minus sign in the middle of the green square bracket should be a plus.

[@Miftahul_786]

This was really confusing me then I went back and realised yeah it should’ve been a plus

[@Beaglow]

yep, you are right.

[@jamesedward9306]

I've watched a ton of videos trying to wrap my head around this topic and your video is by far the best. Subscribed.

[@guyedwards22]

So I've been familiar with some of the "flashier" consequences of SR and GR for years, and I've loved watching more general public oriented physics channels like PBS Spacetime and The Science Asylum, but here very recently I've taken the dive and have started watching Leonard Susskind's lectures on Relativistic Physics. Being able to follow the line of thinking from the formulation of Maxwell's Equations from basic vector calculus all the way up to this result from treating Energy with the gamma factor from Lorentz Transformations has been incredibly rewarding. It's honestly not as opaque of a field as it seemed at first, everything is starting to make wonderful sense really quickly, and you're a big part of that for me! Thank you!

[@JamesKaramath]

Can't thank you enough for these! Give them to my A level students as engaging beyond the course material knowing that they can follow along and enjoy their physics more! Please keep up the great work!

[@novicapetrovic8026]

Absolutely love how you explained this, it was so clear and concise!

[@PhysicsExplainedVideos]

Glad you liked it!

[@cHVF]

Thank you for making this video. If only you and youtube existed when I was grinding my way through college physics in nineties.. I had a great time back then, but it didn't have to take semesters to 'see it'. You are so very good at explaining, I can only hope my kids in college get a chance to feel what I'm feeling now. ..and now, I'm going to rewind and replay and try to follow the math and enjoy every time you say 'when the dust settles', hopefully my scribbles end up in the same place. :)

[@jasjitsingh5457]

Amazing video. For the first time, I truly understand the derivation and it was so simply explained. You are a genius and we are all blessed to have access to your KZbin videos. Thanks a lot

[@edwardjcoad]

Awesome! It's the simplicity of the explain which I enjoy so much. Its calm, smooth and seamless approach which i enjoy. I have a PhD in Physics and I learn alot from these videos especially about the initial stages of a derivation which I appreciate. Another great video.

[@PhysicsExplainedVideos]

Thanks for the kind feedback, I am glad you enjoyed it!

[@ksskaria361]

@@PhysicsExplainedVideos It's really wrong.

[@krzysztofciuba271]

? from 8:00 to 13:00 a typical junk on "time dilation"+ he (surprisingly) got a right conclusion on the "mass" but there is a more simple derivation: just multiply both sites of space-time "distance formula" by a constant "m".

[@ahmedrafea8542]

Just finished watching and thinking about your marvelous presentation. Amazingly clear, comprehensible and definitely useful. Thanks very much. And, forgot to say, you managed to nicely get away of the mo. Please keep up the great work.

[@stevenschilizzi4104]

Once again, a brilliant, crystal clear and most inspiring video, presented in a way that would make the sleepiest student sit up and listen, intently. All teachers should take you as an example to emulate. Thanks heaps for sharing your talent, sowing seeds that will certainly bear fruit!

[@PhysicsExplainedVideos]

Thanks for the very kind words, much appreciated!

[@dankuchar6821]

Excellent work! The best derivations and explanations I have seen in my three decades of Physics.

[@PhysicsExplainedVideos]

Great to hear!

[@junkbucket50]

Love this channel high level treatments of topics while equating them with other relevant things. Really good channel

[@PhysicsExplainedVideos]

Thanks!

[@valerianepartenie1013]

Amazing presentation of Special Relativity! The best so far on KZbin! There is one typing mistake at min 20:33... in the green expression, between the 2 fractions there should be "+" sign not a "-" for the expression to be correct. Other than that, all is brilliantly done!

[@SuperAliAR]

This may be one of the best videos I’ve ever seen on KZbin. Thank you.

[@awolgeordie9926]

Again, simply brilliant. A level Physics teacher here - background in Electronic Engineering - your videos are immensely helpful. And enjoyable. Thank you. (Might be a small error at 20:40. The two terms in the green equation should be added - not subtracted.)

[@PhysicsExplainedVideos]

Thanks for the feedback, I really appreciate it. And thanks for spotting the typo!

[@eddownzzz]

Wow, what a great video and clear derivation. Covered more depth on the subject than my 1st year of undergrad while still keeping it remarkably clear. Good job :)

[@robertmontague5650]

This is the best KZbin presentation on this subject I have ever seen--bar none! Hats off to you.

[@georgiiii1000]

I watch tons of educational KZbin videos (PBS space time, philosophy, history, etc) you have blown my mind with how simple you made this. I have a math background so I was able to follow every step of the way. I have never considered contributing to a patreon before but if you have one I’d donate. You are an excellent educator. Thank you.

[@vedantsridhar8378]

This is real physics. PBS Space Time is only physics as entertainment for the general public.

[@georgiiii1000]

@@vedantsridhar8378 totally agree, still interesting to watch.

[@sync4995]

This channel is actually MVP! Thank you sir for the detailed breakdown please keep doing what you do.

[@Euquila]

Wow, you really did the whole derivation without cutting corners. Nice!

[@rickgrech5577]

Amazing video . Absolutely love how you explained this, it was so clear and concise!.

[@wjiuvdjsvhsnsm]

This video is worth me to take screenshots for every page, and the explanation is worth me to listen for dozens of times. What an art work of teaching

[@alexram5267]

Thankfully you'll never run out of topics

[@ozzymandius666]

Also, the mass of fundamental particles arises due to the Higgs field mixing the left-handed and right-handed wave function of a given particle. In QM, this is the origin of mass. Can we expect a video detailing this process, with the SU(2) symmetry transformation that requires it?

[@PhysicsExplainedVideos]

On my list...

[@a.sanaie2460]

Amazing. I have been looking for a clear explanation of this for many years. Thank you so much 🥰

[@PhysicsExplainedVideos]

Glad it was helpful!

[@DuncanIdaho1980]

Great work really, it's cristal clear, both physically and mathematically. I've taken classes in Special Relativity and an intro into Particle Physics this year, and it's quite exciting to see all those fundamental relations linked in a single masteful video. Double, triple like please. I'm pretty sure that it's been said before, but there's a typo at 20:50 in the green expression (a "-" instead of a"+" in the parenthesis).

[@vhawk1951kl]

Did you*believe* what you were told in the classes you mention or did you verify it for yourself by experiment or what some call " science" the physics of which particle hat you could identify as such was what? What exactly were the details of the experiment that you conducted to find out for yourself whether or not E=-mc2? the energy of what were you trying to discover by experiment? Have you yourself ever measured what is called " the speed of light" In your experiments to discover for yourself as opposed to believing what you were told , was the speed of light always *exactly* the same each time you measured it, and you measured it how?

[@FedericoTrentonGame]

Finally a channel worth clicking the bell button for, you’re a champ!

[@adogonasidecar1262]

Very, very well done. I have studied physics quite a bit for 50 years or so. Yours is one of the most accessible approaches. Of course, there was a quick cheat in the process of integrating, but this is physics, not maths!

[@Aloman76]

I wanna thank you so much. All these people saying relativity is extremely hard to understand, and giving analogies. U explained it specifically and with examples and it’s not as bad as I thought. Thank so much bruh

[@qazwsx6340]

the way you introduced the formula for time dilation was amazing. I never thought it was as simple as just using the pythagorean theorem. It was always just told to me with little explanattion

[@Craznar]

40 years after learning the rest + classical energy formula - I finally get it explained so I understand.

[@PhysicsExplainedVideos]

Glad to hear it!

[@snorkfire]

SUBSCRIBED! That’s a fantastic presentation of the actual mathematical derivation. For my part, I really appreciate that first you’ve studied the topic and now’ve done an excellent, stepwise explanation of this amazing revelation. I’ve used the equation to grasp a nuke’s destructive power, but have always wanted to see the math that arrived at its richer meaning. Thank you very much.

[@PhysicsExplainedVideos]

Glad it was helpful!

[@derekgreenacre9530]

I have tried many times to understand these equations but it is only with this video was I able not only to understand the equations but to be able to derive them for myself now. This is wonderful the equations have an intrinsic beauty that transcends space and time and that of course is what they are all about.

[@PhysicsExplainedVideos]

Glad it was useful

[@walterdavey3276]

This is probably the most interesting item I’ve ever seen . I’m finding that I can actually understand what is being said rather than it just being a mystery, as, until now, it has been for the last fifty years to me. Fantastic presentation. Thanks.

[@appleslover]

I was hungry for such content

[@albela0007]

Beautiful. I always thought the derivation of E=mc^2 involved complex math because I don't remember anyone teaching this in school and college. This explanation provides a straightforward derivation using calculus most people learn in junior high school and college. I don't understand why then this explanation is not included either in math/calculus books or books on physics/mechanics. One small correction I see for the green equation at 20:32. It should + instead of - between the two terms in square brackets.

[@vhawk1951kl]

When you verified for yourself the truth of what you call "E=mc^2" the energy of what were you trying to discover? All sorts of things can be called energy can they not?What exactly were the dtails of the experiment to test the truth of the assertion "E=mc^2"? Just as a matter of interest how did you measure the speed of light to use in the verification for yourself of "E=mc^2"?How many times did you measure the speed of light and was it always the same in both directions? I am of course assuming that you would dream of just believing what you were told like a rather slow childand thus insisted on verifying eveything you were told by experiment known as the scientific or not_religious method; science being be the verification of every supposition by experiment and were you just to accept without questionE=mc^2" that would be religion would it not? So no belief for you who insistn verifying everything for yourself do you not? In your experiment to verify " E=mc^2 the energy of exactly what was the same as its mass multiplied bythe speed of light which was what when you last measured it. In units of what were you measuring the energy of exactly what? When did you verify for yourself that "E=mc^2"? Was it simplicity itself for you to satisfy yourself so that you were sure that " E=mc^2, or was it a complex and elaborate experiment to verify for yourself that "E=mc^2"? Purely as a matter of anodyne academic interest why might the speed of something other than X(specifically light)have any bearing on the energy of X? One has visions of chaps all over the world saying:" I'm not going to tote that barge or lift that bale because light seems rather slow today." Presumably before you can say that the energy of X is its mass multiplied by the speed of invisible rabbits, you would have to devise some method of establish the energy of X *before you multiplied its mass by whatever to see whether or not they were the same. When you were using the scientific method of verification by repeatable experiment, the energy of exactly what was the same as its mass multiplied by the speed of light which you stablished *by experiment* to be what? How exactly*did*you go about discovering for yourself?

[@bkahuja123]

Very nice presentation and clear and systematic derivations...I tried so many times doing it myself but got wrapped up. Will look forward to other physics videos.

[@kenhaley4]

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.

[@hansfijlstra5932]

I have never seen a better tutorial about relativity. I am very impressed how you manage to combine ‘complex’ formulas with a ‘simple and easy’ to follow narrative in such a way that you really feel that you understand what is presented. This kind of teachers are the ones that, as a kid, you will remember for the rest of your life. If you allow me one suggestion: consider (where possible) not spell out the same formula every time again because the listener is already reading it and is already familiar with the formula. But maybe that is only my preference and not that of other listeners. Thanks for your passion and effort to explain difficult matter in an easy way 👍!

[@govindankandasamy1662]

A marvellous enunciation of the most enigmatic Mass-Energy relationship , comprehensible even to the uninitiated , seeking to comprehend the underlying principles , ingrained in the ingenious equation E=MC^2 , unravelled by EINSTEIN , our most revered SPACE-TIME GENIUS .

[@Anza_34832]

Thank you for doing the math and explaining the rather complicated basic principles of nature/physics in quite an intuitive way! I am sure you inspire not only me but very many of your viewers to think on, just like the quote by Einstein says with which your video ends. Your didactic skills are very well developed, so I hope you are or will become a teacher/professor 👍🏻🏅

[@jacksimpson-rogers1069]

The fact that even the gravitational waves predicted (I think) by the General Theory of Relativity have been "observed", pretty much reduces even the excellent "Star Trek" stories to Science Fantasy, rather than SF. The extremely original astrophysicist Fred Hoyle even wrote a BBC TV story/series "A For Andromeda" in which the invasion of Earth from a distant star system was accomplished solely by a message that arrived, naturally, at exactly the speed of light. It successfully tempted the lead character to build a computer, and against his advice a living creature and then a woman, with the intent that the computer would rule the world.

[@GabrielHodge]

I love looking at equations and comparing them or looking at the elements and what comprises them it really helps to be able to use the correct ones when its appropriate

[@theshibby1337]

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

[@tom23rd]

I never knew that very last point, and I'm dumbfounded. Thank you.

[@PhysicsExplainedVideos]

My pleasure! The idea that mass is an invariant quantity is often not emphasised in relativity lectures

[@wolfgangeuen9799]

This is by far the best pesentation/explanation of Einsteins formula E=mc2. Thank you very much.

[@stoyannikolov3165]

The best explanation so far! Congrats!

[@kaanasker4914]

This channel is the best, the other channels that claim they teach physics commonly don't mention the underlying principles and derivations of equations; they are concerned with the view count, not physics. But you, my friend, tell us the derivations using simple approaches and make us understand these beautiful equations without memorizing them. Thank you!

[@PhysicsExplainedVideos]

Wow, thanks!

[@gmtoomey]

This is a clear derivation that demonstrates important concepts.

[@lluis-ramonsalescasals9800]

Thanks for sharing, explained in a marvellous way! Congratulations.

[@mattlee3044]

A fantastic video. Way out of my mathematical league with the manipulations, but - overall - lovely to listen to. I shall have a few more runthroughs, to see if I can grasp the main deductions, again. Very uplifting. Matt Lee

[@abricot8794]

keep going!! love your videos from 🇫🇷

[@PhysicsExplainedVideos]

Thank you! Will do!

[@cederveltman]

Amazing video! Thank you for the incredibly detailed explanation! only this missing is how momentum works without mass, you told us that is was, but not why, maybe subject for another video? Just found your channel, this stuff is awesome!

[@PhysicsExplainedVideos]

Yes, I think I could have gone into a bit more detail there. Would make a good video!

[@paulrichards4854]

Awesome! I am blown away both by the result AND by the absolutely superb explanation. Thank you, thank you, thank you.

[@euva209]

NICE! I've seen a similar derivation and approach in Serway's Physics for Scientists and Engineers. Except he differentiates relativistic momentum with respect to velocity. But what I really appreciated from your video is something missing from Serway's: the chain rule involving the derivative of time in a moving frame with respect to that of time in a rest frame. That shows why p= gamma×rest mass× velocity.

[@aliabdaal]

Amazing video ❤

[@kenpee6751]

❤

[@maybepatrick369]

what is bro doing here

[@user-wg3fe3ij2o]

He getting flexed on by other yoo toobers

[@hareecionelson5875]

It's crazy that we learn these equations as separate nuggets when they are all linked together: Newton, Einstein, De Broglie all mixed together

[@feynmanschwingere_mc2270]

De Broglie is overrated

[@neil2444]

Honestly a little surprised we didn't see something from Euler.

[@justanotherguy469]

On the shoulders of Giants we stand!

[@hareecionelson5875]

@@feynmanschwingere_mc2270 We're enemies now

[@rdhunkins]

Well done presentation. I wish I had these videos to review before my classes in Physics back in the old days.

[@MrAbschied]

These lectures are simply outstanding. I hope that in a further lesson you will also discuss the way Einstein addressed the fact that Maxwell equations are not Lorenz-invariant.

[@dmhouse1024]

Perhaps I misunderstood, but I didn't understand the logical leap at 23:18. Why is it the case that gamma*mc^2 can be interpreted as the total energy of the object?

[@bobwhite137]

I had same question - all but that point was fairly clear

[@DSeyit]

Yes that part is not clear to me either. Why total energy is not kimetic energy?

[@DonkoXI]

This step alone is responsible for ending up with E=mc^2, but he doesn't explain it. It's a critical flaw in this derivation.

[@madhunayak165]

I wish I could like this video a million times over. Your teaching style is just so simple and elegant. Wish I had a teacher like this

[@bierundkippen720]

Well, he's explaining it wrong to make it simple.

[@sandstormdarude8125]

@@bierundkippen720 what did he explain wrong

[@ChetSitaula]

One of the best physics video i've ever watched. Amazing explanation🔥💪💪

[@carljames4862]

Brilliant. What i find especially fascinating is how each great name of physics build on eachother's discoveries to develop a new, and productive, understanding of our experiences.

[@eyebee-sea4444]

dp/dt -> enter tunnel -> rule x, trick y ... -> exit tunnel -> E=mc^2

[@__-1234]

Hi, thanks for a very nice video, it was really good to refresh this point. Only one thing I did not get, at 23:18, why can you say simply that E is the total energy of the object, which by the way sounds strange since it could have a potential energy too.

[@Miftahul_786]

Yes this is the only part that I don’t understand can someone help?

[@motokomotoko5988]

This guy is awesome. Great summary.

[@sameerrai4479]

Absolutely Fantastic video, Great work. 👏👏 And Very nice of you to put Sir. Einstein's quote at the end 😇

[@PhysicsExplainedVideos]

Glad you liked it!

[@ShadowZZZ]

awesome video, very illuminating on special relativity :)

[@PhysicsExplainedVideos]

Thank you very much!

[@OSAMAMAful]

What a clean explanation! Thank you very much!!

[@halamish1]

Another excellent video. Thank you very much. Keep up the good work!

[@juliankimmel6511]

Can't wait to watch this when i get home!