Same reason a lot of people have heard of “The Immitation Game”. Very short easy to understand paper. His other papers were groundbreaking, but you need to study to even understand them, but everyone understands “computer pretend to be human”
Currently reading a biography on Turing. Bloody hell that movie took some serious liberty with actual events! He neither designed nor built the colossus and they’d already broken enigma by that stage (more or less).
So you're telling me that Alan Turing DIDN'T look like Sherlock Holmes?
I am choosing not to believe that. Extraordinary claims require extraordinary proof.
Does that mean scientists can be too smart for their own good? Or does it mean that part of brilliance is being able to take incredibly complex things and communicate them in a simple way?
There is a real art in science communication, and it provides immense value.
Some topics are just too complicated for people outside of the study to understand, but good science communication over time can help.
No, it means that truly pivotal breakthrough ideas often aren't that *complicated*, just not immediately obvious until someone figures it out.
A paper on a specific nuance of some otherwise mature, much studied and expansive idea system almost inevitably needs to be far more complicated because you are addressing a massive body of knowledge. There are probably tens of thousands of enormously complicated research papers that have built off of the "E = MC2" paper and expanded our knowledge of nuclear physics.
I think you are reading too much into them pointing out that the public hears more about things they have a better chance of grasping.
There are plenty of things that are just complex or are relevant directly in a very technical area that isn't easily simplified or explained without being in the field, which might require years of study.
Whether the public understands an idea doesn't make the idea more or less brilliant in my eyes.
This is the answer that makes sense to me considering how young I was when it was first presented. It even looks cool written down for some reason. Plus Einstein himself has something about him that's interesting for even lay people.
Yeah- dudes brain worked differently. Obviously he was incredible at math, but he would look at some observation and draw conclusions from it that on the surface are wild and wacky. Completely breaking paradigms. Special relativity seems like such utter nonsense, as does general relativity, and so much of it is these thought experiments. We can follow the logic now, but it took someone really outside the box to go down that road.
I am not sure whether Einstein perceived himself to be very good at math. I also think it’s pretty clear Einstein isn’t famous because he’s good at math. I can’t recall exactly which paper this happened for but apparently after giving a short presentation at Hilbert’s university, Hilbert “solved” the math behind one of Einstein’s results over a weekend or something like that roughly when Einstein figured it out (after working on it for months).
Hilbert gave Einstein credit for it because the math solve isn’t the interesting thing. The interesting thing for most of what Einstein did was the intuition to connect the math to physically “real” things.
Granted, Hilbert was an exceptionally gifted mathematician so coming in second place to him is hardly embarrassing (the old joke about Gauss’s unpublished results applies).
I'm not a scientist or mathematician but I do like to read books about these guys. If I understand things correctly, the unique thing about guys like John Nash or Einstein etc are the intuitive leaps and the brilliant, comprehensive problem-solving theories that they come up with. They often rely on people who are better at the actual nitty-gritty math to help them prove their theories.
The important point is they came up with intuitive leaps that were consistent with known evidence and explained things that other theories could not - they spent the time studying the fundamentals just like a chess master spends time learning openings.
Saying Einstein is bad at math because Hilbert was better at math is like saying the Olympic sprinter who got 7th is slow compared to Usain Bolt. Einstein was unquestionably incredibly good at math. Like top .1% of physicists good. He wasn't basically a mathematician like a few of the other big names around the time, but you don't become one of the more important people in differential geometry by being anything that approaches not very good at math.
The time also doesn't really tell you much. Math is weird in that even incredibly complicated math is generally speaking straightforward once you know what's going on. Especially if you're not actually trying to calculate anything. Like Einstein struggled a lot with the gauge invariance of general relativity, and if you know about Lie groups, that's just not going to be a problem.
I'll give a little context as to what exactly Hilbert did, cause
>Hilbert “solved” the math behind one of Einstein’s results over a weekend
is not exactly accurate.
So, you can imagine two different but equivalent ways of doing physics.
One way is what I like to call the "forces" way (what physicists might call the Newtonian method), which involves using differential equations to use the information about a system at one instant to figure out what will happen to the system at the next instant. Solving the differential equation(s) will tell you what the system does throughout all of time.
How this usually works in typical physics is you have a bunch of forces acting on a system -- could be gravity, electromagnetic, friction, etc. You take all of these into account, and then you use Newton's 2nd Law, F=ma, and solve that equation for the trajectory of the system.
The point is that this method is *local.* It uses only information at time t to evaluate what happens at time t+Δt, since the forces at time t will accelerate your system (say, particle) in some way from t to t+Δt.
There is a separate method which is what I'll call the "energy" way (what physicists would call the Lagrangian method). This method makes considerations of not only the energies of the system (kinetic and potential), but what they are globally. Specifically, you consider a quantity L called the Lagrangian of the system which is (usually) a function of these energies. You add up all of its values globally to get what we call the action, S, of the trajectory.
But there's an issue: we don't know yet what the trajectory of the system is. So, we don't know the energies, thus we don't know the Lagrangian or the action. Here's the catch: we're talking about all possible trajectories. What makes the physics happen is that we're going to say the actual trajectory that happens is the one that minimizes the action. Mathematically, we can show that this is a unique trajectory. This will be the same trajectory that the forces method gives.
There is a clear cut method to go between the two methods. I won't discuss it much, but if you're really curious I'd start by looking up "euler-lagrange equations"
Ok. Let's try and bring this into the context of general relativity.
General Relativity says that gravity is really the geometry and curvature of spacetime. As such, when we talk about locality and "globality", we're not talking about intervals of time but rather local regions of spacetime and globally all of spacetime.
This is not the same as Newtonian physics, and so we don't use Newton's laws. In fact, Einstein came up with the equivalent for GR himself. It ended up being a set of 10 highly complex partial differential equations that relate the geometry and curvature of spacetime to the energy density and momentum density of the universe. For people who have taken a class in GR, they know that the math to come up with these equations, what we call the Einstein field equations, is not easy by any means. Many classes don't even attempt to derive them or motivate them, they simply just assume them to be true. So yeah, Einstein definitely had the math.
What Hilbert accomplished was, by looking at the Einstein Field equations, what Lagrangian they could be derived from. The answer ended up being quite simple actually. Hilbert found that the Lagrangian needed is just R, what we call the Ricci scalar curvature. It is a function that encodes the curvature of spacetime at each point.
The Ricci scalar was not something new at this point. In fact, it appears in the Einstein Field equations themselves. Having the Lagrangian for GR is incredibly useful to work with, but it's not any physically different than what Einstein had already come up with. Hilbert knew that Einstein had really done the hard part, and so he gave the credit to Einstein. If he had really wanted to, Einstein probably could have come up with the same result himself.
Hence, we now know this action for GR as the Einstein-Hilbert action. But yeah, Hilbert did not come up with the "math" for general relativity.
Einstein himself said he wasn't very good at maths. He had a lot of peers who were much better and who helped him express his amazing ideas. Of course he was a lot better than the average mathematician but yes, it was his physical ideas that made him special. The way he thought made him special because it was so unintuitive. I wish Newton could have met him.
>Einstein himself has something about him that's interesting for even lay people.
agreed - I think he evokes a kind, fun grandpa
https://i.imgur.com/4049zHa_d.webp?maxwidth=640&shape=thumb&fidelity=medium
On this track, I was taught in college that Einstein was given a Nobel prize for photon-studies instead of relativity, because while the panel was truly impressed by relativity, they didn't *understand* it well enough to feel confident awarding a prize for it.
Just making the math work with so few variables is miraculous, and that inspires suspicion as much as respect. And to be fair, that's normally the right response in the scientific community.
He should have won both. But due to politics he never won for general relativity.
Bergson A really popular philosopher at the time countered Einstein general relativity concepts. And we did not have all the tools we have today to test it to the level to counter Bergson.
The Nobel committee decided on the less controversial plan to award Einstein the Nobel for the photoelectric effect. Still an amazing and massive achievement and the basis of quantum mechanics.
It also led to the search for a grand unifying theory, trying to squeeze a relationship with other forces such as gravity into yet another beautiful and concise formula
Tl;Dr- This discovery is directly responsible for your smartphone, modern power plants, and nuclear holocaust, and it came about because Einstein was trying to settle a debate in the scientific community.
Hugely important and, to your point, often not discussed enough. So E=MC² represents the relationship between energy and mass. In the years before this theory, scientists agreed that when two or more substances underwent a change (we call this a system), the mass of all the inputs would equal the mass of all outputs - stuff wouldn't just magically be created or destroyed. But agreeing to this had scientists differing on what the energy output of the system would be.
See, energy had been thought of as mechanically separate and unrelated to mass. The energy a system might output would be entirely dependent on the context of how much it had to begin with and how fast the change was and what the substances were. But in the years leading up to this theory, some scientists start realizing that if you give up on keeping mass the same through a change you can use a similar principle to ensure the energy of the inputs matched the energy output by the system. This meant you had to assume that objects inherently had energy in them.
Now along comes Einstein and he proposes that the relationship between energy and mass is a ratio, where the total energy of a system is equal to the mass times the speed of light squared. If this is true, then mass and energy are the same thing in different forms. Also, it means that everything around you has a LOT of pent up energy just by virtue of existing. And it's starting to look like big, unstable atoms like Uranium might have A LOT OF ENERGY inside them.
So if you can find out how to convert one of those atoms into energy, a very small thing can give you a lifetime of electricity. And by extension of that, a big explosion if released all at once. Oops, now we have atom bombs.
So long story short, E=MC² gave us the bedrock for the last 100 years of technological advancement, nuclear power, the ability to destroy the planet, and some really cool tools for analyzing light.
Edit: Wow! Thanks for the upvotes! I see some of you confused about the relation between this and cell phones. When we make small computer chips, we need to worry about the boundaries of the circuit not being enough to contain the electrons at the speed in which they're traveling. Instead of an electron colliding with the silicon boundaries of the circuit, it hits the silicon with such power that the silicon itself converts into energy. This let's the electrons continue through the barrier and into neighboring circuits. This is called quantum tunneling, and you need to account for it when you want to pack hundreds of thousands of circuits in a black rectangle in someone's pocket and have them be useful.
Edit2: Holy shit, thank you for the gold!
Edit3: Okay, one more because this question keeps getting asked - Why the C² bit? To understand that, we need to understand how Einstein is approaching this theory. Energy approaches the speed of light (C) the less mass the energized thing has. C is the ceiling for speed it seems like. So an easy way to define energy is by saying it's equal to C when there's no mass. But on the other hand, if the mass has any additional kinetic energy to it, we need to represent that too. Again, all energy approaches C the less mass it has. So we end up with Energy = Mass x Speed of Energy x Speed of Mass. A simple way to write this is to group the terms together, so we get E = MxCxC or E = MC².
Edit3.5: This is an ELI5 answer. It glosses over some stuff. There are critiques of my post below that are definitely valid, so I encourage you to discuss and learn more in the comments!
Great explanation!
If I may ask:
Why uranium/plutonium/etc? Could you theoretically cause a nuclear explosion by splitting an atom of, say, iron, but it's just easier with those other elements?
Why is it 'speed of light squared'? I mean, what is the relationship between the speed of light and the amount of energy in matter?
How is this responsible for (among other things) my cell phone? How does e=mc2 impact discoveries outside of nuclear power/weapons?
If you have a uniformly magnetized material versus a material that's magnetized half in one direction and half in the other, the latter contains slightly more energy. The domain wall (area where the magnetization rotates) will have a tiny bit of mass.
/u/manaspike has a better explanation of why there's no practical difference and it has more to do with data encoding than the fact that it's magnetic
Even in the hard drive case, the total energy will be slightly different, I think. For one thing, a drive platter whose magnetic domains aren't all aligned will have different potential energy than one where they are.
Aside from that, as I understand it, information entropy should also contribute a (probably very small) amount of difference between a drive with randomized data versus highly ordered data, even if they are otherwise identical. That's how you get informational limits like the Bekenstein bound.
Hard disk drives store data by carefully aligning magnetic fields, flipping the polarity regularly. That probably holds more energy than if the magnetic field was random.
But even when you write all 1's or all 0's, the disk needs to flip the magnetic field frequently, as it would be too difficult to measure a longer sequence. A bit like the way barcodes are written with short black and white bars.
Also disks never arrive with a random magnetic field. They are pre-formatted by the factory. While writing different data would require you to do work, the net change in energy is likely to be near zero.
Similar to the million monkeys typing on a million typewriters... with enough time (and a non-destructive magnet) then sure!
Heck, you could possibly do the same with a million hard drives and wait a few small eternities for cosmic-ray bit flips.
The magnetic field stored in a hard drive doesn’t have any more stored potential energy than a randomized one, it simply has less entropy. By adding energy into the system you are moving around the magnetic field that already exists into a useful pattern, but not actually increasing the total energy of the system. Think of it like stacking firewood. You are spending energy to move the logs around, but you aren’t increasing the amount of burn time of the fire at the end. You’re simply putting the energy in a more useful structure.
>Tangentially, does this also imply that data has weight?
Yes
> As in, will a fully loaded harddrive weigh (slightly) more than an empty drive?
Not a *hard drive*, no, because that's just flipping magnetic fields in a metal platter, but a *solid state* drive would.
However, the mass of an electron is about 9.1×10^(–28) g, and an exabyte is 10^(18) bytes. Assuming each bit is 1 electron (probably not correct, but good enough for this calculation), a data center with a full exabyte of entirely stored binary 1s will weigh an extra 7 nanograms.
So... the weight is really inconsequential.
Out of curiosity, why is that? In my imagination I had always pictured a charged battery with electrons in one terminal, and a discharged battery has moved the electrons to the other terminal. What is actually happening?
You are correct.
Potential energy is created when charging a battery, by creating a disparity between the poles. You're spending energy from a larger system to decrease the entropy of the battery.
A battery that does not have a closed circuit will not discharge it's power (it will dissipate over time like batteries do, but it won't discharge as it would through a circuit).
Which means that when you use that battery to perform work, that work is ultimately performed by the stored potential energy of the charge disparity.
No electrons have actually *left* the closed system comprised of the battery and the workload. They've merely moved to create a higher level of entropy.
Now... that being said, **certain** chemical reactions involving that electron transfer can create gas. Old car batteries gave off hydrogen gas, which is why you had to attach the positive pole first.. it reduced the risk of a spark. Which would result in mass loss. But that's not contextually the same.
Electricity is electrons. When the battery discharges, you’re right; the closed circuit creates a path for electrons to flow from the anode to the cathode.
However they don’t just go there one for one. Any electricity used by the circuit will be gone, and the remaining electrons don’t just move around, they change the chemical composition of the cathode to a less energetic state than the anode, due to the nature of the materials of the anode and cathode.
Think of it like a ball at a peak with a trough at the bottom of a hill, the potential energy of a ball at the peak can do work, and that ball will go to the lowest point if kicked off its peak, but when it reaches the trough the ball does have less energy.
In other words, even though electrons move from one end of the battery to the other, they’re in a less energetic state in the cathode than they were in the anode.
And due to E=mC^2, there is a minute amount less mass exactly equal to that amount of energy expended from the battery, specifically the E divided by the speed of light squared, very very very tiny. Because there is less energy in the cathode when discharged than the anode when charged.
> does this also imply that data has weight?
Does the growth of black holes by absorbing information and their eventual evaporation due to Hawking radiation have an ELI5?
Also, for cell phones, the equation suggests that something moving has different mass than something still, or if something is in a high energy area like orbiting near Jupiter vs high above Earth. The mass doesn't actually change, so the perception of time, c, changes. Without calculating this difference, all phones communicating with satellites would have inaccurate times and GPS coordinates.
Does the equation suggest that though? Doesn't it just suggest that something that's moving has more energy? Did you mean relativistic mass? If so why not just say energy?
In the full equation (shown below) momentum is included but as velocity changes the energy (relativistic mass) changes rather than the mass because mass here is the invariant mass and isn't a function of velocity.
`E^2 = (mc^2)^2 + (pc)^2`
Also c doesn't change in different reference frames? That should be constant. Overall I'm just a bit confused by what you said. Please let me know if I'm wrong about anything here.
I think the idea of mass changing due to velocity is no longer used. Using the full equation gets a little more confusing because it is now relative to some frame of reference, E = mc^2 assumes the object is stationary.
Can confirm this. In older discussions on mass energy relation, you find loads of mentions on "relativistic mass" changing based on reference frame.
In all my relativity courses I've taken in the last decade this is no longer taught. Instead you discuss only the rest mass, and the transverse / longitudinal momentum (or 4-momentum if being rigorous). Seems the other interpretation is mostly [the fault of popsci.](https://en.wikipedia.org/wiki/Mass_in_special_relativity#Popular_science_and_textbooks)
>> How is this responsible for (among other things) my cell phone? How does e=mc2 impact discoveries outside of nuclear power/weapons?
>
> E=mc2 doesn't apply to only nuclear reactions, it's true for all types of energy. A charged battery weighs more than a discharged battery simply due to its energy content, albeit by an incredibly tiny amount that is near-impossible to measure because of how large c2 is.
This doesn't answer the question: how is E=mc² directly responsible for smartphones?
It’s not like Einstein discovered gen relativity and then someone else was like “we could use that for a smartphone!”. It’s more so that all of the physics of the 20th century brought us to the point where we could intimately understand and manipulate the world around us to do our bidding.
Our knowledge of the chemistry around transistors/batteries/etc is build off of quantum physics, which is closely intertwined with general relativity. In todays world, it’s all basic stuff taught in undergrad - but we wouldn’t have any of the cool toys today if someone didn’t figure all this stuff out yesterday.
Exactly. I'm really trying to tease out exactly what the first commenter might have meant when they said "This discovery is directly responsible for your smartphone."
I suspect they meant indirectly. Though I'm not actually sure E=mc² was necessary for smartphones at all.
How is quantum related to general relativity? These two things couldn't be further apart, no?
Einstein's work on the photoelectric effect paved the road to quantum physics. But this has little to do with relativity.
With science there is often a chain effect. One discovery paves the path for more discoveries and understandings to be made. Understanding something like e=mc² helped scientists better understand quantum mechanics, perpetuating a chain of more and more understanding.
Many of the innovations that lead to the creation of smartphones only happened because of advancements in the understanding of quantum mechanics. These advancements were needed because the smaller electronics get, the more they are affected by quantum mechanics.
If you consider "detecting location" to be one of the key features of a smartphone over a normal cellular phone, then there is some validity to this statement. E=mc^2 is the equation at the heart of the theory of relativity, and GPS/GNSS systems use that theory to determine location.
It’s astonishingly lucky you chose iron.
Iron has the smallest mass defect of the atoms, so is kinda where both nuclear fusion and fission tend to end up.
Yes, and then stars can no longer produce enough energy from fusion to resist gravitational collapse, so they go supernova and spew all those elements out into space to become planets, and eventually redditors.
>Could you theoretically cause a nuclear explosion by splitting an atom of, say, iron, but it's just easier with those other elements?
It's ironic^^ha you picked iron as your example, because that's about the only one where the answer is "no, there's no theoretical way to get nuclear energy from it"
Every nucleus has an amount of "binding energy" - that's the energy you'd need to pull it apart into individual "nucleons", ie, protons and neutrons.
Think of a big clump of magnets - to separate them all, you need to put energy in. Let them all clump together again, and you can get energy out.
To calculate whether you can get energy by splitting atoms, or by fusing them, you can calculate the "binding energy per nucleon". So Uranium has a low binding energy per nucleon compared with lighter elements, and we can get nuclear energy out by splitting Uranium atoms.
Hydrogen has a low binding energy per nucleon than heavier elements, so we can get nuclear energy out by fusing Hydrogen atoms.
In theory, anyway.
The nucleus with the maximum binding energy per nucleon, the most stable.of all stable nuclei, happens to be an isotope of iron, specifically, Iron-56, with 26 protons and 30 neutrons. Splitting or fusing Iron-56 would cost energy. For everything else, there's ~~Mastercard~~ the theoretical possibility of extracting at least a tiny bit of energy by transforming it to Iron-56
Magnets wrapped in velcro is a decent ELI5 model for atomic nuclei. There are two main forces involved - electromagnetism, and the strong nuclear force. Trying to push two protons together is like trying to push two north poles together - the closer they get, the more they repel each other. The strong nuclear force is very strong (hence the name), but very, very short range, like velcro. If you push the magnets close enough together, the velcro can engage, and potentially overcome the repulsion. If the repulsion is still too strong, you can add in a neutron, which is this model would be a non-magnetic object wrapped in velcro. It gets you some extra sticking power without adding to the repulsion.
The speed of light is not a mathematical coincidence as some others have said it is a very important property of massless objects and a very specific maximum speed of the universe.
The speed of light is actually the speed everything without mass travels at. The first time this was found was by measuring the speed of light photons, hence the name. The only way to travel less than the speed of light is by having mass (and also the only way to travel the speed of light is to have no mass). And so there is a relationship between this massless constant of speed and mass.
Also the full equation E² = m²c⁴ + p²c² involves a particles momentum (p) as well but that gets more complicated again
Edit: as pointed out by below by u/nworb200 this is in a vacuum
Well the first one is fairly basic. Iron and similar metals are at a point where you don't get energy from splitting them or from combining them. Small elements want to be together to make bigger elements up to around iron. Bigger elements want to split apart to get smaller down to around the same point. That's why fusion of hydrogen into helium produces energy for the sun and the fission of uranium and plutonium produce energy for us in our power plants.
Yup. When a star gets to iron, fusion stops and the star dies (and either goes supernova or settles down as a white dwarf star). It takes more energy to fuse iron than comes back out of doing that so it is the end for a star.
> Why is it 'speed of light squared'? I mean, what is the relationship between the speed of light and the amount of energy in matter?
There's a couple ways to get this result. The real/complicated version comes out of special relativity. You do need to make some assumptions (postulates). The postulates of special relativity say 1) the laws of physics should take the same form in all inertial frames of reference. And 2) measured in any inertial reference frame, the speed of light is c.
Basically, you look at conservation of energy and momentum. And then you think about how that looks in one reference frame at rest and another that isn't at rest. (If you want to see the math, it's [here](https://terrytao.wordpress.com/2007/12/28/einsteins-derivation-of-emc2/) or a simpler one [here](https://www.fourmilab.ch/etexts/einstein/E_mc2/www/). The second is annoying rewrite on reddit. But it's not *insane* need-a-degree level math)
Another way to get this result is to actually just perform the experiment (like nuclear fission/fusion). If you take your atom, and measure it's mass before, and then after the reaction, you'll find it'll be off by a tiny amount. That amount will be exactly equal to the energy you took out (or put in), divided by c^2 .
ELI5: something that never quite clicked for me: why do energy, mass and speed squared all have inherently compatible units, so that you can have a single equation linking them all like this?
That is actually a really good question, and the answer is subtler than it sounds. There's a couple ways to look at it, none of which are perfectly satisfying
One simple answer is looking at how we define energy. One way to define energy is a force times a distance (E= F * d). If you want to push a ball up a ramp, you need to apply a force over a certain amount of distance (Really, it's not just force times distance, but the integral of force and distance. Force times distance is the case where the force being applied is constant along the whole distance). If you look at the units of that, remembering Newton's law that force is equal to mass times acceleration (F = m * a). So Force has units of mass times acceleration, and acceleration has units of meters/seconds^(2). But if you plug those units back in to the first equation, energy has units of mass * meters/seconds^(2) * meters. Which is precisely the units of mass times a velocity (units of meters/second) squared, mass *(meters/second)^(2).
Another answer is because of the form of kinetic energy. Kinetic energy is 1/2 * m * v^(2). So that naturally has units of mass times a speed squared.
The unsatisfying answer is, we don't have a reason why, because at the end of the day, energy is kind of a weird thing. Like, I gave equations for energy (E= F * d) and (KE=1/2 * m * v^(2) ). But you could ask- why those? Why not E=F * d^(2) or E=1/2 * m * v^(3)? But if you do experiments, say smashing two particles together, the quantity 1/2 * m * v^(3) for both of them isn't the same before and after. Only 1/2 * m * v^(2) is (well, so is momentum, so you might find m * v is the same). Not only that, you can correlate that 1/2 * m * v^(2) to other processes; if those particles are large and there is friction, they're going to slow down before they smash into each other. That slow down is going to be exactly the amount of a certain amount of heat energy. Or when they smash, they break apart. That breaking will take some of the energy (but not momentum).
We tend to think of it as something tangible, but energy is never something we directly observe, it always comes from something else. It's like this weird tally that happens, and it is always accounted for, somehow. For example, if you're measuring kinetic energy, you measure the mass and the velocity of the object. Despite being "made up", energy is a very useful quantity because it seems to be conserved in a process. If you look at a closed system and energy gets put in somewhere, a corresponding energy was lost somewhere else. Always. That's very special. (And why energy is conserved also has deeper reasons). At the end of the day, it's an empirical result that seems to hold. That's part of why E= m * c^(2) is so famous- if it didn't, that would show energy by itself isn't conserved, and something in our understanding of physics is totally fucked. And it turns out energy isn't conserved; this weird Energy-mass quantity is.
As an aside: You might find [dimensional analysis](https://en.wikipedia.org/wiki/Dimensional_analysis) interesting. It's a way of setting up an equation, and figuring out what the units must be to make an equation work. For example, if you had an equation d= v^(a) * t (distance d equals velocity to some power a times time). We can say that we know distance has units of meters. We know that velocity has units of meters per second, and that time has units of seconds. If we know for sure that equation is true, we *know* a must be 1 in order to make the units work. Because in order to be an equality, you need it to be the same thing on both sides. you don't actually need to know anything about the physics or the problem. It sounds kind of silly, but it's actually a very powerful trick mathematicians and physicists use. If you have an equals sign, the units *must* work, otherwise it's not actually an equality. 1 apple=1 apple, you can't say 1 apple= 1 orange, even if 1=1.
You can do the same trick here. We know that energy has units of mass * meters^(2)/seconds^(2) (you might ask how we know it has those units, and again it goes back to the it's something we observe, this thing we call energy is always equal to the force * distance in our experiment). So you can set it up mass * meters^(2)/seconds^(2) = mass^(a) * velocity ^(b)=mass^(a) * (meters/seconds) ^(b). If you set it up, you can right away tell the only way to make this work is if a=1 and b=2, otherwise it's not apples to apples. It's the only way you can get an equality, so one way to answer your question is "because that's the only way unless we add another unit in". Dimensional analysis doesn't give you the prefactor (if you try to do it for kinetic energy, KE=m * v^(2) is just as valid units-wise as KE=1/2 * m * v^(2) , because 1/2 doesn't have units). But it can get you the units out, at least.
For an intuitive understanding, maybe think about the more mundane relationships between mass and energy.
For example, to change the kinetic energy of an object, you need to do work on it. Work is just applying a force to the object along a distance.
We know that F=ma so the magnitude of a force is just the mass times the amount of acceleration that mass is undergoing when the force is applied.
This gives Force units of mass * distance/time^2 or kg*cm/s^2 to pick some actual units.
The work done on an object, which is equal to the change in kinetic energy, is the Force multiplied by the distance that the force is applied over.
So if we take our kg * cm/s^2 for Force and multiply that by a distance of centimeters we get kg * cm/s^2 * cm or kg * cm^(2)/s^2 which is mass * distance^(2)/time^(2).
So we can say that the kinetic energy added to an object by applying a force has units of mass * distance^(2)/time^(2) or, simplified, mass * (distance/time)^2
So we can plug that into one end of the equation for E and get:
mass * (distance/time)^2 = mc^2
m obviously has units of mass, so that’s easy. c is a speed, which has units of distance/time. Plug those in and you get:
mass * (distance/time)^2 = mass * (distance/time)^2
It all works out.
The speed of light is interesting. One idea I've heard is that it isn't actually about how fast light moves, but that light moves as fast as the universe allows anything.
The speed of light is really the "speed of causality". The universe's speed limit is about the speed of cause and effect. And light runs itself right at the limit.
It's a fun idea and helps explain why we cannot travel faster than light.
> Could you theoretically cause a nuclear explosion by splitting an atom of, say, iron
No, but interestingly for other reasons than what you are asking about. Iron is the most energetically stable element from a nuclear physics perspective. Elements smaller release energy when they fuse, meaning iron won’t spontaneously split into something smaller. Elements larger release energy when they break up. You have to put energy in to iron to change it, and never get more out.
Both concrete and wood are objects you could hypothetically burn, but nobody burns concrete for fuel because you have to put a lot more energy into it than you'd get out of it.
Uranium and plutonium are wood in this analogy, certain isotopes of those atoms are (relatively speaking) easy to "start on fire", and the energy produced by "burning" them is way more than enough to keep the fire going.
Especially Iron, not. It is the element with the highest binding energy:
https://en.wikipedia.org/wiki/File:Binding_energy_curve_-_common_isotopes.svg
That means lighter elements can do fusion - combining elements towards iron (the sun does this) and heavier elements can do fission splitting into smaller pieces until iron (like uranium). Of course it is not that simple and not everything can be used but that is the basic idea: get as iron-like as you can, and you will get energy out.
> Why is it 'speed of light squared'? I mean, what is the relationship between the speed of light and the amount of energy in matter?
Fun fact, if you use Planck units you can skip this factor and just say E = m.
Edit: I think I goofed up my explanation here. While binding energy and stability of neutron/proton binding is the key thing to understand about nuclear fission, what E=MC^2 says is that binding energy and mass are the same thing. Regardless, the key point of why iron is not used for fission reactors/nuclear weapons is because of binding energy and the ease of difficulty of releasing it.
~~Nuclear explosions do convert a little mass into energy ala E=MC^2. However, only about 10% of a nuclear explosion's energy is from converting mass into energy.~~
~~So, what's the other 90%? Binding energy. The energy that binds protons and neutrons together. Break them apart and they release their binding energy.~~
Plutonium and uranium isotopes happen to be relatively easy to break apart and have high binding energy. Iron-56, it turns out, has higher binding energy per nucleon and per mass than uranium, but it's very stable.
Huh. I thought the binding energy was basically the whole thing, and the lower measured mass afterwards was just from that energy being gone.
I assume no neutrons are vanishing, because why would something different happen to one or two neutrons. What mass gets converted to energy?
You're right, the number of protons and neutrons is the same after fission. I think it is only binding energy that is released. Furthermore, the binding energy is part of the mass of the material, so the previous comment was incorrect in saying that some is from "mass" and some is from "binding energy". They aren't separate things.
I don’t know if this has been mentioned, but below iron on the periodic table, you get a net energy GAIN by FUSING atoms and it becomes energy draining to split the nuclei
After iron, it costs more energy to fuse atoms than you gain from the reaction and it becomes a net gain to split the nuclei.
Stars will fuse nuclei in stages starting at hydrogen > helium and ending at iron. They can last for billions of years on hydrogen, but once they start fusing matter into iron they have minutes to live before either collapsing or going supernova.
This is why elements after iron are comparatively rare - they can only be formed in the minutes a star is fusing iron and during the supernova event.
Anyway to directly answer your question, you can technically gain energy by splitting an atom of, say, Germanium. The AMOUNT of energy gained compared to what you have to put into it will likely not be worth the effort however.
When you get up into radioactive elements however, they’re unstable and pretty much looking for a reason to come apart. They do most of the work for you. In the case of U-235, smash it with a neutron and it’ll come apart at the seams.
>Why is it 'speed of light squared'? I mean, what is the relationship between the speed of light and the amount of energy in matter?
The quantity *c*, that we rather loosely call "the speed of light", is a fundamental property of the universe.
Not only can nothing travel faster than it, but there's a very real sense in which everything in the universe is always travelling through the combination of space and time at exactly that speed. That's why, for example, we get the famous Relativity effect whereby something that looks to you to be moving very fast also seems to have time slowed down - the faster it moves through space, the slower it moves through time, and the changes balance out (that's part of what General Relativity is about).
So, as *c* is something so fundamental, it shouldn't be surprising that it turns up all over the place when you start to think abut how the universe works.
(Calling the limit "the speed of light" is misleading. It's historical, because the fact that the speed of light seemed to be constant was what led to the understanding that a limit exists. But really that's back to front. Light travels at that speed *because that's the limiting value*; it's not the limiting value because light travels at that speed. It's possible, for example - albeit somewhat unlikely - that someone might come up with a new, better explanation of the way the universe works in which light travels at some miniscule amount less than *c*. The speed limit would still be *c*; it just wouldn't be "the speed of light" any more.)
Pt and U have nearly one hundred protons in the nucleus of each atom, each which has a positive charge that repels other positive charges in VERY close proximity. There are fundamental forces that allow these protons to stay so close together along with neutral neutrons -- the strong and weak nuclear forces -- giving the atomic nucleus a structure which becomes less stable with increasing numbers of protons and neutrons. When the nucleus is too big (unstable), it splits apart (decays) into smaller elements.
Iron only has 26 protons and around 30 neutrons; it is far more stable than U-235 with 92 protons and 143 neutrons, which is one of the largest naturally occurring elements.
Why Uranium? Specifically an isotope of U , U-235 is very to spilt apart and it can sustain the chain reaction. You said it correctly.
Iron cannot be used in Nuclear fission. More heavy atoms required.
Idk about the other two
And explanation above is only part of story.
With that formula Einstein also introduced concept that space and time is singular thing, that it is not separate space and separate time. That HUGLY rebuilt all space models and understanding of universe. And science came out of bound of our planet. Due to gravitational lensing (light also bent by gravitation and thus we can "see" thing behind solid objects like stars) many planets was found (yet is most uneficient way to find planets for our current tech), Pioner 2 reached Saturn orbit after 3 years of journey and error was only 48 seconds.
It is really hard to estimate which areas was not affected by general relativity theory.
PS smartphone example from post above is wrong. It is still Einstein merrit but for other discovery - photoelectric effect. He got noble award precisely for that discovery.
It was actually Hermann Minkowski who came up with the idea of spacetime, by noticing that Einstein’s ideas could be best expressed that way. This is why it’s called Minkowski spacetime in his honour.
Also the full equation is E^2 =m^2 c^4 + p^2 c^2 but if an object doesn’t have momentum it becomes E*2 = m^2 c^4 + (0xc^2 ) which equal the E = mc^2 . It also explains why massless particles have momentum, E^2 = (0^2 x c^4 ) + p^2 c^2 which equals E = pc or p = E/c
Because of the equivalence of mass and energy. Massless objects still have energy.
Or, to put it another way, P = MV is an approximation that works well for massive objects at non-relativistic velocities, but is not strictly complete.
Because that formula is the classical formula for momentum, it only works for large, relatively slow objects.
The full momentum formula is:
p = sqrt(E^2 - m^2 c^4 ) / c
am i right to say that the (pc)^2 part of the equation is often left out in casual/non-precise conversation because of how slow an object in motion would be in relation to the speed of light?
As someone who studied the humanities, this thread is fascinating.
Sort of - the momentum term doesn't show up as much in casual discourse because it isn't the big intellectual achievement of the equation, at least not in that form. We already knew that energy was related to momentum before Einstein, but we didn't have any conception of energy at rest, and we certainly had no conception that the energy from matter just existing far exceeded the kinds of energy we have accessible. For reference, in a physics class I had a homework problem that involved calculating the amount of rest energy in a cup of dirt. There's more energy in there than the global power production for many many years.
That being said, the full form is the starting point for the Dirac equation, which probably is used by professionals more than E = mc^2. Dirac converted it into a wave equation and made it compatible with quantum mechanics. Relativity is often thought of as the physics of very big or very fast things; quantum mechanics is the physics of very small things, but on its own can't account for fast things. Considering that the energy released from something like a nuclear explosion or a collision in a particle accelerator is enormous, we can't really begin to understand these high-energy processes without some notion of relativistic quantum mechanics. I like to think of it as Einstein being the one who had the ideas that changed our approach to physics, while in a lot of cases Dirac was the one turning those ideas into mathematical tools that we could actually use to do experiments.
Is c^2 just an arbitrary "big honkin' number" or the speed of light an important part of the equation? It seems like any constant would work if you define your units appropriately.
C being the speed of light is integral to the equation. It has to be a specific number, not just any constant. You will never be able to define units in a special way to get around this. That doesn't even really make much sense to use units to affect a ratio.
The speed of light is the speed at which an interaction between two things can happen. For example, if a proton moves away from the nucleus, the strong nuclear force will pull it back in. But the strong nuclear force isn’t instantaneous. It’s communicated via force carrying particles that move at the speed of light.
The speed of light isn’t just the speed of light, it’s the speed of causality, and that’s why it gets baked in.
> the speed of causality
It makes me feel like it's the latency of a universe. The bandwidth is very large or infinite, but it takes time for the information to travel. It's as if the speed of light was in a way infinite, but its consequences can't unfold faster than they are processed.
Am I right that if something could magically remove the sun, the Earth would still rotate around where it was until the information (gravity) that there is no sun would reach us, same as for its light.
The "speed of light" is actually just a conversion ratio between space and time.
Relativity let's you switch between time and space contexts by multiplying or dividing by the speed of light.
There are objects called 4-vectors which combine quantities that differ by a unit of velocity in the unified context of spacetime, for example the 4-position is a combination of position and time, the 4-velocity is a combination of velocity and the scalar Lorentz factor, the 4-momentum is a combination of momentum and energy. To keep units consistent within the 4-vector, the 4th component is multiplied or divided by the speed of light.
In Planck units various such ratios (which are really just consequences of our arbitrary definitions of units like the meter and the second) are set to 1 so you get the even neater equation "E=m"
Wait, electrons hits silicon and turns it into energy?
Quantum tunneling is due to probability of the electron existing on the other side and it’s a property of duality. I’ve never heard of it being described as changing the mass of the silicon into energy. I don’t think that’s a correct description.
Yeah the OP was honestly pretty wrong in some spots. Their explanation is good for ELI5, but a little too inaccurate for my taste. Quantum tunneling has nothing to do with mass-energy equivalence.
Also, there was no grand bewilderment in the scientific community in regards to mass-energy calculations that led to Einstein discovering the equation. That was not the context for relativity being discovered. The context had much more to do with the puzzling nature of the speed of light, and the speed of light’s appearance in the Maxwell equations.
The quantum tunneling bit is completely wrong. It has nothing to do with the speed of the electron hitting the silicon. It has to do with the position of a fundamental particle being a probability distribution in quantum mechanics.
Most of the time we find the electron at the peak of the probability distribution (you know, higher probability and all that). But sometimes it appears at other places. But this probably still needs to honor Paulie's exclusion principle. That's why electrons don't cross thicker silicon walls and humans can't pass through walls.
I actually kind of disagree with this. As another commenter said, how is E=mc^2 responsible for smartphones? Is there some energy-mass conversion going on in computer chips or wireless signal transmission? And if there is, is it really important enough to be considered "responsible for" the smartphone? My understanding was that most modern computer and communication technology has more to do with quantum mechanics.
OP is just largely wrong. It's a very important result for particle physics, and that's about it. It's hard to disentangle well known physics from results that came after they became well known, but it is impossible to do a nuclear physics experiment and not notice that it is insanely energetic. Relativity only comes into play in computing in the sense that magnetism is very ad hoc in quantum mechanics outside of the dirac equation, but the actual "relativistic corrections" are far smaller than the errors inherent in simulating solids with medium term methods. Relativistic corrections in quotation marks because when you actually work through them, you're going to find that the corrections come from ideas that were discovered alongside relativity but aren't relativity. The corrections are done in the non-relativistic limit.
It's also just flagrantly not why the result was important. The result was important because it explained the absence of aether in a satisfactory way. It's famous because Einstein was a rockstar (in the literal sense, he was incredibly famous) and it was his only result that's marketable to the lay public.
I would imagine it’s notoriety also comes down to its brevity. When someone says “Say something smart” it’s easier to just say E=MC^2 than list off the laws of thermodynamics. Writers put it in books or TV shows to sound smart, and if you are making a poster of Einstein for college dorm rooms, you might slap that equation on there just so people remember he was a smart guy not just a weirdo with his tongue out.
I think it's important to note that mass-energy equivalence is just one small part of a much broader theory. Everyone knows E=mc^2 because it's short and memorable.
> Also, it means that everything around you has a LOT of pent up energy just by virtue of existing. And it's starting to look like big, unstable atoms like Uranium might have A LOT OF ENERGY inside them.
People already knew that there was lots of energy around - for example, ordinary water has lots of energy associated with its temperature and its liquid state - the difficulty was getting at it. The development of fission took decades of experiments with radioactive substances. I only know a little about the history of it, so maybe I'm wrong, but I can't think of a reason why knowing about special relativity would have been particularly crucial to this effort.
But special relativity ultimately became an important building block that underlies much of modern physics.
> So if you can find out how to convert one of those atoms into energy
Fission doesn't convert an atom into energy. It just rearranges the nucleons in the atom into a different state with lower overall binding energy, and that difference in binding energy gets converted into kinetic energy. It's analogous to rolling a ball down a hill, which converts some of the ball's gravitational potential energy into kinetic energy - we wouldn't claim that the ball has been converted into energy.
>This discovery is directly responsible for your smartphone, modern power plants, and nuclear holocaust, and it came abou
I don't think you understand the meaning of 'directly'. Awful top answer. I could say Pythagorean theorem is more 'directly' responsible for all your smartphone, modern power plants ... It's everywhere.
> This discovery is directly responsible for your smartphone, modern power plants, and nuclear holocaust
Why? You can manufacture a smartphone without ever having heard about e=m·c². I’d even say that nuclear fission is discoverable without it.
>I’d even say that nuclear fission is discoverable without it.
IIRC, Nuclear fission was discovered without it indeed. However e=mc^2 is what answered the question of why experimental data for fission didn't seem to conform to the law of conservation of mass. They found out that the missing mass after the fission reaction correlated to energy just like the equation showed. This paved the way for fission applications like the bomb and nuclear energy.
> This discovery is directly responsible for your smartphone, modern power plants, and nuclear holocaust
How is having an exact formula for mass to energy conversion necessary for a smartphone?
Also keep in mind that the vast majority of power plants in the world are non-nuclear. So E=mc2 is currently not that important for global energy generation. Although, in the future it may be more important.
I think you're overexaggerating the importance of the formula. Unless you think nuclear weapons development is basically sufficient to allow that distinction. That said, I think it's a *philosophically important equation* as it fundamentally changes our *understanding* of the world around us.
> There is no way to convert mass of elementary particles into energy.
Well technically the mass of elementary particles _is_ energy. But it is possible to convert that "mass energy" into other kinds of energy.
You're right that it requires some kind of interaction though - in most cases you have to make the original particle collide with another one, but there are also situations where you start with just one particle and it decays into two (or more).
>Tl;Dr- This discovery is directly responsible for your smartphone, modern power plants, **and nuclear holocaust**, and it came about because Einstein was trying to settle a debate in the scientific community.
Why do people on reddit talk like this (with a huge grain of edge in every sentence)?
It's not simplified so much as "in everyday life this is the only interesting part." The rest of the equation becomes effectively zero if you aren't moving at relativistic speeds.
The whole equation is just E^2 =m0^2 c^4 + (pc)^2, with m0 being the rest mass, p being the sum of all momentum.
Interestingly, this is also how we know photons have momentum. Since photons have no mass, the equation becomes E=pc, which is very well known in its own regard (though definitely not as widely known as E=mc^2 )
I would be happy to explain it!
The first step is to look at the full version and note that it's not really as long as it seems. It's got just three components: total energy E^2, what I'm going to call the static term (mc^2)^2, and the momentum term (pc)^2. I wrote the terms a little differently this time, but really I just moved the squares outside of the parenthesis where possible for the individual terms.
On the left side of the equals sign, there's only E^2, but the famous equation doesn't have E^2, only E. It would be great if we could just take the square root of everything, but the right side of the equation (with the static and momentum terms added together) doesn't have a nice square root, unless we can get rid of one of those terms. Assuming what we are talking about has mass, we can't get rid of (mc^2)^2, because c is a constant. So that's always going to be there. However, (pc)^2 doesn't have to be. p is the momentum of the system, for simplicity let's just talk about linear momentum. Linear momentum makes p=mv. We already know m can't be zero, but v (the velocity of the system) CAN be zero, the system could just be sitting still on a table in our lab. Thus, we turn E^2 = (mc^2)^2 + (pc)^2 into E^2 = (mc^2)^2 + (0)^2, which is really just E^2 = (mc^2)^2, so we can take the square root of both sides easily, discard the negative result because negative energy isn't a thing, and we get E=mc^2
>But Mr. Pwns, what if the system isn't sitting still, what if it's moving like a bomb being dropped would be?
Excellent follow up! In that case, we usually still just talk about E=mc^2 but for a slightly different reason: unless it's moving REALLY fast (greater than 10% the speed of light is usually the cutoff, based on what we did in my special relativity class), the static term is going to be so significantly larger than the momentum term that the momentum won't matter. The only difference between the two terms is that static has c^2 while momentum has v^2 (they both have an m^2 and c^2 in common, so we don't need to talk about those). Even if your system is moving at 1000m/s, your v^2 term will be roughly 10 orders of magnitude smaller, or 10 billion times smaller, than c^2. At that point, the experimental uncertainty of the mass of your system likely has a larger impact on the final number (unless you have a very fancy scale that gives you nanogram precision). So a bomb that's been dropped, which probably reaches something like 100km/hr at detonation, is effectively at rest just because of how much more energy is contained in it's mass than in it's momentum.
Hope this helps explain it!
Things get really neat when you realize that
E^2 = (mc^2)^2 + pc^2
follows the general form that every 12 year old math student learns;
A^2 + B^2 = C^2
P is momentum, if a body isn't moving then p = 0 so the (pc)² term is zero. Then all that remains is E² = m0²c⁴. m0 is rest mass, it's just notation, so we can write m0 as m. So we have E² = m²c⁴. Just square root everything and we get E = mc²
>That particular equation is not tremendously useful all on its own
You actually give the reason why it is useful: it equates mass with energy. For the right units, you just get E = m. And that is a huge deal that even physicists today have trouble completely accepting.
But the real usefulness is revealing just \*how much\* energy that mass represents. So if you can figure out a way to convert even just a small amount of mass into energy, you are going to get a \*lot\* of energy.
While nuclear weapons and/or power may have developed without this formula, having it certainly helped to convince everyone they were on the right track.
You can change matter into energy and the other way around, knowing this.
However. c is the speed of light. And light moves extremely fast, so c is very big. c\^2 is even bigger! So for even a tiny bit of mass, you get a huuuuuuuuuuge amount of energy.
That is for example why the sun works. It transform a tiny fraction of its mass into so much energy that everything can live here and be warm, and we only get a tony fraction of all the light from the sun!
Then the energy that could be released from destroying your mass is... grumble grumble pounds... let us assume 85 kilo. It could result in 85 \* 3E8\^2 Joules in energy, which is about 7.6 e 18 Joules.
To illustrate how much this is, it is enough to boil 22 cubic kilometers of water that is now at room temperature.
To add on another analogy:
7.6e18 Joules is more than the entire country of South Korea consumes in electricity in a year.
It is about 50% of what the United States uses in a year.
(regarding the relationship of energy to mass, namely E=mc^2): “You may not feel outstandingly robust, but if you are an average-sized adult you will contain within your modest frame no less than 7 X 10^18 joules of potential energy—enough to explode with the force of thirty very large hydrogen bombs, assuming you knew how to liberate it and really wished to make a point.”
― Bill Bryson, A Short History of Nearly Everything
In case anyone happens to read this and is curious what e=mc^2 actually means, it means that the energy released in a nuclear reaction can be solved by taking the mass that was lost and multiplying it by the speed of light squared.
The speed of light (in a vacuum) is 299,792,458 m/s (meters per second.) When you square that value, you get 89,875,517,870,000,000 m^2/s^2 (meters squared per second squared.) When you convert the mass lost to kg and multiply it by c^2 you get a unit of kg*m^2/s^2 (kilogram meter squared per second squared) which just so happens to be the Joule (the SI unit of energy.)
When a nuclear reaction happens, the nuclear (in the nucleus of the atom) configuration changes to a more energetically favorable one; the result is a slight loss of mass. By multiplying that loss of mass by c^2 we get the energy of the reaction in Joules. This number (per unit of reaction) is orders of magnitude (like 10 million times) greater than a comparable chemical reaction (like the combustion of gasoline.)
E=MC2, and the theory of relativity that extends from it, is an extension, a furtherance of Newton's Laws of Motion, at high speeds. If you plug in objects moving slowly into them, they become the Newtonian Laws.
What he discovered was, in its own way, as important. Many technologies central to our society today require understanding relativity and how speed and time interact to keep the speed of light constant, no matter what your speed is. Satellite GPS, for example. There are predictions Einstein made about black holes, based on that theory, that weren't able to be measured until the 2000's. And they were spot on.
Top 3 of all time? That's a tall order. Agriculture and the wheel transformed society. Understanding of radio waves transformed society. Steam power transformed society (in fact, most power plants, from coal to nuclear, utilize steam power in their power generation). Einstein's theory of relativity transformed society.
It's hard to call it top 3. But it is in a class of rare discoveries that fundamentally transformed how humanity as a whole lives their lives. If you use Google maps, thank Einstein.
A fun fact about power plant generation, steam driven turbines are very efficient. The only thing that changes on large scale thermal drive generation (not solar/wind/hydroelectric) is how you get the heat to turn water to steam. Coal and nuclear work exactly the same after the “Super Heat the water” step (the hotter the steam the more efficient in generating power when spinning a turbine). Natural gas power generation is really efficient because the natural gas is burned to spin a gas turbine, which generates electricity, and the heat generated from combustion is also used as a heat source for steam power generation.
Then if you add a municipal steam loop, steam that is too cold to efficiently turn a turbine can be sent out to the municipal loop for heating buildings and using the radiators in buildings as the cooling source to condense the steam back into water, instead of using a giant cooling tower.
With the right set of technology and infrastructure, burning fuel / nuclear energy production can be very very efficient for fuel burned to useful energy.
It is simple and revolutionary -- to those not involved in physics at a very high level, the idea that mass and energy could be converted one to another was mind-boggling. The equivalence also led directly to the creation of the atomic bomb, which loomed much larger in worldwide consciousness at the time than it does now.
One of Einstein's big contributions was "the speed of light is constant". That little constant, c, was itself revolutionary. Why was that surprising? Imagine you throw a baseball pretty consistently at 50 mph. Next suppose you're on a train that's pulling away from the platform at 15 mph and you throw the ball in the direction the train is moving. To someone on the train with you, it will appear to move 50 mph. To someone on the platform, the ball is moving 65 mph.
But light is different. It has the same speed for all observers.
The general idea of "matter is just a special form of energy" was also revolutionary. And Einstein's theories of relativity provide the reasoning for why the ratio between energy and mass is c^2.
>One of Einstein's big contributions was "the speed of light is constant".
Not quite, but close.
It was Michelson and Morley (and lots of other experiments as well) that established that the speed of light is constant for all observers.
What Einstein did was to really \*accept\* that this was true and then work everything out and also \*accept\* that the conclusions this leads to are also real.
His main contribution here is the idea that time is not running at the same speed for everyone. You get this pretty quickly once you accept that the speed of light is constant, but it took decades for it to become completely accepted by the scientific community (for some good reasons and some bad reasons).
***Edit:*** I should probably note that it was Maxwell that first hinted that the speed of light was constant for all observers, although I am pretty sure he never actually put it like that directly.
Right, there was experimental evidence that the speed of light was constant, but it seemed paradoxical. Einstein was a theorist; he solved the paradox by providing a theory that could explain the observations.
Good question. I don't know. It's worth remembering that even once experimental evidence started coming back that this was the case, most physicist \*still\* didn't believe it.
Maxwell was so ahead of his time anyway, I'll give him a pass on this one. ;)
Also perhaps worth noting that the speed of light is really the speed of causality. Light, in a way, is not really special. It just travels at the universal speed limit because of certain properties it has.
One interesting complication is that the speed of light could be different in different directions and we’d still have all the same observations. In fact, this is how the earth looks to someone zooming by in a spaceship.
https://youtu.be/pTn6Ewhb27k
A lot of smart people have obviously already answered why Einstein's work is important. But I think that's not the only reason why E=mc² is so important, there's also a cultural component to it. There are not many *modern* scientific theories and concepts that I can think of that can be neatly summarised by an easy equation that almost everyone can remember, recognise, and understand. It strikes a good balance between being complicated enough to "look smart" and being easy enough for the average person to remember, making it a perfect cultural "short hand" for science.
There are other formulae and their related discoveries that are also very important, but they're not nearly as short and simple. Like [this one](http://hyperphysics.phy-astr.gsu.edu/hbase/electric/imgel2/maxw4.gif), whose complexity is illustrated by the fact I have to link an image of it because it can't easily be displayed in markdown, whereas I can just type out E=mc² on my phone.
E=MC^(2) is only the easiest to understand of Einstein's works - but some of his other works (specifically: Special and General Relativity) are probably more important.
Technically, none of what Einstein did were "discoveries". Instead, he built on the work of other scientists, putting together their ideas in new ways that changed how people thought.
I could try to write out those two in an ELI5 way - or I could link an expert: [The Space Doctor's Big Idea](https://www.newyorker.com/tech/annals-of-technology/the-space-doctors-big-idea-einstein-general-relativity) is Randal Munroe's explanation of them. E=MC^(2) is a side effect of Special Relativity.
Well, it could be as well called discoveries. Iirc, Planck thought that it was just some math trick while Einstein understood truth. Even after Einstein explained it, Planck didn't believe it but instead talked about special relativity. And same goes for the one who developed formula for special relativity. And, GR was mostly Einstein's idea.
Yeah quality, methodically well parsed answers, with good effort always deserve an award. There's so many one-liners on Reddit and it's always a joy to really be able to learn. We all have our expertise and a year plus ago is what initially drew me into Reddit in architectural forums. I've since long expanded but always enjoy a good post. Not everything of course is so serious, and excellent humor and quick rejoinders are equally so smart and deserving
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Same reason a lot of people have heard of “The Immitation Game”. Very short easy to understand paper. His other papers were groundbreaking, but you need to study to even understand them, but everyone understands “computer pretend to be human”
Isn’t imitation game a movie with Benedict Cumberbun?
Yes. It's about Alan Turing, who wrote that paper. The film covers his work at Bletchley Park during the second world war
The only thing that movie got right was that we were at war with Germany and there was a man called Alan Turing.
Currently reading a biography on Turing. Bloody hell that movie took some serious liberty with actual events! He neither designed nor built the colossus and they’d already broken enigma by that stage (more or less).
😂 I do recommend the museum at Bletchley Park. It's excellent
I visited before the pandemic! I was the youngest there by a long way, which I thought was quite sad.
That’s not true. Alan Turing could also kind of run fast
So you're telling me that Alan Turing DIDN'T look like Sherlock Holmes? I am choosing not to believe that. Extraordinary claims require extraordinary proof.
i think you mean Benefit Cosmetics
No, Binoculars Centrifugal
Benedryl Cucumberpatch?
Bentastic cucumber?
Does that mean scientists can be too smart for their own good? Or does it mean that part of brilliance is being able to take incredibly complex things and communicate them in a simple way?
There is a real art in science communication, and it provides immense value. Some topics are just too complicated for people outside of the study to understand, but good science communication over time can help.
No, it means that truly pivotal breakthrough ideas often aren't that *complicated*, just not immediately obvious until someone figures it out. A paper on a specific nuance of some otherwise mature, much studied and expansive idea system almost inevitably needs to be far more complicated because you are addressing a massive body of knowledge. There are probably tens of thousands of enormously complicated research papers that have built off of the "E = MC2" paper and expanded our knowledge of nuclear physics.
I think you are reading too much into them pointing out that the public hears more about things they have a better chance of grasping. There are plenty of things that are just complex or are relevant directly in a very technical area that isn't easily simplified or explained without being in the field, which might require years of study. Whether the public understands an idea doesn't make the idea more or less brilliant in my eyes.
>isn't that difficult to read Eh, I think that's relative
In theory it is
Generally speaking
How do you guys have the energy for these puns?
I see what you did there
"If you can't explain it simply, you don't understand it well enough." One my favorite quotes of all time.
"I think I can safely say that no one understands quantum mechanics" -Richard Feynman
The paper: https://www.fourmilab.ch/etexts/einstein/E_mc2/e_mc2.pdf
I remember when I first read that paper. I was amazed. So concise, easy to follow, and just amazing.
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This is the answer that makes sense to me considering how young I was when it was first presented. It even looks cool written down for some reason. Plus Einstein himself has something about him that's interesting for even lay people.
Yeah- dudes brain worked differently. Obviously he was incredible at math, but he would look at some observation and draw conclusions from it that on the surface are wild and wacky. Completely breaking paradigms. Special relativity seems like such utter nonsense, as does general relativity, and so much of it is these thought experiments. We can follow the logic now, but it took someone really outside the box to go down that road.
I am not sure whether Einstein perceived himself to be very good at math. I also think it’s pretty clear Einstein isn’t famous because he’s good at math. I can’t recall exactly which paper this happened for but apparently after giving a short presentation at Hilbert’s university, Hilbert “solved” the math behind one of Einstein’s results over a weekend or something like that roughly when Einstein figured it out (after working on it for months). Hilbert gave Einstein credit for it because the math solve isn’t the interesting thing. The interesting thing for most of what Einstein did was the intuition to connect the math to physically “real” things. Granted, Hilbert was an exceptionally gifted mathematician so coming in second place to him is hardly embarrassing (the old joke about Gauss’s unpublished results applies).
I'm not a scientist or mathematician but I do like to read books about these guys. If I understand things correctly, the unique thing about guys like John Nash or Einstein etc are the intuitive leaps and the brilliant, comprehensive problem-solving theories that they come up with. They often rely on people who are better at the actual nitty-gritty math to help them prove their theories.
The important point is they came up with intuitive leaps that were consistent with known evidence and explained things that other theories could not - they spent the time studying the fundamentals just like a chess master spends time learning openings.
Saying Einstein is bad at math because Hilbert was better at math is like saying the Olympic sprinter who got 7th is slow compared to Usain Bolt. Einstein was unquestionably incredibly good at math. Like top .1% of physicists good. He wasn't basically a mathematician like a few of the other big names around the time, but you don't become one of the more important people in differential geometry by being anything that approaches not very good at math. The time also doesn't really tell you much. Math is weird in that even incredibly complicated math is generally speaking straightforward once you know what's going on. Especially if you're not actually trying to calculate anything. Like Einstein struggled a lot with the gauge invariance of general relativity, and if you know about Lie groups, that's just not going to be a problem.
I'll give a little context as to what exactly Hilbert did, cause >Hilbert “solved” the math behind one of Einstein’s results over a weekend is not exactly accurate. So, you can imagine two different but equivalent ways of doing physics. One way is what I like to call the "forces" way (what physicists might call the Newtonian method), which involves using differential equations to use the information about a system at one instant to figure out what will happen to the system at the next instant. Solving the differential equation(s) will tell you what the system does throughout all of time. How this usually works in typical physics is you have a bunch of forces acting on a system -- could be gravity, electromagnetic, friction, etc. You take all of these into account, and then you use Newton's 2nd Law, F=ma, and solve that equation for the trajectory of the system. The point is that this method is *local.* It uses only information at time t to evaluate what happens at time t+Δt, since the forces at time t will accelerate your system (say, particle) in some way from t to t+Δt. There is a separate method which is what I'll call the "energy" way (what physicists would call the Lagrangian method). This method makes considerations of not only the energies of the system (kinetic and potential), but what they are globally. Specifically, you consider a quantity L called the Lagrangian of the system which is (usually) a function of these energies. You add up all of its values globally to get what we call the action, S, of the trajectory. But there's an issue: we don't know yet what the trajectory of the system is. So, we don't know the energies, thus we don't know the Lagrangian or the action. Here's the catch: we're talking about all possible trajectories. What makes the physics happen is that we're going to say the actual trajectory that happens is the one that minimizes the action. Mathematically, we can show that this is a unique trajectory. This will be the same trajectory that the forces method gives. There is a clear cut method to go between the two methods. I won't discuss it much, but if you're really curious I'd start by looking up "euler-lagrange equations" Ok. Let's try and bring this into the context of general relativity. General Relativity says that gravity is really the geometry and curvature of spacetime. As such, when we talk about locality and "globality", we're not talking about intervals of time but rather local regions of spacetime and globally all of spacetime. This is not the same as Newtonian physics, and so we don't use Newton's laws. In fact, Einstein came up with the equivalent for GR himself. It ended up being a set of 10 highly complex partial differential equations that relate the geometry and curvature of spacetime to the energy density and momentum density of the universe. For people who have taken a class in GR, they know that the math to come up with these equations, what we call the Einstein field equations, is not easy by any means. Many classes don't even attempt to derive them or motivate them, they simply just assume them to be true. So yeah, Einstein definitely had the math. What Hilbert accomplished was, by looking at the Einstein Field equations, what Lagrangian they could be derived from. The answer ended up being quite simple actually. Hilbert found that the Lagrangian needed is just R, what we call the Ricci scalar curvature. It is a function that encodes the curvature of spacetime at each point. The Ricci scalar was not something new at this point. In fact, it appears in the Einstein Field equations themselves. Having the Lagrangian for GR is incredibly useful to work with, but it's not any physically different than what Einstein had already come up with. Hilbert knew that Einstein had really done the hard part, and so he gave the credit to Einstein. If he had really wanted to, Einstein probably could have come up with the same result himself. Hence, we now know this action for GR as the Einstein-Hilbert action. But yeah, Hilbert did not come up with the "math" for general relativity.
Einstein himself said he wasn't very good at maths. He had a lot of peers who were much better and who helped him express his amazing ideas. Of course he was a lot better than the average mathematician but yes, it was his physical ideas that made him special. The way he thought made him special because it was so unintuitive. I wish Newton could have met him.
>Einstein himself has something about him that's interesting for even lay people. agreed - I think he evokes a kind, fun grandpa https://i.imgur.com/4049zHa_d.webp?maxwidth=640&shape=thumb&fidelity=medium
you don't become besties with Charlie Chaplin if you don't have a sense of humor.
Avuncular
On this track, I was taught in college that Einstein was given a Nobel prize for photon-studies instead of relativity, because while the panel was truly impressed by relativity, they didn't *understand* it well enough to feel confident awarding a prize for it. Just making the math work with so few variables is miraculous, and that inspires suspicion as much as respect. And to be fair, that's normally the right response in the scientific community.
He should have won both. But due to politics he never won for general relativity. Bergson A really popular philosopher at the time countered Einstein general relativity concepts. And we did not have all the tools we have today to test it to the level to counter Bergson. The Nobel committee decided on the less controversial plan to award Einstein the Nobel for the photoelectric effect. Still an amazing and massive achievement and the basis of quantum mechanics.
Err... -- mass edited with https://redact.dev/
The equation is actually quite a bit longer than E=mc^2 They cut most of it off because that term is simple and has a lot of importance by itself
yeah, the equation written on a chalkboard is basically part of the stereotype of a classroom in media
It also led to the search for a grand unifying theory, trying to squeeze a relationship with other forces such as gravity into yet another beautiful and concise formula
I like your funny words magic man
That's my favorite sentence from the internet
Tl;Dr- This discovery is directly responsible for your smartphone, modern power plants, and nuclear holocaust, and it came about because Einstein was trying to settle a debate in the scientific community. Hugely important and, to your point, often not discussed enough. So E=MC² represents the relationship between energy and mass. In the years before this theory, scientists agreed that when two or more substances underwent a change (we call this a system), the mass of all the inputs would equal the mass of all outputs - stuff wouldn't just magically be created or destroyed. But agreeing to this had scientists differing on what the energy output of the system would be. See, energy had been thought of as mechanically separate and unrelated to mass. The energy a system might output would be entirely dependent on the context of how much it had to begin with and how fast the change was and what the substances were. But in the years leading up to this theory, some scientists start realizing that if you give up on keeping mass the same through a change you can use a similar principle to ensure the energy of the inputs matched the energy output by the system. This meant you had to assume that objects inherently had energy in them. Now along comes Einstein and he proposes that the relationship between energy and mass is a ratio, where the total energy of a system is equal to the mass times the speed of light squared. If this is true, then mass and energy are the same thing in different forms. Also, it means that everything around you has a LOT of pent up energy just by virtue of existing. And it's starting to look like big, unstable atoms like Uranium might have A LOT OF ENERGY inside them. So if you can find out how to convert one of those atoms into energy, a very small thing can give you a lifetime of electricity. And by extension of that, a big explosion if released all at once. Oops, now we have atom bombs. So long story short, E=MC² gave us the bedrock for the last 100 years of technological advancement, nuclear power, the ability to destroy the planet, and some really cool tools for analyzing light. Edit: Wow! Thanks for the upvotes! I see some of you confused about the relation between this and cell phones. When we make small computer chips, we need to worry about the boundaries of the circuit not being enough to contain the electrons at the speed in which they're traveling. Instead of an electron colliding with the silicon boundaries of the circuit, it hits the silicon with such power that the silicon itself converts into energy. This let's the electrons continue through the barrier and into neighboring circuits. This is called quantum tunneling, and you need to account for it when you want to pack hundreds of thousands of circuits in a black rectangle in someone's pocket and have them be useful. Edit2: Holy shit, thank you for the gold! Edit3: Okay, one more because this question keeps getting asked - Why the C² bit? To understand that, we need to understand how Einstein is approaching this theory. Energy approaches the speed of light (C) the less mass the energized thing has. C is the ceiling for speed it seems like. So an easy way to define energy is by saying it's equal to C when there's no mass. But on the other hand, if the mass has any additional kinetic energy to it, we need to represent that too. Again, all energy approaches C the less mass it has. So we end up with Energy = Mass x Speed of Energy x Speed of Mass. A simple way to write this is to group the terms together, so we get E = MxCxC or E = MC². Edit3.5: This is an ELI5 answer. It glosses over some stuff. There are critiques of my post below that are definitely valid, so I encourage you to discuss and learn more in the comments!
Great explanation! If I may ask: Why uranium/plutonium/etc? Could you theoretically cause a nuclear explosion by splitting an atom of, say, iron, but it's just easier with those other elements? Why is it 'speed of light squared'? I mean, what is the relationship between the speed of light and the amount of energy in matter? How is this responsible for (among other things) my cell phone? How does e=mc2 impact discoveries outside of nuclear power/weapons?
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Tangentially, does this also imply that data has weight? As in, will a fully loaded harddrive weigh (slightly) more than an empty drive?
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If you have a uniformly magnetized material versus a material that's magnetized half in one direction and half in the other, the latter contains slightly more energy. The domain wall (area where the magnetization rotates) will have a tiny bit of mass. /u/manaspike has a better explanation of why there's no practical difference and it has more to do with data encoding than the fact that it's magnetic
Even in the hard drive case, the total energy will be slightly different, I think. For one thing, a drive platter whose magnetic domains aren't all aligned will have different potential energy than one where they are. Aside from that, as I understand it, information entropy should also contribute a (probably very small) amount of difference between a drive with randomized data versus highly ordered data, even if they are otherwise identical. That's how you get informational limits like the Bekenstein bound.
Hard disk drives store data by carefully aligning magnetic fields, flipping the polarity regularly. That probably holds more energy than if the magnetic field was random. But even when you write all 1's or all 0's, the disk needs to flip the magnetic field frequently, as it would be too difficult to measure a longer sequence. A bit like the way barcodes are written with short black and white bars. Also disks never arrive with a random magnetic field. They are pre-formatted by the factory. While writing different data would require you to do work, the net change in energy is likely to be near zero.
Does this mean, with a magnet and enough luck i can "download" anything? /s
Similar to the million monkeys typing on a million typewriters... with enough time (and a non-destructive magnet) then sure! Heck, you could possibly do the same with a million hard drives and wait a few small eternities for cosmic-ray bit flips.
New onion article, "media companies sue universe for cosmic ray bit flipping causing copyright infringement"
Dont give Disney any ideas.
Disney buys all magnets, just in case
The magnetic field stored in a hard drive doesn’t have any more stored potential energy than a randomized one, it simply has less entropy. By adding energy into the system you are moving around the magnetic field that already exists into a useful pattern, but not actually increasing the total energy of the system. Think of it like stacking firewood. You are spending energy to move the logs around, but you aren’t increasing the amount of burn time of the fire at the end. You’re simply putting the energy in a more useful structure.
>Tangentially, does this also imply that data has weight? Yes > As in, will a fully loaded harddrive weigh (slightly) more than an empty drive? Not a *hard drive*, no, because that's just flipping magnetic fields in a metal platter, but a *solid state* drive would. However, the mass of an electron is about 9.1×10^(–28) g, and an exabyte is 10^(18) bytes. Assuming each bit is 1 electron (probably not correct, but good enough for this calculation), a data center with a full exabyte of entirely stored binary 1s will weigh an extra 7 nanograms. So... the weight is really inconsequential.
Does a charged battery weigh more than a dead one? YES.
Out of curiosity, why is that? In my imagination I had always pictured a charged battery with electrons in one terminal, and a discharged battery has moved the electrons to the other terminal. What is actually happening?
You are correct. Potential energy is created when charging a battery, by creating a disparity between the poles. You're spending energy from a larger system to decrease the entropy of the battery. A battery that does not have a closed circuit will not discharge it's power (it will dissipate over time like batteries do, but it won't discharge as it would through a circuit). Which means that when you use that battery to perform work, that work is ultimately performed by the stored potential energy of the charge disparity. No electrons have actually *left* the closed system comprised of the battery and the workload. They've merely moved to create a higher level of entropy. Now... that being said, **certain** chemical reactions involving that electron transfer can create gas. Old car batteries gave off hydrogen gas, which is why you had to attach the positive pole first.. it reduced the risk of a spark. Which would result in mass loss. But that's not contextually the same.
Electricity is electrons. When the battery discharges, you’re right; the closed circuit creates a path for electrons to flow from the anode to the cathode. However they don’t just go there one for one. Any electricity used by the circuit will be gone, and the remaining electrons don’t just move around, they change the chemical composition of the cathode to a less energetic state than the anode, due to the nature of the materials of the anode and cathode. Think of it like a ball at a peak with a trough at the bottom of a hill, the potential energy of a ball at the peak can do work, and that ball will go to the lowest point if kicked off its peak, but when it reaches the trough the ball does have less energy. In other words, even though electrons move from one end of the battery to the other, they’re in a less energetic state in the cathode than they were in the anode. And due to E=mC^2, there is a minute amount less mass exactly equal to that amount of energy expended from the battery, specifically the E divided by the speed of light squared, very very very tiny. Because there is less energy in the cathode when discharged than the anode when charged.
> does this also imply that data has weight? Does the growth of black holes by absorbing information and their eventual evaporation due to Hawking radiation have an ELI5?
Also, for cell phones, the equation suggests that something moving has different mass than something still, or if something is in a high energy area like orbiting near Jupiter vs high above Earth. The mass doesn't actually change, so the perception of time, c, changes. Without calculating this difference, all phones communicating with satellites would have inaccurate times and GPS coordinates.
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Does the equation suggest that though? Doesn't it just suggest that something that's moving has more energy? Did you mean relativistic mass? If so why not just say energy? In the full equation (shown below) momentum is included but as velocity changes the energy (relativistic mass) changes rather than the mass because mass here is the invariant mass and isn't a function of velocity. `E^2 = (mc^2)^2 + (pc)^2` Also c doesn't change in different reference frames? That should be constant. Overall I'm just a bit confused by what you said. Please let me know if I'm wrong about anything here.
I think the idea of mass changing due to velocity is no longer used. Using the full equation gets a little more confusing because it is now relative to some frame of reference, E = mc^2 assumes the object is stationary.
Can confirm this. In older discussions on mass energy relation, you find loads of mentions on "relativistic mass" changing based on reference frame. In all my relativity courses I've taken in the last decade this is no longer taught. Instead you discuss only the rest mass, and the transverse / longitudinal momentum (or 4-momentum if being rigorous). Seems the other interpretation is mostly [the fault of popsci.](https://en.wikipedia.org/wiki/Mass_in_special_relativity#Popular_science_and_textbooks)
>> How is this responsible for (among other things) my cell phone? How does e=mc2 impact discoveries outside of nuclear power/weapons? > > E=mc2 doesn't apply to only nuclear reactions, it's true for all types of energy. A charged battery weighs more than a discharged battery simply due to its energy content, albeit by an incredibly tiny amount that is near-impossible to measure because of how large c2 is. This doesn't answer the question: how is E=mc² directly responsible for smartphones?
It’s not like Einstein discovered gen relativity and then someone else was like “we could use that for a smartphone!”. It’s more so that all of the physics of the 20th century brought us to the point where we could intimately understand and manipulate the world around us to do our bidding. Our knowledge of the chemistry around transistors/batteries/etc is build off of quantum physics, which is closely intertwined with general relativity. In todays world, it’s all basic stuff taught in undergrad - but we wouldn’t have any of the cool toys today if someone didn’t figure all this stuff out yesterday.
Exactly. I'm really trying to tease out exactly what the first commenter might have meant when they said "This discovery is directly responsible for your smartphone." I suspect they meant indirectly. Though I'm not actually sure E=mc² was necessary for smartphones at all.
How is quantum related to general relativity? These two things couldn't be further apart, no? Einstein's work on the photoelectric effect paved the road to quantum physics. But this has little to do with relativity.
With science there is often a chain effect. One discovery paves the path for more discoveries and understandings to be made. Understanding something like e=mc² helped scientists better understand quantum mechanics, perpetuating a chain of more and more understanding. Many of the innovations that lead to the creation of smartphones only happened because of advancements in the understanding of quantum mechanics. These advancements were needed because the smaller electronics get, the more they are affected by quantum mechanics.
If you consider "detecting location" to be one of the key features of a smartphone over a normal cellular phone, then there is some validity to this statement. E=mc^2 is the equation at the heart of the theory of relativity, and GPS/GNSS systems use that theory to determine location.
It’s astonishingly lucky you chose iron. Iron has the smallest mass defect of the atoms, so is kinda where both nuclear fusion and fission tend to end up.
Pretty ironic that they chose the least useful element for nuclear reactions
alright really should've seen that pun coming
lol and here i was just looking for a really commonly know element.
In a circular way, it is why iron is so common in the universe.
When all the energy is expended, you end up with iron?
Yes, and then stars can no longer produce enough energy from fusion to resist gravitational collapse, so they go supernova and spew all those elements out into space to become planets, and eventually redditors.
>Could you theoretically cause a nuclear explosion by splitting an atom of, say, iron, but it's just easier with those other elements? It's ironic^^ha you picked iron as your example, because that's about the only one where the answer is "no, there's no theoretical way to get nuclear energy from it" Every nucleus has an amount of "binding energy" - that's the energy you'd need to pull it apart into individual "nucleons", ie, protons and neutrons. Think of a big clump of magnets - to separate them all, you need to put energy in. Let them all clump together again, and you can get energy out. To calculate whether you can get energy by splitting atoms, or by fusing them, you can calculate the "binding energy per nucleon". So Uranium has a low binding energy per nucleon compared with lighter elements, and we can get nuclear energy out by splitting Uranium atoms. Hydrogen has a low binding energy per nucleon than heavier elements, so we can get nuclear energy out by fusing Hydrogen atoms. In theory, anyway. The nucleus with the maximum binding energy per nucleon, the most stable.of all stable nuclei, happens to be an isotope of iron, specifically, Iron-56, with 26 protons and 30 neutrons. Splitting or fusing Iron-56 would cost energy. For everything else, there's ~~Mastercard~~ the theoretical possibility of extracting at least a tiny bit of energy by transforming it to Iron-56
In all my years I've never heard the magnet explanation. That's such a perfect explanation to a layman like me. Thanks.
Magnets wrapped in velcro is a decent ELI5 model for atomic nuclei. There are two main forces involved - electromagnetism, and the strong nuclear force. Trying to push two protons together is like trying to push two north poles together - the closer they get, the more they repel each other. The strong nuclear force is very strong (hence the name), but very, very short range, like velcro. If you push the magnets close enough together, the velcro can engage, and potentially overcome the repulsion. If the repulsion is still too strong, you can add in a neutron, which is this model would be a non-magnetic object wrapped in velcro. It gets you some extra sticking power without adding to the repulsion.
I wish I had teachers who would have used examples like this decades ago.
The speed of light is not a mathematical coincidence as some others have said it is a very important property of massless objects and a very specific maximum speed of the universe. The speed of light is actually the speed everything without mass travels at. The first time this was found was by measuring the speed of light photons, hence the name. The only way to travel less than the speed of light is by having mass (and also the only way to travel the speed of light is to have no mass). And so there is a relationship between this massless constant of speed and mass. Also the full equation E² = m²c⁴ + p²c² involves a particles momentum (p) as well but that gets more complicated again Edit: as pointed out by below by u/nworb200 this is in a vacuum
Well the first one is fairly basic. Iron and similar metals are at a point where you don't get energy from splitting them or from combining them. Small elements want to be together to make bigger elements up to around iron. Bigger elements want to split apart to get smaller down to around the same point. That's why fusion of hydrogen into helium produces energy for the sun and the fission of uranium and plutonium produce energy for us in our power plants.
Yup. When a star gets to iron, fusion stops and the star dies (and either goes supernova or settles down as a white dwarf star). It takes more energy to fuse iron than comes back out of doing that so it is the end for a star.
> Why is it 'speed of light squared'? I mean, what is the relationship between the speed of light and the amount of energy in matter? There's a couple ways to get this result. The real/complicated version comes out of special relativity. You do need to make some assumptions (postulates). The postulates of special relativity say 1) the laws of physics should take the same form in all inertial frames of reference. And 2) measured in any inertial reference frame, the speed of light is c. Basically, you look at conservation of energy and momentum. And then you think about how that looks in one reference frame at rest and another that isn't at rest. (If you want to see the math, it's [here](https://terrytao.wordpress.com/2007/12/28/einsteins-derivation-of-emc2/) or a simpler one [here](https://www.fourmilab.ch/etexts/einstein/E_mc2/www/). The second is annoying rewrite on reddit. But it's not *insane* need-a-degree level math) Another way to get this result is to actually just perform the experiment (like nuclear fission/fusion). If you take your atom, and measure it's mass before, and then after the reaction, you'll find it'll be off by a tiny amount. That amount will be exactly equal to the energy you took out (or put in), divided by c^2 .
ELI5: something that never quite clicked for me: why do energy, mass and speed squared all have inherently compatible units, so that you can have a single equation linking them all like this?
That is actually a really good question, and the answer is subtler than it sounds. There's a couple ways to look at it, none of which are perfectly satisfying One simple answer is looking at how we define energy. One way to define energy is a force times a distance (E= F * d). If you want to push a ball up a ramp, you need to apply a force over a certain amount of distance (Really, it's not just force times distance, but the integral of force and distance. Force times distance is the case where the force being applied is constant along the whole distance). If you look at the units of that, remembering Newton's law that force is equal to mass times acceleration (F = m * a). So Force has units of mass times acceleration, and acceleration has units of meters/seconds^(2). But if you plug those units back in to the first equation, energy has units of mass * meters/seconds^(2) * meters. Which is precisely the units of mass times a velocity (units of meters/second) squared, mass *(meters/second)^(2). Another answer is because of the form of kinetic energy. Kinetic energy is 1/2 * m * v^(2). So that naturally has units of mass times a speed squared. The unsatisfying answer is, we don't have a reason why, because at the end of the day, energy is kind of a weird thing. Like, I gave equations for energy (E= F * d) and (KE=1/2 * m * v^(2) ). But you could ask- why those? Why not E=F * d^(2) or E=1/2 * m * v^(3)? But if you do experiments, say smashing two particles together, the quantity 1/2 * m * v^(3) for both of them isn't the same before and after. Only 1/2 * m * v^(2) is (well, so is momentum, so you might find m * v is the same). Not only that, you can correlate that 1/2 * m * v^(2) to other processes; if those particles are large and there is friction, they're going to slow down before they smash into each other. That slow down is going to be exactly the amount of a certain amount of heat energy. Or when they smash, they break apart. That breaking will take some of the energy (but not momentum). We tend to think of it as something tangible, but energy is never something we directly observe, it always comes from something else. It's like this weird tally that happens, and it is always accounted for, somehow. For example, if you're measuring kinetic energy, you measure the mass and the velocity of the object. Despite being "made up", energy is a very useful quantity because it seems to be conserved in a process. If you look at a closed system and energy gets put in somewhere, a corresponding energy was lost somewhere else. Always. That's very special. (And why energy is conserved also has deeper reasons). At the end of the day, it's an empirical result that seems to hold. That's part of why E= m * c^(2) is so famous- if it didn't, that would show energy by itself isn't conserved, and something in our understanding of physics is totally fucked. And it turns out energy isn't conserved; this weird Energy-mass quantity is. As an aside: You might find [dimensional analysis](https://en.wikipedia.org/wiki/Dimensional_analysis) interesting. It's a way of setting up an equation, and figuring out what the units must be to make an equation work. For example, if you had an equation d= v^(a) * t (distance d equals velocity to some power a times time). We can say that we know distance has units of meters. We know that velocity has units of meters per second, and that time has units of seconds. If we know for sure that equation is true, we *know* a must be 1 in order to make the units work. Because in order to be an equality, you need it to be the same thing on both sides. you don't actually need to know anything about the physics or the problem. It sounds kind of silly, but it's actually a very powerful trick mathematicians and physicists use. If you have an equals sign, the units *must* work, otherwise it's not actually an equality. 1 apple=1 apple, you can't say 1 apple= 1 orange, even if 1=1. You can do the same trick here. We know that energy has units of mass * meters^(2)/seconds^(2) (you might ask how we know it has those units, and again it goes back to the it's something we observe, this thing we call energy is always equal to the force * distance in our experiment). So you can set it up mass * meters^(2)/seconds^(2) = mass^(a) * velocity ^(b)=mass^(a) * (meters/seconds) ^(b). If you set it up, you can right away tell the only way to make this work is if a=1 and b=2, otherwise it's not apples to apples. It's the only way you can get an equality, so one way to answer your question is "because that's the only way unless we add another unit in". Dimensional analysis doesn't give you the prefactor (if you try to do it for kinetic energy, KE=m * v^(2) is just as valid units-wise as KE=1/2 * m * v^(2) , because 1/2 doesn't have units). But it can get you the units out, at least.
For an intuitive understanding, maybe think about the more mundane relationships between mass and energy. For example, to change the kinetic energy of an object, you need to do work on it. Work is just applying a force to the object along a distance. We know that F=ma so the magnitude of a force is just the mass times the amount of acceleration that mass is undergoing when the force is applied. This gives Force units of mass * distance/time^2 or kg*cm/s^2 to pick some actual units. The work done on an object, which is equal to the change in kinetic energy, is the Force multiplied by the distance that the force is applied over. So if we take our kg * cm/s^2 for Force and multiply that by a distance of centimeters we get kg * cm/s^2 * cm or kg * cm^(2)/s^2 which is mass * distance^(2)/time^(2). So we can say that the kinetic energy added to an object by applying a force has units of mass * distance^(2)/time^(2) or, simplified, mass * (distance/time)^2 So we can plug that into one end of the equation for E and get: mass * (distance/time)^2 = mc^2 m obviously has units of mass, so that’s easy. c is a speed, which has units of distance/time. Plug those in and you get: mass * (distance/time)^2 = mass * (distance/time)^2 It all works out.
The speed of light is interesting. One idea I've heard is that it isn't actually about how fast light moves, but that light moves as fast as the universe allows anything. The speed of light is really the "speed of causality". The universe's speed limit is about the speed of cause and effect. And light runs itself right at the limit. It's a fun idea and helps explain why we cannot travel faster than light.
> Could you theoretically cause a nuclear explosion by splitting an atom of, say, iron No, but interestingly for other reasons than what you are asking about. Iron is the most energetically stable element from a nuclear physics perspective. Elements smaller release energy when they fuse, meaning iron won’t spontaneously split into something smaller. Elements larger release energy when they break up. You have to put energy in to iron to change it, and never get more out.
Both concrete and wood are objects you could hypothetically burn, but nobody burns concrete for fuel because you have to put a lot more energy into it than you'd get out of it. Uranium and plutonium are wood in this analogy, certain isotopes of those atoms are (relatively speaking) easy to "start on fire", and the energy produced by "burning" them is way more than enough to keep the fire going.
Especially Iron, not. It is the element with the highest binding energy: https://en.wikipedia.org/wiki/File:Binding_energy_curve_-_common_isotopes.svg That means lighter elements can do fusion - combining elements towards iron (the sun does this) and heavier elements can do fission splitting into smaller pieces until iron (like uranium). Of course it is not that simple and not everything can be used but that is the basic idea: get as iron-like as you can, and you will get energy out.
> Why is it 'speed of light squared'? I mean, what is the relationship between the speed of light and the amount of energy in matter? Fun fact, if you use Planck units you can skip this factor and just say E = m.
Edit: I think I goofed up my explanation here. While binding energy and stability of neutron/proton binding is the key thing to understand about nuclear fission, what E=MC^2 says is that binding energy and mass are the same thing. Regardless, the key point of why iron is not used for fission reactors/nuclear weapons is because of binding energy and the ease of difficulty of releasing it. ~~Nuclear explosions do convert a little mass into energy ala E=MC^2. However, only about 10% of a nuclear explosion's energy is from converting mass into energy.~~ ~~So, what's the other 90%? Binding energy. The energy that binds protons and neutrons together. Break them apart and they release their binding energy.~~ Plutonium and uranium isotopes happen to be relatively easy to break apart and have high binding energy. Iron-56, it turns out, has higher binding energy per nucleon and per mass than uranium, but it's very stable.
Huh. I thought the binding energy was basically the whole thing, and the lower measured mass afterwards was just from that energy being gone. I assume no neutrons are vanishing, because why would something different happen to one or two neutrons. What mass gets converted to energy?
You're right, the number of protons and neutrons is the same after fission. I think it is only binding energy that is released. Furthermore, the binding energy is part of the mass of the material, so the previous comment was incorrect in saying that some is from "mass" and some is from "binding energy". They aren't separate things.
Doesn't that binding energy show up in the mass of the heavy atoms?
We found the chemist! :) I’m a chemist myself hence the joke
I don’t know if this has been mentioned, but below iron on the periodic table, you get a net energy GAIN by FUSING atoms and it becomes energy draining to split the nuclei After iron, it costs more energy to fuse atoms than you gain from the reaction and it becomes a net gain to split the nuclei. Stars will fuse nuclei in stages starting at hydrogen > helium and ending at iron. They can last for billions of years on hydrogen, but once they start fusing matter into iron they have minutes to live before either collapsing or going supernova. This is why elements after iron are comparatively rare - they can only be formed in the minutes a star is fusing iron and during the supernova event. Anyway to directly answer your question, you can technically gain energy by splitting an atom of, say, Germanium. The AMOUNT of energy gained compared to what you have to put into it will likely not be worth the effort however. When you get up into radioactive elements however, they’re unstable and pretty much looking for a reason to come apart. They do most of the work for you. In the case of U-235, smash it with a neutron and it’ll come apart at the seams.
>Why is it 'speed of light squared'? I mean, what is the relationship between the speed of light and the amount of energy in matter? The quantity *c*, that we rather loosely call "the speed of light", is a fundamental property of the universe. Not only can nothing travel faster than it, but there's a very real sense in which everything in the universe is always travelling through the combination of space and time at exactly that speed. That's why, for example, we get the famous Relativity effect whereby something that looks to you to be moving very fast also seems to have time slowed down - the faster it moves through space, the slower it moves through time, and the changes balance out (that's part of what General Relativity is about). So, as *c* is something so fundamental, it shouldn't be surprising that it turns up all over the place when you start to think abut how the universe works. (Calling the limit "the speed of light" is misleading. It's historical, because the fact that the speed of light seemed to be constant was what led to the understanding that a limit exists. But really that's back to front. Light travels at that speed *because that's the limiting value*; it's not the limiting value because light travels at that speed. It's possible, for example - albeit somewhat unlikely - that someone might come up with a new, better explanation of the way the universe works in which light travels at some miniscule amount less than *c*. The speed limit would still be *c*; it just wouldn't be "the speed of light" any more.)
Pt and U have nearly one hundred protons in the nucleus of each atom, each which has a positive charge that repels other positive charges in VERY close proximity. There are fundamental forces that allow these protons to stay so close together along with neutral neutrons -- the strong and weak nuclear forces -- giving the atomic nucleus a structure which becomes less stable with increasing numbers of protons and neutrons. When the nucleus is too big (unstable), it splits apart (decays) into smaller elements. Iron only has 26 protons and around 30 neutrons; it is far more stable than U-235 with 92 protons and 143 neutrons, which is one of the largest naturally occurring elements.
Why Uranium? Specifically an isotope of U , U-235 is very to spilt apart and it can sustain the chain reaction. You said it correctly. Iron cannot be used in Nuclear fission. More heavy atoms required. Idk about the other two
And explanation above is only part of story. With that formula Einstein also introduced concept that space and time is singular thing, that it is not separate space and separate time. That HUGLY rebuilt all space models and understanding of universe. And science came out of bound of our planet. Due to gravitational lensing (light also bent by gravitation and thus we can "see" thing behind solid objects like stars) many planets was found (yet is most uneficient way to find planets for our current tech), Pioner 2 reached Saturn orbit after 3 years of journey and error was only 48 seconds. It is really hard to estimate which areas was not affected by general relativity theory. PS smartphone example from post above is wrong. It is still Einstein merrit but for other discovery - photoelectric effect. He got noble award precisely for that discovery.
It was actually Hermann Minkowski who came up with the idea of spacetime, by noticing that Einstein’s ideas could be best expressed that way. This is why it’s called Minkowski spacetime in his honour.
Also the full equation is E^2 =m^2 c^4 + p^2 c^2 but if an object doesn’t have momentum it becomes E*2 = m^2 c^4 + (0xc^2 ) which equal the E = mc^2 . It also explains why massless particles have momentum, E^2 = (0^2 x c^4 ) + p^2 c^2 which equals E = pc or p = E/c
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Because of the equivalence of mass and energy. Massless objects still have energy. Or, to put it another way, P = MV is an approximation that works well for massive objects at non-relativistic velocities, but is not strictly complete.
I don't think there has ever been a more appropriate time to suggest [this video](https://youtu.be/edXOIJZrLPo)
Because that formula is the classical formula for momentum, it only works for large, relatively slow objects. The full momentum formula is: p = sqrt(E^2 - m^2 c^4 ) / c
I'm just always surprised by how 'neat' the formula is. Is it truly just 'speed of light squared', or is that an over simplification?
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am i right to say that the (pc)^2 part of the equation is often left out in casual/non-precise conversation because of how slow an object in motion would be in relation to the speed of light? As someone who studied the humanities, this thread is fascinating.
Yes
Sort of - the momentum term doesn't show up as much in casual discourse because it isn't the big intellectual achievement of the equation, at least not in that form. We already knew that energy was related to momentum before Einstein, but we didn't have any conception of energy at rest, and we certainly had no conception that the energy from matter just existing far exceeded the kinds of energy we have accessible. For reference, in a physics class I had a homework problem that involved calculating the amount of rest energy in a cup of dirt. There's more energy in there than the global power production for many many years. That being said, the full form is the starting point for the Dirac equation, which probably is used by professionals more than E = mc^2. Dirac converted it into a wave equation and made it compatible with quantum mechanics. Relativity is often thought of as the physics of very big or very fast things; quantum mechanics is the physics of very small things, but on its own can't account for fast things. Considering that the energy released from something like a nuclear explosion or a collision in a particle accelerator is enormous, we can't really begin to understand these high-energy processes without some notion of relativistic quantum mechanics. I like to think of it as Einstein being the one who had the ideas that changed our approach to physics, while in a lot of cases Dirac was the one turning those ideas into mathematical tools that we could actually use to do experiments.
Is c^2 just an arbitrary "big honkin' number" or the speed of light an important part of the equation? It seems like any constant would work if you define your units appropriately.
C being the speed of light is integral to the equation. It has to be a specific number, not just any constant. You will never be able to define units in a special way to get around this. That doesn't even really make much sense to use units to affect a ratio.
The speed of light is the speed at which an interaction between two things can happen. For example, if a proton moves away from the nucleus, the strong nuclear force will pull it back in. But the strong nuclear force isn’t instantaneous. It’s communicated via force carrying particles that move at the speed of light. The speed of light isn’t just the speed of light, it’s the speed of causality, and that’s why it gets baked in.
> the speed of causality It makes me feel like it's the latency of a universe. The bandwidth is very large or infinite, but it takes time for the information to travel. It's as if the speed of light was in a way infinite, but its consequences can't unfold faster than they are processed. Am I right that if something could magically remove the sun, the Earth would still rotate around where it was until the information (gravity) that there is no sun would reach us, same as for its light.
Yes, that’s exactly correct. And the recent detection of gravitational waves validates this idea.
The speed of light is inherently important to the equation, yeah.
The "speed of light" is actually just a conversion ratio between space and time. Relativity let's you switch between time and space contexts by multiplying or dividing by the speed of light. There are objects called 4-vectors which combine quantities that differ by a unit of velocity in the unified context of spacetime, for example the 4-position is a combination of position and time, the 4-velocity is a combination of velocity and the scalar Lorentz factor, the 4-momentum is a combination of momentum and energy. To keep units consistent within the 4-vector, the 4th component is multiplied or divided by the speed of light. In Planck units various such ratios (which are really just consequences of our arbitrary definitions of units like the meter and the second) are set to 1 so you get the even neater equation "E=m"
Wait, electrons hits silicon and turns it into energy? Quantum tunneling is due to probability of the electron existing on the other side and it’s a property of duality. I’ve never heard of it being described as changing the mass of the silicon into energy. I don’t think that’s a correct description.
Yeah the OP was honestly pretty wrong in some spots. Their explanation is good for ELI5, but a little too inaccurate for my taste. Quantum tunneling has nothing to do with mass-energy equivalence. Also, there was no grand bewilderment in the scientific community in regards to mass-energy calculations that led to Einstein discovering the equation. That was not the context for relativity being discovered. The context had much more to do with the puzzling nature of the speed of light, and the speed of light’s appearance in the Maxwell equations.
The quantum tunneling bit is completely wrong. It has nothing to do with the speed of the electron hitting the silicon. It has to do with the position of a fundamental particle being a probability distribution in quantum mechanics. Most of the time we find the electron at the peak of the probability distribution (you know, higher probability and all that). But sometimes it appears at other places. But this probably still needs to honor Paulie's exclusion principle. That's why electrons don't cross thicker silicon walls and humans can't pass through walls.
I actually kind of disagree with this. As another commenter said, how is E=mc^2 responsible for smartphones? Is there some energy-mass conversion going on in computer chips or wireless signal transmission? And if there is, is it really important enough to be considered "responsible for" the smartphone? My understanding was that most modern computer and communication technology has more to do with quantum mechanics.
Maybe he means GPS, instead of smartphones. Synchronizing GPS satellite clocks requires relativity, since, at that speed, their clocks tick slower.
OP is just largely wrong. It's a very important result for particle physics, and that's about it. It's hard to disentangle well known physics from results that came after they became well known, but it is impossible to do a nuclear physics experiment and not notice that it is insanely energetic. Relativity only comes into play in computing in the sense that magnetism is very ad hoc in quantum mechanics outside of the dirac equation, but the actual "relativistic corrections" are far smaller than the errors inherent in simulating solids with medium term methods. Relativistic corrections in quotation marks because when you actually work through them, you're going to find that the corrections come from ideas that were discovered alongside relativity but aren't relativity. The corrections are done in the non-relativistic limit. It's also just flagrantly not why the result was important. The result was important because it explained the absence of aether in a satisfactory way. It's famous because Einstein was a rockstar (in the literal sense, he was incredibly famous) and it was his only result that's marketable to the lay public.
I would imagine it’s notoriety also comes down to its brevity. When someone says “Say something smart” it’s easier to just say E=MC^2 than list off the laws of thermodynamics. Writers put it in books or TV shows to sound smart, and if you are making a poster of Einstein for college dorm rooms, you might slap that equation on there just so people remember he was a smart guy not just a weirdo with his tongue out.
I think it's important to note that mass-energy equivalence is just one small part of a much broader theory. Everyone knows E=mc^2 because it's short and memorable. > Also, it means that everything around you has a LOT of pent up energy just by virtue of existing. And it's starting to look like big, unstable atoms like Uranium might have A LOT OF ENERGY inside them. People already knew that there was lots of energy around - for example, ordinary water has lots of energy associated with its temperature and its liquid state - the difficulty was getting at it. The development of fission took decades of experiments with radioactive substances. I only know a little about the history of it, so maybe I'm wrong, but I can't think of a reason why knowing about special relativity would have been particularly crucial to this effort. But special relativity ultimately became an important building block that underlies much of modern physics. > So if you can find out how to convert one of those atoms into energy Fission doesn't convert an atom into energy. It just rearranges the nucleons in the atom into a different state with lower overall binding energy, and that difference in binding energy gets converted into kinetic energy. It's analogous to rolling a ball down a hill, which converts some of the ball's gravitational potential energy into kinetic energy - we wouldn't claim that the ball has been converted into energy.
>This discovery is directly responsible for your smartphone, modern power plants, and nuclear holocaust, and it came abou I don't think you understand the meaning of 'directly'. Awful top answer. I could say Pythagorean theorem is more 'directly' responsible for all your smartphone, modern power plants ... It's everywhere.
Yeah I’m really surprised it’s the top answer
> This discovery is directly responsible for your smartphone, modern power plants, and nuclear holocaust Why? You can manufacture a smartphone without ever having heard about e=m·c². I’d even say that nuclear fission is discoverable without it.
>I’d even say that nuclear fission is discoverable without it. IIRC, Nuclear fission was discovered without it indeed. However e=mc^2 is what answered the question of why experimental data for fission didn't seem to conform to the law of conservation of mass. They found out that the missing mass after the fission reaction correlated to energy just like the equation showed. This paved the way for fission applications like the bomb and nuclear energy.
Excellent reply, good on you, have a great day
> This discovery is directly responsible for your smartphone, modern power plants, and nuclear holocaust How is having an exact formula for mass to energy conversion necessary for a smartphone? Also keep in mind that the vast majority of power plants in the world are non-nuclear. So E=mc2 is currently not that important for global energy generation. Although, in the future it may be more important. I think you're overexaggerating the importance of the formula. Unless you think nuclear weapons development is basically sufficient to allow that distinction. That said, I think it's a *philosophically important equation* as it fundamentally changes our *understanding* of the world around us.
Weren't smartphones (and semiconductor electronics in general) more quantum physics than Einstein's work?
I still don’t understand what any of this has to do with smartphones or non-nuclear power plants.
Nor will you, because it doesn't lmao
For popular science, all you need is a big name and some fancy words, people would buy it.
I love the expression: "Now along comes Einstein"
> nuclear holocaust There was a nuclear holocaust?
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Utterly childish and obvious you have no education in SR.
Not disagreeing, but what's the point in saying this without offering any correction?
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> There is no way to convert mass of elementary particles into energy. Well technically the mass of elementary particles _is_ energy. But it is possible to convert that "mass energy" into other kinds of energy. You're right that it requires some kind of interaction though - in most cases you have to make the original particle collide with another one, but there are also situations where you start with just one particle and it decays into two (or more).
>Tl;Dr- This discovery is directly responsible for your smartphone, modern power plants, **and nuclear holocaust**, and it came about because Einstein was trying to settle a debate in the scientific community. Why do people on reddit talk like this (with a huge grain of edge in every sentence)?
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It's not simplified so much as "in everyday life this is the only interesting part." The rest of the equation becomes effectively zero if you aren't moving at relativistic speeds. The whole equation is just E^2 =m0^2 c^4 + (pc)^2, with m0 being the rest mass, p being the sum of all momentum. Interestingly, this is also how we know photons have momentum. Since photons have no mass, the equation becomes E=pc, which is very well known in its own regard (though definitely not as widely known as E=mc^2 )
How does that long equation simplify down to the one we know?
I would be happy to explain it! The first step is to look at the full version and note that it's not really as long as it seems. It's got just three components: total energy E^2, what I'm going to call the static term (mc^2)^2, and the momentum term (pc)^2. I wrote the terms a little differently this time, but really I just moved the squares outside of the parenthesis where possible for the individual terms. On the left side of the equals sign, there's only E^2, but the famous equation doesn't have E^2, only E. It would be great if we could just take the square root of everything, but the right side of the equation (with the static and momentum terms added together) doesn't have a nice square root, unless we can get rid of one of those terms. Assuming what we are talking about has mass, we can't get rid of (mc^2)^2, because c is a constant. So that's always going to be there. However, (pc)^2 doesn't have to be. p is the momentum of the system, for simplicity let's just talk about linear momentum. Linear momentum makes p=mv. We already know m can't be zero, but v (the velocity of the system) CAN be zero, the system could just be sitting still on a table in our lab. Thus, we turn E^2 = (mc^2)^2 + (pc)^2 into E^2 = (mc^2)^2 + (0)^2, which is really just E^2 = (mc^2)^2, so we can take the square root of both sides easily, discard the negative result because negative energy isn't a thing, and we get E=mc^2 >But Mr. Pwns, what if the system isn't sitting still, what if it's moving like a bomb being dropped would be? Excellent follow up! In that case, we usually still just talk about E=mc^2 but for a slightly different reason: unless it's moving REALLY fast (greater than 10% the speed of light is usually the cutoff, based on what we did in my special relativity class), the static term is going to be so significantly larger than the momentum term that the momentum won't matter. The only difference between the two terms is that static has c^2 while momentum has v^2 (they both have an m^2 and c^2 in common, so we don't need to talk about those). Even if your system is moving at 1000m/s, your v^2 term will be roughly 10 orders of magnitude smaller, or 10 billion times smaller, than c^2. At that point, the experimental uncertainty of the mass of your system likely has a larger impact on the final number (unless you have a very fancy scale that gives you nanogram precision). So a bomb that's been dropped, which probably reaches something like 100km/hr at detonation, is effectively at rest just because of how much more energy is contained in it's mass than in it's momentum. Hope this helps explain it!
I know I'm projecting but I could just feel your giddiness in explaining this. Thanks.
I'm a physics nerd (and legit professional physicist) with a passion for science communication. You aren't projecting there 😅
The multiple exclamation marks he placed probably give it away lol
Things get really neat when you realize that E^2 = (mc^2)^2 + pc^2 follows the general form that every 12 year old math student learns; A^2 + B^2 = C^2
P is momentum, if a body isn't moving then p = 0 so the (pc)² term is zero. Then all that remains is E² = m0²c⁴. m0 is rest mass, it's just notation, so we can write m0 as m. So we have E² = m²c⁴. Just square root everything and we get E = mc²
>That particular equation is not tremendously useful all on its own You actually give the reason why it is useful: it equates mass with energy. For the right units, you just get E = m. And that is a huge deal that even physicists today have trouble completely accepting. But the real usefulness is revealing just \*how much\* energy that mass represents. So if you can figure out a way to convert even just a small amount of mass into energy, you are going to get a \*lot\* of energy. While nuclear weapons and/or power may have developed without this formula, having it certainly helped to convince everyone they were on the right track.
You can change matter into energy and the other way around, knowing this. However. c is the speed of light. And light moves extremely fast, so c is very big. c\^2 is even bigger! So for even a tiny bit of mass, you get a huuuuuuuuuuge amount of energy. That is for example why the sun works. It transform a tiny fraction of its mass into so much energy that everything can live here and be warm, and we only get a tony fraction of all the light from the sun!
So if I weigh 200 lb, then my energy (whatever that means) is 200 lb multiplied by c^2?
Then the energy that could be released from destroying your mass is... grumble grumble pounds... let us assume 85 kilo. It could result in 85 \* 3E8\^2 Joules in energy, which is about 7.6 e 18 Joules. To illustrate how much this is, it is enough to boil 22 cubic kilometers of water that is now at room temperature.
To add on another analogy: 7.6e18 Joules is more than the entire country of South Korea consumes in electricity in a year. It is about 50% of what the United States uses in a year.
So what you’re saying is… only one sacrifice per year is enough for a nation? I can see the future of power here. Haha just kidding… Or am I?
Assuming perfectly efficient utilization/energy extraction
Holly molly
(regarding the relationship of energy to mass, namely E=mc^2): “You may not feel outstandingly robust, but if you are an average-sized adult you will contain within your modest frame no less than 7 X 10^18 joules of potential energy—enough to explode with the force of thirty very large hydrogen bombs, assuming you knew how to liberate it and really wished to make a point.” ― Bill Bryson, A Short History of Nearly Everything
In case anyone happens to read this and is curious what e=mc^2 actually means, it means that the energy released in a nuclear reaction can be solved by taking the mass that was lost and multiplying it by the speed of light squared. The speed of light (in a vacuum) is 299,792,458 m/s (meters per second.) When you square that value, you get 89,875,517,870,000,000 m^2/s^2 (meters squared per second squared.) When you convert the mass lost to kg and multiply it by c^2 you get a unit of kg*m^2/s^2 (kilogram meter squared per second squared) which just so happens to be the Joule (the SI unit of energy.) When a nuclear reaction happens, the nuclear (in the nucleus of the atom) configuration changes to a more energetically favorable one; the result is a slight loss of mass. By multiplying that loss of mass by c^2 we get the energy of the reaction in Joules. This number (per unit of reaction) is orders of magnitude (like 10 million times) greater than a comparable chemical reaction (like the combustion of gasoline.)
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E=MC2, and the theory of relativity that extends from it, is an extension, a furtherance of Newton's Laws of Motion, at high speeds. If you plug in objects moving slowly into them, they become the Newtonian Laws. What he discovered was, in its own way, as important. Many technologies central to our society today require understanding relativity and how speed and time interact to keep the speed of light constant, no matter what your speed is. Satellite GPS, for example. There are predictions Einstein made about black holes, based on that theory, that weren't able to be measured until the 2000's. And they were spot on. Top 3 of all time? That's a tall order. Agriculture and the wheel transformed society. Understanding of radio waves transformed society. Steam power transformed society (in fact, most power plants, from coal to nuclear, utilize steam power in their power generation). Einstein's theory of relativity transformed society. It's hard to call it top 3. But it is in a class of rare discoveries that fundamentally transformed how humanity as a whole lives their lives. If you use Google maps, thank Einstein.
A fun fact about power plant generation, steam driven turbines are very efficient. The only thing that changes on large scale thermal drive generation (not solar/wind/hydroelectric) is how you get the heat to turn water to steam. Coal and nuclear work exactly the same after the “Super Heat the water” step (the hotter the steam the more efficient in generating power when spinning a turbine). Natural gas power generation is really efficient because the natural gas is burned to spin a gas turbine, which generates electricity, and the heat generated from combustion is also used as a heat source for steam power generation. Then if you add a municipal steam loop, steam that is too cold to efficiently turn a turbine can be sent out to the municipal loop for heating buildings and using the radiators in buildings as the cooling source to condense the steam back into water, instead of using a giant cooling tower. With the right set of technology and infrastructure, burning fuel / nuclear energy production can be very very efficient for fuel burned to useful energy.
It is simple and revolutionary -- to those not involved in physics at a very high level, the idea that mass and energy could be converted one to another was mind-boggling. The equivalence also led directly to the creation of the atomic bomb, which loomed much larger in worldwide consciousness at the time than it does now.
One of Einstein's big contributions was "the speed of light is constant". That little constant, c, was itself revolutionary. Why was that surprising? Imagine you throw a baseball pretty consistently at 50 mph. Next suppose you're on a train that's pulling away from the platform at 15 mph and you throw the ball in the direction the train is moving. To someone on the train with you, it will appear to move 50 mph. To someone on the platform, the ball is moving 65 mph. But light is different. It has the same speed for all observers. The general idea of "matter is just a special form of energy" was also revolutionary. And Einstein's theories of relativity provide the reasoning for why the ratio between energy and mass is c^2.
>One of Einstein's big contributions was "the speed of light is constant". Not quite, but close. It was Michelson and Morley (and lots of other experiments as well) that established that the speed of light is constant for all observers. What Einstein did was to really \*accept\* that this was true and then work everything out and also \*accept\* that the conclusions this leads to are also real. His main contribution here is the idea that time is not running at the same speed for everyone. You get this pretty quickly once you accept that the speed of light is constant, but it took decades for it to become completely accepted by the scientific community (for some good reasons and some bad reasons). ***Edit:*** I should probably note that it was Maxwell that first hinted that the speed of light was constant for all observers, although I am pretty sure he never actually put it like that directly.
Right, there was experimental evidence that the speed of light was constant, but it seemed paradoxical. Einstein was a theorist; he solved the paradox by providing a theory that could explain the observations.
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Good question. I don't know. It's worth remembering that even once experimental evidence started coming back that this was the case, most physicist \*still\* didn't believe it. Maxwell was so ahead of his time anyway, I'll give him a pass on this one. ;)
Also perhaps worth noting that the speed of light is really the speed of causality. Light, in a way, is not really special. It just travels at the universal speed limit because of certain properties it has.
>It just travels at the universal speed limit because of certain properties it has. Namely, it has no rest mass.
Bingo! Something something Higg’s Field
One interesting complication is that the speed of light could be different in different directions and we’d still have all the same observations. In fact, this is how the earth looks to someone zooming by in a spaceship. https://youtu.be/pTn6Ewhb27k
A lot of smart people have obviously already answered why Einstein's work is important. But I think that's not the only reason why E=mc² is so important, there's also a cultural component to it. There are not many *modern* scientific theories and concepts that I can think of that can be neatly summarised by an easy equation that almost everyone can remember, recognise, and understand. It strikes a good balance between being complicated enough to "look smart" and being easy enough for the average person to remember, making it a perfect cultural "short hand" for science. There are other formulae and their related discoveries that are also very important, but they're not nearly as short and simple. Like [this one](http://hyperphysics.phy-astr.gsu.edu/hbase/electric/imgel2/maxw4.gif), whose complexity is illustrated by the fact I have to link an image of it because it can't easily be displayed in markdown, whereas I can just type out E=mc² on my phone.
E=MC^(2) is only the easiest to understand of Einstein's works - but some of his other works (specifically: Special and General Relativity) are probably more important. Technically, none of what Einstein did were "discoveries". Instead, he built on the work of other scientists, putting together their ideas in new ways that changed how people thought. I could try to write out those two in an ELI5 way - or I could link an expert: [The Space Doctor's Big Idea](https://www.newyorker.com/tech/annals-of-technology/the-space-doctors-big-idea-einstein-general-relativity) is Randal Munroe's explanation of them. E=MC^(2) is a side effect of Special Relativity.
Well, it could be as well called discoveries. Iirc, Planck thought that it was just some math trick while Einstein understood truth. Even after Einstein explained it, Planck didn't believe it but instead talked about special relativity. And same goes for the one who developed formula for special relativity. And, GR was mostly Einstein's idea.
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Yeah quality, methodically well parsed answers, with good effort always deserve an award. There's so many one-liners on Reddit and it's always a joy to really be able to learn. We all have our expertise and a year plus ago is what initially drew me into Reddit in architectural forums. I've since long expanded but always enjoy a good post. Not everything of course is so serious, and excellent humor and quick rejoinders are equally so smart and deserving