Help me understand E=mc²

halkun

Ok. I thought tonight I'm going to try and get my brain around the math behind Einstein's Equivalency Principle. Now here comes the laughable bit....

I failed basic algebra three times in a row and got thrown out of my university. (I later got a degree in business at another school)

Now to preface this. I know what the Equivalency Principle is, and I'm going to demonstrate a part of it visually, because my math sucks and I have no other way to express it. After I lay down that framework, I'm going to make a laughable attempt to "read" the equation.

Say I'm floating in space, and I have two pieces of mass floating along with me. One is 1kg and has a T-handle on it so a I can grab it and spin it around an axis. The second piece of mass is 10,000kg and also has a T-handle for me to grab. Now, what I do is I go to the first piece of mass, grab the handle and spin it. It then goes spinning off into space like a top. No effort at all on my part. Then I go to the huge piece of mass, grab it's handle and give it a spin. Then then launch *myself* spinning off into space and the chunk of mass imperceptibly moves. The idea is the energy needed to move that mass is great, while the little piece of mass needed very little energy to get it to spin. I'm going to start faltering a little bit, so sorry if my terminology is wrong, but "standing still inertia" of the giant mass is an energy, and my attempt to use my energy to move it was way too little. I know that sounds dumb, but that's the best I can explain it.

Particle accelerators work on the principle too. You take two protons and smash them together near the speed of light. In the collision, energy is converted to mass in the form of the particulate spew that comes out of the explosion. However, all energy/mass is conserved in the conversion.

Now that I have my shaky framework down, I'm going to make you all laugh at my terrible attempt to apply the equation. Now, I know I'm reading this wrong (and I know it's gonna be hella wrong), but it's the only starting point I can think of. I warn you, this is not for those with weak stomachs. I'm going to do some pretty obscene shit here with these numbers.

Here we go...

E=mc²

Energy equals mass times the speed of light squared. This means that if you take mass and accelerate to twice the speed of light, it becomes energy. How is that even possible? How does that even relate to inertia?

For fun, let's solve for different parts of the equation. Before we go further, let me tell you about my math skills for a bit. You see, when I solve for a variable in an equation. I actually have no idea what I'm doing, but I know I did it right. My teacher taught me that when you want to keep an equation balanced, think of the equals sign as a transporter from Star Trek. When Good Spock went in, Evil Spock came out the other side. So you use the equals sign to "flip" the "polarity" of the item making it the "evil" version on the other side.

Positive becomes negative.
Multiply becomes divide.
Exponents become square roots.

Now *why* does this happen? I couldn't even tell you. I seriously can not tell you why that mechanic works. I just know it's a "rule". It served me well but then the rules started to compound and I didn't understand what I was doing, other then "teleporting" numbers. When I got into the harder math (and heaven forbid, word problems, that required you to understand the number's meaning, I completely failed.

Anywho... continuing....

Let's try it again, I'm going to my piss-poor math to move the variables around so I can convert energy into mass...

m = E/c²

Now that makes even less sense. Mass equals energy divided by the speed of light squared. So you slow the energy... down? Is that what happens in the particle accelerators? The energy all comes to a stop in the collision?

What happens when you try and get the known constant by itself?
` ____
√(E/m) = c

This is completely indecipherable to me.

Anyone want to take a stab explaining this to me?

japan

halkun wrote:

E=mc²

Energy equals mass times the speed of light squared. This means that if you take mass and accelerate to twice the speed of light, it becomes energy. How is that even possible? How does that even relate to inertia?

It doesn't mean this. Mass can't be accelerated to twice the speed of light (or in fact, to the speed of light) in any case.

You probably need to start with the understanding that mass in the Newtonian physics sense (like your example with a large mass and a small mass floating in space), and mass in a relativistic sense are not quite the same thing. According to wiki Einstein didn't like the expression "relativistic mass" precisely because it's misleading.

The equivalence is partly a way to reconcile the fact that something without mass can still have momentum (which is classically mass multiplied by velocity).

At this point I'm not sure how to explain any further without resorting to calculus, which I'm not sure will help.

Paladin

Before you can learn anything else in physics, you learn dimensional analysis.

Energy is measured in mass * distance^2/time^2. One joule, which is a metric unit of energy, is kg*m^2/s^2. Energy is kind of a function of force: the potential energy an apple on a table has that you can unleash by knocking it off the table is E(apple) = mass * the force of gravity * height of the table. One newton is a metric unit of force: kg*m/s^2. We can take that back further to Newton's second law: Force = mass*acceleration. When you push a toy car from rest, you're sending it from a velocity of zero to some higher velocity, which means you're accelerating it. If it's one of those micro machines with a penny in the back, or it's made of metal or something, it has more mass and you must use more force to push it, hence F = ma, which when you use dimensional analysis, is kg*m/s^2. Therefore, if you understand what mass and acceleration are, you understand what energy is, and why the units of energy are like that.

So, mass*distance^2/time^2. We broke it down into mass*force*distance before, but notice that distance^2/time^2 basically is (distance/time)^2. Distance/time is velocity. V^2. So the remaining thing is mass. We can rewrite the units of energy, therefore, to be mass * velocity^2.

The speed of light is a velocity, and the mass of whatever is a mass. E is still energy. E = mc^2.

Now to put it better in context. There are two popular formulas for energy. The first one I've already told you about: potential energy. E = mgh. (g is gravity). That's an object at a height that obviously has more energy than an object on the ground, because you can drop it from that height and it will fall down, crack the pavement, hit someone on the head, etc. The other type of energy is kinetic energy: the energy of an object already in motion. A car isn't suspended hundreds of feet off the ground usually, but you still wouldn't want to get hit by a speeding car because it's got a lot of energy it's just dying to impart to you.

The formula for that is E=1/2 * mv^2. This is much closer to the E=mc^2 recipe you saw before, and illustrates that as long as you know that m=mass and v=distance/time, you can mess with physics equations purely mathematically and get equally valid and applicable formulas to real life.


As far as why the speed of light seems to be the magic velocity number for the conversion of energy to matter and vice versa, that's pretty advanced stuff and you should learn the basics of physics before you do that kind of thing.


I was going to look for a graph to explain really the only layman's takeaway from E=mc^2, which is basically the speed of light is an unattainable and impassable speed, and the first image result links to a web page that basically explains the fundamentals much better than I did.

tarnok

The bit that Japan quoted also contains a common math error. c^2 is not twice the speed of light, it is the speed of light times itself. Twice the speed of light would be the speed of light times two.

The easiest way to visualize the meaning of the equation, in my opinion, is that it describes the amount of energy contained in a given mass. Moreover, when you consider that the speed of light is about 300,000,000 m/s, and that the speed of light squared is 90,000,000,000,000,000 m^2/s^2, it quickly becomes apparent that even a very tiny bit of matter contains an astonishing amount of energy. This is, of course, where the idea for the nuclear bomb comes from. In a nuclear bomb a very tiny amount of matter is converted almost instantly into energy. As a consequence of this equation that tiny amount of matter contains sufficient energy to completely destroy a large city.

To put it in terms more applicable to particle accelerators try solving for m; you'll get m = E/c^2. In other words, the amount of mass produced in the collisions you mentioned would be equal to the amount of energy put into the particles divided by the speed of light squared (which remember, is the ridiculously huge number 90,000,000,000,000,000). So to get even the handful of sub-atomic particles they're looking for requires a staggering amount of energy.

The third one is relatively simple to understand if you remember what each variable stands for; E is the amount of energy contained in a given mass, m is the amount of mass and c is the speed of light. So if you divide the amount of energy by the amount of mass and then take the square root you will find that it is equal to the speed of light. It's a statement of the universal relationship between mass and energy, just as pi is a statement of the universal relationship between diameter and circumference in circles. No matter what the mass is made of or where it is in the universe, the amount of energy in that mass is related to the amount of mass by E = mc^2.

From reading your post I get the impression that your confusion arises from trying to hold onto the meaning of the individual factors in the equation. In math, variables don't really have "meaning" while you're working on them. Once you plug it into the equation the speed of light is just a number. You cannot divide energy by a speed, you can only divide numbers. It is not the case that the equation is saying that a mass accelerated to twice the speed of light will turn into energy. Multiplication is not the same thing as acceleration, they are not interchangeable. Similarly dividing something by a speed does not mean slowing it down. You mentioned that you don't have a very firm grasp of math and that's really hurting you here. This describes the numerical relationship between mass, energy, and the speed of light; it does not include instructions for accelerating or slowing or any physical thing or act.

Hahnsoo1

E = mc^2 is an application of special relativity (the fact that you cannot apply Force to an object and accelerate it past the speed of light, which is a Lorentz factor applied to Force). It's really a more elegant way of writing the definition of the rest energy of a particle (compared to writing it as the difference between the relativistic mass of a particle moving close to the speed of light and the rest energy of the mass when it is not in motion). Einstein didn't write up E = mc^2 and then work from there; he thought up special relativity at speeds approaching the speed of light first, then applied those equations to Newton's Laws of motion (F=ma... leading to Energy or Work as W=.5mv^2) to determine the energy of a particle. The definition is more of an elegant result... the method by which he determined it is far more complicated and interesting.

You might want to pick up the book "Einstein for Beginners". It's a mini-graphic novel which lays down the foundation for both special relativity and how it can be applied to Newton's Laws of Motion to finally determine the mass-energy equivalence.

badger2d

I read a fascinating book some years ago about this equation (a book written for laymen, which I certainly am one with regard to physics), and the one mind-bending fact that I really took away from it was that this very simple equation is just a conversion factor, which means that in some sense, matter is energy, and vice versa. That's a much simpler fact than the way you're trying to work with it, the way you're trying to think in terms of mass "speeding up" or "slowing down". Multiplying a mass by a particular number (c^2, which is a constant) tells you how much energy that mass is, if you want to look at in terms of energy.

Someone who knows more about physics may well tell me I'm wrong to look at it this way. But so far, as one layman to another, I think this way makes more sense than what you're struggling with, at least.

RadicalTurnip

Yeah, that's just an equation. The "C" in the equation is a given constant, that happens to be the same number as if you square the speed of light. It doesn't mean that you have to do anything with the speed of light, it's just telling you how to convert from energy to mass or vice-versa.

As for the teleporter with evil Spock: Lets take the equation 2x=6. You want to solve the equation for x, and so the real idea is that you're trying to get x alone on its side of the Equals sign. Numbers don't become their "evil" versions, *you* the mathematician get to do whatever you like to the equation; but remember: if you do something to one side of the equals sign, then you have to do it to the other as well. So how would you recommend getting that x alone on its side? Well, you have to get rid of the two. To do that, you can divide by two, so 2x/2=6/2 which, if you simplify it, means that x=3. Yay, you've solved it!

If you wanted to (but there would be no reason to...) you could also do silly things like add one to both sides. So we also know that 2x+1=6+1. Heck, we could even square both sides. (2x)^2=6^2...but there is no reason to do this if you're trying to solve for x. There are other instances that you may want to do that in, but not for 2x=6. I would recommend that if you want to get a better understanding in algebra, forget the evil teleporter thing because it's just hindering you. Instead, just realize that with an equation you can do anything you want to it, as long as you do it to both sides (the equation is your bitch, and it's a slutty bitch at that).

drfog

This thread is fucking dope! Very interesting read.

Reric

RadicalTurnip wrote:

Instead, just realize that with an equation you can do anything you want to it, as long as you do it to both sides (the equation is your bitch, and it's a slutty bitch at that).

You do have to be somewhat careful with what you are doing to the equation though you can sometimes cause issues, like when you square both sides of something simply like x=3 making xx=9 you suddenly have the equation having x = 3 or -3. There are also times when you may not realizing that you are diving by zero during the whole process, you'll sometimes see fake "proofs" of things like 2=3

GotrM15

To boil this down into a very simple phrase: matter is just another form of energy.

There is heat energy, light (electromagnetic energy), kinetic energy, potential energy (say if I have a ball at the top of a hill; relative to the bottom of the hill, the ball has potential energy which can be converted into kinetic energy if I roll the ball), and other kinds of energy. One such form of energy is matter.

E=mc^2 tells us that the amount of energy (E) in a given object with mass (m) is the product of that mass and the speed of light squared -- the fact that c^2 is a part of this equation actually doesn't have anything directly to do with speed.

So if I could convert all of the matter energy in a human person (of average weight 59 kg) into energy, I would have 59 kg * (299,792,458 m/s)^2 worth of energy, or 18 TJ of energy (for reference, that means that 3 people can be converted into as much energy as the yield of the bomb the US dropped on Hiroshima in 1945).

Khavall

Yeah, the simplest problem involved is that you're thinking of the variable E, which describes an amount of energy, as the noun Energy, describing a state of being.

Also you're thinking of the variable m, which describes an amount of mass, as the noun Mass, describing a specific object in a state of being.

E=mc^2 doesn't talk about transitioning between states. In other words, it doesn't talk about mass becoming energy. It simply exists to describe the amount of energy in specific amounts of mass.

tinwhiskers

GotrM15 wrote:

So if I could convert all of the matter energy in a human person (of average weight 59 kg) into energy, I would have 59 kg * (299,792,458 m/s)^2 worth of energy, or 18 TJ of energy (for reference, that means that 3 people can be converted into as much energy as the yield of the bomb the US dropped on Hiroshima in 1945).

good explanation, but your maths wrong on this. A TJ is 10^12 joules, C^2 is 8.987*10^18


2 kg of mass, completely converted(anti-mater annihilation), would be about 100,000 more powerful than the Hiroshima bomb. Or, put the other way, if you could annihilate .2g of mater, you'd release the force of 1 Hiroshima bomb.

The Count Of Midget Fisto

http://www.youtube.com/watch?v=mkiCPMjpysc

Syphyre

The Count Of Midget Fisto wrote:

http://www.youtube.com/watch?v=mkiCPMjpysc

This is excellent for my level of mathematical knowledge. Thank you!