10 ms·

Three Meanings of E=mc²

149 points by LisaDziuba 9y ago

goldenkey 9y ago
The most important part is the conversion of a vector quantity into a scalar. Energy packets, photons, always have a prescribed velocity or direction. When mass is created, this direction is lost. It shows us a fundamental anisotropy of our universe. Total momentum is conserved..but since 0 momentum is degenerate and has no direction, we get the mixing of isotropic emissions, that should in theory, cancel out any biases that may have once existed. Our universe cares about conservation globally - but as scattering shows, at the quantum scale, our universe could care less about you switching a left and a right for an up and a down.
- millstone 9y ago
  Huh? If you have a system with nonzero momentum, it can't later have zero momentum. The "prescribed velocity" can't be lost, it must be conserved.
  - goldenkey 9y ago
    We are talking about a net momentum of zero. It can be in all kinds of different forms without violating conservation. Exchanging a left and right moving particle for up and down ones, changes the system but retains global conservation
millstone 9y ago
E=mc^2 is surely one of the most misunderstood equations in physics. The best interpretation is as part of the full momentum-energy equation (letting c=1): E^2 = p^2 + m^2. This simply says that the energy E of a system is a combination of energy due to movement (p) and energy due to mass (m). At rest (p=0) this reduces to E=m, or E=mc^2 if you kept track of units. > Even masses at rest have an energy inherent to them. This is a real insight. > Mass can be converted into pure energy. This is the second meaning of the equation, where E = mc² tells us exactly how much energy you get from converting mass This is a pop-sci explanation, but it falls apart when you dig a bit. Does F=ma tell you that "force can be converted into acceleration?" Of course not; it tells you that force implies acceleration, and vice versa. Or: if mass can be converted into energy, then you would have more energy and less mass, so E=mc^2 would no longer hold. It can't be both an equivalence and an exchange ratio. > If you take a photon and and electron and smash them together, you get a photon and an electron out. But if you smash them together with enough energy, you’ll get a photon, and electron, and a new matter-antimatter pair of particles out. In other words, you will have created two new massive particles. Later we learn that mass is determined by energy and momentum, both of which are conserved, so mass must be conserved too.
- grondilu 9y ago
  > Does F=ma tell you that "force can be converted into acceleration?" Of course not; it tells you that force implies acceleration, and vice versa. I thought that F=ma is actually a definition of the concept of force. I mean, I may be wrong but it seems to me that before Newton, people only had a vague notion of what a force is. I suppose people considered it to be, in modern terms, a vector with a magnitude, and the direction of the vector was obvious, but I doubt they had any meaningful idea of what the magnitude was. When Newton stated that F=ma, he defined the concept of force precisely. Honestly, I suspect something similar is true about E=mc^2, except Einstein discovered that formula instead of positing it. After all, we know what Energy is, at least from quantum mechanics (E=ihd/dt), but do we know what mass is? We can't say it's the ratio between force and acceleration, since that would be circular. I suspect one can say mass is just a very dense form of energy. So dense that its different order of magnitude makes it look like it's a different thing. As for the distinction between impulsion and mass, well can't it be said that Energy (or mass, since I'm arguing it's the same thing) is actually a four-vector, and as such it has different projections on time and space depending on the frame of reference?
  - semi-extrinsic 9y ago
    > It seems to me that before Newton, people only had a vague notion of what a force is. > When Newton stated that F=ma, he defined the concept of force precisely. Newton wasn't the first to define the concept of force, nor the first to formulate the basic laws of motion. E.g. Galilei had formulated both the first and second laws in some sense, restricted to cases such as straight line motion on a plane. The laws came out of experiments. Newton's brilliant work was in collecting the three laws, generalizing them and realizing their combination applied to celestial mechanics as well as the mundane brick sliding down a plane.
  - millstone 9y ago
    Yes, it is not immediately obvious how mass ought to be undershood in the context of relativity. First attempts treated mass as a directed vector, with transverse and longitudinal components for an accelerated particle (now we would say that the mass is fixed, it is the momentum that changes non-linearly and just use F=dp/dt). Mass should rightly be considered more inherent in SR that energy. Energy in relativity is coordinate dependent, but mass is not: all observers agree on a system’s mass, because it is invariant. Re: the last paragraph, yes you can project the four vector onto time and space, and you get energy and momentum respectively. So energy is not the four vector itself but its projection onto the time axis (and mass is of course its magnitude).
deleted 9y ago
[deleted]
ocfnash 9y ago
One corollary of E=mc^2 is that the rest mass of a bound state of fundamental particles will not equal the sum of the rest masses of the individual constituents. For example, the rest mass of a Helium-4 nucleus turns out to be less than the sum of the rest masses of the four nucleons of which it is composed. The difference is only about 0.7% but is famously quite a lot of energy when interpreted using E=mc^2. It's a lot more fun if we go one level down though: protons are far HEAVIER than the sum of their constituents. Apparently almost 99% of the rest mass of a proton is the binding energy of the zero-rest-mass gluons that hold their three quarks together. I think it's fun that the vast majority of the mass of ordinary matter, of which we are all constituted, is best explained as gluon binding energy via E=mc^2.
- cohomologo 9y ago
  I think you've got that last part backwards. When you have constituents with binding energy, that reduces the total energy of the system and reduces the mass. So it seems that the mass of the proton is quark mass + gluon kinetic energy - bonding energy, and most of the mass comes from gluon kinetic energy.
  - ocfnash 9y ago
    Yes I do indeed; thank you for the correction!
  - hjaved 9y ago
    Thank you. Would like to understand that why does having constituents with binding energy reduce the total energy of the system instead of adding to it?
    - davrosthedalek 9y ago
      You have to think about it the other way around: The constituents don't have binding energy. That energy becomes available when you let them bind, and you have to pay it back if you want to unbind them. Example: If you have a proton and an electron, making a hydrogen atom will release about 13.6 eV (if it goes to the lowest energy state). You have to spend that 13.6eV to ionize the hydrogen, i.e. to get the electron far away from the proton again.
akubera 9y ago
The use of the term "pure energy" always grates me; it gives the impression that energy is an entity unto itself, which is not the case. I realize that it's used as a proxy for photons most of the time, but I still always think it's misleading.
- cousin_it 9y ago
  Oh, this is a pet peeve of mine. Jim Butcher does this constantly in the Dresden Files series. I keep coming up with lampoons like "the witch countered with a blast of pure angular momentum that sent him spinning" or "after a visit to the bank, my wallet was bursting with pure price".
- JabavuAdams 9y ago
  Yes! I just volunteered as a parent helper to a scientist in the school program. I was gritting my teeth as the "scientist" explained that light was "pure energy". I think people use this to mean "insubstantial" and to contrast versus matter. But that still leaves the problem of light pressure. I.e. a flashlight is like a very very weak rocket engine.
plopilop 9y ago
As I explained in an other article here on HN, another meaning (and the most important one imo) is the one Einstein actually wrote about: the inertia of a particle depends on its energy. The thing is, you don't read E = mc² but rather (as Einstein wrote in his surprisingly easy to understand 2-pages paper [0]), m = E / c². The direct interpretation of this formulation is that inertial mass is actually just a side effect of the energy of a particle. Put into a catchy phrase, mass is energy at rest. Edit : [0]: https://www.fourmilab.ch/etexts/einstein/E_mc2/e_mc2.pdf https://www.fourmilab.ch/etexts/einstein/E_mc2/e_mc2.pdf, translated in english. Note the understatement of the sentence before the last one: "It is not impossible that with bodies whose energy-content is variable to a high degree (e.g. with radium salts) the theory may be successfully put to the test."
- gus_massa 9y ago
  The problem is that the "mass" in the direction of the movement is different than the "mass" in the perpendicular direction. So most modern books try to avoid the relativistic mass "m" and use only the rest mass of the particle "m0". (Still, the "relativistic mass" can be useful to make a few back of the envelope calculations, but you must be careful.) More details: https://en.wikipedia.org/wiki/Mass_in_special_relativity#The_relativistic_mass_concept https://en.wikipedia.org/wiki/Mass_in_special_relativity#The...
  - plopilop 9y ago
    I did not know about this relativistic mass, but doesn't wikipedia says it is only true if you consider the mass as m = F/a rather than m = p/v?
contravariant 9y ago
>But under the laws of special relativity, mass simply couldn’t be the ultimate conserved quantity, since different observers would disagree about what the energy of a system was. Maybe I'm being pedantic, but not only does classical mechanics already tell us that the energy of a system different for different for observers, but this by itself doesn't mean it's not conserved, merely that it's not a universal quantity. Furthermore E = mc^2 doesn't break conservation of mass in any way, it's more accurate to say that it equates energy with mass, meaning conservation of energy and conservation of mass are one and the same. And in fact both still hold, when the energy and mass are replaced with their relativistic counterparts (E = mc^2 is obviously false using the classical notions of mass and energy).
ramzyo 9y ago
Fascinating. After first taking a crack at learning about this stuff decades ago, I still have trouble retaining it to the degree that I can succinctly explain its profoundness. This article does a great job at doing just that, and reigniting the sense of wonder and amazement at Einstein's achievements!
enugu 9y ago
What isnt usually mentioned is that this equation is a special case valid in some circumstances. The more general relation is E^2=(mc^2)^2+(pc)^2 which includes momentum term, and applies to photons which dont have a mass. Here's a simple video about this from Fermi Lab, https://www.youtube.com/watch?v=eOCKNH0zaho https://www.youtube.com/watch?v=eOCKNH0zaho Also interesting is how conservation laws for collisions are handled much better in relativistic setting.
nyc111 9y ago
" But even plain, old, regular mass at rest has energy inherent to it..." This sentence is so typical of sloppy so-called science writing in the US. What is "plain mass?" What is "old mass?" Or maybe he put the comma in the wrong place and he meant "plain old regular mass?" If so what is "plain old regular mass?" There is no such thing. But more importantly, nothing is ever at rest in the known world. It is absurd to discuss the properties of something when it is at rest. It will never be at rest. There is no absolute rest.
- ythn 9y ago
  This comment reads like the ravings of a pedant who is unable to relate to the layman.
  - nyc111 9y ago
    You may be right about my comment about his positioning of the comma but my statement that all is motion and nothing but motion is not pedantic or trivial.
    - betenoire 9y ago
      Motion is relative, no? I see things at rest from where I'm standing.
      - nyc111 9y ago
        No. Motion is absolute. Rest is relative. That is, rest is only an appearance. Since you are observing from earth everything you see is in motion since the earth is moving. So there is no absolute rest. And relative rest is just another name for motion. Therefore, no object can have a property that exists only when the object is at rest. Because there is no absolute rest.
        cgmg 9y ago
        > Motion is absolute. What are you talking about? Motion is definitely not absolute, and only Galilean invariance is needed to see that.
ythn 9y ago
If mass created from "pure energy" is always in matter/antimatter pairs, why is the ratio of matter/antimatter so skewed in favor of matter?
- davrosthedalek 9y ago
  That is indeed the one million dollar question. Physicist search for a reaction which breaks this balance to explain the matter/anti-matter imbalance of the observed universe.
  - 100ideas 9y ago
    Most of the antimatter ended up as early supermassive black holes.
- pixl97 9y ago
  The anthropological answer to this question is: Because we wouldn't be here to ask the question if it was balanced.
ucaetano 9y ago
> one of the simplest but most powerful equations ever to be written down I always felt that E=mc² is essentially the Euler's identity of physics: a basic, simple, beautiful equation relating some core components of the entire field.
deleted 9y ago
[deleted]
doctoboggan 9y ago
The YouTube channel PBS SpaceTime gives an excellent overview of these same ideas. The presenter is great at making things as simple as possible, but not simpler. https://www.youtube.com/watch?v=gSKzgpt4HBU https://www.youtube.com/watch?v=gSKzgpt4HBU https://www.youtube.com/watch?v=fHRqibyNMpw https://www.youtube.com/watch?v=fHRqibyNMpw https://www.youtube.com/watch?v=kixAljyfdqU https://www.youtube.com/watch?v=kixAljyfdqU
stcredzero 9y ago
"E = mc^2 is Wrong" (Actually, just incomplete.) https://www.youtube.com/watch?v=eOCKNH0zaho https://www.youtube.com/watch?v=eOCKNH0zaho

149 points by LisaDziuba 9y ago

The most important part is the conversion of a vector quantity into a scalar. Energy packets, photons, always have a prescribed velocity or direction. When mass is created, this direction is lost. It shows us a fundamental anisotropy of our universe. Total momentum is conserved..but since 0 momentum is degenerate and has no direction, we get the mixing of isotropic emissions, that should in theory, cancel out any biases that may have once existed. Our universe cares about conservation globally - but as scattering shows, at the quantum scale, our universe could care less about you switching a left and a right for an up and a down.

millstone 9y ago
Huh? If you have a system with nonzero momentum, it can't later have zero momentum. The "prescribed velocity" can't be lost, it must be conserved.
- goldenkey 9y ago
  We are talking about a net momentum of zero. It can be in all kinds of different forms without violating conservation. Exchanging a left and right moving particle for up and down ones, changes the system but retains global conservation

millstone 9y ago

E=mc^2 is surely one of the most misunderstood equations in physics. The best interpretation is as part of the full momentum-energy equation (letting c=1): E^2 = p^2 + m^2. This simply says that the energy E of a system is a combination of energy due to movement (p) and energy due to mass (m). At rest (p=0) this reduces to E=m, or E=mc^2 if you kept track of units. > Even masses at rest have an energy inherent to them. This is a real insight. > Mass can be converted into pure energy. This is the second meaning of the equation, where E = mc² tells us exactly how much energy you get from converting mass This is a pop-sci explanation, but it falls apart when you dig a bit. Does F=ma tell you that "force can be converted into acceleration?" Of course not; it tells you that force implies acceleration, and vice versa. Or: if mass can be converted into energy, then you would have more energy and less mass, so E=mc^2 would no longer hold. It can't be both an equivalence and an exchange ratio. > If you take a photon and and electron and smash them together, you get a photon and an electron out. But if you smash them together with enough energy, you’ll get a photon, and electron, and a new matter-antimatter pair of particles out. In other words, you will have created two new massive particles. Later we learn that mass is determined by energy and momentum, both of which are conserved, so mass must be conserved too.

grondilu 9y ago
> Does F=ma tell you that "force can be converted into acceleration?" Of course not; it tells you that force implies acceleration, and vice versa. I thought that F=ma is actually a definition of the concept of force. I mean, I may be wrong but it seems to me that before Newton, people only had a vague notion of what a force is. I suppose people considered it to be, in modern terms, a vector with a magnitude, and the direction of the vector was obvious, but I doubt they had any meaningful idea of what the magnitude was. When Newton stated that F=ma, he defined the concept of force precisely. Honestly, I suspect something similar is true about E=mc^2, except Einstein discovered that formula instead of positing it. After all, we know what Energy is, at least from quantum mechanics (E=ihd/dt), but do we know what mass is? We can't say it's the ratio between force and acceleration, since that would be circular. I suspect one can say mass is just a very dense form of energy. So dense that its different order of magnitude makes it look like it's a different thing. As for the distinction between impulsion and mass, well can't it be said that Energy (or mass, since I'm arguing it's the same thing) is actually a four-vector, and as such it has different projections on time and space depending on the frame of reference?
- semi-extrinsic 9y ago
  > It seems to me that before Newton, people only had a vague notion of what a force is. > When Newton stated that F=ma, he defined the concept of force precisely. Newton wasn't the first to define the concept of force, nor the first to formulate the basic laws of motion. E.g. Galilei had formulated both the first and second laws in some sense, restricted to cases such as straight line motion on a plane. The laws came out of experiments. Newton's brilliant work was in collecting the three laws, generalizing them and realizing their combination applied to celestial mechanics as well as the mundane brick sliding down a plane.
- millstone 9y ago
  Yes, it is not immediately obvious how mass ought to be undershood in the context of relativity. First attempts treated mass as a directed vector, with transverse and longitudinal components for an accelerated particle (now we would say that the mass is fixed, it is the momentum that changes non-linearly and just use F=dp/dt). Mass should rightly be considered more inherent in SR that energy. Energy in relativity is coordinate dependent, but mass is not: all observers agree on a system’s mass, because it is invariant. Re: the last paragraph, yes you can project the four vector onto time and space, and you get energy and momentum respectively. So energy is not the four vector itself but its projection onto the time axis (and mass is of course its magnitude).

deleted 9y ago

[deleted]

ocfnash 9y ago

One corollary of E=mc^2 is that the rest mass of a bound state of fundamental particles will not equal the sum of the rest masses of the individual constituents. For example, the rest mass of a Helium-4 nucleus turns out to be less than the sum of the rest masses of the four nucleons of which it is composed. The difference is only about 0.7% but is famously quite a lot of energy when interpreted using E=mc^2. It's a lot more fun if we go one level down though: protons are far HEAVIER than the sum of their constituents. Apparently almost 99% of the rest mass of a proton is the binding energy of the zero-rest-mass gluons that hold their three quarks together. I think it's fun that the vast majority of the mass of ordinary matter, of which we are all constituted, is best explained as gluon binding energy via E=mc^2.

cohomologo 9y ago
I think you've got that last part backwards. When you have constituents with binding energy, that reduces the total energy of the system and reduces the mass. So it seems that the mass of the proton is quark mass + gluon kinetic energy - bonding energy, and most of the mass comes from gluon kinetic energy.
- ocfnash 9y ago
  Yes I do indeed; thank you for the correction!
- hjaved 9y ago
  Thank you. Would like to understand that why does having constituents with binding energy reduce the total energy of the system instead of adding to it?
  - davrosthedalek 9y ago
    You have to think about it the other way around: The constituents don't have binding energy. That energy becomes available when you let them bind, and you have to pay it back if you want to unbind them. Example: If you have a proton and an electron, making a hydrogen atom will release about 13.6 eV (if it goes to the lowest energy state). You have to spend that 13.6eV to ionize the hydrogen, i.e. to get the electron far away from the proton again.

akubera 9y ago

The use of the term "pure energy" always grates me; it gives the impression that energy is an entity unto itself, which is not the case. I realize that it's used as a proxy for photons most of the time, but I still always think it's misleading.

cousin_it 9y ago
Oh, this is a pet peeve of mine. Jim Butcher does this constantly in the Dresden Files series. I keep coming up with lampoons like "the witch countered with a blast of pure angular momentum that sent him spinning" or "after a visit to the bank, my wallet was bursting with pure price".
JabavuAdams 9y ago
Yes! I just volunteered as a parent helper to a scientist in the school program. I was gritting my teeth as the "scientist" explained that light was "pure energy". I think people use this to mean "insubstantial" and to contrast versus matter. But that still leaves the problem of light pressure. I.e. a flashlight is like a very very weak rocket engine.

plopilop 9y ago

As I explained in an other article here on HN, another meaning (and the most important one imo) is the one Einstein actually wrote about: the inertia of a particle depends on its energy. The thing is, you don't read E = mc² but rather (as Einstein wrote in his surprisingly easy to understand 2-pages paper [0]), m = E / c². The direct interpretation of this formulation is that inertial mass is actually just a side effect of the energy of a particle. Put into a catchy phrase, mass is energy at rest. Edit : [0]: https://www.fourmilab.ch/etexts/einstein/E_mc2/e_mc2.pdf https://www.fourmilab.ch/etexts/einstein/E_mc2/e_mc2.pdf, translated in english. Note the understatement of the sentence before the last one: "It is not impossible that with bodies whose energy-content is variable to a high degree (e.g. with radium salts) the theory may be successfully put to the test."

gus_massa 9y ago
The problem is that the "mass" in the direction of the movement is different than the "mass" in the perpendicular direction. So most modern books try to avoid the relativistic mass "m" and use only the rest mass of the particle "m0". (Still, the "relativistic mass" can be useful to make a few back of the envelope calculations, but you must be careful.) More details: https://en.wikipedia.org/wiki/Mass_in_special_relativity#The_relativistic_mass_concept https://en.wikipedia.org/wiki/Mass_in_special_relativity#The...
- plopilop 9y ago
  I did not know about this relativistic mass, but doesn't wikipedia says it is only true if you consider the mass as m = F/a rather than m = p/v?

contravariant 9y ago

>But under the laws of special relativity, mass simply couldn’t be the ultimate conserved quantity, since different observers would disagree about what the energy of a system was. Maybe I'm being pedantic, but not only does classical mechanics already tell us that the energy of a system different for different for observers, but this by itself doesn't mean it's not conserved, merely that it's not a universal quantity. Furthermore E = mc^2 doesn't break conservation of mass in any way, it's more accurate to say that it equates energy with mass, meaning conservation of energy and conservation of mass are one and the same. And in fact both still hold, when the energy and mass are replaced with their relativistic counterparts (E = mc^2 is obviously false using the classical notions of mass and energy).

ramzyo 9y ago

Fascinating. After first taking a crack at learning about this stuff decades ago, I still have trouble retaining it to the degree that I can succinctly explain its profoundness. This article does a great job at doing just that, and reigniting the sense of wonder and amazement at Einstein's achievements!

enugu 9y ago

What isnt usually mentioned is that this equation is a special case valid in some circumstances. The more general relation is E^2=(mc^2)^2+(pc)^2 which includes momentum term, and applies to photons which dont have a mass. Here's a simple video about this from Fermi Lab, https://www.youtube.com/watch?v=eOCKNH0zaho https://www.youtube.com/watch?v=eOCKNH0zaho Also interesting is how conservation laws for collisions are handled much better in relativistic setting.

nyc111 9y ago

" But even plain, old, regular mass at rest has energy inherent to it..." This sentence is so typical of sloppy so-called science writing in the US. What is "plain mass?" What is "old mass?" Or maybe he put the comma in the wrong place and he meant "plain old regular mass?" If so what is "plain old regular mass?" There is no such thing. But more importantly, nothing is ever at rest in the known world. It is absurd to discuss the properties of something when it is at rest. It will never be at rest. There is no absolute rest.

ythn 9y ago
This comment reads like the ravings of a pedant who is unable to relate to the layman.
- nyc111 9y ago
  You may be right about my comment about his positioning of the comma but my statement that all is motion and nothing but motion is not pedantic or trivial.
  - betenoire 9y ago
    Motion is relative, no? I see things at rest from where I'm standing.
    - nyc111 9y ago
      No. Motion is absolute. Rest is relative. That is, rest is only an appearance. Since you are observing from earth everything you see is in motion since the earth is moving. So there is no absolute rest. And relative rest is just another name for motion. Therefore, no object can have a property that exists only when the object is at rest. Because there is no absolute rest.
      - cgmg 9y ago
        > Motion is absolute. What are you talking about? Motion is definitely not absolute, and only Galilean invariance is needed to see that.

ythn 9y ago

If mass created from "pure energy" is always in matter/antimatter pairs, why is the ratio of matter/antimatter so skewed in favor of matter?

davrosthedalek 9y ago
That is indeed the one million dollar question. Physicist search for a reaction which breaks this balance to explain the matter/anti-matter imbalance of the observed universe.
- 100ideas 9y ago
  Most of the antimatter ended up as early supermassive black holes.
pixl97 9y ago
The anthropological answer to this question is: Because we wouldn't be here to ask the question if it was balanced.

ucaetano 9y ago

> one of the simplest but most powerful equations ever to be written down I always felt that E=mc² is essentially the Euler's identity of physics: a basic, simple, beautiful equation relating some core components of the entire field.

deleted 9y ago

[deleted]

doctoboggan 9y ago

The YouTube channel PBS SpaceTime gives an excellent overview of these same ideas. The presenter is great at making things as simple as possible, but not simpler. https://www.youtube.com/watch?v=gSKzgpt4HBU https://www.youtube.com/watch?v=gSKzgpt4HBU https://www.youtube.com/watch?v=fHRqibyNMpw https://www.youtube.com/watch?v=fHRqibyNMpw https://www.youtube.com/watch?v=kixAljyfdqU https://www.youtube.com/watch?v=kixAljyfdqU

stcredzero 9y ago

"E = mc^2 is Wrong" (Actually, just incomplete.) https://www.youtube.com/watch?v=eOCKNH0zaho https://www.youtube.com/watch?v=eOCKNH0zaho