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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

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).
  - davrosthedalek 9y ago
    (Relativistic) mass is totally frame dependent in SR. That's why SR normally talks about the rest or invariant mass, which is the mass in a special frame, the rest frame of the object.
    - Retric 9y ago
      You can't construct a reference frame such that a rotating object is at rest.
      - AnimalMuppet 9y ago
mnsc 9y ago
> if mass can be converted into energy, then you would have more energy and less mass, so E=mc^2 would no longer hold I'm confused why this wouldn't hold? IANAP so I don't know if E=mc2 comes into play in a "non-nuclear" mass -> force conversion. Say a rocket with x kg of fuel mass lifting off with y J of energy spent. Naively I would say that before liftoff there was x kg of fuel which is equivalent to xc^2 energy and no(?) energy y = 0. At a later time (b) When the rocket has spent half of the fuel, the energy yb spent this far could be calculated as (x/2)c^2 and energy left in mass (xb) would (x/2)*c^2. So that the total amount of energy/mass is the same at start and at time b. Hmm, this got really confused, so... I just hit "reply".
- plopilop 9y ago
  The conversion you are talking about is true only if you annihilate your fuel (which creates photons), and then you convert photons into speed in some way with 100% efficiency. But really this never happens, rockets work by ejecting mass at a tremendous speed. Thus not all the mass of the fuel is not converted into energy. But otherwise, yeah, converting mass to energy completely works. This is what happens in stars and nuclear bombs, actually.
  - davrosthedalek 9y ago
    It also happens in all chemical reactions. Or: If you pump water up a mountain, the earth-water system has more rest mass.
    - plopilop 9y ago
      Sure, but it's less noticeable. The Sun loses 4.4 millions of tons every second (out of a mass of 10^30kg, but still)
  - mnsc 9y ago
    > The conversion you are talking about is true only if you annihilate your fuel (which creates photons), and then you convert photons into speed in some way with 100% efficiency. Maybe it's hard to explain from a real world example like a rocket but if mass isn't converted to energy in that case, in the E=mc2 sense, where does the energy come from?
cygx 9y ago
Later we learn that mass is determined by energy and momentum, both of which are conserved, so mass must be conserved too. Not in a meaningful sense - the square of the total momentum 4-vector is of course conserved (and even frame-invariant), but it's not equal to the sum of the masses of the constituents: M² = (E₁ + E₂)² - (p₁ + p₂)² = E₁² - p₁² + E₂² - p₂² + 2E₁E₂ - 2p₁p₂ = m₁² + m₂² + 2(E₁E₂ - p₁p₂)

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).
- davrosthedalek 9y ago
  (Relativistic) mass is totally frame dependent in SR. That's why SR normally talks about the rest or invariant mass, which is the mass in a special frame, the rest frame of the object.
  - Retric 9y ago
    You can't construct a reference frame such that a rotating object is at rest.
    - AnimalMuppet 9y ago

mnsc 9y ago

> if mass can be converted into energy, then you would have more energy and less mass, so E=mc^2 would no longer hold I'm confused why this wouldn't hold? IANAP so I don't know if E=mc2 comes into play in a "non-nuclear" mass -> force conversion. Say a rocket with x kg of fuel mass lifting off with y J of energy spent. Naively I would say that before liftoff there was x kg of fuel which is equivalent to xc^2 energy and no(?) energy y = 0. At a later time (b) When the rocket has spent half of the fuel, the energy yb spent this far could be calculated as (x/2)c^2 and energy left in mass (xb) would (x/2)*c^2. So that the total amount of energy/mass is the same at start and at time b. Hmm, this got really confused, so... I just hit "reply".

plopilop 9y ago
The conversion you are talking about is true only if you annihilate your fuel (which creates photons), and then you convert photons into speed in some way with 100% efficiency. But really this never happens, rockets work by ejecting mass at a tremendous speed. Thus not all the mass of the fuel is not converted into energy. But otherwise, yeah, converting mass to energy completely works. This is what happens in stars and nuclear bombs, actually.
- davrosthedalek 9y ago
  It also happens in all chemical reactions. Or: If you pump water up a mountain, the earth-water system has more rest mass.
  - plopilop 9y ago
    Sure, but it's less noticeable. The Sun loses 4.4 millions of tons every second (out of a mass of 10^30kg, but still)
- mnsc 9y ago
  > The conversion you are talking about is true only if you annihilate your fuel (which creates photons), and then you convert photons into speed in some way with 100% efficiency. Maybe it's hard to explain from a real world example like a rocket but if mass isn't converted to energy in that case, in the E=mc2 sense, where does the energy come from?

cygx 9y ago

Later we learn that mass is determined by energy and momentum, both of which are conserved, so mass must be conserved too. Not in a meaningful sense - the square of the total momentum 4-vector is of course conserved (and even frame-invariant), but it's not equal to the sum of the masses of the constituents: M² = (E₁ + E₂)² - (p₁ + p₂)² = E₁² - p₁² + E₂² - p₂² + 2E₁E₂ - 2p₁p₂ = m₁² + m₂² + 2(E₁E₂ - p₁p₂)