Confusion about $E = mc^2$

Question

Sometimes I hear people speak of mass converting to energy so following that if there is an increase in mass there is a decrease in energy if there are no other factors. That implies that because there is a mass deficit in an atom when a proton and neutron combine there is an increase in energy. However when they coalesce they could only have less potential energy than when they were separate as the strong force compels them towards each other. So there should be a mass addition? Where is the flaw in my logic?

Answer 1

Keep in mind that energy is conserved.

When protons and neutrons come together to form a nucleus, the attractive strong force makes them accelerate towards one another over the last femto-metre distance before they coalesce. They thus arrive next to each other with some non-negligible kinetic energy. If they do not have any means to get rid of that kinetic energy then indeed the total mass of the system has not changed and the nucleons are not yet bound to each other. What happens next is typically that the system emits some gamma-rays. These carry off some energy and enable the ball of nucleons to settle into a lower energy state. Only after this has a stable nucleus formed.

The nucleus that eventually results therefore does have less total energy than the things that came together to make it, because some energy has been emitted as gamma rays. Furthermore, the total energy of the stable nucleus is not just smaller than the total energy of the separated nucleons before they formed it, but also smaller than the rest energy of those nucleons. It follows that its rest mass is smaller than the sum of the rest masses of the constituents when they are separate. It is this final statement that is the more surprising and could not have been predicted using Newtonian physics.

Answer 2

Sometimes I hear people speak of mass converting to energy so following that if there is an increase in mass there is a decrease in energy if there are no other factors.

This is not a good way to think of the equivalence of mass and energy, and leads to confusion. It is better to think of mass and energy as being different ways of measuring the same thing, with the conversion factor between units of mass and units of energy being equal to the speed of light squared. Thus we can calculate the total mass of a system as long as we include the mass equivalent of its energy; or, equivalently, we can calculate the total energy of a system as long as we include the energy equivalent of its mass.

An atomic nucleus has negative binding energy because we must add energy to break the nucleus apart into its constituent protons and neutrons - this negative binding energy is known as the "mass deficit" or "mass defect" of the nucleus (note how the terms mass and energy are being used interchangeably here). Conversely, if an atomic nucleus is assembled from separate protons and neutrons, it has a total mass that is smaller than the masses of the individual protons and neutrons - this difference in mass is its binding energy (converted into units of mass using $E = mc^2$). It is this binding energy that is released in a nuclear fusion reaction.

Answer 3

First we should understand what "mass" is. Mass was invented by ancient people for the purpose of measuring amounts of various things, e.g. potatoes. They did not use the word "mass", but they measured the masses of things,so obviously they had invented mass.

Isaac Newton used mass to talk about motion of matter.

Energy was invented in the 19th century.

Then Einstein found out how much energy must be put onto a scale so that the scale reads 1kg. That is 90000000000000000 Joules of energy. Or one kilogram of energy, in this case it's easier to measure energy in kilograms.

Oh yes, the question was about conversions.

If we have one kilogram of energy in the form of one kilogram of potatoes, then when we convert that energy to an energy that is in the form of light, then we get one kilogram of light.

It is true that people say confused things like e.g. this: If we have one kilogram of energy in the form of one kilogram of mass, then when we convert that mass to an energy that is in the form of light, then the mass of that light is 1 kg.

Answer 4

The full equation is actually $$E^2=(mc^2)^2+(pc)^2,$$ where $p$ is the (relativistic) momentum. Here, $E_k=E-mc^2$ is the kinetic energy of the system. Only in the rest frame of the system do we get $E=mc^2$. This equation is relevant, because it tells us that not all the energy is "massive". Some can be in the kinetic energy of the particle. For example, for photons $m=0$ and so the energy is always $E=pc$.

The mass-energy equivalence tells us that any system that has energy, also has mass. Mass is energy. With one caveat that kinetic energy doesn't count as mass, which you can tell from the first formula.

Potential energy counts too. Your situation is similar to the potential energy in two magnets, or in a massive object some distance above the ground. When you let the magnets collide, or when you drop the object, energy is released. The magnets may be spinning wildly for a second. The object that is dropped will be moving quickly and may bounce back. But at some point this energy will be dissipated. At that point the energy will be lower. When a proton and a neutron become bound, they release energy. The potential energy is converted to kinetic energy for a bit. But once that extra energy is dissipated, the total energy will be lower. In that case the total mass will also be lower.

Interesting side note: the relativistic momentum equals $$p=\gamma m v=\frac{mv}{\sqrt{1+\frac{v^2}{c^2}}}.$$ When $v$ is much smaller than $c$, the kinetic energy becomes approximately $\tfrac 1 2m v^2$