# Real life example of the energy contained at E=γmc^2

• I
• Foruer
In summary, the energy present in a keyboard is due to the rest mass of the constituents. Most of that energy is due to the rest mass of the constituents, but if you throw a baseball toward home plate, the difference in kinetic energy gives the baseball kinetic energy. If you have a strong enough cable, you could extract the entire amount of energy from the keyboard, but any sort of realistic cable would break.f

#### Foruer

We've just begun studying about relativity, and I find it amazing that bodies have the energy of E=γmc^2. Even at rest they have E=mc^2.
But where exactly is this energy present in real life? For example the keyboard I am currently typing this post with has a huge amount of energy, according to this equation, but how is it usable?

But where exactly is this energy present in real life? For example the keyboard I am currently typing this post with has a huge amount of energy,
Most of that energy is due to the rest mass of the constituents.

but how is it usable?
Why does it have to be usable for anything?

• Foruer and russ_watters
Because it's a big amount of energy in small mass bodies, so you'd think that if it was really usable somehow, you would could provide energy to the entire world by using small amount of matter. I just wonder if it can be expressed somehow in real life?

Because it's a big amount of energy in small mass bodies, so you'd think that if it was really usable somehow, you would could provide energy to the entire world by using small amount of matter. I just wonder if it can be expressed somehow in real life?
If you throw a baseball toward home plate, the difference, ##\gamma mc^2 - mc^2## gives the kinetic energy of the baseball.

• Foruer
For example the keyboard I am currently typing this post with has a huge amount of energy, according to this equation, but how is it usable?
Well, you could collide it with an anti-keyboard...

In reality, this isn't usable energy, but rather an energy equivalence. However, if you've studied nuclear energy yet, you'll see that there are small differences in the starting and ending mass that you can evaluate.

• Sean Nelson, nitsuj and Foruer
Aside from colliding your keyboard with an anti-keyboard, you could extract energy from it by lowering it into a black hole. You could extract this energy in principle the same way as you could extract energy from a dropping weight, by having it turn a gearshaft that cranks a generator or whatever.

If you had a strong enough cable, you could in theory extract the entire amount of energy from it, but any sort of realistic cable would break.

• russ_watters
A quarter teaspoon of water plus a quarter teaspoon of anti water would yield an explosion approximately twice the energy of the atomic bomb dropped on Nagasaki in WW II.

But where exactly is this energy present in real life?
If you look closely at any reaction that releases energy, you will find that the mass of the material coming out is very slightly less than the mass of the material that went in (and the other way around for a reaction that consumes energy, like charging a battery). For example, if I burn a chunk of wood... the weight of the wood plus the weight of the oxygen from the air will be very slightly more than the weight of the ashes and the gases released by combustion. The difference is exactly what you'd calculate from the energy change using the ##E=\gamma{m}c^2## formula (easiest if everything is at rest so ##\gamma=1##, of course).

So the best answer for where the energy is will be something along the lines of "the mass is the energy, just in a different form".

As a practical matter, only nuclear reactions involve enough energy for the effect to be detectable; ##c^2## is a very big number. For example, a lead-acid automobile battery weighs only a few nanograms more when charged than discharged.

Because it's a big amount of energy in small mass bodies,

No, it's not. The rest energy is equivalent to the mass, so it doesn't make sense to say that one is larger than the other.

No, it's not. The rest energy is equivalent to the mass, so it doesn't make sense to say that one is larger than the other.

I found it clear that they were referring to energy density, not a stress energy tensor

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But where exactly is this energy present in real life? For example the keyboard I am currently typing this post with has a huge amount of energy, according to this equation, but how is it usable?

Read about the constituents of an atom. It's really neat, allot of it (mass in "real life") is kinetic energy of quarks (things that form protons neutrons) bound by gluons; this is the strong nuclear force.

there's also such a thing as a nuclear battery, weak nuclear not strong...using tritium beta decay! That's hydrogen turning into helium "exactly" over time.

with both fission and fusion we can (potentially) do work.

there was allot of added info in the replies.

Yes to "power the world"; interesting political topic regarding nuclear power...far to complicated for discussion in a physics thread :D

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I found it clear that they were referring to energy density, not a stress energy tensor
May I ask how that matters with respect to the equation in question? If the rest energy is equivalent to the mass, then they are one and the same. It's not like the left side of the equation takes up more volume than the right side, or something like that. I mean, does it make sense to ask if mass*acceleration is larger than force? So yeah, your response has confused me somewhat. Would you care to elaborate why Mr. T is wrong?

A pet peeve of mine about ##E = mc^2## is that people often say that nuclear reactions are demonstrations of that formula, that the reason that so much energy is produced in an atomic explosion is because matter is being turned into energy. I consider that both true and misleading.

The facts for nuclear fusion is that there is a transformation that looks like this (yes, I know, that is not the way that fusion happens, directly, but I'm just simplifying for illustration)

##4H \Rightarrow He + 2 \bar{\nu_e}##

The fact that this reaction is exothermic (the total energy of the right-hand side is less than the total energy of the left hand side) is what makes it possible for fusion to produce energy. I don't see that it essentially involves ##E = mc^2## any more than the chemical reaction

##2H + O \Rightarrow H_2 O##

The relevance of ##E = mc^2## is that if you measure the total mass of the products on the right-hand side, it will be slightly less than the total mass of the constituents on the left-hand side. But that's as true for the chemical reaction as it is for the nuclear reaction: Water has slightly less mass than the hydrogen and oxygen atoms it was made of. I don't see why the nuclear process is an example of matter being turned into energy any more than the chemical process is.

• vsv86 and SiennaTheGr8
I don't see why the nuclear process is an example of matter being turned into energy any more than the chemical process is.
A pet peeve of mine is that matter is not being turned into energy. It is converted from one form of energy (mass energy) to another (kinetic energy in the products that in the end goes to heat).

• Sorcerer and SiennaTheGr8
A pet peeve of mine is that matter is not being turned into energy. It is converted from one form of energy (mass energy) to another (kinetic energy in the products that in the end goes to heat).

But I would say that it's not really doing that, either. I would say that in both nuclear and chemical processes, it's a matter of binding energy, rather than mass energy. If you have a nucleus with N protons and M neutrons, in some cases a neutron can transform to a proton + electron + anti-neutrino + kinetic energy, and sometimes a proton can transform to a neutron + positron + neutrino + kinetic energy. The reason one or the other is favored is because of the binding energy for nucleons (which involves both strong forces and electromagnetic forces). So it's really a matter of converting potential energy into kinetic energy, just like chemical processes.

May I ask how that matters with respect to the equation in question? If the rest energy is equivalent to the mass, then they are one and the same. It's not like the left side of the equation takes up more volume than the right side, or something like that. I mean, does it make sense to ask if mass*acceleration is larger than force? So yeah, your response has confused me somewhat. Would you care to elaborate why Mr. T is wrong?

their statement wasn't wrong at all, the interpretation of the question was. Mister T has allot of great replies to questions and has enlightened me a number of times.

To be even more clear, surely the OP is, like the vast majority of people, used to "chemical energy", and in turn the concept of energy density.

They are not "one and the same", they are equivalent...clearly on opposite sides of the equation.

But I would say that it's not really doing that, either. I would say that in both nuclear and chemical processes, it's a matter of binding energy, rather than mass energy. If you have a nucleus with N protons and M neutrons, in some cases a neutron can transform to a proton + electron + anti-neutrino + kinetic energy, and sometimes a proton can transform to a neutron + positron + neutrino + kinetic energy. The reason one or the other is favored is because of the binding energy for nucleons (which involves both strong forces and electromagnetic forces). So it's really a matter of converting potential energy into kinetic energy, just like chemical processes.

So the distinction between one being a nuclear force and the other being em force is moot, because with either we use the work in the same way ?

So the distinction between one being a nuclear force and the other being em force is moot, because with either we use the work in the same way ?

The big distinction that I see between common nuclear reactions and chemical reactions is that in the case of chemical reactions, you can think of the transformation as a rearrangement of constituent particles. In contrast, when a neutron transforms into a proton + electron + anti-neutrino, it's not literally the case that it's just a rearrangement of constituent particles. (There is no proton or electron or neutrino inside a neutron).

That's something new with nuclear physics: transformations that are not just rearrangements of constituents. But I would still say that in nuclear reactions, it's the binding energy that makes the reaction go, rather than converting rest mass energy into kinetic energy.

Let's take two different nuclear reactions:
1. ##^{14}C \Rightarrow ^{14}N + e + \bar{\nu_e}##: Carbon 14 decays into Nitrogen
2. ##^{23}Mg \Rightarrow ^{23}Na + e^+ + \nu_e##. Magnesium 23 decays into Sodium
Now, let's combine the two reactions into one:

##^{14}C +\ ^{23}Mg \Rightarrow\ ^{14}N +\ ^{23}Na##

That's a nuclear reaction which is possible (though extremely unlikely) that is simply a rearrangement of protons and neutrons. In this case, it's much more like a chemical reaction: A rearrangement of constituent particles results in a lower total energy, so it releases some kinetic energy. But to me, it doesn't make sense to say that the first two reactions involve transmuting rest mass energy into kinetic energy, but the last one does not. But if you say that the latter illustrates converting rest mass energy into kinetic energy, then I don't see how that description wouldn't apply equally well to chemical reactions.

To me, the lesson from ##E=mc^2## is that if you have a reaction that releases kinetic energy, then the rest mass of the products will be less than the rest mass of the original ingredients. That's true of chemical reactions as well as nuclear reactions.

• nitsuj
Since we are talking about peeves, I have this one. It bothers me that people say things like "matter is turned in this or that" or "matter is equivalent/equal to so and so". The ##m## in the equation stands for mass, matter is not mentioned at all.

Since we are talking about peeves, I have this one. It bothers me that people say things like "matter is turned in this or that" or "matter is equivalent/equal to so and so". The ##m## in the equation stands for mass, matter is not mentioned at all.

Matter is just an informal word meaning stuff that has mass, isn't it?

Matter is just an informal word meaning stuff that has mass, isn't it?
If that is the case, then matter can have energy but cannot be energy.

• weirdoguy
I would say that in both nuclear and chemical processes, it's a matter of binding energy, rather than mass energy.
That depends on what you mean by "mass energy". As the mass of the original systems it is certainly a mass energy of, e.g., the incoming O2 molecule. If you further subdivide that you can say that the mass energy of the molecule has several contributions. It is all a question of what level you do your bookkeeping at.

If that is the case, then matter can have energy but cannot be energy.
Nothing can be energy. Energy is a property, not a substance.

• russ_watters and SiennaTheGr8
That depends on what you mean by "mass energy". As the mass of the original systems it is certainly a mass energy of, e.g., the incoming O2 molecule. If you further subdivide that you can say that the mass energy of the molecule has several contributions. It is all a question of what level you do your bookkeeping at.

My point is to dispute that nuclear reactions involve transforming mass energy into kinetic energy any more than chemical reactions do.

Nothing can be energy. Energy is a property, not a substance.
That is exactly my point.

That depends on what you mean by "mass energy". As the mass of the original systems it is certainly a mass energy of, e.g., the incoming O2 molecule. If you further subdivide that you can say that the mass energy of the molecule has several contributions. It is all a question of what level you do your bookkeeping at.

"Bookkeeping" is a great word to use here.

I don't view the mass of a composite system as a form of energy, per se. Rather, it's the total energy of the system as measured in its rest frame, equal to the sum of the kinetic, potential, and rest energies (masses) of the system's constituents (as measured in that frame)—and the same applies to the masses of the constituents, and to the masses of the constituents' constituents, and so on, until you get to the elementary particles, whose mass is fundamental and certainly a "form" of energy.

So yes, I'd say that the mass of a composite system is more a "bookkeeping" device than a "form" of energy. But it's also convenient when taking an "outside" view to speak of a system's mass as energy that can be converted to other types of energy. For this purpose, I think "type" is a better word than "form."

My take on the pet peeves:

-"Mass is converted to energy." No, mass already is energy (rest energy). Rest energy can be converted to other types of energy. In matter/antimatter annihilation, for example, the mass of elementary particles is converted to kinetic/electromagnetic energy (and the elementary particles cease to exist). In chemical and nuclear reactions, we can take an "outside" view and say that mass of the composite system is converted to kinetic energy, or we can take an "inside" view and say that potential energy is converted to kinetic energy.

-"Matter is converted to energy." No, the energy in question was already a property of the "matter." Closer to the mark is "matter is converted to radiation," but even then one needs to carefully define both "matter" and "radiation."

"Bookkeeping" is a great word to use here.
Feynman's metaphor for energy as blocks resonates with me.
http://www.feynmanlectures.caltech.edu/I_04.html
Especially the punch line: "The most remarkable aspect that must be abstracted from this picture is that there are no blocks."

My take on the pet peeves:

-"Mass is converted to energy." No, mass already is energy (rest energy). Rest energy can be converted to other types of energy. In matter/antimatter annihilation, for example, the mass of elementary particles is converted to kinetic/electromagnetic energy (and the elementary particles cease to exist). In chemical and nuclear reactions, we can take an "outside" view and say that mass of the composite system is converted to kinetic energy, or we can take an "inside" view and say that potential energy is converted to kinetic energy.

-"Matter is converted to energy." No, the energy in question was already a property of the "matter." Closer to the mark is "matter is converted to radiation," but even then one needs to carefully define both "matter" and "radiation."

Mass is NOT "already energy". The dimensions are wrong! Not only that, a "mass" may carry charge, while energy can't! You are looking at it simply from ONE characteristic (energy equivalent) and then making wholesale conclusion that they are one of the same thing while ignoring other differences between mass and energy. There are OTHER things involved in the "bookkeeping", not just amount of energy.

To the OP and the original question, the "missing mass" that we get when we measure the mass of atoms is our everyday illustration of energy-mass conversion. Take the individual mass of an electron, a proton, a neutron, and then add them all together in the appropriate amount to form all the various elements in the periodic table. Then compare those masses that you have added to the actual masses of each individual element. They are not identical!

Zz.

Mass is NOT "already energy". The dimensions are wrong! Not only that, a "mass" may carry charge, while energy can't! You are looking at it simply from ONE characteristic (energy equivalent) and then making wholesale conclusion that they are one of the same thing while ignoring other differences between mass and energy. There are OTHER things involved in the "bookkeeping", not just amount of energy.
I strongly disagree with essentially everything in this paragraph. First of all, ”mass” is not a thing and does not in any sense relate to being able to carry charge. It is a property of a system, just as energy is. Second, it does have the correct dimensions as long as you stick to a reasonable system of dimensions where spacetime coordinates all share the same dimension such that speeds are dimensionless. That it happens to be convenient for everyday use to use a different unit for it such that the conversion factor is c^2 is a different matter. I agree that there are more things to bookkeep, but the mass goes into the energy bookkeeping and nowhere else. Charge is a separate property. Both mass and energy are properties of a system, not anything that in itself can carry other properties.

I strongly disagree with essentially everything in this paragraph. First of all, ”mass” is not a thing and does not in any sense relate to being able to carry charge. It is a property of a system, just as energy is. Second, it does have the correct dimensions as long as you stick to a reasonable system of dimensions where spacetime coordinates all share the same dimension such that speeds are dimensionless. That it happens to be convenient for everyday use to use a different unit for it such that the conversion factor is c^2 is a different matter. I agree that there are more things to bookkeep, but the mass goes into the energy bookkeeping and nowhere else. Charge is a separate property. Both mass and energy are properties of a system, not anything that in itself can carry other properties.

And I disagree with your assessment that they are. That's like saying water in liquid form is identical to water in ice form, while ignoring the phase transition between them. The mass of an electron is more than just an energy content. It's presence in itself immediately puts a limit to how fast it can move in any frame. It is not a merely a superficial difference.

Because of that (i.e. they are not identical), it should not be wrong or even be someone's pet peeve to say that there is a form of "conversion" when it goes from one form to another, especially when now, it is a matter of semantics! And that was what I was addressing.

Zz.

Do you agree that "rest energy" ##E_0## is "already energy"?

Do you agree that "rest energy" ##E_0## is "already energy"?

Yes, and do you agree that this is not mathematically equal to the rest mass?

Zz.

Yes, and do you agree that this is not mathematically equal to the rest mass?

Zz.
No. It is just a matter of unit conversion in terms of the actual SR definitions, which defines mass as the rest energy up to the unit conversion factor ##c^2##. That the rest mass then happens to be the inertia in the rest frame is the mass-energy equivalence.

Yes, and do you agree that this is not mathematically equal to the rest mass?

Zz.

Yes, but the ##c^2## in the equation ##E_0 = mc^2## is no more than a unit-conversion factor, which we're free to set equal to ##1##. The distinction between mass and rest energy is merely an artifact of unnecessarily using different units for energy and mass in the first place. There is no conceptual difference between the quantities at all. They are one and the same property, measured in different units.

The mass of an electron is more than just an energy content.
It really isn't in terms of how mass is defined in SR.

• SiennaTheGr8
No. It is just a matter of unit conversion in terms of the actual SR definitions, which defines mass as the rest energy up to the unit conversion factor ##c^2##. That the rest mass then happens to be the inertia in the rest frame is the mass-energy equivalence.

Yes, but the ##c^2## in the equation ##E_0 = mc^2## is no more than a unit-conversion factor, which we're free to set equal to ##1##. The distinction between mass and rest energy is merely an artifact of unnecessarily using different units for energy and mass in the first place. There is no conceptual difference between the quantities at all. They are one and the same property, measured in different units.

Then we are at a disagreement with what is meant by "conversion", because there is a physical difference between "energy" and "mass" from my perspective. Simply calling c2 "merely" a conversion factor doesn't remove the fact that there IS a "conversion" between mass and energy.

Please keep in mind that you're talking to someone who is very familiar with calling "ω" or "1/k" as "binding energy". So I get it! But how this fits into the thread at the level that the OP is asking is puzzling, and invoking advanced ideas of mass and energy into something like this is very confusing.

Zz.