Not only spiritual roots of conservation laws
Energy has been talked about for a very long time, thousands of years before the scientific method was born. The word "ἐνέργεια" [energeia] was used by the Greeks long before they became a socialist nation and acquired the debt of 150% of their GDP. In their language, the word really means "activity" or "operation". You could say that linguistically, it's the same word as "work".
However, in physics, energy isn't the work itself but either the motion betraying that some work is underway, or "something" that you may "store" if you want to do some work later. I don't want to get to the technically accurate scientific stuff too quickly, so let us first look how the nonscientists are using the word "energy".
Ghosts and energy
By nonscientists, I don't mean engineers who design power plants or cars. I mean real nonscientists. Nonscientist nonscientists. Most of them believe in various religious, spiritual, or assorted superstitions in which the concept of "energy" plays an important role.
For example, there's a consensus among 97%98% of treehuggers and religious people from a list of 10 churches I may enumerate that if you hug a tree, you may extract some of the living "energy" out of the tree which will make you healthier, happier, and more productive. ;)
This procedure will pump "something" into the bound state (=union held together by an attractive force) of your body and your soul which will allow you to do things in the future, more of them, and better things, too. It's important in this prescientific understanding that this "something" has a different, less material character than food and beer that you normally eat and drink. Later, we will challenge the "difference" between these two things, of course.
(In some of the laymen's conception of soul, it's hard to distinguish soul and gas or a fart but I don't want to go too deeply into nonscientists' solid state physics.)
If you divide your identity to your "body" and your "soul", "energy" is surely meant to have the character of "soul". Some soul is found inside every living body, except for zombies :), which finally brings us to the concept of conservation laws. People understood some conservation law for energy long before they started to look at the world scientifically. They believed in the conservation of a prescientific version of energy which was nothing else than the amount of soul or number of souls. Note that it was also a soul, and not e.g. muscles driven by sugars, that was putting you in motion.
The oldest conservation law that people believed – and many people still believe – is the conservation law for the number of souls. Note that this conservation law predicts that you can't die; this prediction often holds for long years but at some point, it can apparently break. However, it's not a problem for the religious people including Christians and Buddhists. They solved the problem in the same way as Wolfgang Pauli solved the apparent violation of energy conservation in the decay of neutrons: the energy goes to a new particle, the neutrino.
Analogously, the religious people say that when you die, your soul goes somewhere – to a counterpart of the neutrino – which is either a newborn baby skunk according to the Indian traditions; or some suburbs of the European Union of Heavens and Hells according to the Christians. Religions usually invent sophisticated worlds of unobserved phenomena that occur after the death in order to save the conservation law for souls. Interestingly enough, they don't spend too much time by solving the apparently analogous and symmetric problem how the souls appeared in the first place when Adam or his foreign competitors were created.
God may just create the souls, can't She? You see that the prescientific thinking is not only incompatible with the evidence but it also uses double standards. Creation and annihilation operators are obviously analogous (and Hermitian conjugate) to each other and if you're fair, you can't possibly say that one of them is a problem and the other is not. One may see that the prescientific people don't care too much about being fair. It's the death that is the real problem they want to get rid, for purely egotist reason, so they don't spend much time in explaining fairytales about the kindergarten of souls that are waiting to jump into a fertilized egg. The Buddhists at least recycle the souls but it still fails to work if the number of animals on Earth is increasing or decreasing – and it almost certainly has been.
Getting from soul(e)s to Joul(e)s
I wonder how many TRF readers had the nerves to get through all of this superstitious nonsense. But I actually guess that many more people are willing to read nonsense of this kind than to read about somewhat more serious science (but not too serious science) that will follow. After all, you are among them as well because you managed to get here so you confirm my prediction, too. By an anthropic selection, I could have been sure in advance that my prediction would be considered correct by most readers of this paragraph. :)
If you measure the validity of propositions by the consensus, you may make it certain that certain preconceived things will end up being valid. That's how consensus "science" always works.
Kinetic energy
Instead of preparing the reader for the transition from superstitions to quantitative science in dozens of extra vacuous sentences, I decided that the shock therapy was what was needed. Well, you are suddenly in the quantitative part of the article. The oldest form of energy that people would actively think about was kinetic energy, \(T=E_{\rm kin}\). This is the "activity" from the Greek word. The "amount of motion", if you wish.
However, how should we measure the "amount of motion"? The method should satisfy certain criteria. If you have two independent and separated moving objects, their total kinetic energy should be the sum of the kinetic energy of each – the sum of two numbers that you would calculate separately if the other object were absent. We say that energy is and should be an extensive quantity: if objects are added, the energy just adds up.
(In local field theory, this extensive property of energy becomes an even stronger condition and energy must be generally calculable as the integral of some local function, the energy density, over space.)
But this still doesn't answer the question how the kinetic energy depends, for example, on the speed \(v\) or the angular frequency \(\omega\) of rotation of an object. You ultimately know that it's the second power (in nonrelativistic physics) and it's a natural choice. But how can you really show that the energy isn't proportional to the fourth power of the velocity or any other function? (Later, we will see that it is a different function in relativity.)
So what's the rule that determines the dependence of the kinetic energy on the speed or mass or other things? Could you choose a different dependence? Would your alternative choice be equally valid or justified? You know it wouldn't. What would be wrong about it?
The key condition is that the energy is conserved.
For example, if you have two hard balls whose mass is \(M\) in each case, you may observationally prove that
\[ \frac{Mv_A^2}{2} + \frac{Mv_B^2}{2} = {\rm const}. \] That's true even if the two balls collide and change their speed. And it wouldn't be right if you replaced the second powers of the speed by other powers or other functions; or if you included a nonlinear dependence on the mass, or made another drastic modification.
The energy conservation in elastic collisions is an observation you could prove in an ad hoc way, by looking at collisions. Alternatively, you may prove that the energy conservation holds according to Newton's equations of motion.
More general systems
But not all objects are hard balls. What happens if you have springs, rubber bands, alkaline, batteries, nuclear power plants, thermonuclear bombs, and many other things? The world is getting complicated, isn't it?
Let's try one example: a vibrating spring. The total energy will be
\[ E = \frac{kx^2}{2} + \frac{mv^2}{2} \] where \(k\) is the spring constant. You may prove that the position \(x\) of the ball attached to the spring will periodically depend on time,
\[ x(t) = x_{\rm max} \cos (\omega t +\phi_0) \] where \(\omega=\sqrt{k/m}\). The amplitude \(x_{\rm max}\) and the phase shift \(\phi_0\) are arbitrary real numbers.
The energy is conserved because the kinetic term is
\[ T = \frac{mv^2}{2} = \frac{1}{2} x_{\rm max}^2 m\omega^2 \sin^2 (\omega t+\phi_0) \] while the potential energy is
\[ E_{\rm pot} = \frac{kx^2}{2} = \frac{1}{2} x_{\rm max}^2 k\cos^2(\omega t + \phi_0).\] Their sum simplifies because \(m\omega^2=k\) so the coefficients are the same in both terms, \(k\), and
\[\sin^2\beta+\cos^2\beta = 1\] which simplifies even more dramatically. So the total energy is
\[ E = \frac{k}{2} x_{\rm max}^2 .\] I wanted to go through at least one exercise explicitly. It's important that we have found a quantity that is conserved. And if you actually study more complicated systems that may include nuclear power plants and batteries aside from hard balls and springs, you will find out that it is always possible to define a formula for energy that is conserved. Is it a coincidence?
Energy is abstract but less vague than soul
Before I answer, it is useful to notice one point. Energy is a number with some units – for example 20 kilowatthours. And this number may be calculated for every physical object (or "system") according to some formula whose formal prescription depends on the system but doesn't depend on its particular state (on the position or velocities). However, when you substitute the positions, velocities, and other observable properties, the resulting energy will of course depend on them.
So the energy is a number that may depend on everything we may in principle know about the system. Well, Nature doesn't really guarantee that we will be able to "see" or "measure" everything so the energy may also depend on things that are very hard to measure – and in principle, it could also depend on things that are completely invisible and unmeasurable even though the principles of physics would considerably weaken. As far as we can say, all things that the energy may depend on are measurable at least in principle. Even the energy of neutrinos is however very hard to measure because almost all neutrinos escape just like ghosts and we can't ever see them again.
Energy is as abstract as a mathematical structure – but it is not really as mysterious as a soul. It's a very different kind of abstractness. Indeed, the purpose of using energy – and many other things – is to remove the fog and mystery from the world, not to increase it.
Hamiltonian in classical and quantum physics
Why Newton's or Maxwell's equations allow us to define a quantity like energy at all? There are many objects where energy is apparently lost. For example, if you have a system with friction, the motion will stop. For example when a train moves on the tracks, it will stop after some time. But we know that the energy is actually not quite lost. It is being transferred to some other properties of the system – well, it is converted to heat, another form of energy, by the friction forces.
Heat is the mostly kinetic energy of individual atoms (or other microscopic constituents whose number is large) in an object. However, heat may only be used as "stored work" if your objects have different temperatures. If the temperature of everything is uniform, you can't get any work out of the heat because that would violate the second law of thermodynamics (which says that you can't do work just by cooling another object): the entropy of an object with a uniform temperature is maximized and you can't decrease it.
(The amount of "useful work" that the system can do is given by "free energy" which differs from the normal energy by a \(TS\) term. Needless to say, the term "free energy" is being abused by the nonscientists in an even more atrocious way.)
(Heat used to be considered "matter": the hypothetical material contained in hot objects was known as "phlogiston". Such an idea is only legitimate if you generalize the notion of "matter" so that all original predictions of the "phlogiston theory" are invalidated and the notion becomes useless. The idea that the heat may be represented by a simple material was debunked many centuries ago. Heat is much closer to the "soul" type of energy. The equivalence between energy and heat was demonstrated by Joule who also found the conversion factor in the old units of his time. And that's why we use the unit of Joule for both energy and heat in our more unified modern systems of units.)
Energy is abstract but comprehensible and welldefined. And it may be found in systems such as objects obeying Newton's classical mechanics because... because Newton's mechanics may actually be defined in a way that promotes energy to the most important player. It's the Hamiltonian mechanics, a pretty modern way to derive the laws of mechanics that William Rowan Hamilton invented in 1833.
Instead of writing individual differential equations for each position or velocity, Hamilton instructs you to write down the total energy of the system. Because he was a modest guy who could quantify the importance of his contributions, he also wants you to use a different word for energy. The name doesn't really matter, it's just a stupid word, and calm down, he won't tell you that you should call it Wakalixes. Instead, you should call it the Hamiltonian. It's a coincidence that the beginning of the word agrees with Hamilton's last name.
Fine, so energy should be called \(H\). As you may guess, this is not the most important contribution by Hamilton. What is more important is that time evolution of any quantity \(L\) which may be position \(x\), momentum \(p\), or any other quantity (well, typically a function of many \(x,p\) variables) may be prescribed by a simple and completely universal formula,
\[ \frac{{\rm d} L}{{\rm d}t} = \{L,H\} + \frac{\partial L}{\partial t}. \] The last term on the right hand side contains partial derivatives and it is only nonzero if \(L\) is defined to explicitly depend on time: positions and momenta never do.
This is a remarkably simple formula but the simplicity is partly fake because I used a new symbol, the curly bracket, which is known as the Poisson bracket. The Poisson bracket of any two objects is defined by
\[ \{f,g\} = \sum_{i=1}^N \left[ \frac{\partial f}{\partial q_i} \frac{\partial g}{\partial p_i}  \frac{\partial f}{\partial p_i} \frac{\partial g}{\partial q_i} \right] \] where \(q_i=x_i\) is Hamilton's new symbol for all the positions (coordinates) of parts of the system and where \(p_i\) are the canonical momenta which coincide with the simple momenta, \(p_i=mv_i\), in the simplest mechanical examples. The number of different coordinates that describe the position of the system is \(N\).
I leave it to investigative comparative historians among the TRF readers to figure out how it's possible that the most critical portion of Hamiltonian mechanics is named after Siméon Denis Poisson. Just to give you a hint, Poisson wrote similar formulae before Hamilton but he didn't have the full set of laws.
Things have gotten even more bizarre. Why can the physical laws for mechanics, electromagnetism, and other classes of physical phenomena be formulated in terms of some bizarrely contrived Poisson bracket that was made productive by Hamilton? The answer is that those 19th century physicists actually discovered something that only became natural with the arrival of quantum mechanics 100 years later.
Quantum mechanics
In quantum mechanics, the Hamiltonian still exists and still encodes the total energy but much like all other quantities associated with a system – which were renamed as observables to make quantum mechanics less philosophically ambiguous – the Hamiltonian has become an operator.
(The nicely constructed term "observable" turned out to be useless because many people, including some very selfconfident ones, completely misunderstood that quantum physics was all about observables anyway, which is why they invented many things that don't exist and can't exist in quantum physics such as "beables". If someone talks about "beables" instead of "observables", it means that he is only capable of understanding classical physics, not quantum physics, and is severely intellectually limited regardless of the selfconfidence that such a person displays. Our quantum world allows quantities to be "observed" but it doesn't allow them to "be" at fixed values before the observations because the results of the observations have a stochastic component and the random generator is only activated during the measurement.)
What dramatically simplifies is that the Poisson bracket may be defined as or replaced by
\[ \{f,g\} = \frac{1}{i\hbar} [f,g] = \frac{1}{i\hbar} (fggf). \] It gets replaced by the simple "commutator" which is just the difference between two products \(fg\) and \(gf\). They were equal for ordinary numbers because the multiplication of numbers is commutative. But observables in quantum mechanics are no longer numbers. They're operators i.e. (generalized) matrices that don't commute when you multiply them. However, the deviation of the commutator from zero is small (for systems that behave nearly classically, i.e. heavy ones) which is why you have to divide it by \(\hbar\) to get a "finite" value that may be interpreted classically. In quantum mechanics, \(\hbar\) always measures the magnitude of firstorder quantum effects i.e. the most important effects that are zero in classical physics but nonzero in quantum physics.
The commutator also has to be divided by \(i\) to get a Hermitian ("real") operator if the operators \(f,g\) themselves are Hermitian ("real"): the commutator of two Hermitian operators is antiHermitian which would be bad. At any rate, the commutator is the simplest nontrivial operation you may do with two operators \(f,g\) that is of order \(\hbar\) i.e. that vanishes in the classical limit – the \(\hbar\to 0 \) limit. You should better study such "commutators". And because of various identities, you may reduce the commutators of complicated functions of the coordinates and momenta to the commutators of \(x,p\) themselves which finally leads you to to the basic commutator behind Werner Heisenberg's uncertainty principle,
\[ [x_i,p_j] = i\hbar \delta_{ij} .\] The coordinates commute with each other, the momenta commute with each other, but the commutator of a coordinate and its complementary momentum is equal to \(i\hbar\). When you put all these things together, you will be able to prove that the "simple" commutator reduces to the apparently "complicated" Poisson bracket in the classical limit.
If you learn some group theory and I really mean the theory of (continuous) Lie groups, you will understand that the commutator has something to do with various generalized "rotations around different axes" for which it matters which of them is done first. So commutators are closely linked to continuous symmetries. Well, commutators with a "symmetry generator", a particular operator, may be represented as small transformations of the other operator. Under the infinitesimal generalized angle \(\delta \phi\),
\[ \delta_L M = M_{\rm after}  M_{\rm before} = \delta\phi [L,M]. \] What I mean is that \([H,L]\) is the symmetry transformation of the operator or object \(L\) – imagine e.g. the angular momentum – by the symmetry associated with the operator \(H\), the Hamiltonian. What is the symmetry associated with the Hamiltonian? Well, it is the translation in time.
Noether's theorem
When we say a "symmetry", we usually imagine some rotation or leftright reflection. You face looks almost the same in the mirror; a snowflake remains unchanged if you rotate it by 60°. We usually imagine operations that return you to the same state if you perform them \(N\) times. But that doesn't have to be the case. You may have symmetries that never return you back. Like translations.
The translations in time (much like translations in space) are symmetries of the normal laws of physics. This is just a different way of saying that the laws of physics themselves don't depend on time. If you prepare a classical system in the same initial state once again, it will evolve in the same way as it did before. Well, there can be some chaotic behavior that may produce different results – e.g. if you throw dice. But in principle, if you arranged the die exactly as you did yesterday, you will get the same number.
In quantum mechanics, it's not guaranteed. Nature which exploits the free will given to Her by the postulates of quantum mechanics will give you different results even if you prepare the initial state as identically as you can. Randomness of individual results is an inherent property of Nature, we learned in the mid 1920s.
Emmy Noether, one of the best female mathematicians of all time, discovered a remarkable relationship between symmetries and consrevation laws. For every conserved quantity, there is a symmetry, and vice versa. She realized that if the laws are timetranslationally symmetric, there will be a conserved quantity that deserves to be called "energy". If the system is symmetric under translations in space i.e. if the laws of physics don't depend on the location, they will conserve the total "momentum". If they are rotationally invariant under rotations around an axis, they will conserve the component of the "angular momentum" with respect to this axis, and so on. (Leftright symmetric laws of physics preserve "parity" but "parity" may only be +1 or –1, if you forget about subtleties with \(\pm i\).)
We will see a trivial reason why Noether's insight is right, using the elegant maths of quantum mechanics. However, you must appreciate that she discovered those things at the level of classical mechanics where you must replace all the simple and elegant commutators by the distasteful and awkward Poisson brackets and similar objects. In fact, Noether's papers were even more inelegant than just the Poisson brackets but that doesn't change the fact that she clearly discovered the relationship.
So why are the conservation laws linked to symmetries? It's all about the commutator \([H,L]\) where \(H\) is the Hamiltonian and \(L\) is the generator of the symmetry corresponding to the conserved quantity \(L\). You see that there's some twotoone notation in my description of the situation but it's just fine. Why?
It's because the commutator \([H,L]=[L,H]\) may be interpreted in two ways. It's either the thing you get by transforming \(L\) by the symmetry generated by \(H\), i.e. by translations in time, or vice versa, i.e. it may be the thing that you get by transforming \(H\), the total energy, by the symmetry genenerated by the operator \(L\).
The funny thing is that the first definition is simply the time evolution of \(L\), because of Hamilton's equations that replaced Newton's and all other dynamical equations in 1833: if the commutator is zero, \(L\) will not depend on time because its time derivative is zero.
On the other hand, the other, equivalent definition may be interpreted as the transformation of the Hamiltonian by symmetries generated by \(L\). If the Hamiltonian is invariant under these symmetries, then the commutator is zero. But we may also say that in this case, the laws of physics are invariant under these symmetries because all the laws of physics are really encoded in the Hamiltonian and the Hamiltonian is invariant.
So the vanishing of the commutator – and it doesn't matter which one because up to a sign, they're the same – may be interpreted either as the conservation of \(L\) in time or the symmetry of the laws of physics under transformations by \(L\). The same operator \(L\) remembers how the infinitesimal transformations work; and what is the conserved quantity at the same moment.
In classical physics, symmetries were "operations" and looked "abstract", "soullike", and very different from "material" objects such as the angular momentum. However, in quantum mechanics, the difference between these two types of objects is eliminated. Objects themselves determine i.e. generate symmetries.
Once you understand the linear algebra underlying quantum mechanics, this point and Noether's theorem becomes really trivial. And yes, it's the right and modern way to deal with deep insights such as Noether's theorem. The woman figured those things in a more awkward classical formalism which you may consider to be more painful or ingenious, depending on your viewpoint. But I am sure that there's no reason to teach her contrived derivation in the future. Certain things really become simpler in quantum mechanics and I am sure that future physics textbooks will exploit these simplifications as effectively as you can get.
In this context, we learned that energy is either the generator of translations in time, or the quantity that is conserved because the laws are symmetric under translations in time.
The two definitions in the previous sentence are equivalent because \([H,H]=[H,H]=0.\) This is seemingly a vacuous tautology but what is not tautology and what doesn't have to be true is the assumption that the laws of physics may be defined in the Hamiltonian way. Very general laws of physics don't have to be derivable from a Hamiltonian: for example, think about systems with frictions where you don't remember anything about the heat and temperature of the objects. In such systems, any prescription for the energy fails to be conserved and you won't be able to derive the equations of motion from a Hamiltonian.
But whenever the energy is conserved, the equations of motion may be defined in the Hamiltonian way and the Hamiltonian itself is therefore the generator of translations in time.
Energy in relativity and field theory
You should understand that these Noether's games are not just fancy and redundant exercises. They're totally necessary for you to understand what the energy is in the case of a general enough system. The energy didn't scale as the second power of the velocity just because an obnoxious elementary school teacher forced you to repeat this dogma (the same teacher who may want you to parrot that Allah created humans or that global warming is dangerous).
The actual reason why the formula has the form you see is that this is the right formula that leads to a conserved quantity and the conservation law is nothing less than the defining criterion that tells you what energy may be and what it cannot be. Nonscientists may have been told about "energy" by God and then they could marvel at its properties (e.g. that it can be obtained from the trees) but physicists only introduced the term with a reason, namely that it is a conserved quantity that tells us certain things, and there is no Godgiven formula whose origin remains inaccessible to the human mind. And Noether's theorem gives you a simple way to find what the energy is: it's whatever generates the translations in time via commutators or their classical limits, the Poisson brackets.
In special relativity, the kinetic energy has the form
\[ E_{\rm kin} = \frac{m_0c^2}{\sqrt{1v^2/c^2}} = m_0c^2 +\frac{m_0v^2}{2}+\dots .\] It contains a new velocityindependent (and therefore dynamically irrelevant: its derivatives in Hamilton's equations vanish) term, Einstein's famous \(E=mc^2\) whose 0.1% is used in nuclear power plants, as well the usual kinetic energy scaling as the squared velocity and higherorder terms arising from the Taylor expansion of the square root which produces the exact formula.
Why is there the square root in special relativity? Well, it's because relativity unifies space and time into a spacetime (where 3+1 coordinates may be mixed by Lorentz transformations). And because the energy and the momentum are the generators of translations in time and space, energy and momentum are also unified into the energymomentum vector which also transforms under the Lorentz transformations. The Lorentz transformations must however preserve the "squared Minkowskian length" of the 4vector unchanged,
\[ m_0^2 c^4 = E^2  p^2 c^2 \] which allows you to see that
\[ E = \sqrt{m_0^2 c^4+p^2c^2}. \] Together with the formula determining the slope of the world lines in spacetime,
\[ \frac{v}{c} = \frac{pc}{E},\] you become capable of deriving the dependence on the velocity. Check that these equations are consistent with each other and any three of them imply the fourth.
In similar ways, one may derive the prescription for the energy of electromagnetic field or any other field. The Hamiltonian knows everything about the laws of physics. In most cases, one may also replace the Hamiltonian mechanics by the Lagrangian mechanics. The Lagrangian and the Hamiltonian are related objects (with the same units and related by the Legendre transform) but the methods how to formulate the laws of physics using either of them are very different. The Lagrangian mechanics builds on the "principle of least action" in classical physics which is identified as the limit of "Feynman's path integral" in quantum mechanics, but that's a topic for another article (very analogous to this one).
Every physical system has some definition of the energy which pretty much knows everything about the laws how the system evolves in time, the socalled dynamical laws.
Energy as mass; soul as body
We can finally return to our superstitious friends and relatives. One of the key reasons why those people talk about the soul is that the soul is not the body. In fact, it's meant to be a nonbody.
If the body gets promoted to the mass \(m\) in physics and the soul becomes the energy \(E\), you could think that the energy has to be something completely different than the mass. However, we have already seen the hints that physics has a surprise. The energy and the mass is really the same thing, because of Einstein's formula \(E=mc^2\).
Before relativity was discovered, people thought that there were separate conservation laws, one for mass and one for energy. However, in proper physics, conservation laws are linked to symmetries and there's only one symmetry group that could give you a law independent of any directions in space: the translations in time. So it gives you just one conservation law and it is the energy/mass conservation law.
What we used to call the mass, like a few pounds of uranium, may be converted to energy (e.g. one needed to destroy a city), and vice versa: energy 2 times 3.5 TeV pumped by electromagnetic fields into fast protons may be converted to the mass of dozens of new protons and pions that are created during the collision of two protons at the LHC. Only the total energy including the latent energy of the mass is conserved. In other words, only the total mass \(m\) that is appropriately increased if the object is moving – so it is not the rest mass \(m_0\) – is conserved. The rest mass is not conserved.
In this sense, physics and especially the nuclear bomb erases the difference between the body and the soul. Before 1945, believers and atheists could have hated each other because one of them focused on the soul and the others focused on the body and the matter. However, the nuclear bomb allowed us to establish eternal peace between both groups: the body of a little boy or a fat man can be converted to the soul which may destroy many other bodies which emit extra souls and some of these souls may paircollide and create new bodies (particles), and so on.
Different manifestations of the mass
Because the mass and the energy have been shown to be the same thing, even though we thought they were different, this article should now include everything we know about the mass as well. However, this attitude could ultimately expand this article indefinitely so I have to exponentially shrink what I write.
Let me just say that the mass manifests itself by its inertia (resistance against acceleration), as encoded in Newton's
\[F=m_{\rm inertial}a\] or by the strength of its gravitational field, as encoded by Newton's law for the acceleration of another test object:
\[a_{\rm grav} = \frac{Gm_{\rm gravitational}}{r^2} \] It is possible to choose units – and you are encouraged to choose units – in which
\[ m_{\rm inertial} = m_{\rm gravitational} .\] Note that at this moment, we have at least three different "operational definitions" of the total energy/mass. One of them is the thing conserved because of timetranslational symmetry (which generates the translations in time according to the Hamiltonian mechanics); the second one is the inertial mass; the third one is the gravitational mass.
Why are all of them the same thing?
Well, the inertial mass is the same thing as the conserved total mass/energy because of the conservation of momentum. The acceleration of an object is encoded in the curvature of the world line in the spacetime (relativistically) and the precise relationship may be written down in terms of the force (time derivative of the momentum) and the total energy (the other component of the conserved 4vector).
The gravitational mass has to coincide with the conserved mass because the metric tensor \(g_{\mu\nu}\), a kind of a gauge field, must be coupled to a conserved quantity \(T_{\mu\nu}\) (whose divergence vanishes) much like the electromagnetic field \(A_\mu\) must be sourced by a conserved electric current \(j_\mu\). So one may completely demystify arguments why all the three aspects of the total mass/energy actually define the same quantity.
Summary
To summarize, physics has demnystified the notion of energy. It became a conserved quantity linked to translations in time (and the symmetry group composed of them). It became abstract but totally unambiguous, extensive yet dependent on all degrees of freedom in the system, and it was linked to the total inertial or gravitational mass.
Energy is not an oldfashioned soul; energy is not just a potato you can touch in a standardized way, either. It is a flexible animal that is able to change its form in many ways – from more visible ones to less visible ones; from more material to more spiritual ones; and back. Everything you may know about the energy is a single number with a unit; everything that you imagine beyond this number is just your fantasy that may help you to visualize things but that isn't necessarily real (or valid, in the sense of making valid predictions) in any physical way.
Energy has many possible forms and the sum of all these terms is conserved. The only useful and fully realistic context where the conservation law becomes invalid or vacuous is the case of cosmology – general relativity applied to a very general spacetime background. In particular, inflation is the ultimate free lunch as its cofather Alan Guth once said. It can create – and it probably did create – large amounts of mass/energy in a large space out of (almost?) nothing, thus franchising one of God's trademark skills.
Comments of the Week #138: from the time dimension to going beyond the Standard Model

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