# Energy a mathematical expression?

1. May 3, 2013

### Phztastic

So, energy is simply a mathematical expression of how much work is done in the system?
Since energy is not a "physical" object that has mass. It's not matter at all.
If so, the laws of conservation makes sense, it can't be created nor destroyed.
But I can't imagine how a mathematical expression be converted from one "form" to another.

I tend to avoid using the term "energy" because I get confused a lot and can't interpret the idea at all.
Yet I do use "Work" a lot. As W= Fd, its much easier.
I treat energy as a number thats it. That can't be created/destroyed because it's a number that measures the work.

Here is one of the questions that confuses me. What does it mean that energy is conserved as being the sum of PE and KE?
Is energy a constant value in the universe?

How should I deal with energy... As what exactly?

2. May 3, 2013

### espen180

I mordern classical mechanics, dynamical systems are described by so-called Lagrangian and Hamiltonian functions. A Lagranigan or Hamiltonian function is a mathematical expression in certain variables, which gives rise to the equations of motion of the system in a canonical way.

Purely formally, energy is a certain "constant of motion" in closed systems. It arises naturally from the mathematical formalism of dynamical systems whose Hamiltonian or Lagrangian function does not evolve with time. The exact expression for the energy, e.g. kinetic + potential etc., also arises from the Lagrangian and Hamiltonian formalisms.

Classical mechanics itself does not, as far as I can tell, give a mechanism for the transfer of energy, so for all intents and purposes, it is simply a formal constraint that you put on the system.

3. May 3, 2013

### A.T.

There are different mathematical expressions for different forms of energy. The definition of work is just one specific one.

4. May 3, 2013

### DEvens

[snippers]

What sort of answer would satisfy you? You already have the idea of work. When you do work you are moving energy from one location to another, or changing it's form from one type to another.

As to your other questions: energy is conserved (mass-energy when you include relativity). When you get a bit farther in physics you meet Noether's theorm and you learn that conservation laws are (always?) associated with symmetry. This is part of why symmetry is one of the most powerful methods of obtaining results that we have in physics.

The symmetry for energy conservation is time translation invariance. Similarly, the symmetry for momentum conservation is space translation invariance, and the symmetry for angular momentum conservation is rotational invariance. Physical laws look the same now as 10 seconds from now. Or when moved 5 meters to the left. Or when rotated 16.45 degrees clockwise. And so there are conserved quantities that correspond.

When you add relativity you add invariance under changes of velocity, and the conservation becomes conservation of mass-energy.
Dan

5. May 3, 2013

6. May 4, 2013

### Popper

No. How much work is done in a system relates to changes in kinetic energy of particles in the system and/or a change in potential energy in the system. Energy is not identical to work. Energy is conserved while work is not.

Think of a particle falling in a uniform gravitational field. The total energy E of the particle is given by

E = K + V = mv^2/2 + mgy

When the particle is falling the v increases while the y decreases while the two quantity K(v) and V(y) add to give a constant. This is precisely what is meant to mean that energy is being converted from one form to another.

People often mistakenly define energy as the ability to do work. But this attempt at a definition is flawed for reasons I won't state just now. There actually is no definition of energy which is not flawed since energy cannot be defined. Certain quantities in physics are like that. As the Dutch physicist H.A. Kramers once said
As Richard Feynman said in his Lectures

That's incorrect. You seem to be confusing energy with work. They are not the same thing. Think of a photon. When an atom emits a photon there is no work involved in creating it yet it has energy. Yet a photon has energy. Consider an object at rest in an inertial frame of reference. The object has energy since it has mass by virtue of the mass-energy relation E = mc2. But an object at rest in an inertial frame of reference, a frame where no other matter exists, cannot do work. Yet it has energy.

It means that there is a relationship between the physical parameters of the system which when plugged into the expressions of K and V and then summed result in a constant of motion (also known as an integral of motion).

Yes. Most cosmologists believe that the total energy of the universe is zero. Although I believe they are speaking of a spatially closed universe when they say that since the total energy of an infinite universe can't be defined unless we assume that the total of all the energies in this infinite universe sums to zero. But I find that an odd thing to talk about like that.

Think of energy as Feynman explains in the quoted I gave above. Think of it as bookkeeping and that the books always balance. This idea was how the neutrino was first postulated to exist. There was energy missing from certain decay processes so they assumed that there was a particle carrying the energy away as kinetic energy. They called that particle the "neutrino" (which they later realized that they should call it an antineutrino).

Hope that helps.

7. May 4, 2013

### WannabeNewton

Do you have a reference for this? Otherwise it is just nonsense. You are trivializing conservation of energy in GR. If $(M,g_{ab})$ is a space-time with a time-like killing vector field $\xi^{a}$ i.e. a time-like vector field satisfying $\nabla_{(a} \xi_{b)} = 0$ then we can construct a conserved energy current $J_{a} = T_{ab}\xi^{b}$, where $T_{ab}$ is the stress-energy tensor, because $\nabla^{a}J_{a} = T_{ab}\nabla^{a}\xi^{b} + \xi^{b}\nabla^{a}T_{ab} = 0$ hence by Stokes theorem $\int _{\Sigma}J_{a}n^{a}\epsilon$ is independent of the space-like hypersurface $\Sigma$. This implies global energy conservation for space-times with time-like killing vector fields (also called stationary space-times).

The RW metric used for the RW cosmological model is not a stationary metric and it is one of the simpler of the standard cosmological models out there; this is independent of what geometry we choose for the space-like foliations (i.e. what class of constant sectional curvature 3-manifolds).

EDIT: Just to add, "total energy" is also something that is very non-trivial in GR. The simplest notion of total energy comes from the Komar mass which can be defined for stationary, asymptotically flat space-times with time-like killing field $\xi^{a}$ as $M = -\frac{1}{8\pi}\int _{S}\epsilon_{abcd}\nabla^{c}\xi^{d}$ where $S$ is a topological 2-sphere but again this makes no sense for non-stationary space-times such as the RW universe.

Last edited: May 4, 2013
8. May 4, 2013

### Popper

Yes. Please see The Inflationary Universe by Alan H. Guth. Addison-Wesley (1997) pages 11-12 in chapter 1 - The Ultimate Free Lunch (want to take a guess what the free lunch refers to?)

A statement does not depend on what reference a poster has but in what has been said and what it implies. Rather than assert that if I don’t have a reference at my very side at the moment I posted that explanation it does not mean that one does not exist or that I can’t elaborate as to why most cosmologists believe it. Instead it’s much better to ask why such a thing is true or if I can clarify why that is so it’s much better to wait for their response before suggesting that what they said is nonsense merely because they don’t have a reference ready in hand. I know a great deal of many areas in physics and I’d be hard pressed to readily post a textual or journal reference for each bit of knowledge/fact that I know/post.

Whether something is nonsense or not depends on the reasoning behind it, i.e. the reasons that they used to come to such a conclusion. In this case it seems that you didn’t take into account the energy associated with gravitational potential energy. It seems that you were only considering the energy associated with matter.

In he first chapter entitled on pages 11-12 Guth writes
Alan Guth and Edward Farhi from MIT believe it may be possible to create a universe from a small amount of exotic matter. Using advanced methods of physics they published a paper called An obstacle to creating a universe in the laboratory, Physics Letters B, Volume 183, Issue 2, p. 149-155. They presented a derivation wherein they proposed that in order to create a universe one needs about 25 grams of false vacuum matter, which is equal to roughly one ounce of matter!

That is incorrect too. Does what are you basing that assertion on based entirely of what you posted in your post?

And nowhere in your post did you include the energy from the gravitational field? Why is that?

9. May 4, 2013

### WannabeNewton

Before you start making grand claims, you should first understand how energy works in general relativity. Nothing in your quotes claims there exists a notion of total energy for the FLRW universe. Your claim was that there was indeed a way to define a physically meaningful expression for the total energy of a non-stationary non-asymptotically flat space-time. Prove to me that you can define a notion of total energy for the RW metric and we can take it from there.

Again, please learn general relativity before you start making claims like this. First note that there is no meaningful notion for the local stress-energy of the gravitational field in GR. Things like gravitational potential energy in the case of stationary space-times, on the other hand, are taken into account in the Komar integral. In fact, using a bit of tensor calculus and Stokes theorem, $M = -\frac{1}{8\pi}\int _{S}\epsilon_{abcd}\nabla^{c}\xi^{d} = -\frac{3}{8\pi}\int _{\Sigma}\nabla_{[e}(\epsilon_{ab]cd}\nabla^{c}\xi^{d}) = \\ -\frac{1}{4\pi}\int _{\Sigma}R^{d}{}{}_{f}\xi^{f}\epsilon_{deab} = 2\int _{\Sigma}(T_{ab} - \frac{1}{2}Tg_{ab})n^{a}\xi^{b}dV$

where in the very last step Einstein's equations were used.

10. May 4, 2013

### WannabeNewton

Gravitational potential energy doesn't even make sense for non-stationary space-times such as the FLRW universe. In the case of stationary space-times, see the above post. Komar integrals are standard topics you will find in any decent GR text e.g. Carroll. as well as ADM energy.

11. May 4, 2013

### Popper

Since this is the Classical Physics sub forum and not the Special & General Relativity sub forum I responded using non-relativistic mechanics. The only reason I touched on gravitational potential energy is because the OP asked Is energy a constant value in the universe? and since I know what cosmologists believe on this issue I stated it. If you have a problem with Guth and what those other cosmologists believe then I suggest you take it up with them, not me. I believe what Guth says because I know him know how well he knows his stuff. I spoke to him two weeks ago about this very subject in fact. That's why I posted what I did. I don't know the derivation of this he was basing it on but by knowing him I trust him very much and if he says that is what most cosmologist believe that I take that as fact contrary to what the wannabe above would like to insult me with.

You referred to the Komar mas and ADM mass. Those terms cannot be applied to the total energy of the universe since their defined in terms of finite gravitational distributions of matter. As Wald explains, those integrals pertain to asymptotically flat spacetimes and thus cannot be applied to the entire universe

[Mentor's note: Off-topic stuff deleted]

I suggest that the OP send me a PM for any further proofs/questions/comments etc. if he'sinterested in the details.

Last edited by a moderator: May 4, 2013
12. May 4, 2013

### micromass

Staff Emeritus
Wait, your reference is a pop-sci book??? I don't think it's good practice to take things in a pop-sci book very seriously.

I'm afraid this is not how this forum works. Every statement here should be backed up by a valid reference if people ask for it. So WannabeNewton had every right to ask for a reference. If you were right, then he would just read the reference and argue from there. Otherwise, he just has to take your word for it.

13. May 4, 2013

### Popper

I disagree. If the author of a book states that the energy of the total energy of the universe is zero then he doens't mean that it's undefined, or has such and such a value. He means quite literally that it's zero. How do I know? Because I know him and asked him this exact question.

What I said is precisely true. Recall what I was trying to say:

(1) If I don't have a reference it doesn;t mean that its nonsense. It only means I don't have a reference handy.

(2) If you want a refernce then ask for it and then complain. Don't complain and then ask for it.

(3) The statement I made is backed up with a referance. It explains how the total energy of the universe could be zero even though the sum of mass-energy is positive. So he asked for a referene and I gave it to him. So instead of stating why Guth is wrong, i.e. what part of his arguement was erroneous then he should have said so. But that wasn't the case. Instead he chose to be rude. His comments had nothing to do with what Guth said. He made a serious error of addressing the conservation of energy of a particle moving in a gravitational field as well as tossing out mentioning ADM and Komar mass which he doesn't seem to undertand what they are. They pertain to the mass of a closed system for which the spacetime is asymptotically flat. Those have absolutely nothing[/n] to do with the subject of the total mass of the universe. Such a derivation would include gravitational energy which is a pseudo tensor. Did you see him mention that? No. Instead he referred only to particles moving in spacetime.

[Mentor's note: Off-topic stuff deleted]

Last edited by a moderator: May 4, 2013
14. May 4, 2013

### micromass

Staff Emeritus
[Mentor's note: Off-topic stuff deleted]

Please, in the future, when engaged in a technical discussion, never reference to a pop-sci book. You may think the information in there is true (and it may very well be true), but we have no way to read the actual paper and to understand the actual science. We have no access to the derivation, proofs or experiments. Instead, we have to rely to some sentences in a book written for laymen. This is not helpful to the discussion.

Last edited by a moderator: May 4, 2013
15. May 4, 2013

### WannabeNewton

Exactly, so either you bring up a reference that can define a physically realizable notion of total energy of the FLRW universe or prove by yourself that such a concept exists. Otherwise, you've just gone on a long discussion about, as I originally said, nonsensical things. You said most cosmologists "believe" in such things but never gave an authoritative reference to a journalistic paper that agreed with your claim. It's one thing to attribute something to yourself but to claim most cosmoligists believe in something definitely warrants an authoritative backup don't you think friend ?

Last edited: May 4, 2013
16. May 4, 2013

### Fredrik

Staff Emeritus
I wouldn't put it that way. It's not that the term "energy" can't be defined at all, it's just that it can't be defined in a theory-independent way. It has to be defined by a theory of physics, and different theories can (and do) define it in different ways. I wouldn't say that the idea that "energy is the ability to do work" is wrong. I would just say this:

1. It's not a definition.
2. It's a good way of thinking about the various types of energy that are defined in pre-relativistic classical mechanics.
3. It's not a good way of thinking about energy in quantum mechanics.
4. It's a guideline that we can use to choose which mathematical concepts in a new theory that should be given names that have the word "energy" in them.

17. May 4, 2013

### Fredrik

Staff Emeritus
Guys, can we cut the bickering and stick to the facts?

The idea that the total energy is (in some sense) zero is something I've heard many times before, since before I even studied physics. So it appears to be part of some physicists' attempt to communicate an idea that they think is at least interesting, to an audience that doesn't know a lot of mathematics. The reference shows that Alan Guth is one such physicist.

As far as I can tell (after reading page 10 at amazon.com and the quote from pages 11-12 posted above), the book is saying roughly this:

1. In Newton's theory of gravity, a mass in a gravitational field has potential energy.
2. In special relativity, matter has energy as indicated by $E=mc^2$ or $E^2=\mathbf p^2c^2+m^2c^4$.
3. The former is negative, and the latter is positive, so maybe it all adds up to zero.

Since the argument isn't even made within the framework of a single theory, it's not strong. But Guth doesn't say that is. He just says that "it's conceivable" that the total energy is zero. This can't be disproved using GR, because Guth doesn't say what theory's concept of total energy he's referring to. Even if total energy doesn't make sense in GR, it still could make sense in a quantum theory of gravity. (I wouldn't count on it though).

Discussions about this sort of thing are close to being too speculative to this forum, which is intended to be a place where we can help each other understand the established theories.

18. May 4, 2013

### pumila

From a practical viewpoint in classical physics, my experience is that energy is converted from one form to another via forces. So energy of type 'a' generates a force from F=dU/dl as it is consumed. Force is anonymous so can then create energy of type 'b' through U= integral of F.dl.

19. May 5, 2013

### Phztastic

1. Then what is it?
2. Energy? Or the definition?
3. Why? What about energy in QM?
4. Do explain!

In the common world... What is energy in both CM and QM? An ability? A property? What?
Or just a measuring tool for the amount of work?
Why use the idea of energy? Why do we need if?
Why not calculate the Work done and so be it?
What's the point of Potential energy?

What is our purpose of building mathematical models for energy and not knowing what "energy" is?!

The conservation principles are really mind boggling...

20. May 5, 2013

### Fredrik

Staff Emeritus
1. Explained by 2-4 (in the post you quoted), especially 4.
2. The statement that's not a definition.
3. In QM, the Hamiltonian is defined as the generator of translations in time, and we refer to its eigenvalues as "energy". "Work" is a less useful concept in the foundations of QM at least.
4. I thought that my statement was sort of self-explanatory, or at least that what I said before the list made the meaning clear.

As I said, such terms are defined by theories. They are not defined "in the common world".

I'm not going to write 20 pages.

You're missing the point, those mathematical models (plus the experiments that tell us how good or bad their predictions are) are how we understand what energy is. If you want to understand energy, the only way to do it is to study its mathematical definitions in the relevant theories, and do some exercises.

Last edited: May 6, 2013