Re: energy, work, heat

[Daniel Schroeder <DSCHROEDER@CC.WEBER.EDU>]

Dear Leigh and anyone else who thinks he might be right:

> The discussion in Chapter 5 [in MTW] seems to be about 4-momentum, one
> component of which is the energy. While Taylor and Wheeler do speak
> (somewhat inelegantly) of the flow of 4-momentum in a fluid-like way:
>
>          Total flux of 4-momentum outward across a closed
>          three-dimensional surface must vanish.
>
> They make no such claim for energy *per se*.

Can you construct a syllogism?  Energy is one component of the
four-momentum.  Each component of the four-momentum is locally
conserved.  Therefore energy is locally conserved.
(Are you bothered by the fact that energy is frame-dependent?
That's true for any component of a vector, and has no effect
on conservation, so long as the other components are also
conserved so the conservation law will be true in all (inertial)
frames.)

> There is a prevailing opinion that energy is not localizable.
> (See Feynman Volume 1, Chapter 4.)

I don't see anything in Feynman I:4 to support this view at all.
In fact, the language of blocks "going in and out" seems to
imply rather strongly a local viewpoint, where energy travels
from place to place rather than just disappearing in one place
and appearing in another.

> Neither energy nor translational momentum are conserved in general
> in an accelerated frame. Charge and atoms are conserved in such
> frames. The attribution of common fluidical properties to energy
> which apply to charges and atoms is false.

In an accelerated frame, there's not even a *global* law of
conservation of energy.  Nor is Newton's second law true.
Should we stop teaching Newton's second law?  I would prefer
that we stick to (local) inertial frames.

> > Again, GR *requires* local conservation of
> > all (nongravitational) energy.
>
> Well, since you cite MTW as your source for that statement I'm
> afraid you'll have to direct me to the particular passage where
> that statement is made.

Good grief, Leigh, that's like asking for a reference to Maxwell's
equations in Jackson.  It's practically the theme of the whole
book!  Just as Maxwell's equations *require* local conservation
of charge, the Einstein field equation *requires* (via the
Bianchi identity) local conservation of four-momentum (i.e., energy
and momentum).  In MTW, this idea is the basic motivation that leads
to the correct field equation.  The beginning of Chapter 17 is where
the lines of reasoning all come together.  (If you haven't already,
you'll also want to read Chapter 20, where the nonlocalizability
of *gravitational* energy is discussed.)

> The concept of "nongravitational energy"
> could also use a rigorous definition, especially in light of the
> principle of equivalence.

The principle of equivalence is what allows us to define nongravitational
energy.  Go to a locally inertial frame in which there are no fictitious
gravitational forces.  In this frame, measure the energy of all the
stuff that's present (in some small region).  That's the nongravitational
energy.  Divide by the volume of your region and you have
the time-time component of the stress-energy tensor, which is the
right-hand side of the field equation.  Remember, of course, that
if your system is a star or something else big, you have to divide it
into little chunks and carry out this procedure chunk by chunk, because
there's no global inertial frame--only local ones.

> There never arises a *need* to talk about *local* conservation of
> energy in introductory physics courses. To do so gratuitously injects
> the caloric idea into the students' concept store, and I believe it
> would be wrong to do so. Give me any energy flow argument you think
> is obligatory and I will translate it to what I think is a physically
> respectable form which is just as readily assimilable by beginning
> students.

I agree that we *could* teach introductory physics without local
conservation laws (energy or otherwise).  We could also teach E+M
without fields.  I feel that to do either would be a disservice
(though I don't expect to persuade you).  You're right about the
fundamental nonlocalizability of *gravitational* energy, but this
is a rather advanced issue that I would prefer not to raise in an
introductory course.  In any case, gravitational energy is the
exception, not the rule.  And as I said in my previous post,
you can often, in weak-field situations, get away with pretending
that gravitational energy can be localized; for students who haven't
studied GR, I have no qualms about doing so.  The one thing you
*cannot* do is understand GR while ignoring local conservation of
nongravitational energy.

Dan