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Re: energy, work, heat



At 12:51 PM -0700 9/3/99, Daniel Schroeder wrote:

Leigh,

Local conservation of energy and momentum is discussed at length
in Chapter 5 of Misner, Thorne, and Wheeler. It is dynamically
guaranteed by the Einstein field equations. However, the
energy and momentum referred to here does not include gravitational
contributions; in GR it's most natural to give gravity special
treatment and, if possible, not even think about gravitational
energy.

The discussion in Chapter 5 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*. Indeed such claims do
appear in the literature (an AJP paper by Peterson about 1972 comes
to mind), but they are controversial to say the least, which is why
they are publishable in the first place. There is a prevailing
opinion that energy is not localizable. (See Feynman Volume 1,
Chapter 4.)

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.

I repeat my claim that energy is, in general, not locally conserved.
It may appear to be pedagogically helpful to describe physical
phenomena in this way (e.g. "the potential energy of the falling
object is converted to kinetic energy") but in my opinion it is not
helpful to the student to do so.

When we switch to a Newtonian viewpoint and treat gravity as a
force, it is often possible (and convenient) to talk about gravitational
energy and treat it like other forms of energy. So we can talk about
a binary pulsar losing gravitational energy in the form of gravitational
waves. We're in the weak-field limit here and everything is consistent,
to a very good approximation. At a more fundamental level, of course,
the Newtonian viewpoint breaks down and we shouldn't treat gravitational
energy in this way. In that case, please see the previous paragraph.

To suggest that we should *never* talk about local conservation of
energy just because the concept can't be applied to gravitational
energy in all situations is most certainly throwing the baby out
with the bathwater. 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. The concept of "nongravitational energy"
could also use a rigorous definition, especially in light of the
principle of equivalence.

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.

Leigh