Re: the energy

[Hugh Logan <hloganPHY@CFL.RR.COM>]

Leigh Palmer wrote:

> I'm sorry to be so slow in my own discussion. I do appreciate all the
> comments that are being made. They will inform my thinking, I'm sure.
>
> To my note:
>
> > > The orthodox view, which I hold to be correct: The energy is not
> > > substantial; it is not a real entity. The energy is an abstraction.
> > > The
> > > energy did not exist before it was invented. The energy is a state
> > > function, a quantity which may be calculated for any isolated physical
> > > system from the values of all the parameters that characterize its
> > > state.
>
> John Clement replied:
>
> > Yes, but is this how energy should be taught?  The Modeling people
> > firmly
> > come down against this view as one that should be taught in the intro.
> > course.  They model energy as something which can be transferred sort
> > of
> > like a fluid.
There is a nice article reflecting the view of the Modeling people on
the teaching of  energy by Mark Schober at
<http://www.jburroughs.org/science/mschober/physteach/energy03/index.html>
(Then click on "a handout... .") In this article, Mark promotes the idea
of energy flowing as a substance. Quantities are said to be
subtance-like if they have density and flow through space. Such
quantities are supposed to obey conservation laws as in the case of
charge. Their are other abstract quantites in physics that can be
described using the idea of flow and density such as probability density
in quantum mechanics. Texts on quantum mechanics (_Quantum Mechanics_,
Schiff, 3rd ed., p. 26-27 derive an equation analogous to the equation
of continuity in fluid mechanics and for charge in electricity and
magnetism. This equation states

DP(_r_, t)/Dt + div _S(_r_, t)=0,

where "D" denotes partial differentiation, _P(_r_, t) =probability
density=(psi(_r_, t)*)(psi(_r_, t)), and _S_(_r_,t)=probability current
density. As Schiff points out, it would be misleading to reify _S(_r_,
t) as the average particle flux at (_r_, t) as that would be
inconsistent with the uncertainty principle. Still physicists do not
hesitate to speak of the flow of probability, an abstract man-made
mathematical quantity. It is not "stuff" in any physical sense.

I do not see any objection to using the idea of flow of energy on the
grounds that energy is a man-made concept that cannot be reified as a
substance. The question, for me, is whether or not
the idea of energy flow makes the teaching and learning of energy
conservation easier or better.
In cases like the energy of of a system of charged objects or
gravitationally attracting objects, one can
assign an energy density to the electric and gravitational fields,
respectively. If, for example, a system consists of two electrons, the
energy of the electric field of the system would decrease while the
energy of motion of the electrons would increase by the same amount. If
I understand the Modelers correctly, I think they would say that energy
would be transferred from the electric field to the energy of motion of
the electrons. With that I would agree. I think they would also say that
energy flows from
the electric field to the energy associated with the motion of the
electrons. It seems to me that the use of the word "flow" is a little
ambiguous Is it a flow of a substance (or the flow of of something that
behaves analogously to a substance as in the case of probability in QM),
or rather is it a figurative
use of the word "flow" such as used in a "flow chart." The Modelers use
bar graphs to show the different "containers" of the internal energy
within a system before and after an interaction. Between the two graphs
there is an energy transfer diagram or flow diagram showing transfers of
energy into or out of the system through the processes of work, heat, or
radiation. Again, I feel more comfortable with "transfer" than with
"flow." Mark mentions the analogy between energy and money that appears in
Randy Knight's _Physics - A Conteporary Perspective_. I don't have this
book, so I don't know if it uses the idea of energy flow as well.
According to Mark, energy has a universal nature. He prefers to
speak of the transfer of energy from one storage mechanism to another
rather than the transformation of one form of energy to another. It is
analogous to the idea that money is essentially the same thing wherever
it is stored. I like the money transfer analogy better than the substnce
flow idea. The money
analogy is not new for energy, the idea of attaching a value to a a
physical entity in which it can be compared with another entity in
regard to some characteristic. I mentioned previously that Synge and
Griffith used the analogy of money with mass in their book, _Principles
of Mechanics_. One can speak of the flow of money, but the idea of money
flow is not very precise. For example, if I want to transfer money from
my bank account to a store, I can drive to the bank, withdraw the money,
and then take it to the store, or  I can pay the bill with a credit
card, perhaps from my computer. The transfer of money is the same in
either case (negecting gasoline expenses or any service charges which
would correspond to dissipated energy in the analogy), and I might say
that money flows from the bank to the store, but this would be a
figurative use of the word "flow." It seems that the same could be said
for the flow of energy. It puts a strain on my intuition to think of
energy flowing into a system consisting of a spring as a consequence of
work done on it by an external force if the flow is like that of a
substance. However, it does not bother me to think of the energy
transfer diagrams as flow charts in which the word "flow" is not taken
too literally. I don't know what the Piagetians would think about
replacing substance flow with the money analogy.

I think Mark's paper is a very valuable resource in teaching about
energy except for a couple of reservations such as too literal an
interpretation of energy flow as a flow of a substance. Regarding the
universal nature of energy, I don't agree that a distinction should be
made between mass (rest mass) and rest energy. They are essentially the
same according to E_0=mc^2 or E=m in units in which c=1.
E_0 is rest energy, and m is mass (which used to be called "rest mass."
Any energy transferred to a system in its rest frame will increase its
inertial mass by that amount of energy if c=1 (or that amount divided by
c^2 if conventional units are used) no matter what storage mechanism is
used for the energy in the system. (Einstein is explicit about this on
pp. 46-47 of _Relativity, The Special and the General Theory_, 15th ed.)
In that sense, energy has inertia because it is the same as inertial
mass. By the principle of equivalence, inertial mass implies a
proportional gravitational mass -- equal if the same standard is used
for each. Mass is something like a common measure of energy no matter
what the storage mechanism. Unfortunately, the change of mass with
change in rest energy cannot be measured except in a thought experiment,
since the change is too small to be measurable.

The Modelers (following Dr. Arnold Arons) emphasize enlarging the system
to avoid using the work-energy theorem, which inappropriately applied,
leads to incorrect results -- for example in the treatment of a block
regarded as the system being pushed across the floor at constant speed.
If one assumed that the frictional force lust balanced the force pushing
the block, the work done by the net external force would be zero, and
there would be nothing to account for the increased temperature of the
block. Their method of treating the problem would be to extend the
system to include the surface of the floor as well as the block.Their
solution shows the work done by the external force pushing the block
increasing the thermal (dissipated) part of the internal energy, which
eventually leaves the system
by the process of (negative) heating Q across the boundary in the
context of the first law of thermodynamics.

One thing I noticed was that the internal energy of their extended
systems was not limited to microscopic energies more typical of
thermodynamics. For example it might include the kinetic energy
of a a ball released from a spring-loaded gun as well as the
gravitational potential energy.

The idea of non-microscopic internal energy occurred to me in connection
with an isolated system in special relativity. Imagine such a system the
size of a room isolated from any transfer of energy across the system
boundaries by working, heating, or radiation. The system could include
the earth as well to allow for gravitational potential energy as part of
the energy. Imagine two baseball players playing catch inside the room.
The chemical energy of one player would be transferred to kinetic energy
as he throws the ball. There would be some transfer of kinetic energy to
gravitational potential energy and vice-versa as the ball travels to the
other player, and the kinetic energy would be dissipated as the ball was
brought to rest. Throughout all this, the internal energy of the system
would be constant, and hence its rest mass would be constant -- at least
according to the theory, since measurement wouldn't be practical. The
constituents of the internal energy (rest energy) of the the entire
system would change in such a way that their sum E_0, and hence its rest
mass  M remained constant.  If the room were placed on a train that
could travel at a relativistic speed,  the total energy of the room and
its contents would be  gamma*M  in units with c=1. Still the  (rest)
mass and rest energy of the train would be conserved, at least if the
train moved uniformly.   (I seem to recall that  the  total energy, as
seen by an inertial observer outside the train would be gamma*M whether
or not the  the train was moving uniformly -- as would be the case for a
charged particle in a circular particle accelerator. I am assuming that
the gravitational potential energy doesn't change as the train moves.)
The point of this was to be more explicit about the assertion that
mass-energy is conserved in special relativity by means of an example of
a more complex system.

On 19-Oct-04 Michael D. Edmiston wrote:

> And what exactly does E=mc2 mean?  Doesn't it mean mass and energy are
> equivalent?


No, it does not. It is one term (which must be suitably qualified)
in a function that quantifies the energy of an isolated physical
system. The terms in this function have been fashioned over a long
period of time by several contributors. This one, of course, is due
to Einstein. In its most useful form it should probably have a
nought subscript appended to the "m", denoting the rest energy.
Otherwise it is a trivial statement about the energy of any
isolated physical system, and no other terms are even necessary.

Leigh

Although I have used m_0 to denote rest mass since m=gamma*m_0, now out
of fashion, was "relativistic mass." However, I think most recent texts
use "m" for mass -- what used to be called rest mass. I am resigned to
the fact that I shouldn't use "relativistic mass." So "m" is mass. It is
the invariant norm of the momentum-energy 4-vector (E, p_x, p_y, p_z)
where E=gamma*m in units where c=1.
Einstein used "m" for (rest) mass in both books, _Relativity, The
Special and the General Theory_ and in the more advanced _The Meaning of
Relativity_. In the latter, he writes "E_0=m*c^2" on p.46. Taylor and
Wheeler also use "m" for mass in _Spacetime Physics_. They no longer
call it "rest mass" as in earlier editions, because students might think
it changes with speed. There is a lengthy discussion of this among the
references posted by Antti Savinainen (Item #5 in particular.)

I am not sure what Leigh means when he says that E=mc^2 doesn't mean
that energy and mass are equivalent. I have thought that E_0=m*c^2 means
that rest energy and (rest) mass are equivalent.
However, if one means relativistic mass by "m," then I would think that
E=m*c^2 means that total energy and relativistic mass are equivalent.
This formulation is not favored, but it can be argued that it is not
wrong. In texts like Taylor and Wheeler, a system of units is used in
which c=1, length and time are in the same units, as are mass, energy
and momentum. Even so, momentum can't be the same as mass or energy
because 3-momentum is a vector and mass and energy are scalars.
However,  mass and rest energy are equivalent. One can't change without
the other changing in the same proportion, and they are equal in units
where c is a dimensionless 1.

As I see it, E_0=m is the conserved quantity in an isolated physical
system. I see E_0 as a term including the terms previously developed
such as those for electrostatic and gravitational potential energy, and
also kinetic energy. It occurred to me that if the speeds of particles
within a system were relativistic,  then the contribution of each such
particle within the system would have to be  the relativistic value --
like a box within a box.

Hugh Logan.

In writing the first part of this message, I referred to the paper
"Making Work Work" by Gregg Swackhamer on the ASU Modeling Instruction
web site in addition to the paper by Mark Schober.
Swackhamer's paper is along the same lines, but it emphasizes energy
stored in a field. The word "flow" is used in connection with energy
transfer. There are references to papers that are not readily available,
at least here.