[Phys-l] power, energy flow, vectors, tensors, et cetera.

[John Denker <jsd@av8n.com>]

On 07/27/2011 03:41 PM, Bernard Cleyet wrote:
> [power meters] run backward when the "power" direction reversed. 
> (Does this make power a (pseudo?) vector?)

That's an interesting question.  It's a good sign when people
ask questions like that.  It means they are paying attention
... paying attention to the fundamentals.

The short answer is no, power is not any kind of vector.  Depending
on how you want to define things, it might be a scalar (not a
Lorentz scalar) or it might be the 0,0 component of a 4x4 tensor,
but there is precisely no chance of it being a vector.

Power is /related/ to the flow of energy, but "related to" is
not the same as "same as".

There is such a thing as energy flow, and (by conservation of energy)
we know that reversing the direction of energy flow does in fact 
reverse the power, but again, that doesn't mean that power and 
energy flow are the same thing.

According to a modern (post-1908) understanding of how the
universe is put together, we should look at such things from
the spacetime point of view.  That leads us to consider the
stress-energy tensor.

We can start with the following, from the highly quotable John Baez:

> > The stress-energy tensor, aka energy-momentum tensor, T_{ab}, where
> > a,b go from 0 to 3. This tells you everything about what energy and
> > momentum are doing at your given point of spacetime, as follows:
> >
> > T_{ab} is the flow in the "a" direction of momentum in the "b"
> > direction!
> >
> > To understand this, remember that a,b=0,1,2,3 correspond to t,x,y,
> > and z respectively. Also, remember that "energy" is the same as
> > "momentum in the time direction", and that "density" is the same as
> > "flow in the time direction". Thus the top row of the stress-energy
> > tensor keeps track of the density of energy --- that's T_{00} --- and
> > the density of momentum in the x,y, and z directions --- those are
> > T_{01}, T_{02}, and T_{03} respectively. On the other hand, T_{10},
> > T_{20}, and T_{30} represent the flow of energy in the x, y and z
> > directions, respectively. The other entries keep track of the flow of
> > spatial momentum in various spatial directions. For example, T_{12}
> > keeps track of the flow in the x direction of momentum in the y
> > direction.

  http://math.ucr.edu/home/baez/gr/old/stress.energy.html

Conservation of energy (and momentum) can be expressed as the
divergence of the stress-energy tensor:

   0 = ∇i Tij		                  [1]

As always, there is an implicit sum over i=0,1,2,3.

The j=0 piece of equation [1] expresses conservation of energy.
The j=1,2,3 pieces express conservation of the various components
of momentum.

The j=0 piece says that the power is equal to -1 times the 3D
(spacelike) divergence of the flow of energy.

On the topic of conservative flow, a more-detailed discussion,
with diagrams, can be found at:
  http://www.av8n.com/physics/conservative-flow.htm

To repeat:

> > "density" is the same as
> > "flow in the time direction".

(That is, stuff just sits at the same x,y,z location and moves in
the t direction at the rate of 60 minutes per hour.)

> > "energy" is the same as
> > "momentum in the time direction",

> > T_{ab} is the flow in the "a" direction of momentum in the "b"
> > direction

The first time I heard those three statements they sounded insane,
and it took me the better part of a year to figure out what they
meant.  OTOH I don't know of any better way to summarize things.
If you want an actual explanation (not just a summary) you can
google for "stress energy tensor" and/or read a book on general
relativity.  I like Misner / Thorne / Wheeler _Gravitation_.


Note that the 3x3 stress tensor is heavily used in fluid dynamics
and mechanical engineering.  Pedagogically speaking, I'm not sure 
what to make of that, because nowadays there are AFAICT more people 
who know a little about relativity than people who know anything 
about fluid dynamics.  So we might as well start with the 4x4
version.

====================

As for "pseudo vectors" aka axial vectors, you don't want to go
there.  Ever.  Most of the things that used to be thought of as
"axial vectors" -- e.g. angular momentum -- are better thought 
of as bivectors.
  http://www.av8n.com/physics/clifford-intro.htm