Can someone please explain or provide some links which explain how the
physics of a non-symmetric energy tensor would be different from that of
a symmetric energy tensor.

Especially, with the Poynting components

T^0k T^k0, k=1,2,3,

how would one interpret T^0k versus T^k0 and the physics of the energy
flux associated with each?

Thanks,

Jay,
____________________________
Jay R. Yablon
Email:
co-moderator: sci.physics.foundations
Weblog: http://jayryablon.wordpress.com/
Web Site: http://home.nycap.rr.com/jry/FermionMass.htm

Jay R. Yablon wrote:
Can someone please explain or provide some links which explain how the
physics of a non-symmetric energy tensor would be different from that of
a symmetric energy tensor.

A non-symmetric energy tensor makes precisely the same amount of sense
as a non-symmetric metric tensor. Remember that the energy tensor is
defined as the variational derivative of the (scalar) Lagrangian with
respect to the metric.

While some people think a non-symmetric metric makes sense, no useful
results have resulted from such metrics. C.f. Einstein's long and
fruitless efforts at a "unified field theory", in which non-symmetric
metrics played a major role. From a purely geometrical standpoint, a
non-symmetric metric makes no sense (remember that the essence of the
metric is to describe the geometry).


Tom Roberts

Jay R. Yablon wrote on Fri, 21 Mar 2008 10:47:44 -0400:

Hi, Jay

Can someone please explain or provide some links which explain how the
physics of a non-symmetric energy tensor would be different from that of
a symmetric energy tensor.

Matter spin and geometric torsion

http://en.wikipedia.org/wiki/Spin_tensor

http://en.wikipedia.org/wiki/Stress-energy_tensor


Especially, with the Poynting components

T^0k T^k0, k=1,2,3,

how would one interpret T^0k versus T^k0 and the physics of the energy
flux associated with each?

Thanks,

Jay,
____________________________
Jay R. Yablon
Email:
co-moderator: sci.physics.foundations Weblog:
http://jayryablon.wordpress.com/ Web Site:
http://home.nycap.rr.com/jry/FermionMass.htm



--
http://canonicalscience.org/en/misce...guidelines.txt

"Tom Roberts" wrote in message
et...
Jay R. Yablon wrote:
Can someone please explain or provide some links which explain how
the
physics of a non-symmetric energy tensor would be different from that
of
a symmetric energy tensor.

A non-symmetric energy tensor makes precisely the same amount of sense
as a non-symmetric metric tensor. Remember that the energy tensor is
defined as the variational derivative of the (scalar) Lagrangian with
respect to the metric.

While some people think a non-symmetric metric makes sense, no useful
results have resulted from such metrics. C.f. Einstein's long and
fruitless efforts at a "unified field theory", in which non-symmetric
metrics played a major role. From a purely geometrical standpoint, a
non-symmetric metric makes no sense (remember that the essence of the
metric is to describe the geometry).


Tom Roberts

Tom, what about the links that Juan posted after your post, at:

http://en.wikipedia.org/wiki/Spin_tensor

http://en.wikipedia.org/wiki/Stress-energy_tensor

which shows how the a non-symmetric energy tensor T is related to the
spin tensor S according to (&=partial derivative):

&_uS^abu = T^ba-T^ab 0 ?

Jay.

Tom Roberts says...

Jay R. Yablon wrote:
Can someone please explain or provide some links which explain how the
physics of a non-symmetric energy tensor would be different from that of
a symmetric energy tensor.

A non-symmetric energy tensor makes precisely the same amount of sense
as a non-symmetric metric tensor. Remember that the energy tensor is
defined as the variational derivative of the (scalar) Lagrangian with
respect to the metric.

I'm not sure that I agree with that. According to Wikipedia,
http://en.wikipedia.org/wiki/Stress-energy_tensor,
asymmetry in the stress-energy tensor is to expected
if there is a nonzero spin density.

Classical General Relativity makes the assumption that the total
angular momentum in a small region of space goes to zero as the
volume of the space goes to zero, but that doesn't hold if there
are point-particles with intrinsic spin.

--
Daryl McCullough
Ithaca, NY

On Mar 21, 7:55*am, Tom Roberts wrote:
Jay R. Yablon wrote:
Can someone please explain or provide some links which explain how the
physics of a non-symmetric energy tensor would be different from that of
a symmetric energy tensor.

A non-symmetric energy tensor makes precisely the same amount of sense
as a non-symmetric metric tensor. Remember that the energy tensor is
defined as the variational derivative of the (scalar) Lagrangian with
respect to the metric.

There is an implicit assumption made about some of the components made
when doing the variation that squish out torsion.


While some people think a non-symmetric metric makes sense, no useful
results have resulted from such metrics. C.f. Einstein's long and
fruitless efforts at a "unified field theory", in which non-symmetric
metrics played a major role. From a purely geometrical standpoint, a
non-symmetric metric makes no sense (remember that the essence of the
metric is to describe the geometry).

Tom Roberts

On Mar 21, 5:55*pm, (Daryl McCullough)
wrote:
Tom Roberts says...



Jay R. Yablon wrote:
Can someone please explain or provide some links which explain how the
physics of a non-symmetric energy tensor would be different from that of
a symmetric energy tensor.

A non-symmetric energy tensor makes precisely the same amount of sense
as a non-symmetric metric tensor. Remember that the energy tensor is
defined as the variational derivative of the (scalar) Lagrangian with
respect to the metric.

I'm not sure that I agree with that. According to Wikipedia,http://en.wikipedia.org/wiki/Stress-energy_tensor,
asymmetry in the stress-energy tensor is to expected
if there is a nonzero spin density.

I was just discussing something related with Eric Gisse.

Our discussion got derailed early by some semantics involving a spin
fluid, which was, in some contexts, attached to an important sounding
Name which escapes me just now.

The problems seems to be (although Eric never agreed with me) that,
classically, we can envisage a fluid having distributed (intensive,
specific, a 3-volume density... etc/) intrinsic angular momentum.
This classical idea is sometimes called, with justice, a "spin fluid",
as in "its wee little bits are spinning round 'n round" without
reference to quantum spin. A physical realization of such a fluid
however might be a fluid _with_ a significant density of unpaired
quantum spins.

We never could get past the idea that each time I wrote "spin" I must
be mistakenly invoking the quantum concept in an otherwise classical
discussion, no matter how strenuously I asserted and reasserted the
seemingly transparent gloss above.

Anyway, as you say, one of the tentative conclusions seems to be that
a metric with torsion, in some expanded thing called Einstein/Cartan
theory, may take as a source term a spin (classical intensive angular
momentum density!!!) density -- which (ad nauseum here) might be
physically conceived by a little semi-classical extension, as a
classical universe with some otherwise unpaired electrons wondering
about.

This idea also seems to be intimately related to what might form a
source term for Weyl metric, which, also seemingly having something to
do with rotating stuff, seems a little shy about living with a source.

I also had a cockamamie half-baked idea of the kind I normally get
when I learn 0.5 % about some complex topic, that the Weyl solution,
while possibly a self-consistent vacuum solution to the orindary
Einstein equations, must have some new kind of source, because the
conventional stress energy tensor -- even _without_ considering the
exotica of possible (classical!!!) spin fluids and intensive angular
momentum, seems to have a real problem with even a spinning massive
ball: it can't represent it as spinning one way vs. the other (stress
is symmetric under this transformation).

Now it's appealing to think that (classical!!!) spin fluid to the
rescue, and that _it_ is going to be represented by an antisymmetric
stress-energy tensor, thank you very much, but I share some of Tom
Roberts' doubts (though of course none of his standing). To wit,
purely classically classically (as opposed to relativistically
classically), "intrinsic angular momentum" doesn't seem to make much
sense for massive objects -- it requires an angular velocity which
blows up at fine enough levels.

Since then I have thought of at least three outs: the most obvious one
(perhaps) -- the kids are already jumping out of their seats singing
"I know!" "I know!", is to let a wee bit o' quantum into our classical
theory, via (this time quantum) spin, which doesn't have to obey no
stinkin' classical rules. The second is to consider a swarm of
pebbles each spinning (classically!) about their own axes, and then to
look at the swarm on a scale where the pebbles are unresolved. You
know have this mysterious concentration of intrinsic angular momentum
out there, with no obvious source, which must be accounted for as
"distributed intensive specifice and etc." angular momentum.

Finally, I thought of cirularly polarized light, which carries
intrinsic angular momentum: when you move to classical fields, you
don't have far to look. But is cicularly polarized light described by
an anti-symmetric stress tensor? I'm not sure. Maybe the formalism
is better (necessarily?) to just introduce another vector density --
the density of intrinsic angular momentum.

Hmm... now, if ordinary stress is the flux of momentum vector density,
what is the flux of _angular_ momentum (axial) vector desnity? It's
something the same rank as the ordinary stress tensor: perhaps it is
the antisymmetric part thereof? Hmmm.... damn it all.

Classical General Relativity makes the assumption that the total
angular momentum in a small region of space goes to zero as the
volume of the space goes to zero, but that doesn't hold if there
are point-particles with intrinsic spin.

Or if "as the volume of the space goes to zero" on our scale of
analysis, there are unresolved (classically!!!) spinning bits of
matter -- or perhaps if there are the right kinds of EM fields?

Enlighten me with your enlightened enlightenedment in a sea of inky
darkness, Daryl.

"Edward Green" wrote in message
...
.. . .

Finally, I thought of cirularly polarized light, which carries
intrinsic angular momentum: when you move to classical fields, you
don't have far to look. But is cicularly polarized light described by
an anti-symmetric stress tensor? I'm not sure. Maybe the formalism
is better (necessarily?) to just introduce another vector density --
the density of intrinsic angular momentum.

[start Jay]

I thought about exactly the same question. Is there a covariant
formulation for circularly polarized light this essentially adds one or
more terms to the Maxwell tensor and renders it non-symmetric? If so,
where might I find it?

Jay.

[end Jay]

Daryl McCullough wrote on Fri, 21 Mar 2008 14:55:43 -0700:

Tom Roberts says...

Jay R. Yablon wrote:
Can someone please explain or provide some links which explain how the
physics of a non-symmetric energy tensor would be different from that
of a symmetric energy tensor.

A non-symmetric energy tensor makes precisely the same amount of sense
as a non-symmetric metric tensor. Remember that the energy tensor is
defined as the variational derivative of the (scalar) Lagrangian with
respect to the metric.

I'm not sure that I agree with that. According to Wikipedia,
http://en.wikipedia.org/wiki/Stress-energy_tensor, asymmetry in the
stress-energy tensor is to expected if there is a nonzero spin density.

There exists undergrad textbooks explaining that. E.g.

Gravitación. 2005. Editorial URSS. Ivanenko, Dmitri DmÃ*trievich;
Sardanashvili, Guenadi Alexándrovich.

Classical General Relativity makes the assumption that the total angular
momentum in a small region of space goes to zero as the volume of the
space goes to zero, but that doesn't hold if there are point-particles
with intrinsic spin.

One of conditions of GR *connections* is precisely g_ab = g_ba, which
does not hold for particles rotating around.


--
http://canonicalscience.org/en/misce...guidelines.txt

On Mar 22, 7:33*am, "Juan R." González-Álvarez
wrote:

One of conditions of GR *connections* is precisely g_ab = g_ba, which

Let me shoot for the moon and hit my foot.

Can we describe the meaning of "g_ab" in words?

E.g., it is the number which tells us, in terms of a coordinate x^a
and a coordinate x^b, exactly what happens to first order when we....

First bonus question: what would it mean, operationally if g_ab /=
g_ba ?

Second bonus question: rephrase both answers for the remaining
permutations of raised and lowered indices.

Extra-extra credit: I am handed a ruler, and told that it measures a
coordinate x. How can I tell, on operational grounds, whether I am
measuring a coordinate with a raised or lowered index?