| Topic: |
Science > Physics |
| User: |
"Jason Pawloski" |
| Date: |
14 Dec 2006 04:51:15 PM |
| Object: |
Jackson Question |
I was trolling aimlessly through Jackson yesterday and I found a
question that piqued my curiosity, because I had never noticed this
before. Its problem 4.3:
The lth term in the multipole expansion of the potential is specified
by the (2l + 1) multipole moments q_lm. On the other hand, the
Cartesian multipole moments
Q_{alpha beta gamme}^(l) = int p(x) x^alpha y^beta z^gamma d^x
with alpha, beta and gamme nonnegative integers subject to the
constraint alpha + beta + gamma = l, are (l + 1)(l + 2)/2 in number.
Thus for l>1, there are more Cartesian multipole moments than seem
necessary to describe the term in the potential whose radial dependence
is r^(-l-1).
Show that while the q_lm transform under rotations as irreducible
spherical tensors of rank l, the Cartesian multipole moments correspond
to reducible spherical tensors of ranks l, l-2, l-4, ... l_min, where
l_min = 0 or 1 for l even or odd respectively. Check that the number of
different tensorial components adds up to the total number of Cartesian
tensors. (et cetera)
I'll be the first to admit my Jackson E&M class was very shoddy and
didn't follow the book at all, and this question does not seem
self-contained as I can find no place Jackson where describes what
(ir)reducible tensors are, or spherical tensors, and I don't see an
easy generalization of what I know the adjective irreducible can mean
(irreducible polynomial, for instance).
So can someone explain to me what the significance is with all of these
tensor things and why there are more Cartesian multipole moments?
.
|
|
| User: "Sorcerer" |
|
| Title: Re: Jackson Question |
14 Dec 2006 08:33:19 PM |
|
|
"Jason Pawloski" <jpawloski@gmail.com> wrote in message =
news:1166136675.516133.65330@t46g2000cwa.googlegroups.com...
|I was trolling aimlessly through Jackson yesterday and I found a
| question that piqued my curiosity, because I had never noticed this
| before. Its problem 4.3:
|=20
| The lth term in the multipole expansion of the potential is specified
| by the (2l + 1) multipole moments q_lm. On the other hand, the
| Cartesian multipole moments
|=20
| Q_{alpha beta gamme}^(l) =3D int p(x) x^alpha y^beta z^gamma d^x
|=20
| with alpha, beta and gamme nonnegative integers subject to the
| constraint alpha + beta + gamma =3D l, are (l + 1)(l + 2)/2 in number.
| Thus for l>1, there are more Cartesian multipole moments than seem
| necessary to describe the term in the potential whose radial =
dependence
| is r^(-l-1).
|=20
| Show that while the q_lm transform under rotations as irreducible
| spherical tensors of rank l, the Cartesian multipole moments =
correspond
| to reducible spherical tensors of ranks l, l-2, l-4, ... l_min, where
| l_min =3D 0 or 1 for l even or odd respectively. Check that the number =
of
| different tensorial components adds up to the total number of =
Cartesian
| tensors. (et cetera)
|=20
| I'll be the first to admit my Jackson E&M class was very shoddy and
| didn't follow the book at all, and this question does not seem
| self-contained as I can find no place Jackson where describes what
| (ir)reducible tensors are, or spherical tensors, and I don't see an
| easy generalization of what I know the adjective irreducible can mean
| (irreducible polynomial, for instance).
|=20
| So can someone explain to me what the significance is with all of =
these
| tensor things and why there are more Cartesian multipole moments?
Sure.=20
Divisions by zero, singular matrices. The rest you can figure
out for yourself (or not).
.
|
|
|
|
| User: "Sue..." |
|
| Title: Re: Jackson Question |
15 Dec 2006 02:08:25 AM |
|
|
Jason Pawloski wrote:
I was trolling aimlessly through Jackson yesterday and I found a
question that piqued my curiosity, because I had never noticed this
before. Its problem 4.3:
The lth term in the multipole expansion of the potential is specified
by the (2l + 1) multipole moments q_lm. On the other hand, the
Cartesian multipole moments
Q_{alpha beta gamme}^(l) = int p(x) x^alpha y^beta z^gamma d^x
with alpha, beta and gamme nonnegative integers subject to the
constraint alpha + beta + gamma = l, are (l + 1)(l + 2)/2 in number.
Thus for l>1, there are more Cartesian multipole moments than seem
necessary to describe the term in the potential whose radial dependence
is r^(-l-1).
Show that while the q_lm transform under rotations as irreducible
spherical tensors of rank l, the Cartesian multipole moments correspond
to reducible spherical tensors of ranks l, l-2, l-4, ... l_min, where
l_min = 0 or 1 for l even or odd respectively. Check that the number of
different tensorial components adds up to the total number of Cartesian
tensors. (et cetera)
I'll be the first to admit my Jackson E&M class was very shoddy and
didn't follow the book at all, and this question does not seem
self-contained as I can find no place Jackson where describes what
(ir)reducible tensors are, or spherical tensors, and I don't see an
easy generalization of what I know the adjective irreducible can mean
(irreducible polynomial, for instance).
So can someone explain to me what the significance is with all of these
tensor things and why there are more Cartesian multipole moments?
<< The Lorenz gauge is incomplete in the sense that there
remains a subspace of gauge transformations which preserve
the constraint. These remaining degrees of freedom correspond
to gauge functions which satisfy the wave equation... >>
http://en.wikipedia.org/wiki/%CE%9E_gauge
<<A Lorentz transformation or any other coordinate
transformation will convert electric or magnetic
fields into mixtures of electric and magnetic fields,
but no transformation mixes them with the
gravitational field. >>
http://www.aip.org/pt/vol-58/iss-11/p31.html
http://www.research.ibm.com/grape/grape_ewald.htm
http://rc.uits.iu.edu/rats/research/grapes/grapes.shtml
Sue...
http://farside.ph.utexas.edu/teaching.html
http://web.mit.edu/8.02t/www/802TEAL3D/visualizations/light/index.htm
.
|
|
|
| User: "Sorcerer" |
|
| Title: Re: Jackson Question |
15 Dec 2006 02:37:41 AM |
|
|
"Sue..." <suzysewnshow@yahoo.com.au> wrote in message =
news:1166170105.568325.151050@f1g2000cwa.googlegroups.com...
[...]
|=20
| Sue...
I've never seen an aether, have you?
|
.
|
|
|
| User: "Sue..." |
|
| Title: Re: Jackson Question |
15 Dec 2006 02:43:52 AM |
|
|
Sorcerer wrote:
"Sue..." <suzysewnshow@yahoo.com.au> wrote in message news:1166170105.568325.151050@f1g2000cwa.googlegroups.com...
[...]
|
| Sue...
I've never seen an aether, have you?
Not in this thread I haven't
Wikipedia:Requests for comment/Der alte Hexenmeister
http://en.wikipedia.org/wiki/Wikipedia:Requests_for_comment/Der_alte_Hexenmeister
Try:
Propagation in a dielectric medium
http://farside.ph.utexas.edu/teaching/em/lectures/node98.html
http://en.wikipedia.org/wiki/Wave_impedance
http://en.wikipedia.org/wiki/Free_space
http://www-ssg.sr.unh.edu/ism/what.html
http://farside.ph.utexas.edu/teaching.html
http://web.mit.edu/8.02t/www/802TEAL3D/visualizations/light/index.htm
Sue...
|
.
|
|
|
| User: "Sorcerer" |
|
| Title: Re: Jackson Question |
15 Dec 2006 03:18:09 AM |
|
|
"Sue..." <suzysewnshow@yahoo.com.au> wrote in message =
news:1166172232.193592.30020@16g2000cwy.googlegroups.com...
|=20
| Sorcerer wrote:
| > "Sue..." <suzysewnshow@yahoo.com.au> wrote in message =
news:1166170105.568325.151050@f1g2000cwa.googlegroups.com...
| > [...]
| > |
| > | Sue...
| >
| > I've never seen an aether, have you?
|=20
| Not in this thread I haven't
Yes, but have you EVER seen an aether anywhere?=20
I saw a photon, you pointed it out to me, smarty pants.
.
|
|
|
|
|
|
| User: "Edward Green" |
|
| Title: Re: Jackson Question |
15 Dec 2006 07:10:34 AM |
|
|
Sue... wrote:
<<A Lorentz transformation or any other coordinate
transformation will convert electric or magnetic
fields into mixtures of electric and magnetic fields,
but no transformation mixes them with the
gravitational field. >>
http://www.aip.org/pt/vol-58/iss-11/p31.html
The links you post are invariably interesting, but none, so far as I
can see, are remotely near addressing the question that was asked.
.
|
|
|
| User: "Sue..." |
|
| Title: Re: Jackson Question |
15 Dec 2006 07:30:06 AM |
|
|
Edward Green wrote:
Sue... wrote:
<<A Lorentz transformation or any other coordinate
transformation will convert electric or magnetic
fields into mixtures of electric and magnetic fields,
but no transformation mixes them with the
gravitational field. >>
http://www.aip.org/pt/vol-58/iss-11/p31.html
The links you post are invariably interesting, but none, so far as I
can see, are remotely near addressing the question that was asked.
That is because you seldom bother to learn the physics
that motivates the development of any particular maths
discpline.
To be critical about how formalism represents phenomena,
you need several several points of view. Not just rote
manipulation of symbols.
I won't elaborate for this thread, because the original post
is not included in your reply.
This will give you some relevant papers.
http://www.google.com/search?q=%22j.+d.+jackson%22+inertia&hl=en&lr=&safe=off&start=10&sa=N
Sue...
.
|
|
|
| User: "Edward Green" |
|
| Title: Re: Jackson Question |
15 Dec 2006 08:10:16 AM |
|
|
Sue... wrote:
Edward Green wrote:
Sue... wrote:
<<A Lorentz transformation or any other coordinate
transformation will convert electric or magnetic
fields into mixtures of electric and magnetic fields,
but no transformation mixes them with the
gravitational field. >>
http://www.aip.org/pt/vol-58/iss-11/p31.html
The links you post are invariably interesting, but none, so far as I
can see, are remotely near addressing the question that was asked.
That is because you seldom bother to learn the physics
that motivates the development of any particular maths
discpline.
That's a pretty bold statement, considering how little you know about
me. A little bizarre.
To be critical about how formalism represents phenomena,
you need several several points of view. Not just rote
manipulation of symbols.
Fine mom and apple pie sentiment. Now, the question involved some
aspects of tensorial definition and manipulation in reference to
multipoles, if I understand it correctly. Just to give a "different
point of view", you cite an essay by Steven Weinberg on Einstein's
errors (should I euphemize that phrase?), discussing, among other
things, the cosmological constant, attempts to generalize GR to handle
other forces, and well known dissensions regarding quantum mechanics.
The quote included related to an attempt on the second front.
Suggesting that the answer will be found there is like fielding a
question on real estate law by handing the person an article on legal
philosophy, and suggesting he may be able to work things out on general
principles -- but here's a hint.
You share memetic genes with Al Schwartz. You cite material which may
be vaguely related to the question -- in so much as tree shrews bear
some relation to amoebas -- and imply that the clever reader will be
able to follow your profound lead. You're like Al without the social
rants and insults, or Jeff Relf plus citations.
Still, you pick excellent irrelevant essays.
.
|
|
|
| User: "Sue..." |
|
| Title: Re: Jackson Question |
15 Dec 2006 08:19:54 AM |
|
|
Edward Green wrote:
Sue... wrote:
Edward Green wrote:
Sue... wrote:
<<A Lorentz transformation or any other coordinate
transformation will convert electric or magnetic
fields into mixtures of electric and magnetic fields,
but no transformation mixes them with the
gravitational field. >>
http://www.aip.org/pt/vol-58/iss-11/p31.html
The links you post are invariably interesting, but none, so far as I
can see, are remotely near addressing the question that was asked.
That is because you seldom bother to learn the physics
that motivates the development of any particular maths
discpline.
That's a pretty bold statement, considering how little you know about
me. A little bizarre.
To be critical about how formalism represents phenomena,
you need several several points of view. Not just rote
manipulation of symbols.
Fine mom and apple pie sentiment. Now, the question involved some
aspects of tensorial definition and manipulation in reference to
multipoles, if I understand it correctly.
You understand only a piece of it then.
Just to give a "different
point of view", you cite an essay by Steven Weinberg on Einstein's
errors (should I euphemize that phrase?), discussing, among other
things, the cosmological constant, attempts to generalize GR to handle
other forces, and well known dissensions regarding quantum mechanics.
The quote included related to an attempt on the second front.
Suggesting that the answer will be found there is like fielding a
question on real estate law by handing the person an article on legal
philosophy, and suggesting he may be able to work things out on general
principles -- but here's a hint.
You share memetic genes with Al Schwartz. You cite material which may
be vaguely related to the question -- in so much as tree shrews bear
some relation to amoebas -- and imply that the clever reader will be
able to follow your profound lead. You're like Al without the social
rants and insults, or Jeff Relf plus citations.
Still, you pick excellent irrelevant essays.
If you don't like my references then try Timo's:
M.E. Rose, Elementary theory of angular momentum, Dover, gives a
reasonably thorough coverage of irreducible tensors and their point.
I have no doubt it too will still say:
<<A Lorentz transformation or any other coordinate
transformation will convert electric or magnetic
fields into mixtures of electric and magnetic fields,
but no transformation mixes them with the
gravitational field. >>
http://www.aip.org/pt/vol-58/iss-11/p31.html
Sue...
.
|
|
|
| User: "Edward Green" |
|
| Title: Re: Jackson Question |
15 Dec 2006 08:47:49 AM |
|
|
Sue... wrote:
Edward Green wrote:
Sue... wrote:
<=2E..>
To be critical about how formalism represents phenomena,
you need several several points of view. Not just rote
manipulation of symbols.
Fine mom and apple pie sentiment. Now, the question involved some
aspects of tensorial definition and manipulation in reference to
multipoles, if I understand it correctly.
You understand only a piece of it then.
No doubt... since I have not worked through the problem, and have so
far been spared the aching general insights that await those who have.
At issue is not my understanding, but yours: you implicitly allege
profound understanding, which you drop tantalizing hints of, but I have
my doubts there is anything behind the curtain.
<=2E..>
If you don't like my references then try Timo's:
M.E. Rose, Elementary theory of angular momentum, Dover, gives a
reasonably thorough coverage of irreducible tensors and their point.
I have no doubt it too will still say:
<<A Lorentz transformation or any other coordinate
transformation will convert electric or magnetic
fields into mixtures of electric and magnetic fields,
but no transformation mixes them with the
gravitational field. >>
http://www.aip.org/pt/vol-58/iss-11/p31.html
Let's put that in context (I'll steal your quoting convention):
<<Even before Klein's work, Einstein had started on a different
approach, based on a simple bit of counting. If you give up the
condition that the 4 =D7 4 metric tensor should be symmetric, then it
will have 16 rather than 10 independent components, and the extra 6
components will have the right properties to be identified with the
electric and magnetic fields. Equivalently, one can assume that the
metric is complex, but Hermitian. The trouble with this idea, as
Einstein became painfully aware, is that there really is nothing in it
that ties the 6 components of the electric and magnetic fields to the
10 components of the ordinary metric tensor that describes gravitation,
other than that one is using the same letter of the alphabet for all
these fields. **A Lorentz transformation or any other coordinate
transformation will convert electric or magnetic fields into mixtures
of electric and magnetic fields, but no transformation mixes them with
the gravitational field.** This purely formal approach, unlike the
Kaluza-Klein idea, has left no significant trace in current research.
The faith in mathematics as a source of physical inspiration, which had
served Einstein so well in his development of general relativity, was
now betraying him.>>
<emphasis added>
Actually... I was wrong. This quote is not about ideas for extending
GR via extra dimensions; it's about extending GR by dropping a
constraint on the existing metric tensor. This attempt suffers from
a defect, Weinberg says, that the electromagnetic and gravitational
components don't talk to each other -- this is the quote. I've found
you out: I've actually looked at your quote in context. You can't
explain how this is relevant to the question, though you will no doubt
drops continued hints of a profound connection.
The quoted passage however _is_ fascinating -- it's one of those "too
profound a conincidence to be altogether a coincidence, even if we
don't altogether understand what it means" things. Like the
relativisitically correct behavior of deBroglie waves. You drop a
constraint from GR, and you get EM riding along for free, as an
inevitable consequence? Wow.
You are well read in web-science, and have excellent taste in quotes.
But instead of suggesting you are answering questions, why not just
call them "Sue's web picks"? Or just present your choices as topics
for discussion -- they are all good.
.
|
|
|
|
| User: "Edward Green" |
|
| Title: Re: Jackson Question |
15 Dec 2006 08:49:20 AM |
|
|
Sue... wrote:
Edward Green wrote:
Sue... wrote:
<=2E..>
To be critical about how formalism represents phenomena,
you need several several points of view. Not just rote
manipulation of symbols.
Fine mom and apple pie sentiment. Now, the question involved some
aspects of tensorial definition and manipulation in reference to
multipoles, if I understand it correctly.
You understand only a piece of it then.
No doubt... since I have not worked through the problem, and have so
far been spared the aching general insights that await those who have.
At issue is not my understanding, but yours: you implicitly allege
profound understanding, which you drop tantalizing hints of, but I have
my doubts there is anything behind the curtain.
<=2E..>
If you don't like my references then try Timo's:
M.E. Rose, Elementary theory of angular momentum, Dover, gives a
reasonably thorough coverage of irreducible tensors and their point.
I have no doubt it too will still say:
<<A Lorentz transformation or any other coordinate
transformation will convert electric or magnetic
fields into mixtures of electric and magnetic fields,
but no transformation mixes them with the
gravitational field. >>
http://www.aip.org/pt/vol-58/iss-11/p31.html
Let's put that in context (I'll steal your quoting convention):
<<Even before Klein's work, Einstein had started on a different
approach, based on a simple bit of counting. If you give up the
condition that the 4 =D7 4 metric tensor should be symmetric, then it
will have 16 rather than 10 independent components, and the extra 6
components will have the right properties to be identified with the
electric and magnetic fields. Equivalently, one can assume that the
metric is complex, but Hermitian. The trouble with this idea, as
Einstein became painfully aware, is that there really is nothing in it
that ties the 6 components of the electric and magnetic fields to the
10 components of the ordinary metric tensor that describes gravitation,
other than that one is using the same letter of the alphabet for all
these fields. **A Lorentz transformation or any other coordinate
transformation will convert electric or magnetic fields into mixtures
of electric and magnetic fields, but no transformation mixes them with
the gravitational field.** This purely formal approach, unlike the
Kaluza-Klein idea, has left no significant trace in current research.
The faith in mathematics as a source of physical inspiration, which had
served Einstein so well in his development of general relativity, was
now betraying him.>>
<emphasis added>
Actually... I was wrong. This quote is not about ideas for extending
GR via extra dimensions; it's about extending GR by dropping a
constraint on the existing metric tensor. This attempt suffers from
a defect, Weinberg says, that the electromagnetic and gravitational
components don't talk to each other -- this is the quote. I've found
you out: I've actually looked at your quote in context. You can't
explain how this is relevant to the question, though you will no doubt
drops continued hints of a profound connection.
The quoted passage however _is_ fascinating -- it's one of those "too
profound a conincidence to be altogether a coincidence, even if we
don't altogether understand what it means" things. Like the
relativisitically correct behavior of deBroglie waves. You drop a
constraint from GR, and you get EM riding along for free, as an
inevitable consequence? Wow.
You are well read in web-science, and have excellent taste in quotes.
But instead of suggesting you are answering questions, why not just
call them "Sue's web picks"? Or just present your choices as topics
for discussion -- they are all good.
.
|
|
|
| User: "Sue..." |
|
| Title: Re: Jackson Question |
15 Dec 2006 09:28:26 AM |
|
|
Edward Green wrote:
Sue... wrote:
Edward Green wrote:
Sue... wrote:
<...>
To be critical about how formalism represents phenomena,
you need several several points of view. Not just rote
manipulation of symbols.
Fine mom and apple pie sentiment. Now, the question involved some
aspects of tensorial definition and manipulation in reference to
multipoles, if I understand it correctly.
You understand only a piece of it then.
No doubt... since I have not worked through the problem, and have so
far been spared the aching general insights that await those who have.
At issue is not my understanding, but yours: you implicitly allege
profound understanding, which you drop tantalizing hints of, but I have
my doubts there is anything behind the curtain.
<...>
If you don't like my references then try Timo's:
M.E. Rose, Elementary theory of angular momentum, Dover, gives a
reasonably thorough coverage of irreducible tensors and their point.
I have no doubt it too will still say:
<<A Lorentz transformation or any other coordinate
transformation will convert electric or magnetic
fields into mixtures of electric and magnetic fields,
but no transformation mixes them with the
gravitational field. >>
http://www.aip.org/pt/vol-58/iss-11/p31.html
Let's put that in context (I'll steal your quoting convention):
Snipped debatingclub material from:
http://www.aip.org/pt/vol-58/iss-11/p31.html
Oh! Your'e checking up on me!
Keep checking: :-)
http://arxiv.org/abs/physics/0204034
The quoted passage however _is_ fascinating -- it's one of those "too
profound a conincidence to be altogether a coincidence, even if we
don't altogether understand what it means" things. Like the
relativisitically correct behavior of deBroglie waves. You drop a
constraint from GR, and you get EM riding along for free, as an
inevitable consequence? Wow.
You are well read in web-science, and have excellent taste in quotes.
But instead of suggesting you are answering questions, why not just
call them "Sue's web picks"? Or just present your choices as topics
for discussion -- they are all good.
OP: So can someone explain to me what the significance is with all of
these
tensor things and why there are more Cartesian multipole moments?
<< The Lorenz gauge is incomplete in the sense that there
remains a subspace of gauge transformations which preserve
the constraint. These remaining degrees of freedom correspond
to gauge functions which satisfy the wave equation... >>
http://en.wikipedia.org/wiki/%CE%9E_gauge
Does Cartesian and Coulomb gauge and QM mean anything
to you?
Is the Tensor calculus working in a Lorenz space?
Would hidden multipole moments explain this?
http://www.esa.int/SPECIALS/GSP/SEM0L6OVGJE_0.html
Sue...
.
|
|
|
| User: "Greg Hansen" |
|
| Title: Re: Jackson Question |
15 Dec 2006 10:02:50 AM |
|
|
Sue... wrote:
Edward Green wrote:
Sue... wrote:
Edward Green wrote:
Sue... wrote:
<...>
To be critical about how formalism represents phenomena,
you need several several points of view. Not just rote
manipulation of symbols.
Fine mom and apple pie sentiment. Now, the question involved some
aspects of tensorial definition and manipulation in reference to
multipoles, if I understand it correctly.
You understand only a piece of it then.
No doubt... since I have not worked through the problem, and have so
far been spared the aching general insights that await those who have.
At issue is not my understanding, but yours: you implicitly allege
profound understanding, which you drop tantalizing hints of, but I have
my doubts there is anything behind the curtain.
<...>
If you don't like my references then try Timo's:
M.E. Rose, Elementary theory of angular momentum, Dover, gives a
reasonably thorough coverage of irreducible tensors and their point.
I have no doubt it too will still say:
<<A Lorentz transformation or any other coordinate
transformation will convert electric or magnetic
fields into mixtures of electric and magnetic fields,
but no transformation mixes them with the
gravitational field. >>
http://www.aip.org/pt/vol-58/iss-11/p31.html
Let's put that in context (I'll steal your quoting convention):
Snipped debatingclub material from:
http://www.aip.org/pt/vol-58/iss-11/p31.html
Oh! Your'e checking up on me!
Keep checking: :-)
http://arxiv.org/abs/physics/0204034
The quoted passage however _is_ fascinating -- it's one of those "too
profound a conincidence to be altogether a coincidence, even if we
don't altogether understand what it means" things. Like the
relativisitically correct behavior of deBroglie waves. You drop a
constraint from GR, and you get EM riding along for free, as an
inevitable consequence? Wow.
You are well read in web-science, and have excellent taste in quotes.
But instead of suggesting you are answering questions, why not just
call them "Sue's web picks"? Or just present your choices as topics
for discussion -- they are all good.
OP: So can someone explain to me what the significance is with all of
these
tensor things and why there are more Cartesian multipole moments?
<< The Lorenz gauge is incomplete in the sense that there
remains a subspace of gauge transformations which preserve
the constraint. These remaining degrees of freedom correspond
to gauge functions which satisfy the wave equation... >>
http://en.wikipedia.org/wiki/%CE%9E_gauge
Does Cartesian and Coulomb gauge and QM mean anything
to you?
Is the Tensor calculus working in a Lorenz space?
Would hidden multipole moments explain this?
http://www.esa.int/SPECIALS/GSP/SEM0L6OVGJE_0.html
Sue...
The question was about a particular application of tensors in classical
electrostatics. Well, tensors are used in general relativity, and there
exists a quantum theory of electromagnetism. Despite that, the Lorentz
transforms, general relativity, unified field theories, and quantum
mechanics really don't address the question at all. It's not even a
wrong answer, it just passes by without interaction. Not every
application of tensors involves relativity theory.
.
|
|
|
|
|
| User: "Sue..." |
|
| Title: Re: Jackson Question |
15 Dec 2006 11:06:54 AM |
|
|
Edward Green wrote:
Sue... wrote:
Edward Green wrote:
Sue... wrote:
<...>
To be critical about how formalism represents phenomena,
you need several several points of view. Not just rote
manipulation of symbols.
Fine mom and apple pie sentiment. Now, the question involved some
aspects of tensorial definition and manipulation in reference to
multipoles, if I understand it correctly.
You understand only a piece of it then.
No doubt... since I have not worked through the problem, and have so
far been spared the aching general insights that await those who have.
At issue is not my understanding, but yours: you implicitly allege
profound understanding, which you drop tantalizing hints of, but I have
my doubts there is anything behind the curtain.
<...>
If you don't like my references then try Timo's:
M.E. Rose, Elementary theory of angular momentum, Dover, gives a
reasonably thorough coverage of irreducible tensors and their point.
I have no doubt it too will still say:
<<A Lorentz transformation or any other coordinate
transformation will convert electric or magnetic
fields into mixtures of electric and magnetic fields,
but no transformation mixes them with the
gravitational field. >>
http://www.aip.org/pt/vol-58/iss-11/p31.html
Let's put that in context (I'll steal your quoting convention):
Snipped debatingclub material from:
http://www.aip.org/pt/vol-58/iss-11/p31.html
Oh! Your'e checking up on me!
Keep checking: :-)
http://arxiv.org/abs/physics/0204034
The quoted passage however _is_ fascinating -- it's one of those "too
profound a conincidence to be altogether a coincidence, even if we
don't altogether understand what it means" things. Like the
relativisitically correct behavior of deBroglie waves. You drop a
constraint from GR, and you get EM riding along for free, as an
inevitable consequence? Wow.
You are well read in web-science, and have excellent taste in quotes.
But instead of suggesting you are answering questions, why not just
call them "Sue's web picks"? Or just present your choices as topics
for discussion -- they are all good.
OP: So can someone explain to me what the significance is with all of
these
tensor things and why there are more Cartesian multipole moments?
<< The Lorenz gauge is incomplete in the sense that there
remains a subspace of gauge transformations which preserve
the constraint. These remaining degrees of freedom correspond
to gauge functions which satisfy the wave equation... >>
http://en.wikipedia.org/wiki/%CE%9E_gauge
Does Cartesian and Coulomb gauge and QM mean anything
to you?
Is the Tensor calculus working in a Lorenz space?
Would hidden multipole moments explain this?
http://www.esa.int/SPECIALS/GSP/SEM0L6OVGJE_0.html
Sue...
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| User: "Sue..." |
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| Title: Re: Jackson Question |
15 Dec 2006 11:07:39 AM |
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Edward Green wrote:
Sue... wrote:
Edward Green wrote:
Sue... wrote:
<...>
To be critical about how formalism represents phenomena,
you need several several points of view. Not just rote
manipulation of symbols.
Fine mom and apple pie sentiment. Now, the question involved some
aspects of tensorial definition and manipulation in reference to
multipoles, if I understand it correctly.
You understand only a piece of it then.
No doubt... since I have not worked through the problem, and have so
far been spared the aching general insights that await those who have.
At issue is not my understanding, but yours: you implicitly allege
profound understanding, which you drop tantalizing hints of, but I have
my doubts there is anything behind the curtain.
<...>
If you don't like my references then try Timo's:
M.E. Rose, Elementary theory of angular momentum, Dover, gives a
reasonably thorough coverage of irreducible tensors and their point.
I have no doubt it too will still say:
<<A Lorentz transformation or any other coordinate
transformation will convert electric or magnetic
fields into mixtures of electric and magnetic fields,
but no transformation mixes them with the
gravitational field. >>
http://www.aip.org/pt/vol-58/iss-11/p31.html
Let's put that in context (I'll steal your quoting convention):
Snipped debatingclub material from:
http://www.aip.org/pt/vol-58/iss-11/p31.html
Oh! Your'e checking up on me!
Keep checking: :-)
http://arxiv.org/abs/physics/0204034
The quoted passage however _is_ fascinating -- it's one of those "too
profound a conincidence to be altogether a coincidence, even if we
don't altogether understand what it means" things. Like the
relativisitically correct behavior of deBroglie waves. You drop a
constraint from GR, and you get EM riding along for free, as an
inevitable consequence? Wow.
You are well read in web-science, and have excellent taste in quotes.
But instead of suggesting you are answering questions, why not just
call them "Sue's web picks"? Or just present your choices as topics
for discussion -- they are all good.
OP: So can someone explain to me what the significance is with all of
these
tensor things and why there are more Cartesian multipole moments?
<< The Lorenz gauge is incomplete in the sense that there
remains a subspace of gauge transformations which preserve
the constraint. These remaining degrees of freedom correspond
to gauge functions which satisfy the wave equation... >>
http://en.wikipedia.org/wiki/%CE%9E_gauge
Does Cartesian and Coulomb gauge and QM mean anything
to you?
Is the Tensor calculus working in a Lorenz space?
Would hidden multipole moments explain this?
http://www.esa.int/SPECIALS/GSP/SEM0L6OVGJE_0.html
Sue...
.
|
|
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| User: "Sue..." |
|
| Title: Re: Jackson Question |
15 Dec 2006 11:10:20 AM |
|
|
Edward Green wrote:
Sue... wrote:
Edward Green wrote:
Sue... wrote:
<...>
To be critical about how formalism represents phenomena,
you need several several points of view. Not just rote
manipulation of symbols.
Fine mom and apple pie sentiment. Now, the question involved some
aspects of tensorial definition and manipulation in reference to
multipoles, if I understand it correctly.
You understand only a piece of it then.
No doubt... since I have not worked through the problem, and have so
far been spared the aching general insights that await those who have.
At issue is not my understanding, but yours: you implicitly allege
profound understanding, which you drop tantalizing hints of, but I have
my doubts there is anything behind the curtain.
<...>
If you don't like my references then try Timo's:
M.E. Rose, Elementary theory of angular momentum, Dover, gives a
reasonably thorough coverage of irreducible tensors and their point.
I have no doubt it too will still say:
<<A Lorentz transformation or any other coordinate
transformation will convert electric or magnetic
fields into mixtures of electric and magnetic fields,
but no transformation mixes them with the
gravitational field. >>
http://www.aip.org/pt/vol-58/iss-11/p31.html
Let's put that in context (I'll steal your quoting convention):
Snipped debatingclub material from:
http://www.aip.org/pt/vol-58/iss-11/p31.html
Oh! Your'e checking up on me!
Keep checking: :-)
http://arxiv.org/abs/physics/0204034
The quoted passage however _is_ fascinating -- it's one of those "too
profound a conincidence to be altogether a coincidence, even if we
don't altogether understand what it means" things. Like the
relativisitically correct behavior of deBroglie waves. You drop a
constraint from GR, and you get EM riding along for free, as an
inevitable consequence? Wow.
You are well read in web-science, and have excellent taste in quotes.
But instead of suggesting you are answering questions, why not just
call them "Sue's web picks"? Or just present your choices as topics
for discussion -- they are all good.
OP: So can someone explain to me what the significance is with all of
these
tensor things and why there are more Cartesian multipole moments?
<< The Lorenz gauge is incomplete in the sense that there
remains a subspace of gauge transformations which preserve
the constraint. These remaining degrees of freedom correspond
to gauge functions which satisfy the wave equation... >>
http://en.wikipedia.org/wiki/%CE%9E_gauge
Does Cartesian and Coulomb gauge and QM mean anything
to you?
Is the Tensor calculus working in a Lorenz space?
Would hidden multipole moments explain this?
http://www.esa.int/SPECIALS/GSP/SEM0L6OVGJE_0.html
Sue...
.
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| User: "Timo Nieminen" |
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| Title: Re: Jackson Question |
14 Dec 2006 09:34:53 PM |
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On Fri, 14 Dec 2006, Jason Pawloski wrote:
I was trolling aimlessly through Jackson yesterday and I found a
question that piqued my curiosity, because I had never noticed this
before. Its problem 4.3:
The lth term in the multipole expansion of the potential is specified
by the (2l + 1) multipole moments q_lm. On the other hand, the
Cartesian multipole moments
Q_{alpha beta gamme}^(l) = int p(x) x^alpha y^beta z^gamma d^x
with alpha, beta and gamme nonnegative integers subject to the
constraint alpha + beta + gamma = l, are (l + 1)(l + 2)/2 in number.
Thus for l>1, there are more Cartesian multipole moments than seem
necessary to describe the term in the potential whose radial dependence
is r^(-l-1).
Show that while the q_lm transform under rotations as irreducible
spherical tensors of rank l, the Cartesian multipole moments correspond
to reducible spherical tensors of ranks l, l-2, l-4, ... l_min, where
l_min = 0 or 1 for l even or odd respectively. Check that the number of
different tensorial components adds up to the total number of Cartesian
tensors. (et cetera)
I'll be the first to admit my Jackson E&M class was very shoddy and
didn't follow the book at all, and this question does not seem
self-contained as I can find no place Jackson where describes what
(ir)reducible tensors are, or spherical tensors, and I don't see an
easy generalization of what I know the adjective irreducible can mean
(irreducible polynomial, for instance).
So can someone explain to me what the significance is with all of these
tensor things and why there are more Cartesian multipole moments?
Landau covers this compactly and quite nicely, or at least your final
question.
For a charge q, the monopole moment is q, the dipole moment is the vector
(qx,qy,qz) where (x,y,z) is the position of the charge. The quadrupole
tensor is then
[ 3qxx-r^2 3qxy 3qxz ]
[ 3qyx 3qyy-r^2 3qyz ]
[ 3qzx 3qzy 3qzz-r^2 ]
which is symmetric. Thus, instead of 9 independent components, you only
have 6. Note that the sum of the diagonal is zero. This reduces the number
to 5, which is the same number as the spherical multipoles. The same works
for the higher order multipoles, giving 2n+1 independent components of the
2^n order multipole tensor.
The quadrupole tensor above is the traceless tensor (Jackson (4.9)), while
Jackson gives the non-traceless version in problem 4.3. The r^2 can be in
there because 1/r satisfies the Laplace equation. This gives the nice
trace=0 property that means that the 3 diagonal elements are not
independent - only 2 of them are. But you have that in the non-traceless
version as well - the trace of the non-traceless version is qr^2, so you
still only have 2 independent elements on the diagonal.
I've seen multiple definitions of reducible and irreducible tensors.
Jackson very briefly mentions them in the text, Landau very briefly
mentions them two, and both seem to have different definitions in mind.
Consider the 9 components of a rank 2 tensor T. We can write the rank 2
tensor as a sum of its trace * I, and antisymmetric tensor (Tij - Tji),
and a symmetric rank 2 tensor with zero trace, which we can obtain by
subtracting the others from T. Thus,
Tij = (1/3) trace(T) I + (1/2) (Tij - Tji) + Sij
where Sij is symmetric and has zero trace.
trace(T) is a scalar, and is therefore invariant under rotations. The
antisymmetric term transforms similarly to a vector (ie rank 1 tensor).
Since we're interested in things that transform under rotations like a
rank 2 tensor, Sij is all we need to have the most compact description.
M.E. Rose, Elementary theory of angular momentum, Dover, gives a
reasonably thorough coverage of irreducible tensors and their point.
--
Timo Nieminen - Home page: http://www.physics.uq.edu.au/people/nieminen/
E-prints: http://eprint.uq.edu.au/view/person/Nieminen,_Timo_A..html
Shrine to Spirits: http://www.users.bigpond.com/timo_nieminen/spirits.html
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