EM "Dirac-Hamiltonian" & Where is E. Forgy?
Urs Schreiber
Originally I wanted to ask Eric Forgy the following question, but his
former email address seems to be outdated. Does anyone know how I can
contact him?
Anyway, this is what I wanted to ask him. (Of course I'd appreciate
comments from others, too, I just thought Eric was the one most likely to
know the answer.) It's really about something that I have played around
with already quite a while ago, but recently I maybe achieved a better
understanding of it:
As everybody knows, classical sourcefree electromagnetism on a
gravitational background is about a 2-form field F on a semi-Riemannian
manifold (M,g) which satisfies the two "constraints" d F = 0 = del F. I am
interested in obtaining a notion of time evolution from these constraints
without invoking phase space and the canonical Hamiltonian formulation of
classical EM. So assume that there exists a timelike Killing vector v of g
and let t be the label of the spatial hyperslices Sigma_t orthogonal to v
such that dt = v_m dx^m. From the usual indefinite Hodge inner product
<a|b> = int_M a/\*b one obtains a positive definite scalar product (.|.)
on forms restricted to Sigma_0 by setting (a|b) = <a|eta|b>, where the
operator eta is defined by eta = delta(t) v.y- v.y+/sqrt(-v.v). Here the
dot "." indicates index contraction with respect to g and y+- = dx/\ +-
dx-> are the usual Clifford generators on the exterior bundle. Note that
(F|F) = int_Sigma0 v.T.v vol_Sigma, where T is the usual Maxwell
energy-momentum tensor of F. Is there an operator on 2-forms F that
generates evolution of F from Sigma_t1 to Sigma_t2?
I finally came up with the following construction, which improves on a
similar previous attempt: With the abbreviation D+- = d +- del the
constraints d F = del F = 0 are equivalent to the equation
({v.y+,D-} - {v.y-,D+})F = -([v.y+,D-] - [v.y-,D+])F
together with a similar equation with reversed signs. Here [,] and {,} are
commutator and anticommutator. The point is that, since v is Killing, it
can be shown that the left hand side of the above equation is
4 L_v F ,
where L_v = {d,v.dx->} is the Lie derivative operator along v. Hence if one
defines the "Hamiltonian"
H = (i/4)([v.y-,D+] - [v.y+,D-])
then the above equation takes the form of a Dirac-Schroedinger equation
i L_v F = H_v F .
It can be (easily) shown that [L_v,H_v] = 0, so that this Hamiltonian is
"time independent", and (with much more work) that H_v is hermitian with
respect to (|) and hence generates a unitary evolution along v. (But
remember that there is a further constraint that this evolution is subject
to.)
I call this a "Dirac-Schroedinger" equation since, for instance, on a
Minkowski background we have
H_v = (v.y- y-^j - v.y+ y+^j)/2 i partial_j
where the term in brackets generates a Clifford algebra. This is really the
sum of an ordinary "left going" and a "right going" Dirac-electron-type
Hamiltonian.
My question is: What am I really doing here? Is that H_v discussed above
known in the literature? Does anyone recognize the wheel I am reinventing
here?
Eric A. Forgy
Hi Urs!!
Urs Schreiber <Urs.Sc...@uni-essen.de> wrote in message news:<b1duvr$119kqd$1...@ID-168578.news.dfncis.de>...
> Originally I wanted to ask Eric Forgy the following question, but his
> former email address seems to be outdated. Does anyone know how I can
> contact him?
I'm still alive :) After a long and grueling six years of grad school,
I FINALLY finished my phd and I'm now working at MIT Lincoln
Laboratory :) One of the things that got me through the harshness of
grad school was the idea that things would be smooth sailing after I
graduated. If I knew that I would remain just as stressed out as
before, I don't know if I would have made it :)
> Anyway, this is what I wanted to ask him. (Of course I'd appreciate
> comments from others, too, I just thought Eric was the one most likely to
> know the answer.) It's really about something that I have played around
> with already quite a while ago, but recently I maybe achieved a better
> understanding of it:
*gulp*
No pressure :)
> As everybody knows, classical sourcefree electromagnetism on a
> gravitational background is about a 2-form field F on a semi-Riemannian
> manifold (M,g) which satisfies the two "constraints" d F = 0 = del F. I am
> interested in obtaining a notion of time evolution from these constraints
> without invoking phase space and the canonical Hamiltonian formulation of
> classical EM. So assume that there exists a timelike Killing vector v of g
> and let t be the label of the spatial hyperslices Sigma_t orthogonal to v
> such that dt = v_m dx^m. From the usual indefinite Hodge inner product
> <a|b> = int_M a/\*b one obtains a positive definite scalar product (.|.)
> on forms restricted to Sigma_0 by setting (a|b) = <a|eta|b>, where the
> operator eta is defined by eta = delta(t) v.y- v.y+/sqrt(-v.v). Here the
> dot "." indicates index contraction with respect to g and y+- = dx/\ +-
> dx-> are the usual Clifford generators on the exterior bundle. Note that
> (F|F) = int_Sigma0 v.T.v vol_Sigma, where T is the usual Maxwell
> energy-momentum tensor of F. Is there an operator on 2-forms F that
> generates evolution of F from Sigma_t1 to Sigma_t2?
I don't know if I can say anything intelligent along the direction
that you've gone, but I can make another simple observation at this
point.
Instead of looking at the entire hyperslice sigma_t, let's look at a
finite smooth amoeba shaped patch of sigma_t and call it patch_t. The
flow generated by v can be used to push patch_t forward to
patch_{t+delta(t)} or even backward to patch_{t-delta(t)}. Let's
denote the hypervolume enclosed by patch_t1 and patch_t2 by N. Now N
is a piecewise smooth manifold with boundary. At the risk of bringing
back the psuedo shmuedo forms nightmare again, let's assume that N is
inner orientable and put an inner orientation on it and denote the
inner oriented piecewise smooth manifold with boundary by +N.
In my thesis I call a "piecewise smooth manifold with boundary" a
"computational domain." :) Therefore, +N is an inner orieted
computational domain.
The boundary of +N is composed of three smooth pieces now:
@N
= patch_t2 - patch_t1 + sides.
Now we know that dF = 0, it follows that
int_@N F = 0.
Therefore,
int_{patch_t2} F
= int_{patch_t1} F - int_{sides} F.
As far as evolving F from t1 to t2 is concerned, I am inclined to say
that we don't care about F so much as we care about int F. So the
integral of F at t2 can be determined by the integral of F at t1 in
addition to the integral of F over the sides. If you REALLY wanted to
find F at t2, then you could probably reduce the volume of patch_1 to
near a point so that N is a thin hypertube and then
int_{patch_t1} F ~= F(t1)*[volume of patch_t1].
We could then partition sigma_t up into a bunch of patches covering
sigma_t and then refine the partition as much as we like and perform
this procedure for each patch.
Another important observation is that in 4d, in a coordinate patch,
there are 6 bases
dy/\dz, dz/\dx, dx/\dy (space-space)
dx/\dt, dy/\dt, dz/\dt (space-time)
Once we have this "time vector field" v, then we can decompose F into
space-space pieces and space-time pieces. This decomposition is given
by
F = B + E/\dt
where B is the space-space piece of F and E/\dt is the space-time
piece of F.
With this we can rewrite the evolution of F from t1 to t2 (since
patch_t1 and patch_t2 are space-space domains and the side is a
space-time domain) via
int_{patch_t2} B =
int_{patch_t1} B - int_{sides} E/\dt
If you squint your eyes when you look at this, you can see Faradays
law :)
dB/dt = -curl(E).
So asking for a way to evolve F from t1 to t2 is the same thing as
asking for an expression of Faradays law on a general semi-Riemannian
manifold. However, it should also be pointed out that this does not
involve the metric at all. Faradays law is purely topological in
nature and results from the topological fact d^2 = 0.
> I finally came up with the following construction, which improves on a
> similar previous attempt: With the abbreviation D+- = d +- del the
> constraints d F = del F = 0 are equivalent to the equation
>
> ({v.y+,D-} - {v.y-,D+})F = -([v.y+,D-] - [v.y-,D+])F
>
> together with a similar equation with reversed signs. Here [,] and {,} are
> commutator and anticommutator. The point is that, since v is Killing, it
> can be shown that the left hand side of the above equation is
>
> 4 L_v F ,
>
> where L_v = {d,v.dx->} is the Lie derivative operator along v.
I am struggling a little with your notation :)
Is v.y+- = (#v)/\ +- i_v, where #v is the 1-form dual to v and i_v is
the interior product?
If that is the case, I can make another simple observation. That is
i_v F = -E.
(I hope!)
If that is true, then
L_v F
= d i_v F - i_v dF
= -dE.
> Hence if one defines the "Hamiltonian"
>
> H = (i/4)([v.y-,D+] - [v.y+,D-])
>
> then the above equation takes the form of a Dirac-Schroedinger equation
>
> i L_v F = H_v F .
>
> It can be (easily) shown that [L_v,H_v] = 0, so that this Hamiltonian is
> "time independent", and (with much more work) that H_v is hermitian with
> respect to (|) and hence generates a unitary evolution along v. (But
> remember that there is a further constraint that this evolution is subject
> to.)
>
> I call this a "Dirac-Schroedinger" equation since, for instance, on a
> Minkowski background we have
>
> H_v = (v.y- y-^j - v.y+ y+^j)/2 i partial_j
>
> where the term in brackets generates a Clifford algebra. This is really the
> sum of an ordinary "left going" and a "right going" Dirac-electron-type
> Hamiltonian.
>
> My question is: What am I really doing here? Is that H_v discussed above
> known in the literature? Does anyone recognize the wheel I am reinventing
> here?
I could be (and probably am) wrong, but it looks to me like you may
have found a curious way to rewrite Faraday's law :)
Moral: Spacetime is beautiful. It is only when you start arbitrarily
chopping it up into space and time do things become heinous :)
Best wishes,
Eric
Urs Schreiber
> I'm still alive :)
Great, and you are still reading spr! Nice to hear from you again!
> After a long and grueling six years of grad school,
> I FINALLY finished my phd and I'm now working at MIT Lincoln
> Laboratory :)
This sure sounds like a very prestigious place to be. What is your work
like at MIT Lincoln Laboratory?
Thanks a lot for looking at my question. You write:
> Now we know that dF = 0, it follows that
>
> int_@N F = 0.
>
> Therefore,
>
> int_{patch_t2} F
> = int_{patch_t1} F - int_{sides} F.
[...]
> int_{patch_t2} B =
> int_{patch_t1} B - int_{sides} E/\dt
>
> If you squint your eyes when you look at this, you can see Faradays
> law :)
>
> dB/dt = -curl(E).
>
> So asking for a way to evolve F from t1 to t2 is the same thing as
> asking for an expression of Faradays law on a general semi-Riemannian
> manifold. However, it should also be pointed out that this does not
> involve the metric at all. Faradays law is purely topological in
> nature and results from the topological fact d^2 = 0.
[...]
> I could be (and probably am) wrong, but it looks to me like you may
> have found a curious way to rewrite Faraday's law :)
OK, yes. You are translating my muttering into more standard terminology so
that it actually looks like electromagnetism. I should have done that
myself to facilitate communication, and I am lucky that you do it for me
instead.
Let me expand on the point that you brought up: If we go through the same
construction that you describe above but now for *F, which is also closed
in the absence of sources, d*F = 0, then we get
int_{patch_t2} *F - int_{patch_t1} *F = -int_{sides} *F
and due to
*F = E_2 - B_1/\dt
this reduces, as above, to the equation
int_{patch_t2} E_2 - int_{patch_t1} E_2 = int_{sides} B_1/\dt ,
whose infinitesimal version is
dE/dt = + curl(B),
which complements Faraday's law. (Of course this one does involve the
metric.)
All in all this gives the time evolution of F in the special case of a
Minkowski background:
dF/dt = d/dt(B + E/\dt) = -curl(E)_2 + curl(B)_1/\dt .
Now one can check that the "Hamiltonian" operator that I defined in my last
post by
H = i/4 ([v.y-,D_+] - [v.y+,D_-])
maps, for a Minkowski background and for v = partial_0, any 2-form
f = b_2 + e_1/\dt
to
H f = i(-curl(e)_2 + curl(b)_1/\dt )
(where I write lower-case f,b,e to indicate that these are arbitrary forms
of the indicated degree, i.e. that the above equation is an identity which
does not depend on f,e,b satisfying Maxwell's equations).
Hence the Schroedinger-like evolution equation
i L_v F = H F ,
that I talked about, says in this special case that
<=> i d/dt F = H F
<=> dF/dt = -curl(E)_2 + curl(B)_1/\dt
and hence is equivalent to 2 of the 4 Maxwell equations. (In the context of
this formalism the other two come from the remaining constraint that I
mentioned in my previous post.)
> Is v.y+- = (#v)/\ +- i_v, where #v is the 1-form dual to v and i_v is
> the interior product?
Yes, exactly.
> If that is the case, I can make another simple observation. That is
>
> i_v F = -E.
Agreed.
> If that is true, then
>
> L_v F
> = d i_v F - i_v dF
> = -dE.
Yes, I think this is just another way to arrive at
d/dt F = L_v F = {d,i_v} F = d i_v F = -dE = -curl(E)_2 + d/dt E/\dt,
and, using dE/dt = curl(B) again, at
... = -curl(E)_2 + curl(B)_1/\dt,
as above. But this makes use of Maxwell's equations. Of course you can
enter my strange equation
i L_v F = H F
with an exact F = dA and hence get dF=0 for free, which takes care of the
"topological" Faraday law, as you mention above, and reduces i L_v F = HF
to
-idE = HF ,
which is then equivalent to the remaining non-topological dE/dt = +
curl(B).
I guess this is where absolutely everybody who has followed this so far is
asking: "What then is the point of writing iL_v = HF??"
Well, it gives an evolution of F along a timelike Killing vector for more
general cases, most notably for arbitrary background metices. Plus, that H
satisfies some nice properties, like being hermitian wrt the (.,.) scalar
product that I mentioned before.
But I don't want to imply that iL_v F = HF is somehow a better way to do EM
than any other way. As one may have guessed, my motivation to look at this
equation comes from outside of ordinary EM, where it has its definite
merits. But since it applies to EM, too, I am curious if maybe something
similar is used there, too. For instance, is anybody studying the
sourcefree evolution of EM-waves on fixed gravitational backgrounds along
a timelike Killing vector? Like scattering EM waves at black holes or the
like?
It seems to me like in such a case a reformulation of dF = d*F = 0 in the
form iL_v F = HF might by useful. But I may be wrong. One problem that I
am aware of is that H, the generator of v-evolution only commutes with the
remaining constraint, which expresses two of Maxwell's equations, in flat
Minkowski background.
> Moral: Spacetime is beautiful. It is only when you start arbitrarily
> chopping it up into space and time do things become heinous :)
I'll try to remember that next time when I evolve into a state that makes
me want to split space and time. ;-)
John Devers
Urs Schreiber <Urs.Sc...@uni-essen.de> wrote in message news:<b1nul8$153eec$1...@ID-168578.news.dfncis.de>...
> For instance, is anybody studying the
> sourcefree evolution of EM-waves on fixed gravitational backgrounds along
> a timelike Killing vector? Like scattering EM waves at black holes or the
> like?
These articles may help.
Laser could scan skies for black holes
Light bounces off black holes into bright concentric rings.
http://www.nature.com/nsu/021014/021014-2.html
LLNL Scientists Create A Virtual Star
http://www.llnl.gov/llnl/06news/NewsReleases/2002/NR-02-01-01.html
Eric A. Forgy
I should probably sit down and try to understand exactly how your
formalism is going to work, but if you don't mind, maybe I can just
ask a few questions :)
Urs Schreiber <Urs.Sc...@uni-essen.de> wrote:
> Eric A. Forgy wrote:
>
> Let me expand on the point that you brought up: If we go through the same
> construction that you describe above but now for *F, which is also closed
> in the absence of sources, d*F = 0, then we get
>
> int_{patch_t2} *F - int_{patch_t1} *F = -int_{sides} *F
>
> and due to
>
> *F = E_2 - B_1/\dt
>
> this reduces, as above, to the equation
>
> int_{patch_t2} E_2 - int_{patch_t1} E_2 = int_{sides} B_1/\dt ,
>
> whose infinitesimal version is
>
> dE/dt = + curl(B),
>
> which complements Faraday's law. (Of course this one does involve the
> metric.)
Ack! I know that it is standard to write things like D = E in physics,
but since we are hoping to do a little better than that with the
differential geometric concepts, I hope that I can convince you to
alter your notation a little bit.
I would write the above as:
*F = D - H/\dt.
Then considering the spacetime tube and taking the appropriate limits
we get
dD/dt = curl(H).
I am a proponent of keeping E,D and B,H as separate notions and
letting all the constitutive components be stuffed into the Hodge
star. This is more geometric in flavor. This is just a personal
preference of course and anyone can feel free to do whatever they
like.
> All in all this gives the time evolution of F in the special case of a
> Minkowski background:
>
> dF/dt = d/dt(B + E/\dt) = -curl(E)_2 + curl(B)_1/\dt .
The expression dF/dt bothers me for some reason. I guess it is fine
though. Have you considered looking at the conglomeration
F^(k) = F + k *F
for some scalar k. Then you'd have
dF^(k)
= d(F + k *F)
= d(B + E/\dt) + k d(D - H/\dt)
= [d(B+kD)/dt + curl(E-kH)]/\dt + [div(B) + k div(D)] vol^3
= 0
You get two separate sets of equations by letting k = +/-1 (maybe
+/-i). This will lead you to electromagnetic duality and all kinds of
neat things in string theory that I have no clue about (I think) :)
> Of course you can enter my strange equation
>
> i L_v F = H F
>
> with an exact F = dA and hence get dF=0 for free, which takes care of the
> "topological" Faraday law, as you mention above, and reduces i L_v F = HF
> to
>
> -idE = HF ,
>
> which is then equivalent to the remaining non-topological dE/dt = +
> curl(B).
Ok. I guess my gut feeling was wrong. I didn't see how you'd get the
non-topological equation from that. I guess it wouldn't be very fun if
all you ended up doing was writing a complicated expression for d^2 =
0 :)
> I guess this is where absolutely everybody who has followed this so far is
> asking: "What then is the point of writing iL_v = HF??"
>
> Well, it gives an evolution of F along a timelike Killing vector for more
> general cases, most notably for arbitrary background metices. Plus, that H
> satisfies some nice properties, like being hermitian wrt the (.,.) scalar
> product that I mentioned before.
>
> But I don't want to imply that iL_v F = HF is somehow a better way to do EM
> than any other way. As one may have guessed, my motivation to look at this
> equation comes from outside of ordinary EM, where it has its definite
> merits. But since it applies to EM, too, I am curious if maybe something
> similar is used there, too. For instance, is anybody studying the
> sourcefree evolution of EM-waves on fixed gravitational backgrounds along
> a timelike Killing vector? Like scattering EM waves at black holes or the
> like?
Studying EM-waves on fixed gravitational backgrounds along a timelike
Killing vector is mathematically equivalent to studying EM-waves in
fixed background inhomogeneous media. That was one of justifications
for teaching myself differential geometry in the first place. I wanted
to understand general relativity so that I could understand Maxwell's
equations in inhomogeneous media :) Engineers do this every day
whether they realize it or not :) Then I wanted to understand quantum
gravity so that I could understand general relativity so that I could
understand Maxwell's equations in inhomogeneous media. Yes, I am a
masochist :) Oddly enough, the problems you face in trying to do
computational electromagnetics "right" are very similar to the
problems you face in trying to do quantum gravity "right." I think
more effort should be placed in doing Maxwell's equations "right." But
that is just me :)
As another aside, I did once implement a "black wall" :) In
computational electromagnetics, if you try to discretize Maxwell's
equations and propagate the fields in a lattice (like a kind of
cellular automata), then eventually those propagating fields are going
to reach the end of your computational domain (due to finite computer
memory). If you do nothing about this, the wave will reflect back into
the grid. This reflection is undesirable. MUCH research has gone into
eliminating this problem. A revolution came in 1994, but that is a
different story. Anyway, I took a Schwarzschild metric in spherical
coordinates and modified it so that it would appear to be a 1d black
hole in one of the spatial dimensions and and the usual Euclidean
space in the other spatial dimensions (I seem to remember checking and
it seemed that the metric I ended up with actually satisfied the field
equations of GR). Essentially, I was trying to create a numerical
"black wall" :) The idea is that any wave that strikes it will get
gobbled up and cannot reflect back into the grid. Well, I implemented
the code and it was actually pretty cool. You could see the wave
approach the wall. As it approached the black wall, it appeared to
slow down. The closer it got, the slower it moved. I was excited. It
appeared to be working! Unfortunately, due to the discretization, the
wave never truly "stops." It will eventually hit the wall. When it
does, it reflects back into the grid. As you would expect, it starts
off slowly in the opposite direction and gradually picks up speed. Oh
well. It was fun! :)
Again, the "black wall" was equivalent to an inhomogeneous material
background. In engineering speak, at each point of the lattice, I
maintanined a matched impedance, which eliminates reflections. Then
epsilon and mu are increased in concert as you approach the wall (this
is a slightly over simplified account, but it is the basic idea.)
As a fun exercise, I suggest taking a Schwarzschild metric, which
defines a Hodge star. Then interpret the resulting Maxwell's equations
as if it were an inhomogeneous media and see what it amounts to in
terms of epsilon and mu.
> It seems to me like in such a case a reformulation of dF = d*F = 0 in the
> form iL_v F = HF might by useful. But I may be wrong. One problem that I
> am aware of is that H, the generator of v-evolution only commutes with the
> remaining constraint, which expresses two of Maxwell's equations, in flat
> Minkowski background.
You might want to look up "symplectic integration" particularly in the
context of computational electromagnetics. You might find that this is
equivalent to what you have done.
Good luck and keep us posted on your progress. I know I'm interested
to see what happens :)
Eric
Urs Schreiber
> I should probably sit down and try to understand exactly how your
> formalism is going to work, but if you don't mind, maybe I can just
> ask a few questions :)
Sure.
> Urs Schreiber <Urs.Sc...@uni-essen.de> wrote:
>> dE/dt = + curl(B),
> Ack! I know that it is standard to write things like D = E in physics,
> but since we are hoping to do a little better than that with the
> differential geometric concepts, I hope that I can convince you to
> alter your notation a little bit.
>
> I would write the above as:
>
> *F = D - H/\dt.
>
> Then considering the spacetime tube and taking the appropriate limits
> we get
>
> dD/dt = curl(H).
All right, I am already convinced :-)
>> dF/dt = d/dt(B + E/\dt) = -curl(E)_2 + curl(B)_1/\dt .
>
> The expression dF/dt bothers me for some reason. I guess it is fine
> though.
It looks better when written as dF/dt = L_v F, where v = partial_0 in this
case.
> Have you considered looking at the conglomeration
>
> F^(k) = F + k *F
>
> for some scalar k. Then you'd have
>
> dF^(k)
> = d(F + k *F)
> = d(B + E/\dt) + k d(D - H/\dt)
> = [d(B+kD)/dt + curl(E-kH)]/\dt + [div(B) + k div(D)] vol^3
> = 0
>
> You get two separate sets of equations by letting k = +/-1 (maybe
> +/-i). This will lead you to electromagnetic duality and all kinds of
> neat things in string theory that I have no clue about (I think) :)
Maybe I should stress that my key motivation is to obtain from a set of
"timeless" constraints similar to dF = 0 = del F an evolution equation iL_v F
= HF plus a set of constraints on the restriction of F to hypersurfaces
orthogonal to v. You are right, of course, that the above is a possible
reformulation of dF = 0 = del F, but it is not quite what I am looking for, I
think.
I am getting the impression, though, that as far as ordinary electromagnetism
is concerned, such a reformulation (I mean in terms of operators acting on
forms and keeping everything adapted to curved space) is of very little, if
any, practical value. That's basically because, as you have mentioned (I'll
address that further below), every gravitational background for ordinary EM
waves can be translated into the presence of a medium, and evolution of EM
waves in the presence of a medium is a standard exercise in using matrix
valued differential operators. For instance, I have talked to a mathematician
working on "boundary value problems in the exterior region" who is doing just
that. He told me to look at
author = {R. Leis},
title = {Initial Value Problems in Mathematical Physics},
publisher = {Teubner},
year = {1985} .
There a "Maxwell operator" A is defined, a matrix with the "curl operator" in
the off-diagonal entries. On 6-component column vectors U time evolution of
an EM wave is described by dU/dt = AU. I have essentially described in my
last post how my operator H is represented on R^6 by this operator A for a
trivial background. So here my H is just a fancy form-language reformulation
of a plain old well known boring matrix valued differential operator. I was
thinking that in curved space my i Lv F = HF on forms might be prettier than
the corresponding dU/dt = AU on matrices (and that mathematician said that
maybe conceptually it is) but apparently there is no practical implication,
since inhomogeneous media are easily handled by means of a straightforward
generalization of dU/dt = AU.
So this seems to answer my original question: The reformulation of dF=0=del F
as iL_v F = HF together with a spatial constraint, while certainly useful
(for me) in generalized settings, is just a notational curiosity in the
context of ordinary electromagnetism. When using the gravity<->inhomogeneous
medium correspondence and a representation of 2-forms on a vector space, it
reduces to the well-known standard evolution equation. In other words, the
problems related to having a nontrivial background metric, which is what I
tried to deal with, are already dealt with by other well-established methods.
(On the other hand, in my formalism the background geometry is manifest and
transparent, while the reformulation in terms of inhomogeneous media might
conceal some physics.)
> Good luck and keep us posted on your progress. I know I'm interested
> to see what happens :)
One little piece of progress I made is this: As I mentioned in my last post,
the operator C that gives the spatial constraint CF = 0 in my formalism
(which accompanies my (iL_v-H)F=0 and which for ordinary EM essentially says
that div D = 0 = div B) does in general not commute with H. But I could now
show (I needed a surprisingly non-obvious calculation to do that) that it
commutes _weakly_ with H, i.e.
[C,H] = OC,
where O is some operator. Hence it follows that any form F_0 on Sigma_0 and
satisfying CF_0 = 0 uniquely extends to a form F on all of spacetime given by
F = exp(-iHt)F_0, which satisfies dF=0=delF.
I am aware that in the case of ordinary EM this is probably an example of
"physics made difficult". :-) The point of all this is that all these
constructions carry over to the case where we deform d and del.
> Studying EM-waves on fixed gravitational backgrounds along a timelike
> Killing vector is mathematically equivalent to studying EM-waves in
> fixed background inhomogeneous media. That was one of justifications
> for teaching myself differential geometry in the first place. I wanted
> to understand general relativity so that I could understand Maxwell's
> equations in inhomogeneous media :) Engineers do this every day
> whether they realize it or not :) Then I wanted to understand quantum
> gravity so that I could understand general relativity so that I could
> understand Maxwell's equations in inhomogeneous media. Yes, I am a
> masochist :)
> Oddly enough, the problems you face in trying to do
> computational electromagnetics "right" are very similar to the
> problems you face in trying to do quantum gravity "right."
This is a surprising remark! I cannot say that I see in which sense both
problems are similar. Could you expand on this point?
> I think
> more effort should be placed in doing Maxwell's equations "right." But
> that is just me :)
So how are Maxwell's equations done "right" in your opinion?
> As another aside, I did once implement a "black wall" :)
[very interesting discussion snipped]
> As a fun exercise, I suggest taking a Schwarzschild metric, which
> defines a Hodge star. Then interpret the resulting Maxwell's equations
> as if it were an inhomogeneous media and see what it amounts to in
> terms of epsilon and mu.
I am cheating and looking it up in the literature: For instance
P. Bergliaffa & K. Hibbard, Electromagnetic waves in a wormhole geometry,
gr-qc/0002036
mention the general formula to translate from EM in a curved spacetime to EM
in flat space with a medium. This seems to go back at least to
J. Plebanski, Phys. Rev. 118, 1396 (1960) .
In the 70s B. Mashhoon makes frequent use of these formulas, for instance in
B. Mashhoon, Influence of gravitation on the propagation of electromagnetic
radiation, Phys. Rev. D. 11(10), 2697-2684 (1975).
For the record, here is how it works (more or less copied from gr-qc/0002036):
Start with the ordinary curved space Maxwell equations (I assume vanishing
sources here):
F^mn_;n = 0 = F_[mn,s].
Now define
H^mn = sqrt(-g)g^mr g^ns F_rs .
In terms of H the divergence equation becomes
H^mn_,n = 0,
where the covariant derivative ";" is replace by the ordinary derivative ",".
Now assume a cartesian coordinate system and decompose the antisymmetric
tensors F and H as usual in terms of four 3-component "vectors" E,B,D,H
F_mn <-> (E,B)
H^mn <-> (-D,H) .
These satisfy the ordinary Maxwell equations as in Minkwoski space:
dB/dt = - curl(B)
dD/dt = curl(H).
By a straightforward comparison of coefficients, one finds the constitutive
relations
D_i = epsilon_ik E_k - (G x H)_i
B_i = mu_ik H_k + (G x E)_i,
where epsilon, mu, and G are given by
epsilon_ik = mu_ik = -sqrt(-g)g^ik/g_00
G_i = -g_0i/g_00 .
This way the effect of every gravitational background field can be translated
into the effect of a medium. (But not every medium can be interpreted in
terms of a curved background.)
I am sure that the above index notation can be replaced by something prettier.
(E.g. Pertti Lounesto once mentioned his way to deal with background metric
plus constitutive relations in Clifford algebraic formalism. Sad that he
won't be able to help us anymore.)
A fairly recent example of an application of this formalism that I have found
is B. Hu and K. Shiokawa, Wave Propagation in Stochastic Spacetime:
Localization, Amplification and Particle Creation, gr-qc/9708023 .
But from my quick search for relevant literature, I got the impression that
the above rewriting of the curved space Maxwell's equations is not
necessarily the favored tool for people working on EM waves on a fixed
background. For instance when studying EM waves on black hole backgrounds
people seem to usually instead mention the "Teukolsky Master Equation". For
instance see section 4 of gr-qc/0203069, where also a reference to the
original paper S. Teukolsky, Astroph. J. 185, 635 (1973) is given. This
equation describes wave propagation of general spin s fields on a Kerr BH
background. By separating the solution into "spin weighted spheroidal
harmonics" the Teukolsky equation essentially reduces to a 1-dimensional ODE
for the radial part. For non-spinning BHs this can be further simplified to
the form of the "generalized Regge-Wheeler equation" (see gr-qc/0002043).
> You might want to look up "symplectic integration" particularly in the
> context of computational electromagnetics. You might find that this is
> equivalent to what you have done.
I have searched the web and the arxive for "symplectic integration" together
with "electromagnetism" but did not get really satisfying results. To my
shame I have to admit that I seem to have lost the reference that you once
gave me. Could you perhaps please dig it out again? Thanks!
--
Urs.Sc...@uni-essen.de
Eric A. Forgy
Urs Schreiber <Urs.Sc...@uni-essen.de> wrote:
> Eric A. Forgy wrote:
>
> > Have you considered looking at the conglomeration
> >
> > F^(k) = F + k *F
> >
> > for some scalar k.
>
> Maybe I should stress that my key motivation is to obtain from a set of
> "timeless" constraints similar to dF = 0 = del F an evolution equation
> iL_v F = HF plus a set of constraints on the restriction of F to
> hypersurfaces orthogonal to v. You are right, of course, that the above is a
> possible reformulation of dF = 0 = del F, but it is not quite what I am
> looking for, I think.
The reason I suggested that comglomeration is that I wanted to
eventually get at the "well known" expression that you wrote
dU/dt = A*U.
Maybe my suggestion was not the most direct route though :)
> author = {R. Leis},
> title = {Initial Value Problems in Mathematical Physics},
> publisher = {Teubner},
> year = {1985} .
> There a "Maxwell operator" A is defined, a matrix with the "curl operator"
> in the off-diagonal entries. On 6-component column vectors U time evolution
> of an EM wave is described by dU/dt = AU. I have essentially described in my
> last post how my operator H is represented on R^6 by this operator A for a
> trivial background. So here my H is just a fancy form-language reformulation
> of a plain old well known boring matrix valued differential operator. I was
> thinking that in curved space my i Lv F = HF on forms might be prettier than
> the corresponding dU/dt = AU on matrices (and that mathematician said that
> maybe conceptually it is) but apparently there is no practical implication,
> since inhomogeneous media are easily handled by means of a straightforward
> generalization of dU/dt = AU.
Note: If you forget their algebraic properties "4 choose 2 = 6", i.e.
a 2-form in 4d may be thought of as an element of R^6 upon choosing a
basis. (I know you know that :))
> So this seems to answer my original question: The reformulation of
> dF=0=del F as iL_v F = HF together with a spatial constraint, while
> certainly useful (for me) in generalized settings, is just a notational
> curiosity in the context of ordinary electromagnetism. When using the
> gravity<->inhomogeneous medium correspondence and a representation of 2-
> forms on a vector space, it reduces to the well-known standard evolution
> equation. In other words, the problems related to having a nontrivial
> background metric, which is what I tried to deal with, are already dealt
> with by other well-established methods. (On the other hand, in my formalism
> the background geometry is manifest and transparent, while the reformulation
> in terms of inhomogeneous media might conceal some physics.)
Well, here you should not give us too much credit. Although you can
formally write down dU/dt = A*U and say, "This is well known...,"
there is still a LOT about this which is NOT known. If there wasn't,
then the field of "computational electromagnetics" would have been
dead long ago. I assure you that the field is alive and kicking :)
For example, how are you going to numerically solve dU/dt = A*U in the
"right" way? You are inevitably going to be forced to make some ad hoc
approximations. These approximations generally break the beautiful
geometry underlying the whole thing. Doing it "right" would involve
respecting the underlying algebraic/geometrical properties. Very
little concern is usually given to this and people tend to be happy
with ad hoc approaches. I tend to "believe" (yes, it is almost
relgious) that there should be a beautiful/natural way to solve this
equation numerically. Part of the reason for this belief is that I
think that combinatorial/numerical methods more closely resonate with
"nature" than the continuum/analytical counterparts, which people
usually refer to as the "exact" theory. I "believe" there is an
"exact" combinatorial theory and the continuum is an approximation to
this. I call it the "bottom up" approach to computation. Rather than
having these god-given equations based on continuum concepts and being
force to "approximate" these (the "top down" approach), I am trying to
build up a discrete theory of space-time from scratch from which the
numerical methods fall out naturally ("bottom up"). Once you start
trying to formulate numerical methods in this way, it begins to pick
up a lot of parallels with what people try to do in quantum gravity
(in particular spin foam models). The grand hurdle for developing this
approach is to come up with the "right" inner product. Coming up with
the right inner product is also a big hurdle in quantum gravity and
has resulted in the present rift between the northern and southern
hemispheres in regard to LQG :)I don't think the two problems are
really as distinct as they may appear at face value. Defining an
appropriate "discrete" inner product is a lot harder than it may seem.
> I am aware that in the case of ordinary EM this is probably an example of
> "physics made difficult". :-) The point of all this is that all these
> constructions carry over to the case where we deform d and del.
Again, don't make the mistake of thinking "ordinary EM" is a closed
issue :)
> > Oddly enough, the problems you face in trying to do
> > computational electromagnetics "right" are very similar to the
> > problems you face in trying to do quantum gravity "right."
>
> This is a surprising remark! I cannot say that I see in which sense both
> problems are similar. Could you expand on this point?
I am referring to the difficulty of defining an appropriate inner
product. I could be way off (I'm no expert for sure), but it is my gut
feeling that one of the difficulties inherent in the spin foam models
is the definition of a meaningful inner product.
You can develop a fairly complete version of exterior calculus on an
"abstract simplicial complex" (see Munkres, "Elements of AT"). But to
define this abstract version of Maxwell's equations, you need an inner
product. Once you have this, you can jump through the hoops and define
a discrete del operator and then you're done. I tried for 6 years and
wasn't able to come up with the "right" inner product (I'm STILL
working on it!).
> So how are Maxwell's equations done "right" in your opinion?
It boils down to the inner product.
> By a straightforward comparison of coefficients, one finds the constitutive
> relations
>
> D_i = epsilon_ik E_k - (G x H)_i
>
> B_i = mu_ik H_k + (G x E)_i,
>
> where epsilon, mu, and G are given by
>
> epsilon_ik = mu_ik = -sqrt(-g)g^ik/g_00
>
> G_i = -g_0i/g_00 .
>
> This way the effect of every gravitational background field can be
> translated into the effect of a medium. (But not every medium can be
> interpreted in terms of a curved background.)
In this way, studying Maxwell's equations in inhomogeneous media is
more difficult than studying EM in fixed background spacetimes in GR
:)
> I have searched the web and the arxive for "symplectic integration" together
> with "electromagnetism" but did not get really satisfying results. To my
> shame I have to admit that I seem to have lost the reference that you once
> gave me. Could you perhaps please dig it out again? Thanks!
All I wanted you to see was the relation dU/dt = A*U, which you found
through other means. This equation results from a Hamiltonian system
and can be solved using symplectic integration techniques, which
basically preserve volumes in phase space even when you discretize the
equations.
I'm guessing the reference I gave you before was:
Numerical Hamiltonian Problems
(Applied Mathematics and Mathematical Computation, No 7)
by J. M. Sanz-Serna, M. P. Calvo (Contributor), J. M. San-Serna
Eric
Urs Schreiber
> For example, how are you going to numerically solve dU/dt = A*U in the
> "right" way? You are inevitably going to be forced to make some ad hoc
> approximations. These approximations generally break the beautiful
> geometry underlying the whole thing. Doing it "right" would involve
> respecting the underlying algebraic/geometrical properties. Very
> little concern is usually given to this and people tend to be happy
> with ad hoc approaches. I tend to "believe" (yes, it is almost
> religious) that there should be a beautiful/natural way to solve this
> equation numerically.
But that dU/dt = AU is not a beautiful or natural formula itself. I'd say it
hides the underlying algebraic/geometrical properties, doesn't it? (See
below.)
[on discrete spacetime]
> The grand hurdle for developing this
> approach is to come up with the "right" inner product.
You are referring to an inner product on (the discrete version of) tangent
vectors here, right? - I mean rather than an inner product on some space of
states?
>> I am aware that in the case of ordinary EM this is probably an example of
>> "physics made difficult". :-) The point of all this is that all these
>> constructions carry over to the case where we deform d and del.
>
> Again, don't make the mistake of thinking "ordinary EM" is a closed
> issue :)
Sorry, I did not mean to imply that what is ordinary is trivial or necessarily
a closed issue, I just tried be be clear about what I was trying to talk
about. Actually I would still like to better understand what the method of
choice today is for numerically evolving an EM field over a curved spacetime.
You seemed to be saying that one would invoke this trick:
>> By a straightforward comparison of coefficients, one finds the constitutive
>> relations
>>
>> D_i = epsilon_ik E_k - (G x H)_i
>>
>> B_i = mu_ik H_k + (G x E)_i,
>>
>> where epsilon, mu, and G are given by
>>
>> epsilon_ik = mu_ik = -sqrt(-g)g^ik/g_00
>>
>> G_i = -g_0i/g_00 .
>>
>> This way the effect of every gravitational background field can be
>> translated into the effect of a medium. (But not every medium can be
>> interpreted in terms of a curved background.)
But now you write:
> In this way, studying Maxwell's equations in inhomogeneous media is
> more difficult than studying EM in fixed background spacetimes in GR
> :)
Now I am confused! Isn't this the construction that you were referring to in
your previous post? The above doesn't look all that pretty, as always when a
covariant thing is broken up into components in some "unnatural" way, but
that's what it was supposed to do. I thought I was the one who was proposing
a prettier way... :-)