E. Noether contra B. Carter

[JanPB]

BTW, I have no idea why the lines are wrapped so badly.
It never happened before.

--
Jan

[The Starmaker]

A side note which is sort of interesting but rather useless: Carter
constant *does* follow from
Noether's theorem after all. OK, let's rewind a bit. Let gamma(s) be
any geodesic in
the Kerr geometry.
(1) q = <gamma', gamma'> ("mass"). This constant is a freebe because in
*any* geometry
*any* geodesic has this property,
order to make
the number E positive for timelike future-pointing geodesics outside
the outer
horizon. This is a constant because the metric does not depend on t
(translational
symmetry in the t direction),
g_ab does not
depend on phi (rotational symmetry).
There does not seem to exist any other symmetry of the Kerr geometry to
yield another
constant. Having four motion constants would be great because it would
reduce the
geodesic equations to first-order equations.
As is well-known, such fourth constant K does exist, it was found as a
separation constant of
a certain PDE, so not related to any symmetry. It can be written e.g. in
this form
K = rho^4 theta'^2 - q a^2 cos^2(theta) - [L - a E
sin^2(theta)]^2/sin^2(theta) (*)
..where rho^2 = r^2 + a^2 cos^2(theta), (a shortcut notation) and the
components of gamma are: (t, r, theta, phi).
Looks unpleasant but it's extremely useful as it allows one to avoid
dealing with
the geodesic equation.
But once the formula is known, one can try to reverse engineer it by
other methods.
One can play a game with Noether's theorem. I'll skip the details but
the (rather useless) result surprised me, I haven't seen it mentioned
assume gamma(s) is a geodesic, with the usual components (t, r, theta,
phi)
(four functions of s). Consider the following, surprisingly simple
*and* totally
t_epsilon = t
r_epsilon = r
theta_epsilon = theta - epsilon * 2 rho^2 * theta'
phi_epsilon = phi
Although the corresponding variated Lagrangian does NOT have the zero
epsilon-derivative
(i.e., the above variation is not a symmetry), it's equal to a total
(d/ds) derivative of...
something. Denoting the epsilon-derivative at epsilon=0 by delta (the
usual
delta(Lagrangian) = d/ds{ -rho^4 theta'^2 - q a^2 cos^2(theta) -
- [L - a E sin^2(theta)]^2/sin^2(theta) }
(since delta(t) = delta(r) = delta(phi) = 0 by the above variation
definition).
- [L - a E sin^2(theta)]^2/sin^2(theta) = const.
(**)
theta')
rho^4 theta'^2 - q a^2 cos^2(theta) - [L - a E
sin^2(theta)]^2/sin^2(theta) = const.
This is the Carter constant.
r_epsilon = r + epsilon * 2 rho^2 r'
OK, so it's totally uninspiring because the Lagrangian variation is
merely a
total derivative, not zero. So the geometric meaning remains as murky
as ever.
K = < r^2 gamma' - rho^2 P(gamma') , gamma' >
..where P denotes the projection onto the principal plane. Still
completely
unexpected.
--
Jan

Your math is correct but doesn't seem to have any meaning.


It's like a Kamala Number Salad...

[JanPB]

[...]

Post by The Starmaker
Your math is correct but doesn't seem to have any meaning.

No geometric meaning (which I was hoping for).

Post by The Starmaker
It's like a Kamala Number Salad...

My previous post has disappeared mysteriously(?) so I'll repeat,
more or less: Kamala is like Basil Fawlty at the White House.
It's amazing it's real life.

We'll see if this post survives.

--
Jan

[The Starmaker]

Post by JanPB
[...]

Post by The Starmaker
Your math is correct but doesn't seem to have any meaning.

No geometric meaning (which I was hoping for).

i see, you have your own approach to doings things no one else does..

Post by JanPB

Post by The Starmaker
It's like a Kamala Number Salad...

My previous post has disappeared mysteriously(?) so I'll repeat,
more or less: Kamala is like Basil Fawlty at the White House.
It's amazing it's real life.
We'll see if this post survives.
--
Jan

[The Starmaker]

Post by The Starmaker

Post by JanPB
[...]

Post by The Starmaker
Your math is correct but doesn't seem to have any meaning.

No geometric meaning (which I was hoping for).

i see, you have your own approach to doings things no one else does..

Also the P isn't standard and needs more clarification

K = < r^2 gamma' - rho^2 P(gamma') , gamma' >




further more, there are too many errors for me to list them all...


you finally reached ...kooksville.

[JanPB]

Typo:

"a principal" --> "a principal geodesic"

--
Jan

[The Starmaker]

Post by JanPB
"a principal" --> "a principal geodesic"
--
Jan

"Typo:"???? Don't you mean...error? a mistake??..error error error


My post! My post is melting! Melting! Oh, what a world, what a world!

[The Starmaker]

Post by The Starmaker

Post by JanPB
"a principal" --> "a principal geodesic"
--
Jan

"Typo:"???? Don't you mean...error? a mistake??..error error error
My post! My post is melting! Melting! Oh, what a world, what a world!

In America we say "I made a typo error."

i don't know how dey say it in London town, but might be "I say o'l
chap, I mauk a typo!"


with a spoon in the mouth...

[JanPB]

Post by The Starmaker

Post by JanPB
[...]

Post by The Starmaker
Your math is correct but doesn't seem to have any meaning.

No geometric meaning (which I was hoping for).

i see, you have your own approach to doings things no one else does..
Also the P isn't standard and needs more clarification

K = < r^2 gamma' - rho^2 P(gamma') , gamma' >

OK, fair point. Surprisingly perhaps, the Kerr spacetime has
an orthogonal moving frame almost everywhere (away from the
usual suspects like the singular set and the horizons):

e0 = (r^2 + a^2)@/@t + a @/@phi
e1 = @/@r
e2 = @/@theta
e3 = a sin^2(theta) @/@t + @/@phi

e2 and e3 are always spacelike while e0, e1 are always of
the opposite causal character, so e0 and e1 span a
Minkowski signature plane, called the principal plane.

The P I used above denotes the orthogonal projection onto
that plane.

BTW, a geodesic is called principal if its tangent vector
lies in the principal plane.

EXERCISE. Let gamma be a timelike geodesic. Then:
K = 0 if and only if gamma is a principal in
the equatorial plane (theta = pi/2).

further more, there are too many errors for me to list them all...

There are no errors.

you finally reached ...kooksville.

Talk is cheap.

--
Jan

[Ross Finlayson]

Post by JanPB

Post by The Starmaker

Post by JanPB
[...]

Post by The Starmaker
Your math is correct but doesn't seem to have any meaning.

No geometric meaning (which I was hoping for).

i see, you have your own approach to doings things no one else does..
Also the P isn't standard and needs more clarification

K = < r^2 gamma' - rho^2 P(gamma') , gamma' >

OK, fair point. Surprisingly perhaps, the Kerr spacetime has
an orthogonal moving frame almost everywhere (away from the
e2 and e3 are always spacelike while e0, e1 are always of
the opposite causal character, so e0 and e1 span a
Minkowski signature plane, called the principal plane.
The P I used above denotes the orthogonal projection onto
that plane.
BTW, a geodesic is called principal if its tangent vector
lies in the principal plane.
K = 0 if and only if gamma is a principal in
the equatorial plane (theta = pi/2).

further more, there are too many errors for me to list them all...

There are no errors.

you finally reached ...kooksville.

Talk is cheap.
--
Jan

Seems you're trying to figure out
"spaghettification" vis-a-vis "suckage".

One such notion is that of "cube wall".
This is that on one side of the boundary,
the horizon, it's cubic, and space terms,
on the other side gradient, and down.

Then it seems you leave one term out to
skate while building up what first would
have to be a "square wall", with regards
to the plane tangent the horizon.

Instead it's sort of that they each scale down,
from cube to wall.

The jets that usually result of course are
exactly anti-podal what gets input.

So anyways what that begins to address is
that in the very small or otherwise the very
extreme, that units start exchanging, like
mass and length. Now these are matters
of projection then as well about singularities,
so most coat-tailing paper-hangers either
pick a normal partial thats deemed to fall out,
blow up the number of dimensions, or assign non-real
interpretations to the values like negative time,
advised as they are that the success of something
like Clebsch-Gordon like what you got there,
in other situations of "what's squoze", functional
freedom, or various inversions in the space.

Instead there's like that in the Planck-ian,
or, at the horizon, is what gets into some
pretty simple notions geometrically, of
this "cube wall", and, "cube spiral".

Now, this involves geometry in the infinitesimal
or as with regards to "infinite shear" and these
kinds of things, so, anyways what you got there
"what's squoze makes something flattened"
instead of "what's squoze is squoze", has that
usual models that result spaghettification after
usual models of what's suckage, either just say
that your patty-cakes can have examples in the
sky survey found where it looks so, and examples
in the sky survey where it don't.

So, you need something like a "cube wall" and
"cube spiral", what you got there is a "patty-cakes".

[JanPB]

Post by Ross Finlayson
Seems you're trying to figure out
"spaghettification" vis-a-vis "suckage".

No, this is just about a motion constant.

--
Jan

[Ross Finlayson]

Post by JanPB

Post by Ross Finlayson
Seems you're trying to figure out
"spaghettification" vis-a-vis "suckage".

No, this is just about a motion constant.
--
Jan

Same difference.

In case I didn't mention "tidal forces"
they're considered part-and-parcel "spaghettification".

[The Starmaker]

Post by JanPB

Post by The Starmaker

Post by JanPB
[...]

Post by The Starmaker
Your math is correct but doesn't seem to have any meaning.

No geometric meaning (which I was hoping for).

i see, you have your own approach to doings things no one else does..
Also the P isn't standard and needs more clarification

K = < r^2 gamma' - rho^2 P(gamma') , gamma' >

OK, fair point. Surprisingly perhaps, the Kerr spacetime has
an orthogonal moving frame almost everywhere (away from the
e2 and e3 are always spacelike while e0, e1 are always of
the opposite causal character, so e0 and e1 span a
Minkowski signature plane, called the principal plane.
The P I used above denotes the orthogonal projection onto
that plane.
BTW, a geodesic is called principal if its tangent vector
lies in the principal plane.
K = 0 if and only if gamma is a principal in
the equatorial plane (theta = pi/2).

further more, there are too many errors for me to list them all...

There are no errors.

you finally reached ...kooksville.

Talk is cheap.
--
Jan

I'ts not clear what "projection onto the principal plane" means in this
context.


"orthogonal projection"???? Now you sound like Einstein throwing in
more words to hide his mistakes!


Einstein said: "Nothing travels faster than light." ...then he throws
in..."in a vacuum".


How about outside the vacuum?


Einstein sez: "Well, no, I mean.. I don't think so...but vacuum sounds
soooo cool!"

[The Starmaker]

Post by JanPB

Post by The Starmaker

Post by JanPB
[...]

Post by The Starmaker
Your math is correct but doesn't seem to have any meaning.

No geometric meaning (which I was hoping for).

i see, you have your own approach to doings things no one else does..
Also the P isn't standard and needs more clarification

K = < r^2 gamma' - rho^2 P(gamma') , gamma' >

OK, fair point. Surprisingly perhaps, the Kerr spacetime has
an orthogonal moving frame almost everywhere (away from the
e2 and e3 are always spacelike while e0, e1 are always of
the opposite causal character, so e0 and e1 span a
Minkowski signature plane, called the principal plane.
The P I used above denotes the orthogonal projection onto
that plane.
BTW, a geodesic is called principal if its tangent vector
lies in the principal plane.
K = 0 if and only if gamma is a principal in
the equatorial plane (theta = pi/2).

further more, there are too many errors for me to list them all...

There are no errors.

you finally reached ...kooksville.

Talk is cheap.

"Talk is cheap."???? Do you think I have time to do ALL your homework???!


Let me give you another example of the error of your ways...


You wrote: "There does not seem to exist any other symmetry of the Kerr geometry to
yield another constant."

but, but..the hidden symmetry leading to the Carter constant is associated with the Killing tensor, which you didn't mention...
i cannot complete your homework for you.

there are too many errors for me to list them all...


I would suggest you take up another hobby like ...playing the piano but. that doesn't seem to
work for you either.

i throw up my hands.

I'm failing for this class. I give you a B minus.




and PLEASE, don't mix Carter with Noether...you're making a mess!



and don't play the violin!



buy ten cats.

Post by JanPB
--
Jan

[Ross Finlayson]

Post by JanPB
BTW, I have no idea why the lines are wrapped so badly.
It never happened before.
--
Jan

I was reading some theory in the spectral analysis
of molecular weights, and they get into a similar
sort of ansaetze with regards to the quantities.

There where for example you get small-angle
approximation and the nil-square and Laplacian,
instead they have resonance theory and molecular chemistry.

So it's sort of like you're missing a constant, or term,
sort of like the cosmological constant, which of course
everybody knows as both vanishing yet non-zero.

I.e., it's sort of like your linearisations and
dimensionless analysis, have sort of resulted
that your coordinate mapping has not-enough-information.

Anyways other fields have similar sorts of setups
and have made do with more sorts of non-linear
and non-standard analysis, yet what all works up
just fine as resonance theory as more than the
sum of harmonic functions, or that Laplacians
don't look good from every angle.

[Ross Finlayson]

Post by Ross Finlayson

Post by JanPB
BTW, I have no idea why the lines are wrapped so badly.
It never happened before.
--
Jan

I was reading some theory in the spectral analysis
of molecular weights, and they get into a similar
sort of ansaetze with regards to the quantities.
There where for example you get small-angle
approximation and the nil-square and Laplacian,
instead they have resonance theory and molecular chemistry.
So it's sort of like you're missing a constant, or term,
sort of like the cosmological constant, which of course
everybody knows as both vanishing yet non-zero.
I.e., it's sort of like your linearisations and
dimensionless analysis, have sort of resulted
that your coordinate mapping has not-enough-information.
Anyways other fields have similar sorts of setups
and have made do with more sorts of non-linear
and non-standard analysis, yet what all works up
just fine as resonance theory as more than the
sum of harmonic functions, or that Laplacians
don't look good from every angle.

https://link.springer.com/article/10.1007/BF01412752

"The idea of consistently averaging the hydrodynamic interaction
and its various consequences for Hookean dumbbells are reviewed."