Gravitomagnetic Field Equations?

In Einstein's book, "The Meaning of Relativity" pg 100
he writes,

"1. The inertia of a body must increase when ponderable
masses are piled up in its neighbourhood.
2. A body must experience an accelerating force when
neighbouring masses are accelerated, and, in fact, the
force must be in the same direction as that acceleration.
3. A rotating hollow body must generate inside of itself
a 'Coriolis field', which deflects moving bodies in the sense
of rotation, and a radial centrifugal field as well."

IMO, these are basic definitions of the terms
(1) is Mach's Principle, (2) is frame-dragging
and (3) is GravitoMagnetic (GM) force.
(2) and (3) seem to be used interchangeably.

This stuff is relevent to the near $1,000,000,000
Gravity Probe B experiment. (For discussion,
let's consider the Earth is spinning in the xy plane,
z points north and the point to study is on the x axis
with r=x).

What I've been researching is how the Einstein
Field Equations (EFE) can in theory predict GM force,
begiining with a basic analysis. ((So Please do not
post analogies to magnetic force or Kerr metrics,
or g_14 components, as these supposed solutions,
are widely available in relativity literature)).

We are looking for some reason to expect a tangential
GM force (perpendicular to radius) from the EFE's.

The conventional EFE solution for a point outside of
Earths atmosphere (vacuum) is G_uv =0.

How can a tangential GM force (as defined by 3
above) be *suggested* by the EFE's?

IMO, G_uv=0 cannot do this, because,
its obvious that G_uv = G_vu =0, so
G_12 = G_21 and G_14 = G_41 and so on.
It's a matter that a tangential GM force in direction y
or -y has no polarization information, (it could if G_12
contained nonsymmetrical elements, but it doesn't)

Next, let's consider G_uv = kT_uv at the location
of the satellite, with it's mass, speed, etc described
by T_uv, so that T_uv is non-zero.

Also lets place a second satellite at the same location
orbiting (CW= ClockWise) in the opposite direction,
to the first satellite so that we create two EFE's,

G_uv (CW) = kT_uv (CW)

G_uv (CCW) = kT_uv (CCW)

that can be compared. GM force predicts a difference
in the effects of the spinning Earth on satelite CW and
CCW because the relative spin of Earth will be different.

Subtracting the RHS's above gives a tensor (I'll use
Y because that is the expected direction of GM force),

Y_uv = T_uv (CW) - T_uv (CCW)

and Y_uv is the GM energy-momentum tensor.

Y_uv must be anti-symmetrical in order to find
the differences in the GM force is in the direction
of +y or -y.

Since Y_uv is a difference relating two CS's, it can
be defined generally by

Y_uv = T_uv;w U^w = DT_uv, (U^w=dx^w/ds),

where DT_uv is the absolute derivative of the energy-
momentum tensor.
In the case of our 2 satellites experiment each at the same
location, a non-zero invariant difference DT_uv would
make GM force invariant and therefore real, in the sense
it cannot be transformed away in any CS.

IMO, when Y_uv is antisymmetrical, then T_uv is
nonsymmetrical, and then G_uv is nonsymmetrical.
This in turn (I think) would require nonsymmetrical
Christoffel symbols, as Einstein included in his book
(ref above), "Including the Relativistic Theory of the
Non-Symmetric Field" on the front cover.

IMHO Einstein had an obsessive desire to incorporate
his point (1) above (Mach's Principle) into General
Relativity, that I find requires a non-symmetric G_uv.

Regards
Ken S. Tucker

(Ken S. Tucker) wrote in message . com...
Gravitomagnetic Field Equations?

In Einstein's book, "The Meaning of Relativity" pg 100
he writes,

"1. The inertia of a body must increase when ponderable
masses are piled up in its neighbourhood.
2. A body must experience an accelerating force when
neighbouring masses are accelerated, and, in fact, the
force must be in the same direction as that acceleration.
3. A rotating hollow body must generate inside of itself
a 'Coriolis field', which deflects moving bodies in the sense
of rotation, and a radial centrifugal field as well."

IMO, these are basic definitions of the terms
(1) is Mach's Principle, (2) is frame-dragging
and (3) is GravitoMagnetic (GM) force.
(2) and (3) seem to be used interchangeably.

This stuff is relevent to the near $1,000,000,000
Gravity Probe B experiment. (For discussion,
let's consider the Earth is spinning in the xy plane,
z points north and the point to study is on the x axis
with r=x).

What I've been researching is how the Einstein
Field Equations (EFE) can in theory predict GM force,
begiining with a basic analysis. ((So Please do not
post analogies to magnetic force or Kerr metrics,
or g_14 components, as these supposed solutions,
are widely available in relativity literature)).

We are looking for some reason to expect a tangential
GM force (perpendicular to radius) from the EFE's.

The conventional EFE solution for a point outside of
Earths atmosphere (vacuum) is G_uv =0.

How can a tangential GM force (as defined by 3
above) be *suggested* by the EFE's?

IMO, G_uv=0 cannot do this, because,
its obvious that G_uv = G_vu =0, so
G_12 = G_21 and G_14 = G_41 and so on.
It's a matter that a tangential GM force in direction y
or -y has no polarization information, (it could if G_12
contained nonsymmetrical elements, but it doesn't)

Next, let's consider G_uv = kT_uv at the location
of the satellite, with it's mass, speed, etc described
by T_uv, so that T_uv is non-zero.

Also lets place a second satellite at the same location
orbiting (CW= ClockWise) in the opposite direction,
to the first satellite so that we create two EFE's,

G_uv (CW) = kT_uv (CW)

G_uv (CCW) = kT_uv (CCW)

that can be compared. GM force predicts a difference
in the effects of the spinning Earth on satelite CW and
CCW because the relative spin of Earth will be different.

Subtracting the RHS's above gives a tensor (I'll use
Y because that is the expected direction of GM force),

Y_uv = T_uv (CW) - T_uv (CCW)

and Y_uv is the GM energy-momentum tensor.

Y_uv must be anti-symmetrical in order to find
the differences in the GM force is in the direction
of +y or -y.

Since Y_uv is a difference relating two CS's, it can
be defined generally by

Y_uv = T_uv;w U^w = DT_uv, (U^w=dx^w/ds),

where DT_uv is the absolute derivative of the energy-
momentum tensor.
In the case of our 2 satellites experiment each at the same
location, a non-zero invariant difference DT_uv would
make GM force invariant and therefore real, in the sense
it cannot be transformed away in any CS.

IMO, when Y_uv is antisymmetrical, then T_uv is
nonsymmetrical, and then G_uv is nonsymmetrical.
This in turn (I think) would require nonsymmetrical
Christoffel symbols, as Einstein included in his book
(ref above), "Including the Relativistic Theory of the
Non-Symmetric Field" on the front cover.

IMHO Einstein had an obsessive desire to incorporate
his point (1) above (Mach's Principle) into General
Relativity, that I find requires a non-symmetric G_uv.

Regards
Ken S. Tucker

xxein: Anybody can and will make math to describe anything they
desire. That is the power of imaginative math. It does not follow
into reality physics.

So make arguements wrt math and expect that to affect reality???

Reality needs more understanding than you or anybody can furnish at
this time. We just pretend with belief that we know anything. Can
you agree with that? If not, join the same math lalaland with the
rest and recognise that it is all a construct that pleases your fancy.

Again, Ken, this raises interesting issues and is not directed at you
per se. You just raise interesting discussion.

(xxein) wrote in message om...

(Ken S. Tucker) wrote in message . com...

[snip stuff xxein doesn't directly ref too]

IMHO Einstein had an obsessive desire to incorporate
his point (1) above (Mach's Principle) into General
Relativity, that I find requires a non-symmetric G_uv.
Ken S. Tucker

xxein: Anybody can and will make math to describe anything they
desire. That is the power of imaginative math. It does not follow
into reality physics.

Given the enormously technical theory of Color TV,
with all these EM waves, antenna and all, well...
I doubt it is possible, I flatly refuse to accept such
trickery, but I smell better after seeing a deodorant
commercial.

So make arguements wrt math and expect that to affect reality???

I bought deodorant!

Reality needs more understanding than you or anybody can furnish at
this time. We just pretend with belief that we know anything. Can
you agree with that?

No, I smell better, because my wife lets me watch her Color TV!
(mine is theoretically impossible :-)

If not, join the same math lalaland with the
rest and recognise that it is all a construct that pleases your fancy.
Again, Ken, this raises interesting issues and is not directed at you
per se. You just raise interesting discussion.

Thanks xxein, I'm sharing my excitement about NASA's
Gravity Probe B experiment. From what I understand this
is by far the most delicate instrument ever created. It's like
a microscope that sees tiny bits of the gravitational field.
Moreover, used wisely, it will test our intellectual limits
and strengths and define our ignorance and weakness.
Like MMX, whatever the GP-B experiment reports
will be of tremendous importance. I see the GP-B
experiment as a modernized version of the MMX.
Regards
Ken S. Tucker

"Ken S. Tucker" wrote in message
om...
Gravitomagnetic Field Equations?

In Einstein's book, "The Meaning of Relativity" pg 100
he writes,

"1. The inertia of a body must increase when ponderable
masses are piled up in its neighbourhood.
2. A body must experience an accelerating force when
neighbouring masses are accelerated, and, in fact, the
force must be in the same direction as that acceleration.
3. A rotating hollow body must generate inside of itself
a 'Coriolis field', which deflects moving bodies in the sense
of rotation, and a radial centrifugal field as well."

IMO, these are basic definitions of the terms
(1) is Mach's Principle, (2) is frame-dragging
and (3) is GravitoMagnetic (GM) force.
(2) and (3) seem to be used interchangeably.

This stuff is relevent to the near $1,000,000,000
Gravity Probe B experiment. (For discussion,
let's consider the Earth is spinning in the xy plane,
z points north and the point to study is on the x axis
with r=x).

What I've been researching is how the Einstein
Field Equations (EFE) can in theory predict GM force,
begiining with a basic analysis. ((So Please do not
post analogies to magnetic force or Kerr metrics,
or g_14 components, as these supposed solutions,
are widely available in relativity literature)).

We are looking for some reason to expect a tangential
GM force (perpendicular to radius) from the EFE's.

The conventional EFE solution for a point outside of
Earths atmosphere (vacuum) is G_uv =0.

How can a tangential GM force (as defined by 3
above) be *suggested* by the EFE's?

Hi Ken. I was under the opinion that Lense-Thirring frame dragging provided
the "gravitomagnetic" component but I mostly got that notion from Ciufollini
and Wheeler's [CW] treatment in "Gravitation and Inertia". That book does
make extensive use of analogies to electrodynamics (as do most every book or
paper I've seen on the subject) as well as the Kerr metric and various
initial value assumptions which you may not find to your liking. CW's
treatment is pretty subtle - I like the paper at
http://xxx.lanl.gov/PS_cache/gr-qc/pdf/0207/0207065.pdf
which gives an excellent treatment of Lense-Thirring and gravitomagnetism.

Regards,
Perion

"Ken S. Tucker" wrote in message

In admiration of Einsteins Field Equations and the high
standard of theory they represent, the gravitomagnetic
field tensor was deduced in the first post in this thread
given by,

Y_uv = DT_uv

where Y_uv is the gravitomagnetic energy-momentum
tensor.

The tensor Y_uv is antisymmetrical like the Electro-
magnetic field tensor F_uv and so analogies comparing
Y_uv to F_uv are abundant. Some people think that
if F_uv has magnetic properties then Y_uv does and is
non-zero! That's what I question.

Let's go back to your first post regarding two (earth) orbiting satellites
orbiting in opposite directions. Actually, I puzzled over this very same
scenario when I first encountered frame dragging as a possible souce for
rotational inertia and am still puzzled. Anyway, you stated:

"We are looking for some reason to expect a tangential
GM force (perpendicular to radius) from the EFE's.
The conventional EFE solution for a point outside of
Earths atmosphere (vacuum) is G_uv =0.
How can a tangential GM force (as defined by 3
above) be *suggested* by the EFE's?"

Let's forget about the two satellites and just take the earth as the
example. The earth spins in relation to the entire universe. Supposedly it's
this relative motion which produces the tangential GM "force" exhibited by
equatorial bulging. If the earth were spinning in the opposite direction we
would still have that same inertial effect. How can frame dragging by the
universe-earth's relative motion produce the same tangential force in both
cases? Answer - beats me. Obviously, for rotational motion of the earth,
each constituent particle is being constantly forced from their geodesics
regardless of any orientation or spin direction. So maybe the answer must
lie in components in Riemann (which defines the local geodesics entirely)
not found in Einstein tensor G that the universe primarily dictates in the
local region of spacetime. Remember - stress-energy tensor T only provides
information that relates to Einstein G - not the rest of Riemann. G = kT is
really about the relative acceleration (geodesic separation vector
acceleration) for very close test particles in free fall and says nothing
about rotational inertial effects - I think.

Anyway, I haven't studied either Kerr or Lense-Thirring enough to know if or
how or to what degree they are relevant to rotational inertia. In other
words, I'm still pretty much in the dark. I'm hoping others can shed some
light.

Best regards,
Perion

"Perion" wrote in message ...

"Ken S. Tucker" wrote in message
. com...
[snip]

What I've been researching is how the Einstein
Field Equations (EFE) can in theory predict GM force,
begiining with a basic analysis. ((So Please do not
post analogies to magnetic force or Kerr metrics,
or g_14 components, as these supposed solutions,
are widely available in relativity literature)).
We are looking for some reason to expect a tangential
GM force (perpendicular to radius) from the EFE's.
The conventional EFE solution for a point outside of
Earths atmosphere (vacuum) is G_uv =0.
How can a tangential GM force be *suggested* by the EFE's?

Hi Ken. I was under the opinion that Lense-Thirring frame dragging provided
the "gravitomagnetic" component but I mostly got that notion from Ciufollini
and Wheeler's [CW] treatment in "Gravitation and Inertia". That book does
make extensive use of analogies to electrodynamics (as do most every book or
paper I've seen on the subject) as well as the Kerr metric and various
initial value assumptions which you may not find to your liking. CW's
treatment is pretty subtle - I like the paper at
http://xxx.lanl.gov/PS_cache/gr-qc/pdf/0207/0207065.pdf
which gives an excellent treatment of Lense-Thirring and gravitomagnetism.
Regards,
Perion

Hi Perion (this is my second post to your reply)
I studied the first part of gr-qc0207/0207065, and using
this ref. I can be very specific about my apphrensions,
which anyone can kindly check and correct.

For brevity I'll sub Eq.(10) into (8) and write this,

h^0i = S^n x^k e^i_nk [-2*G/c^3r^3] Eq.8(10)

e^i_nk is Levi-Cevita's antisymetrical tensor and
the stuff in [] is not germain, latin indices are summed
over 1,2,3.

Eq.8(10) supposedly transmits the information about
the rotating source mass to the field, including
rotation direction information.

I find h^0i =0, here's why...

1) I implicitly assume S^n x^k = x^k S^n,
because the notation implies an outer product.

2) I assume summation over indices 'n' and 'k',
as they are repeated twice in the same term.

3) I assume the permutation e^i_nk = - e^i_kn.

So if we arbitarily set i=1, and []=1 then sum,

h^01 = S^2 x^3 e^1_23 + S^3 x^2 e^1_32 =0

because e^1_23 = - e^1_32.

Apart from this problem, it seems impossible to
encode rotation direction information - which
requires asymetry - into the symmetrical
component h^0i.

Regards
Ken S. Tucker

Ken S. Tucker wrote:
: Gravitomagnetic Field Equations?
: In Einstein's book, "The Meaning of Relativity" pg 100
: he writes,
:[snip]
: IMHO Einstein had an obsessive desire to incorporate
: his point (1) above (Mach's Principle) into General
: Relativity, that I find requires a non-symmetric G_uv.

In some stages of his life, yes, but I don't
think all of his writings support Mach's views.

It is almost certain that there will be
_no_ forces acting on a small satellite, but the
problem is complicated by apparent effects that
relate to a number of things, and those may be
confused with forces acting (precession, etc.).

Joe Fischer

--
3

"Perion" wrote in message ...

"Ken S. Tucker" wrote in message

Hi Perion, read through your post several times.

In admiration of Einsteins Field Equations and the high
standard of theory they represent, the gravitomagnetic
field tensor was deduced in the first post in this thread
given by,

Y_uv = DT_uv

where Y_uv is the gravitomagnetic energy-momentum
tensor.

The tensor Y_uv is antisymmetrical like the Electro-
magnetic field tensor F_uv and so analogies comparing
Y_uv to F_uv are abundant. Some people think that
if F_uv has magnetic properties then Y_uv does and is
non-zero! That's what I question.

Let's go back to your first post regarding two (earth) orbiting satellites
orbiting in opposite directions. Actually, I puzzled over this very same
scenario when I first encountered frame dragging as a possible souce for
rotational inertia and am still puzzled.

Well, great minds think alike! But in all sincerity it
seems that the CW and CCW satellite should operate
in different fields iff gravitomagnetic force is real.
(By real I mean not a CS artifact, I mean invariant).

Anyway, you stated:
"We are looking for some reason to expect a tangential
GM force (perpendicular to radius) from the EFE's.
The conventional EFE solution for a point outside of
Earths atmosphere (vacuum) is G_uv =0.
How can a tangential GM force (as defined by 3
above) be *suggested* by the EFE's?"

Let's forget about the two satellites and just take the earth as the
example. The earth spins in relation to the entire universe.

If you accept there is no absolute Frame of Reference,
even accelerating FoR's as GR denies, then Earth's
spin is relative and does not acquire any absolute nature
relative to the "entire universe". (being a bit picky, but
GR means the mention of Earth's spin is meaningless,
ie. it does not exist in GR).

Supposedly it's
this relative motion which produces the tangential GM "force" exhibited by
equatorial bulging. If the earth were spinning in the opposite direction we
would still have that same inertial effect. How can frame dragging by the
universe-earth's relative motion produce the same tangential force in both
cases? Answer - beats me.

Well sir, we share a common ignorance.

Obviously, for rotational motion of the earth,
each constituent particle is being constantly forced from their geodesics
regardless of any orientation or spin direction. So maybe the answer must
lie in components in Riemann (which defines the local geodesics entirely)
not found in Einstein tensor G that the universe primarily dictates in the
local region of spacetime. Remember - stress-energy tensor T only provides
information that relates to Einstein G - not the rest of Riemann. G = kT is
really about the relative acceleration (geodesic separation vector
acceleration) for very close test particles in free fall and says nothing
about rotational inertial effects - I think.

Agreed, we have parallel thought.

Anyway, I haven't studied either Kerr or Lense-Thirring enough to know if or
how or to what degree they are relevant to rotational inertia. In other
words, I'm still pretty much in the dark. I'm hoping others can shed some
light.

Best regards,
Perion

Thanks, and same to you.
Ken S. Tucker
PS: At time of writing a did a second post to your first post,
but these postings are delayed for some f$*#ing reason.
KST

----- Original Message -----
From: "Ken S. Tucker"
Newsgroups: sci.physics.relativity
Sent: Tuesday, September 16, 2003 3:16 PM
Subject: Gravitomagnetic Field Equations?


"Perion" wrote in message
...

"Ken S. Tucker" wrote in message

Let's go back to your first post regarding two (earth) orbiting
satellites
orbiting in opposite directions. Actually, I puzzled over this very same
scenario when I first encountered frame dragging as a possible souce for
rotational inertia and am still puzzled.

Well, great minds think alike! But in all sincerity it
seems that the CW and CCW satellite should operate
in different fields iff gravitomagnetic force is real.
(By real I mean not a CS artifact, I mean invariant).

1. I was hasty in dismissing the two satellite scenario.
2. What do you mean by "operate in different fields"? Like 3 below?
3. In my opinion, maybe right - maybe wrong - if GM exists then I can't see
how (the earth's) frame dragging action on the two satellites could possibly
have the same directional effect on each of them. If it wouldn't then how
can we use frame dragging as a source for the local GM field that should
exist due to the relative rotational motion of the universe-earth system? I
ask that because that GM field is supposedly the major factor in accounting
for rotational inertial effects - unless I'm reading Ciufolini and Wheeler
wrong. BTW - do have that book ("Gravitation and Inertia")?
4. Regardless of any of the above, Gravity Probe B, costly and time
consuming as it may be, seems like an interesting test of frame dragging.
I've been anxiously waiting for years now....



Let's forget about the two satellites and just take the earth as the
example. The earth spins in relation to the entire universe.

If you accept there is no absolute Frame of Reference,
even accelerating FoR's as GR denies, then Earth's
spin is relative and does not acquire any absolute nature
relative to the "entire universe". (being a bit picky, but
GR means the mention of Earth's spin is meaningless,
ie. it does not exist in GR).

No absolute nature for sure! But that's the very problem. Take some huge
(very massive) hollow shell with a very small spherical mass at its center.
It happens that we live on that sphere. Assume there is some relative
rotional motion between the two which is exibited by measurable interial
effects upon the sphere. We could say the sphere is spinning and the shell
is stationary and account for the rotional inertia in the standard manner.
We should just as easily be able say (via GR) that the shell is spinning and
the sphere is stationary. In that case how do we account for those inertial
effects upon the sphere? Wheeler et al view the GM field as the source of
those effects - the huge mass-shell dictating the local (sphere's) GM
curvature that arises due to the relative rotational motion. That is,
dictating what constitutes deviation from local geodesics for the sphere's
constituent particle masses according to whatever Riemann components that GM
field is embodied. It's interesting to consider the same scenario in
electrodynamics only now we have a heavily charged shell and lightly charged
sphere with no relative motion between them (electric field - no magnetic
field) and then again with relative motion (electric and magnetic field).

Check out the diagrams -
http://www.aip.org/physnews/graphics/html/gravmag.htm - gravitomagnetic
induction producing matter currents. Looks like the standard electric
field/magnetic field stuff with the labels changed. Wheeler is honest
though. He states that the similarities (analogies) between electromagnetism
and gravitomagnetism are very crude and only go so far. He never pushes them
too far.

PS: At time of writing a did a second post to your first post,
but these postings are delayed for some f$*#ing reason.
KST
:-)
Thanks for the discussion Ken. I may have time today to get to the other
post. Not sure.

Adios Amigo,
Perion

(Ken S. Tucker) wrote in message . com...

I GOOFED MAYBE...

My post on 09-16 09:28 to this thread contains an error,
please forgive.

Hi Perion (this is my second post to your reply)
I studied the first part of gr-qc0207/0207065, and using
this ref. I can be very specific about my apphrensions,
which anyone can kindly check and correct.

For brevity I'll sub Eq.(10) into (8) and write this,

h^0i = S^n x^k e^i_nk [-2*G/c^3r^3] Eq.8(10)

e^i_nk is Levi-Cevita's antisymetrical tensor and
the stuff in [] is not germain, latin indices are summed
over 1,2,3.

Eq.8(10) supposedly transmits the information about
the rotating source mass to the field, including
rotation direction information.

I find h^0i =0, here's why...
(( ERROR ))

1) I implicitly assume S^n x^k = x^k S^n,
because the notation implies an outer product.

2) I assume summation over indices 'n' and 'k',
as they are repeated twice in the same term.

3) I assume the permutation e^i_nk = - e^i_kn.

So if we arbitarily set i=1, and []=1 then sum,

h^01 = S^2 x^3 e^1_23 + S^3 x^2 e^1_32

and because e^1_23 = - e^1_32,

the CORRECTED substitution for Eq. 8(10) is,

h^01 = S^2 x^3 - S^3 x^2.

But that can't work either, because we're setting

Symmetrical (0,1) = Antisymmetrical(2,3)

I think the Levi-Cevita tensor requires 4 indices
in space-time, however Eq. 8(10) has only 3.
In 4D one requires, e_uvab where indices u,v,a,b
can take on any dimension 0,1,2,3.

Apart from this problem, it seems impossible to
encode rotation direction information - which
requires asymetry - into the symmetrical
component h^0i.

Regards
Ken S. Tucker