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From: Tunel Vision on 9 Dec 2009 16:38
On Dec 8, 1:43 am, Tom Roberts <tjroberts...(a)sbcglobal.net> wrote:
> Igor wrote:
> > A static metric is defined as one that only depends on the spatial
> > coordinates.
>
> No. It's considerably more complicated than that, because the
> coordinates themselves are arbitrary, and in a small enough region of
> any manifold with metric one could always come up with an outlandish
> coordinate system such that the "metric only depends on the spatial
> coordinates".
>
> Here are the relevant definitions, in geometrical terms; all manifolds
> are Lorentzian (i.e. 4-d para-compact semi-Riemannian manifolds with
> signature 2; they are usually also Hausdorff):

Sir, thanks, this is long definition

i only understand if i visualize, how would i visualize a

"4-d para-compact semi-Riemannian manifolds with
signature 2; they are usually also Hausdorff"

thanks

>
> A region of a manifold is said to be "stationary" iff there is a
> timelike Killing vector throughout the region.

again, how to visualize a

"timelike Killing vector throughout the region"

thanks

>
> Examples: Minkowski spacetime (infinite number of timelike
> killing vectors); the region outside the horizon of Kerr
> spacetime (one timelike Killing vector).
>
> A region of a manifold is said to be "static" iff it is stationary and
> at every point in the region there exists a neighborhood of the point in
> question with a 3-d spatial submanifold orthogonal to the timelike
> Killing vector at every point within the neighborhood.

how a 3d spatial manifold can be orthogonal
to something timelike, Killing vector

and what is a "Killing vector" anyway

thanks

>
> Examples: Minkowski spacetime; the region outside the
> horizon of Schwarzschild spacetime.
>
> NOTE: some authors apply "stationary" and "static" to the metric, some
> to the manifold itself; I do the latter (because in physics we only
> consider a "manifold with metric" and the last two words are
> often/usually omitted).
>
> Tom Roberts

From: Tom Roberts on 9 Dec 2009 22:25
[This thread has been separated from an identical one in
sci.physics.foundations. This is how I replied there.]

Ken S. Tucker wrote:
> At 1st strike, I find two physical instances of a "static"
> g-field, where the g-potential relating two bodies with masses
> M and m, remains constant,
> 1) Circular orbit.
> 2) m on the surface of M, such as we (m) sit in a chair.

Those are NOT instances of a static gravitational field (as being discussed in
this thread). Those are merely instances of locations at which the metric is
constant. The key notion you missed is that a static g-field is static
throughout a REGION of the manifold, not just at a single point, or on a path or
surface.

Need I point out that a region of a manifold necessarily has
the same dimensionality as the manifold itself? Your orbit is
only 1-D, and your surface is only 3-D, but the manifold is of
course 4-D.

Exercise for Ken: explain why your examples are only 1-D and 3-D.

[Ken, this is a VERY simple exercise, intended to permit you
to assess your own understanding. Or lack thereof.]


There is no such thing as a static g-field with two masses, unless they are
rigidly connected (in which case they act as a single object; beware of the
impossibility of truly rigid connections in relativity). "Sitting in a chair"
qualifies, and a universe consisting of just a planet and you sitting motionless
(forever) on a chair could be a static manifold.

Exercise for Ken: I said COULD be -- what other conditions are
required for that to be a static manifold?

[Ken, this is a VERY simple exercise, intended to permit you
to assess your own understanding. Or lack thereof.]


> We can employ the geodesic as ref'd here, [...]

Of course such a geodesic is only a 1-D path (in a 4-D manifold). So you could
"employ" it, but it cannot give any understanding about "static g-field",
because it does not define a REGION of the manifold. Of course if you had
instead discussed a congruence of geodesics, that COULD be used to do so....

Exercise for Ken: explain why a geodesic is 1-D.

[Ken, this is a VERY simple exercise, intended to permit you
to assess your own understanding. Or lack thereof.]

Advanced exercise: explain why a congruence of geodesics can
span a 4-D region of the manifold. Hint1: it need not do so.
Hint2: first explain what "congruence" means in this context.

More advanced exercise: explain how a 4-D congruence of
geodesics can be used to determine whether or not the
region it spans is static or not. Hint: I have an idea,
but have not proven it does this....


Tom Roberts
From: Ken S. Tucker on 10 Dec 2009 11:51
On Dec 9, 7:25 pm, Tom Roberts <tjroberts...(a)sbcglobal.net> wrote:
> [This thread has been separated from an identical one in
> sci.physics.foundations. This is how I replied there.]
>
> Ken S. Tucker wrote:
> > At 1st strike, I find two physical instances of a "static"
> > g-field, where the g-potential relating two bodies with masses
> > M and m, remains constant,
> > 1) Circular orbit.
> > 2) m on the surface of M, such as we (m) sit in a chair.
>
> Those are NOT instances of a static gravitational field (as being discussed in
> this thread). Those are merely instances of locations at which the metric is
> constant. The key notion you missed is that a static g-field is static
> throughout a REGION of the manifold, not just at a single point, or on a path or
> surface.

I'm suggesting obtaining a deeper understanding of
the problem from a study of geodesics.
OTOH, Einstein provides the metrics for a static
field in GR1916, Eq.(70), (available online), but
that should be well known to a GR student.
....
Ken S. Tucker
From: eric gisse on 10 Dec 2009 13:51
Ken S. Tucker wrote:

> On Dec 9, 7:25 pm, Tom Roberts <tjroberts...(a)sbcglobal.net> wrote:
>> [This thread has been separated from an identical one in
>> sci.physics.foundations. This is how I replied there.]
>>
>> Ken S. Tucker wrote:
>> > At 1st strike, I find two physical instances of a "static"
>> > g-field, where the g-potential relating two bodies with masses
>> > M and m, remains constant,
>> > 1) Circular orbit.
>> > 2) m on the surface of M, such as we (m) sit in a chair.
>>
>> Those are NOT instances of a static gravitational field (as being
>> discussed in this thread). Those are merely instances of locations at
>> which the metric is constant. The key notion you missed is that a static
>> g-field is static throughout a REGION of the manifold, not just at a
>> single point, or on a path or surface.
>
> I'm suggesting obtaining a deeper understanding of
> the problem from a study of geodesics.

God, what a horrible idea. You are too braindead to know why, though.

> OTOH, Einstein provides the metrics for a static
> field in GR1916, Eq.(70), (available online), but
> that should be well known to a GR student.
> ...
> Ken S. Tucker

Do you ever get bored of prattling about 'GR1916' over and over when it is
clear you don't understand GR at any level?
From: Ken S. Tucker on 10 Dec 2009 14:21
On Dec 10, 10:51 am, eric gisse <jowr.pi.nos...(a)gmail.com> wrote:
> Ken S. Tucker wrote:
> > On Dec 9, 7:25 pm, Tom Roberts <tjroberts...(a)sbcglobal.net> wrote:
> >> [This thread has been separated from an identical one in
> >> sci.physics.foundations. This is how I replied there.]
>
> >> Ken S. Tucker wrote:
> >> > At 1st strike, I find two physical instances of a "static"
> >> > g-field, where the g-potential relating two bodies with masses
> >> > M and m, remains constant,
> >> > 1) Circular orbit.
> >> > 2) m on the surface of M, such as we (m) sit in a chair.
>
> >> Those are NOT instances of a static gravitational field (as being
> >> discussed in this thread). Those are merely instances of locations at
> >> which the metric is constant. The key notion you missed is that a static
> >> g-field is static throughout a REGION of the manifold, not just at a
> >> single point, or on a path or surface.
>
> > I'm suggesting obtaining a deeper understanding of
> > the problem from a study of geodesics.

> > OTOH, Einstein provides the metrics for a static
> > field in GR1916, Eq.(70), (available online), but
> > that should be well known to a GR student.
> > ...
> > Ken S. Tucker
>
> Do you ever get bored of prattling about 'GR1916'

Yes, only about 20 guys on the planet understand GR
and one or two post, I do so sporadically.

Unlike Gisse and Roberts, real THEORETICIANS take
the metrics to experiment, that involves REAL
measurements of the predicted geodesics based on
the metric quantities, to VERIFY the symmetric and
antisymmetic components that may be available to the
unwashed masses (such as Gisse), in 20-50 years, but
I have no hope of that happening in my life time.
Ken S. Tucker
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