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Misinterpretation of the radial parameter in the Schwarzschild solution?
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Jan Bielawski
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Joined: 08 May 2005
Posts: 388

PostPosted: Fri Jul 21, 2006 6:34 am    Post subject: Re: Misinterpretation of the radial parameter in the Schwarzschild solution? Reply with quote

LEJ Brouwer wrote:
Quote:
Daryl McCullough wrote:

That seems to be the reasoning---that only the region r >= 2m is physically
meaningful, so if the mass is anywhere, it is at r=2m.

No, the mass it at r = r0 by definition, and this corresponds to r*(r0)
= 2m. It is not at all as vague or subjective as you try to make out.

OK, define this precisely. Crothers' paper does a terrible job
explaining this point. I can understand the (unnecessary) rewriting of
the metric in the form of his equation (7):

ds^2 = (sqrt(C)-alpha)/sqrt(C) dt^2 - (sqrt(C)/(sqrt(C)-alpha))
C'^2/(4C) dr^2 - c dO^2

....where C(r) is a positive function.

What I don't understand is his constant shifting of the definitions.
Specifically: how is r0 defined? He says "Let the test particle at r0
acquire mass". So if the particle is at r0 then r0 must be the origin
because of the spherical symmetry. But a bit later he has:
"Furthermore, one can see from (13) and (14) that r0 is arbitrary [?],
i.e. the point-mass can be located at any point [??] and its location
has no intrinsic meaning". How does (13) or (14) [the formula for the
distance from r0 to another radial position r] imply that r0 is
"arbitrary"? If he means by that that coordinate value is of no
physical significance then he is stating a triviality (albeit dressed
up as a revelation), if he means that the actual position of the point
can be anywhere, then he is obviously wrong by the symmetry. What am I
missing?

--
Jan Bielawski
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Sue...
science forum Guru


Joined: 08 May 2005
Posts: 2684

PostPosted: Fri Jul 21, 2006 8:02 am    Post subject: Re: Misinterpretation of the radial parameter in the Schwarzschild solution? Reply with quote

JanPB wrote:

snip

Quote:

As I suspected, at the bottom of this is the confusing "switch" of
coordinates between spacelike and timelike. Even the naming conventions
("r" and "t") are powerful tricksters.

I suspect the same. Especially after seeing posters to this thread
that should know better, defend a position mathematical rigour is
not necessary to remove spatial components from a space-time
interval. The royal invocation of the term 'time-like' is not a
substitute
for the rigourous appplication of imaginary operators to transform
from
the Lorenz gauge (anisotropic where 1/r^2 is meaningless)
back to the Coulomb gauge ( isotropic where 1/r^2 correctly
expresses the attenuation on a radial path).

[correct placement shown near bottom of page]
http://www.nrao.edu/~smyers/courses/astro12/speedoflight.html
"Gauge Transformations"
http://arxiv.org/abs/physics/0204034
"Space time"
http://farside.ph.utexas.edu/teaching/em/lectures/node113.html

Sue...

Quote:
--
Jan Bielawski
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LEJ Brouwer
science forum Guru Wannabe


Joined: 07 May 2005
Posts: 120

PostPosted: Fri Jul 21, 2006 10:59 am    Post subject: Re: Misinterpretation of the radial parameter in the Schwarzschild solution? Reply with quote

Tom Roberts wrote:
Quote:
As is well known...

Well, there's no arguing with that. You win once again.

Quote:
Tom Roberts

- Sabbir.
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LEJ Brouwer
science forum Guru Wannabe


Joined: 07 May 2005
Posts: 120

PostPosted: Fri Jul 21, 2006 11:39 am    Post subject: Re: Misinterpretation of the radial parameter in the Schwarzschild solution? Reply with quote

JanPB wrote:
Quote:
LEJ Brouwer wrote:
Daryl McCullough wrote:

That seems to be the reasoning---that only the region r >= 2m is physically
meaningful, so if the mass is anywhere, it is at r=2m.

No, the mass it at r = r0 by definition, and this corresponds to r*(r0)
= 2m. It is not at all as vague or subjective as you try to make out.

OK, define this precisely. Crothers' paper does a terrible job
explaining this point. I can understand the (unnecessary) rewriting of
the metric in the form of his equation (7):

ds^2 = (sqrt(C)-alpha)/sqrt(C) dt^2 - (sqrt(C)/(sqrt(C)-alpha))
C'^2/(4C) dr^2 - c dO^2

...where C(r) is a positive function.

What I don't understand is his constant shifting of the definitions.
Specifically: how is r0 defined? He says "Let the test particle at r0
acquire mass". So if the particle is at r0 then r0 must be the origin
because of the spherical symmetry. But a bit later he has:
"Furthermore, one can see from (13) and (14) that r0 is arbitrary [?],
i.e. the point-mass can be located at any point [??] and its location
has no intrinsic meaning". How does (13) or (14) [the formula for the
distance from r0 to another radial position r] imply that r0 is
"arbitrary"? If he means by that that coordinate value is of no
physical significance then he is stating a triviality (albeit dressed
up as a revelation), if he means that the actual position of the point
can be anywhere, then he is obviously wrong by the symmetry. What am I
missing?

Okay, Crothers' wording is not great here. Basically, 'r' parametrises
2-spheres in some way (there is an arbitrariness in this choice). Given
any choice of parametrisation, the value of 'r' corresponding to the
position of the point mass, and hence the origin, must correspond to
some value of r, which is taken to be r0. So, yes he is 'stating a
triviality'. I don't agree that he has exaggerated the point because by
taking this little extra care, he has indeed come up with a
'revelation', i.e. that the event horizon and the point mass coincide.

Quote:

--
Jan Bielawski

- Sabbir.

P.S. Apologies for the appalling spelling and grammar in my last post -
it was 4am and my brain was working on impulse power!
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