[Moderator's note: Seņor Tessel, can you limit your lines to 72
characters? This will allow them to be quoted without becoming too long.
I have reformatted your post to have shorter lines. -P.H.]
On Mon, 28 Aug 2006, JanPB wrote:
> Confusingly, the components in the Eddington-Finkelstein basis do change
> upon t->-t.
Not confusing at all once you realize that the coordinate basis vector
field @/@T in the EF chart is distinct from the coordinate basis vector
field @/@t in the Schwarzschild chart, which happens to be the Killing
vector field (timelike in each exterior region). So by definition, @/@t
but not @/@T expresses a symmetry in the metric tensor (a self-isometry
of the Lorentzian spacetime).
To verify this, put
T = t + int 1/sqrt(1-2m/r) dr
where r is the Schwarzschild radial coordinate. Plugging this into the
Schwarzschild line element in the exterior region gives the line element
for the Eddington chart, which is continued into the future interior in
the obvious way (namely, extend the range of r but don't change anything
else). Now you can verify that @/@T does NOT solve the Killing equation
(written down in the new chart). Next, you can transform @/@t into the
new chart and verify that it -is- a Killing vector field.
Even better, put
Tbar = t + int sqrt(2m/r)/sqrt(1-2m/r) dr
to obtain the far superior Painleve chart, in which the hyperslices
orthogonal to the Lemaitre observers have been flattened out so that
they are represented by coordinate hyperplanes, as they should be
because they have the geometry of E^3. This is also just what you want
for the OS model, where you must match to an FRW dust with flat
hyperslices. See for example gr-qc/0001069 or the book by Frolov and
Novikov, Black Hole Physics.
I am getting a bit sick of urging readers of this group to become
familiar with this chart, BTW. It has become widely recognized as
essential material, far more important than the Eddington or Lemaitre
charts, for example.
Sorry, Jan, this remark isn't aimed at you in particular--- I guess I am
a bit sick of observing that obviously even readers who -could- benefit
from my attempts to educate here are not doing so, because they are not
following up on my reading suggestions.
Also, I am very sick of observing that the relativity cranks even in
this moderated newsgroup outnumber those who understand the mainstream
viewpoint, and these folk have no business incorrectly "answering"
questions. This is why I try to suggest from time to time that perhaps
the moderators should stop accepting any posts and limit their activity
to canceling forged posts to this newsgroup.
Query for the tiny subset of the physics world which administrates and
sometimes participates halfheartedly in this newsgroup: Maybe it's time
to shut down s.p.research on the grounds that the name incorrectly
suggests to baffled laypersons/reporters that the majority here has
mastered those areas of contemporary math/physics which they post here
about? Isn't sci.math.research doing more harm than good as a soure of
allegedly "reliable information" [sic]?
If no equally knowledgeable younger generation of fervent sci/math
proselytizers is appearing to take up the pedagogical efforts of Baez &
co, does this not mean that there is no longer any possible rationale,
even pedagogical, far less "research", for continuing this newsgroup? I
certainly don't have sufficient time or energy to continue to try to
perform this task even in regards gtr, but if I don't correct even
blatant misstatements, no-one else steps forward to do so ... So, what
> I always had this problem with white holes in the sense that what
> conceivable principle could determine WHAT comes out of them? A piano? A
> sperm whale? Ketchup?
Right. This shows the limitations of the classical spacetime concept;
it need not be taken as anything more than an extremely convenient and
suggestive way to think about kinematic relations.