| Topic: |
Science > Physics |
| User: |
"Barrow" |
| Date: |
14 May 2007 09:34:49 AM |
| Object: |
FLRW metric related (pseudosphere ) |
Dear all,
I'm reading the FLRW metric of the big bang cosmology. In order to
understand the homogeneous and isotropic metrics of the 3 dimensional
space, I try to find and work out the homogeneous and isotropic ones
in 2 dimensional space.
I found from the book that the homogeneous and isotropic metrics in
2 D can be written as:
ds^2 = a^2 [ dr^2/(1-k r^2) + r^2d\theta^2 ] ---(1)
I know the cases k = 0, 1 corresponding to plane and 2-sphere
respectively. I also found that the k = -1 case is the pseudosphere
which is the revolution of the tractrix. I worked out the metric to
be:
ds^2 = a^2 [ cot^2(u)du^2 + sin^2u dv^2] ---(2)
The problem is, I tried to transform the line element eq(2) to the
line element eq(1) with k = -1 by coordinate transformation, but in
vain. I have tried several change of variables, but they just didn't
work. I am wondering whether the line element of eq(1) with k = -1
equals to line element of eq(2).
I searched so many sites and some differential geometry books, like
do Carmo and O'neill, but they didn't mention about it. Any help will
be appreciated much!
.
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| User: "Barrow" |
|
| Title: Re: FLRW metric related (pseudosphere ) |
15 May 2007 06:25:53 AM |
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On 5=A4=EB14=A4=E9, =A4U=A4=C810=AE=C934=A4=C0, Barrow <GRsemi...@gmail.com=
wrote:
Dear all,
I'm reading the FLRW metric of the big bang cosmology. In order to
understand the homogeneous and isotropic metrics of the 3 dimensional
space, I try to find and work out the homogeneous and isotropic ones
in 2 dimensional space.
I found from the book that the homogeneous and isotropic metrics in
2 D can be written as:
ds^2 =3D a^2 [ dr^2/(1-k r^2) + r^2d\theta^2 ] ---(1)
I know the cases k =3D 0, 1 corresponding to plane and 2-sphere
respectively. I also found that the k =3D -1 case is the pseudosphere
which is the revolution of the tractrix. I worked out the metric to
be:
ds^2 =3D a^2 [ cot^2(u)du^2 + sin^2u dv^2] ---(2)
The problem is, I tried to transform the line element eq(2) to the
line element eq(1) with k =3D -1 by coordinate transformation, but in
vain. I have tried several change of variables, but they just didn't
work. I am wondering whether the line element of eq(1) with k =3D -1
equals to line element of eq(2).
I searched so many sites and some differential geometry books, like
do Carmo and O'neill, but they didn't mention about it. Any help will
be appreciated much!
I found that the following article about the relation of nanotech. and
Hawking radiation (I'm surprised that they are related...) mentioned
about the line elements of my eq(1) and eq(2).
http://arxiv.org/PS_cache/cond-mat/pdf/0510/0510743v2.pdf (check
eq(3) and eq(6))
The author said that the coordinates of my eq(2) is a global angular
coordinates whlie the coordinates in eq(1) is local angular ones. But
I still can't figure out the transformation between line element eq(1)
and eq(2).
By the way, Thanks for Jim Black's help!
Sincerely Barrow
.
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| User: "Jim Black" |
|
| Title: Re: FLRW metric related (pseudosphere ) |
14 May 2007 06:25:36 PM |
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On May 14, 9:34 am, Barrow <GRsemi...@gmail.com> wrote:
Dear all,
I'm reading the FLRW metric of the big bang cosmology. In order to
understand the homogeneous and isotropic metrics of the 3 dimensional
space, I try to find and work out the homogeneous and isotropic ones
in 2 dimensional space.
I found from the book that the homogeneous and isotropic metrics in
2 D can be written as:
ds^2 = a^2 [ dr^2/(1-k r^2) + r^2d\theta^2 ] ---(1)
I know the cases k = 0, 1 corresponding to plane and 2-sphere
respectively. I also found that the k = -1 case is the pseudosphere
which is the revolution of the tractrix.
Sort of. The pseudosphere has an edge, whereas the k=-1 universe does
not. Also, it has a different topology because there are loops on the
pseudosphere that you can't continuously deform to a point, whereas
there are no such loops in a k=-1 universe. It appears that the
pseudosphere is to your 2-dimensional k=-1 universe as half of an
infinitely long cylinder is to the Euclidean plane.
I worked out the metric to
be:
ds^2 = a^2 [ cot^2(u)du^2 + sin^2u dv^2] ---(2)
The problem is, I tried to transform the line element eq(2) to the
line element eq(1) with k = -1 by coordinate transformation, but in
vain. I have tried several change of variables, but they just didn't
work. I am wondering whether the line element of eq(1) with k = -1
equals to line element of eq(2).
Your equations (1) and (2) appear to be correct.
Despite the difference in topology, there ought to be a coordinate
transformation from a section of the k=-1 universe to the
pseudosphere. It just won't be one-to-one; an infinite number of
points on the k=-1 universe will be mapped to the same point on the
pseudosphere. But it might not be something you'd get by guessing.
I haven't solved the problem myself yet, but my strategy would be as
follows. We know that rotating the pseudosphere leaves the u=constant
lines unchanged. So to find these lines on the k=-1 universe, we look
for the corresponding isometric transformation of the k=-1 universe.
This isometry should not change the point at infinity on the k=-1
universe that corresponds to the point at infinity on the
pseudosphere.
What do we know about isometries in the k=-1 universe? Turns out we
know a lot! That's because the k=-1 universe has the same line
element as the slice of 2+1 spacetime that satisfies dt^2 - dx^2 -
dy^2 = a positive constant and t>0. You can check this for yourself
-- it should be a great deal easier than checking the pseudosphere.
And the isometries of this space are just the Lorentz transformations!
If we look at our dt^2 - dx^2 - dy^2 = const. space as the space of
unit 4-vectors, we see that a point at infinity corresponds to a light
ray. So the transformation we're looking for is composed of a boost
that causes aberration of the light ray, followed by a rotation to put
the light ray back where it started.
--
Hope this helps,
Jim E. Black
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