https://www.youtube.com/watch?v=aJZHzEM_khE 2:37
--rwg
On 2014-10-31 14:28, Dan Asimov wrote:
> Sorry, was trying to do too many things at once. Got so wrapped up in
> finding a good picture
> of the (2,3,5) spherical tiling that I forgot about the stereographic
> projection.
>
> Such a picture is about 1/15 of the way down this page, for instance:
>
> < http://westy31.home.xs4all.nl/Geometry/Geometry.html >
>
> -6-^Dan
>
>
> On Oct 31, 2014, at 11:48 AM, Dan Asimov <dasimov(a)earthlink.net> wrote:
>
>> Stereographic projection (from a north-pole-less n-sphere to n-space) is
really quite interesting.
>>
>> Besides the surprising fact that angles are preserved is the even more
surprising fact that
>> (n-1)-spheres are carried to (n-1)-spheres.
>>
>> Stereographic projection helps immensely with visualizing things on the
3-sphere, since (except for
>> its N pole) it gets projected to 3-space. So, all the great circles of
the Hopf fibration of S^3
>> gets projected to circles in R^3 -- though in R^3 they have variously
every radius >= 1.
>>
>> (Except for the Hopf circle through the N pole (0,0,0,1), which is
projected to the z-axis of R^3.)
>>
>> I admit to puzzlement over this passage in the article:
>>
>> -----
>> Henry and Saul then turned to a well-known spherical tiling, called a
(2,3,5) tiling, made up of triangles, but arranged to make diamonds, larger
triangles and pentagons. They discovered that in the nineteenth century,
the German mathematician August Möbius - yes, he of the eponymous strip -
had drawn a sketch of how the tiling would look under a stereographic
projection. Henry and Saul decided to make it.
>>
>> “Maybe the only reason it hadn’t already been done before 3D printing is
that any errors in the geometry get magnified, so you need a very precise
way to make the model,” Henry says.
>> -----
>>
>> I'm really not sure what it is that "hadn't already been done before",
since the (2,3,5) tiling has been
>> depicted by computer graphics innumerable times. One such picture
appears on the page
>>
>> < https://en.wikipedia.org/wiki/Triangle_group >.
>>
>> --Dan
>>
>>
>> On Oct 31, 2014, at 1:49 AM, Alex Bellos <alexanderbellos(a)gmail.com>
wrote:
>>
>>> One for Halloween:
>>>
>>> Stereographic projections as light sculpture...amazing 3D-printed
jack-o-lanterns by Henry Segerman and Saul Schleimer
>>>
>>>
http://www.theguardian.com/science/alexs-adventures-in-numberland/2014/oct/…
>>
A litsearch found this paper, and Meeker sent me a pdf of it:
Urs M. Schaudt and Herbert Pfister:
The Boundary Value Problem for the Stationary
and Axisymmetric Einstein Equations is Generically Solvable,
Phys. Rev. Lett. 77,16 (Oct 1996) 3284-3287
ABSTRACT
We prove existence, uniqueness, and regularity for Dirichlet solutions of
the stationary and axisymmetric Einstein equations in vacuum and in rigidly
rotating ideal fluid matter for data (on a ball) whose absolute values are in a
characteristic way limited by the "diameter" of the ball. These
results have important
consequences for the existence of exterior vacuum and interior matter
solutions for rotating stars.
The mathematical procedures and results have remarkable connections with a
numerical solution technique for rotating stars.
They also cite:
Uwe Heilig: On the existence of rotating stars in general relativity,
Commun Math'l Phys. 166,3 (1995) 457-493
http://projecteuclid.org/euclid.cmp/1104271700
ABSTRACT
Abstract: The Newtonian equations of motion, and Newton's law of
gravitation can be obtained by a limit L=1/c^2-->0 of Einstein's
equations. For a sufficiently small
constant A the existence of a set of solutions (0<L<A) of Einstein's
equations of a stationary, axisymmetric star is proven. This existence
is proven in weighted Sobolev spaces with the implicit function
theorem. Since the value of the causality constant L depends only on
the units used to measure the velocity, the existence of a solution
for any small L is physically interesting.
---
Here is my quick attempt to summarize these papers.
If you postulate some equation-of-state for the matter inside a "star"
and assume it is rigidly rotating blob of matter in hydrostatic equilibrium,
then Einstein GR equations can be viewed as a Dirichlet problem -- given data
on the boundary, can we solve it inside that boundary?
Schaudt & Pfister set up an iteration which "improves" an approximate
solution, and prove this is a "contraction" in a Banach space, and
therefore an attractor exists, and therefore a solution exists.
Hence, they get a theorem saying, "rigidly rotating stars exist"
provided the mass is not too large and the spin is not too large -- to
make their proof work they need masses and spins upper bounded by
about 1% of what they presume would really be the maximum allowed.
So, if the boundary data is chosen to agree with the Kerr exterior, I think
this means that this proves that a matching "rigid rotating" interior
always exists provided
(1) mass and densities never too large
(2) spin not too large
(3) postulated equation of state for the matter (in the right class).
But they do not actually FIND any such interior solution, they merely
prove nonconstructively that at least one exists. It would be more
desirable to have
an explicit example that was not too hard to deal with.
But it seems like the Banach-contaction proof should yield a numerical procedure
for approximating such a solution to arbitrary accuracy.
Heilig starts from a known exact solution in Newtonian gravity
(rotating fluid ellipsoid,
constant density fluid) and then tries to perturb it to work in Einstein gravity
using L=1/c^2, where L=0 for Newton and L>0 for Einstein; he claims he
proves such a perturbation exists if L is not too large.
--
Warren D. Smith
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