| Topic: |
Science > Physics |
| User: |
"" |
| Date: |
26 May 2005 12:28:04 PM |
| Object: |
spinning black hole |
i wanted to calculate the effective potential for a rotating, charge
free blackhole.
By effective potential i simply mean evaluationg, [dr/d(tau)]^2.
using energy at infinity,E and angular momentum,L as constants of
motion i was able to evaluate dphi/d(tau) and dt/d(tau) in terms of E
and L. I still need an expression for d(theta)/d(tau) to plug it in the
metric and get [dr/d(tau)]^2.
how do i evaluate d(theta)/d(tau)???
I wish to calculate the effective potential for all theta and NOT JUST
the equitorial plane.
PLease help.
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| User: "Jim Black" |
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| Title: Re: spinning black hole |
26 May 2005 12:39:13 PM |
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wrote:
i wanted to calculate the effective potential for a rotating, charge
free blackhole.
By effective potential i simply mean evaluationg, [dr/d(tau)]^2.
Have you tried sci.physics.research?
I would take a stab at it, but I have no idea how the coordinate system
you're using is set up.
-- Jim Black
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| User: "" |
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| Title: Re: spinning black hole |
26 May 2005 02:49:52 PM |
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I am using Schwarszchild coordinates.
Jim Black wrote:
karan@iitk.ac.in wrote:
i wanted to calculate the effective potential for a rotating, charge
free blackhole.
By effective potential i simply mean evaluationg, [dr/d(tau)]^2.
Have you tried sci.physics.research?
I would take a stab at it, but I have no idea how the coordinate system
you're using is set up.
-- Jim Black
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| User: "Uncle Al" |
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| Title: Re: spinning black hole |
26 May 2005 04:13:33 PM |
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wrote:
I am using Schwarszchild coordinates.
Jim Black wrote:
wrote:
i wanted to calculate the effective potential for a rotating, charge
free blackhole.
By effective potential i simply mean evaluationg, [dr/d(tau)]^2.
Have you tried sci.physics.research?
I would take a stab at it, but I have no idea how the coordinate system
you're using is set up.
1) Don't top-post.
2) Schwarzschild assumes a non-rotating black hole.
<http://scienceworld.wolfram.com/physics/SchwarzschildBlackHole.html>
3)Don't you want the Kerr treatment instead?
<http://scienceworld.wolfram.com/physics/KerrBlackHole.html>
--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
http://www.mazepath.com/uncleal/qz.pdf
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| User: "Jim Black" |
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| Title: Re: spinning black hole |
26 May 2005 07:46:41 PM |
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wrote:
Jim Black wrote:
wrote:
i wanted to calculate the effective potential for a rotating, charge
free blackhole.
By effective potential i simply mean evaluationg, [dr/d(tau)]^2.
Have you tried sci.physics.research?
I would take a stab at it, but I have no idea how the coordinate system
you're using is set up.
-- Jim Black
I am using Schwarszchild coordinates.
That might be part of the problem. If my thinking is correct,
Schwarzschild coordinates assume spherical symmetry, which is
definitely not true in the case of a rotating black hole. Here's a
website:
http://scienceworld.wolfram.com/physics/Kerr-NewmanBlackHole.html
that gives the metric for a charged, rotating black hole in
Boyer-Lindquist coordinates, which it describes as a generalization of
the Schwarzschild coordinates.
Now that I'm looking at this a bit more closely, I'm beginning to see
what the problem is. The conserved energy should be straightforward to
work out via Noether's theorem, but figuring out the angular momentum
will be a problem. In the case of a Schwarzschild black hole, we have
spherical symmetry, so, without loss of generality, we can take the
orbit to be in the plane where theta = pi/2, and then it's easy to
calculate the conserved angular momentum. But in the case of a
rotating black hole, we don't have spherical symmetry, so there's no
reason to think that angular momentum is conserved at all, except for
the component parallel to the axis on which theta = 0 or pi. I don't
know if I can figure a way around this problem, but I'll be thinking
about it.
-- Jim Black
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| User: "Jim Black" |
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| Title: Re: spinning black hole |
26 May 2005 08:19:00 PM |
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Jim Black wrote:
http://scienceworld.wolfram.com/physics/Kerr-NewmanBlackHole.html
Sorry, somehow I misread "charge free" as "charged." The link you'd
want would be the one Uncle Al provided:
<http://scienceworld.wolfram.com/physics/KerrBlackHole.html>
-- Jim Black
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| User: "" |
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| Title: Re: spinning black hole |
27 May 2005 03:20:10 AM |
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sorry, i've been using Boyer Lindquist coordinates and NOT
schwarzschild.
Regarding the angular momentum, thats pretty straightforward... just
evaluate del(tau)/del(phi).
This must be a constant of motion since the metric is independent of
phi. Far from the black hole , on putting a=0 this gives us the
expression for angular momentum in Schwarzschild coord.
But the real problem is that I still have two differentials, d(theta)
and d(r) and one equation,viz, the metric( after using the two
constants opf motion to eliminate d(phi) and dt)
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| User: "Jim Black" |
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| Title: Re: spinning black hole |
27 May 2005 07:57:21 AM |
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wrote:
sorry, i've been using Boyer Lindquist coordinates and NOT
schwarzschild.
Regarding the angular momentum, thats pretty straightforward... just
evaluate del(tau)/del(phi).
That's a component of the angular momentum.
This must be a constant of motion since the metric is independent of
phi. Far from the black hole , on putting a=0 this gives us the
expression for angular momentum in Schwarzschild coord.
But the real problem is that I still have two differentials, d(theta)
and d(r) and one equation,viz, the metric( after using the two
constants opf motion to eliminate d(phi) and dt)
The differential d(theta) should be related to the other components of
angular momentum that are not necessarily conserved. One can solve for
their evolution via the calculus of variations and hope they'll be
simple; other than that, I can't think of anything else to do yet. At
the worst, you'll have two differential equations to mess with instead
of one.
-- Jim Black
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| User: "" |
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| Title: Re: spinning black hole |
29 May 2005 02:34:37 PM |
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wrote:
sorry, i've been using Boyer Lindquist coordinates and NOT
schwarzschild.
Regarding the angular momentum, thats pretty straightforward... just
evaluate del(tau)/del(phi).
This must be a constant of motion since the metric is independent of
phi. Far from the black hole , on putting a=0 this gives us the
expression for angular momentum in Schwarzschild coord.
But the real problem is that I still have two differentials, d(theta)
and d(r) and one equation,viz, the metric( after using the two
constants opf motion to eliminate d(phi) and dt)
For geodesics in a Kerr geometry, there is an additional, rather
unobvious, constant of motion, called Carter's constant. You will
find a discussion in section 33.5 of Misner, Thorne, and Wheeler,
Its existence is related to the presence of a Killing tensor. But
you'll have to look up the details...
Steve Carlip
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| User: "" |
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| Title: Re: spinning black hole |
03 Jun 2005 02:41:59 AM |
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I tried reading Misner, Wheeler and THorne, but I am not that
comfortable with killing vectors and would like to get Carter's
constant using properties of the metric and calculus .
Evaluating E and Lz as constants of motion was easy(using calculus)
since del(tau)/del(t) = constant and del(tau)/del(phi)= another
constant. This follows from the fact that the metric does not depend on
t or phi. I tried evaluating another function, f(r,theta) such that the
metric does not depend on f(r,theta) explicitly. This would result in
another constant of motion(if any). However, I havent been able to do
so. Could anybody please help me out.
-Karan
P.S. I am a second year undergraduate just reading up G.R. on my own,
so please be lucid.
carlip-nospam@physics.ucdavis.edu wrote:
karan@iitk.ac.in wrote:
sorry, i've been using Boyer Lindquist coordinates and NOT
schwarzschild.
Regarding the angular momentum, thats pretty straightforward... just
evaluate del(tau)/del(phi).
This must be a constant of motion since the metric is independent of
phi. Far from the black hole , on putting a=0 this gives us the
expression for angular momentum in Schwarzschild coord.
But the real problem is that I still have two differentials, d(theta)
and d(r) and one equation,viz, the metric( after using the two
constants opf motion to eliminate d(phi) and dt)
For geodesics in a Kerr geometry, there is an additional, rather
unobvious, constant of motion, called Carter's constant. You will
find a discussion in section 33.5 of Misner, Thorne, and Wheeler,
Its existence is related to the presence of a Killing tensor. But
you'll have to look up the details...
Steve Carlip
.
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| User: "" |
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| Title: Re: spinning black hole |
06 Jun 2005 07:11:31 PM |
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wrote:
I tried reading Misner, Wheeler and THorne, but I am not that
comfortable with killing vectors and would like to get Carter's
constant using properties of the metric and calculus .
I don't know of any especially easy way to do this. You can
try working backwards -- look at the expression for Carter's
constant in Misner, Thorne, and Wheeler, take its derivative,
and check that the geodesic equations imply that the derivative
is zero. Or if you want to avoid Killing tensors, you can look
at the Hamilton-Jacobi approach, which MTW gives as a problem.
Exercise 33.7 goes through this step by step. (If you don't
know the Hamilton-Jacobi equation, it's worth taking a few
days to learn it -- it's useful in many places.)
[...]
P.S. I am a second year undergraduate just reading up G.R. on my own,
Congratulations.
so please be lucid.
I'll try. But you should know that the problem you're asking
about isn't usually addressed even in introductory graduate courses
on GR -- it's not conceptually terribly complicated, but it's a lot
of work.
Steve Carlip
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| User: "" |
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| Title: Re: spinning black hole |
05 Jun 2005 11:42:51 PM |
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wrote:
I tried reading Misner, Wheeler and THorne, but I am not that
comfortable with killing vectors and would like to get Carter's
constant using properties of the metric and calculus .
Evaluating E and Lz as constants of motion was easy(using calculus)
since del(tau)/del(t) = constant and del(tau)/del(phi)= another
constant. This follows from the fact that the metric does not depend on
t or phi. I tried evaluating another function, f(r,theta) such that the
metric does not depend on f(r,theta) explicitly. This would result in
another constant of motion(if any). However, I havent been able to do
so. Could anybody please help me out.
-Karan
P.S. I am a second year undergraduate just reading up G.R. on my own,
so please be lucid.
carlip-nospam@physics.ucdavis.edu wrote:
wrote:
sorry, i've been using Boyer Lindquist coordinates and NOT
schwarzschild.
Regarding the angular momentum, thats pretty straightforward... just
evaluate del(tau)/del(phi).
This must be a constant of motion since the metric is independent of
phi. Far from the black hole , on putting a=0 this gives us the
expression for angular momentum in Schwarzschild coord.
But the real problem is that I still have two differentials, d(theta)
and d(r) and one equation,viz, the metric( after using the two
constants opf motion to eliminate d(phi) and dt)
For geodesics in a Kerr geometry, there is an additional, rather
unobvious, constant of motion, called Carter's constant. You will
find a discussion in section 33.5 of Misner, Thorne, and Wheeler,
Its existence is related to the presence of a Killing tensor. But
you'll have to look up the details...
Steve Carlip
.
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