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pervect
science forum beginner

Joined: 22 Jun 2006
Posts: 1

Posted: Thu Jun 22, 2006 6:42 am    Post subject: Birkhoff's theorem with cosmological constant

Birkhoff's theorem says that any vacuum solution of Einstein's equations
must be static, and asymptotically flat.

One of the consequences of Birkhoff's theorem is that the gravitational
field inside any spherical shell of matter is zero, even if the shell is
expanding.

But what happens if we allow a cosmological constant? Can we still say
that the field inside a spherical shell of matter (including expanding
shells) is zero if we assume that the universe has a non-zero
cosmological constant?

Some context might help explain why I am asking this question. I am
addressing the question of the effect (if any) of the cosmological
expansion on the orbits of the Solar system. I want to justify
ignoring the gravitational effect of the homogeneous part of the
universe on the solar system via Birkhoff's theorem.

I'm a bit unclear about the applicability of Birkhoff's theorem to the
case with the cosmological constant, unfortunately - and the universe
in the latest models does have a cosmological constant.

Ultimately I want to reconcile the approach taken in
http://tinyurl.com/gonxz

mentioned in Ned Wright's cosmology FAQ:
http://tinyurl.com/5lxs

which predicts a very, very small cosmological effect to the arguments
presented in

http://tinyurl.com/fsh4h

which predict no effect at all. I wish to argue that what is important
is the total mass contained within a sphere of radius R of the sun, and
that the bulk of the expanding universe does not contribute at all to
any solar system expansion.

To do this successfully, I need to know if Birkhoff's theorem does work
in the presence of a cosmological constant.

--
pervect

Did you hear about the man who stayed up all night to watch the sun
rise?

AB CADEFFG HEJDKH LD MAN, D=N, H=D
------------------------------------------------------------------------
pervect's Profile: http://www.physicsforums.com/forums/member.php?action=getinfo&userid=14402
Jonathan Thornburg -- rem
science forum beginner

Joined: 06 May 2005
Posts: 41

Posted: Thu Jun 22, 2006 9:38 pm    Post subject: Re: Birkhoff's theorem with cosmological constant

pervect <pervect.29qve4@physicsforums.com> wrote:
 Quote: I am addressing the question of the effect (if any) of the cosmological expansion on the orbits of the Solar system.

You might also find
http://de.arxiv.org/abs/gr-qc/0508052
of interest:

Title: In an expanding universe, what doesn't expand?
Authors: Richard H. Price
Abstract:
The expansion of the universe is often viewed as a uniform stretching
of space that would affect compact objects, atoms and stars, as well
as the separation of galaxies. One usually hears that bound systems
do not take part in the general expansion, but a much more subtle
question is whether bound systems expand partially. In this paper, a
very definitive answer is given for a very simple system: a classical
``atom'' bound by electrical attraction. With a mathemical description
appropriate for undergraduate physics majors, we show that this
bound system either completely follows the cosmological expansion,
or -- after initial transients -- completely ignores it. This
``all or nothing'' behavior can be understood with techniques of
junior-level mechanics. Lastly, the simple description is shown
to be a justifiable approximation of the relativistically correct
formulation of the problem.

ciao,

--
-- "Jonathan Thornburg -- remove -animal to reply" <jthorn@aei.mpg-zebra.de>
Max-Planck-Institut fuer Gravitationsphysik (Albert-Einstein-Institut),
Golm, Germany, "Old Europe" http://www.aei.mpg.de/~jthorn/home.html
"Washing one's hands of the conflict between the powerful and the
powerless means to side with the powerful, not to be neutral."
-- quote by Freire / poster by Oxfam
ebunn@lfa221051.richmond.
science forum beginner

Joined: 21 Sep 2005
Posts: 38

Posted: Fri Jun 23, 2006 3:22 am    Post subject: Re: Birkhoff's theorem with cosmological constant

In article <pervect.29qve4@physicsforums.com>,
pervect <pervect.29qve4@physicsforums.com> wrote:
 Quote: Birkhoff's theorem says that any vacuum solution of Einstein's equations must be static, and asymptotically flat.

It can't say this, because this isn't true. Think of a vacuum
solution consisting of a little pulse of gravitational radiation
whizzing along. It's not static. You need to add the magic words
"spherically symmetric."

Birkhoff's theorem doesn't apply when there's a cosmological
constant. One way to see this is to look at John Baez's and my
formulation of Einstein's equation:

http://math.ucr.edu/home/baez/einstein/

Suppose you've got a spherically-symmetric vacuum solution with
nonzero cosmological constant. In our formulation we treat the
cosmological constant (if any) as a form of "matter" -- that is, we
slide it over to the right side of the Einstein equation rather than
the left. So this vacuum solution will have nonzero energy density
and pressure. In particular, the combination rho + 3P is nonzero.
As we explain, that means that a little ball of test particles
that's initially at rest will change in volume over time. In
this case, it will expand, with (d^2V/dt^2)/V proportional to the
value of the cosmological constant.

In the solar system context, this must end up meaning that there's an
"anti-Hooke's Law" force on the planets due to the cosmological
constant: the effective force on the particles in a shell of radius R
is proportional to R.

In suitable units, the cosmological constant is the same order of magnitude
as 1/D^2 where D is the hubble length. I guess that a planet
with an orbital radius R must have a cosmological-constant-induced
acceleration of order Rc^2/D^2. Compare that with the ordinary gravitational
acceleration: GM/R^2 where M is the mass of the Sun. The ratio is

(cosmological constant term) / (ordinary acceleration) ~ R^3/(D^2 M)

dropping boring terms like G and c. In round numbers, R is 10^(11) m
for the Earth, D is 10^(26) m, and M is 10^3 m. So the cosmological
constant's effect on the Earth's orbit is about a part in 10^(22).

 Quote: http://tinyurl.com/gonxz

Hmm. That gives a ratio of 10^(44), which looks suspiciously like the
square of the value I got. I think that we're actually both right and
that despite different appearances we're calculating two different
things.

 Quote: mentioned in Ned Wright's cosmology FAQ: http://tinyurl.com/5lxs

Ned Wright really knows his stuff, so if he vouches for it that
increases my confidence that they've got things right.

 Quote: which predicts a very, very small cosmological effect to the arguments presented in http://tinyurl.com/fsh4h which predict no effect at all.

It doesn't look to me like that's what that paper says. It looks like
it says that the effect is negligibly small for realistic values of
Lambda, which is true. Are you sure these two papers disagree with
each other?

 Quote: I wish to argue that what is important is the total mass contained within a sphere of radius R of the sun,

That's pretty much right, if you replace the word "mass" with
"mass and pressure," since pressure also gravitates in general relativity,
and if you include the Lambda term in your accounting of mass and pressure.

 Quote: and that the bulk of the expanding universe does not contribute at all to any solar system expansion.

The bottom line is that there is an effect in principle but in practice
it's negligible.

-Ted

--
[E-mail me at name@domain.edu, as opposed to name@machine.domain.edu.]
Phillip Helbig---remove C

Joined: 06 May 2005
Posts: 88

Posted: Fri Jun 23, 2006 3:22 am    Post subject: Re: Birkhoff's theorem with cosmological constant

In article <4fvimnF1ld31aU2@individual.net>, "Jonathan Thornburg --
remove -animal to reply" <jthorn@aei.mpg-zebra.de> writes:

 Quote: pervect wrote: I am addressing the question of the effect (if any) of the cosmological expansion on the orbits of the Solar system. You might also find http://de.arxiv.org/abs/gr-qc/0508052 of interest: Title: In an expanding universe, what doesn't expand? Authors: Richard H. Price Abstract: The expansion of the universe is often viewed as a uniform stretching of space that would affect compact objects, atoms and stars, as well as the separation of galaxies. One usually hears that bound systems do not take part in the general expansion, but a much more subtle question is whether bound systems expand partially. In this paper, a very definitive answer is given for a very simple system: a classical ``atom'' bound by electrical attraction. With a mathemical description appropriate for undergraduate physics majors, we show that this bound system either completely follows the cosmological expansion, or -- after initial transients -- completely ignores it. This ``all or nothing'' behavior can be understood with techniques of junior-level mechanics. Lastly, the simple description is shown to be a justifiable approximation of the relativistically correct formulation of the problem.

Check out as well

http://www.arxiv.org/abs/astro-ph/0104349

http://www.arxiv.org/abs/astro-ph/0310808

for related ideas.
Ben Rudiak-Gould
science forum Guru

Joined: 04 May 2005
Posts: 382

Posted: Fri Jun 23, 2006 10:05 pm    Post subject: Re: Birkhoff's theorem with cosmological constant

Jonathan Thornburg wrote:
 Quote: http://de.arxiv.org/abs/gr-qc/0508052

Surely the result of this paper is (a) obvious, and (b) wrong in practice.

It's obvious because interacting systems always exhibit the all-or-nothing
behavior that the author describes as unexpected. Either the system is bound
or it's unbound, and in the latter case the pieces can be treated as
noninteracting except during a brief interval of close approach.

It's wrong in practice because comoving objects have no tendency whatsoever
to separate in an expanding universe, not even at the extremely slow Hubble
rate. As soon as you've disturbed a particle from rest with respect to the
usual FRW coordinates, its former state of motion becomes irrelevant. It's
totally wrong to treat the expansion as a force, as the author does in
equation (5). Maybe the dark energy works differently, but that's a separate
issue.

Am I wrong about any of this?

-- Ben
science forum beginner

Joined: 02 May 2005
Posts: 21

Posted: Fri Jun 23, 2006 10:05 pm    Post subject: Re: Birkhoff's theorem with cosmological constant

On Thu, 22 Jun 2006 06:42:43 +0000 (UTC), pervect
<pervect.29qve4@physicsforums.com> wrote:

 Quote: Birkhoff's theorem says that any vacuum solution of Einstein's equations must be static, and asymptotically flat. One of the consequences of Birkhoff's theorem is that the gravitational field inside any spherical shell of matter is zero, even if the shell is expanding. But what happens if we allow a cosmological constant? Can we still say that the field inside a spherical shell of matter (including expanding shells) is zero if we assume that the universe has a non-zero cosmological constant?

Perhaps the paper "General Non-Static Spherically Symmetric Solutions of
Einstein Vaccum Field Equations with Lambda" by Soheila Gharanfoli, Amir
H. Abbassi

http://arxiv.org/abs/gr-qc/9906049

will be of some help?

ark
--

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