Question about the path of a particle in Schwarzschild region of ancosmos

[Oh No]

As I understand it we can think of the Schwarzschild geometry
surrounding an isolated star as a static perturbation within an
expanding flat or near flat space-time. I would much appreciate
correction if I am wrong.

We can model elliptical orbits with a Newtonian approximation at
moderate distances from the star, and in this case expansion has no
meaning beyond a change of coordinates. That seems fine.

But if we model a hyperbolic orbit then at some point the gravity of the
star becomes irrelevant, and the orbiting body moves into deep space, so
that Hubble expansion must become the dominant part of its motion. My
question is, How do we model such an orbit and where may I better study
it?




Regards

--
Charles Francis
substitute charles for NotI to email

[Henning Makholm]

Scripsit Oh No <NotI@charlesfrancis.wanadoo.co.uk>

> As I understand it we can think of the Schwarzschild geometry
> surrounding an isolated star as a static perturbation within an
> expanding flat or near flat space-time.

...

> But if we model a hyperbolic orbit then at some point the gravity of the
> star becomes irrelevant, and the orbiting body moves into deep space, so
> that Hubble expansion must become the dominant part of its motion. My
> question is, How do we model such an orbit and where may I better study
> it?

The Schwarzschild metric does not include any Hubble expansion.
If you want to include that, you need to switch to a more realistic
cosmological model than "isolated star in an otherwise empty
universe". Which implies that you have to provide some positive
average matter density in the outer universe.

--
Henning Makholm "I can get fat! I can sing!"

[Oh No]

Thus spake Henning Makholm <henning@makholm.net>
>Scripsit Oh No <NotI@charlesfrancis.wanadoo.co.uk>
>
>> As I understand it we can think of the Schwarzschild geometry
>> surrounding an isolated star as a static perturbation within an
>> expanding flat or near flat space-time.
>
>..
>
>> But if we model a hyperbolic orbit then at some point the gravity of the
>> star becomes irrelevant, and the orbiting body moves into deep space, so
>> that Hubble expansion must become the dominant part of its motion. My
>> question is, How do we model such an orbit and where may I better study
>> it?
>
>The Schwarzschild metric does not include any Hubble expansion.
>If you want to include that, you need to switch to a more realistic
>cosmological model than "isolated star in an otherwise empty
>universe". Which implies that you have to provide some positive
>average matter density in the outer universe.
>
Well, yes. That is the reason for my question. Unless I have overlooked
it, such a model is not given in any of my text books. The example I
gave was merely the simplest instance of such a model I could think of,
since it is perfectly possible to describe an empty flat expanding toy
universe. Just not one that obeys Friedman's equation.


Regards

--
Charles Francis
substitute charles for NotI to email

[mark_horn@sbcglobal.net]

Oh No wrote:
> But if we model a hyperbolic orbit then at some point the gravity of the
> star becomes irrelevant, and the orbiting body moves into deep space, so
> that Hubble expansion must become the dominant part of its motion. My
> question is, How do we model such an orbit and where may I better study
> it?

I believe that, at the distances where the Hubble expansion becomes
significant, a solar system is (presumably) still bound within a
galaxy, so the expansion velocity is always irrelevant, wrt to your
orbiting mass. Beyond that I think you need to consider things like
large-scale bulk flow of whole galaxies in the streaming motion of
clusters; the CMB dipole anisotropy etc. (see e.g. Lauer and Portman,
APJ 425:418-438, 1994 April and Hudson et al APJ 512:70-82, 1999).

Cheers,
mark jonathan horn

[Greg Egan]

Oh No wrote:
> As I understand it we can think of the Schwarzschild geometry
> surrounding an isolated star as a static perturbation within an
> expanding flat or near flat space-time. I would much appreciate
> correction if I am wrong.
>
> We can model elliptical orbits with a Newtonian approximation at
> moderate distances from the star, and in this case expansion has no
> meaning beyond a change of coordinates. That seems fine.
>
> But if we model a hyperbolic orbit then at some point the gravity of the
> star becomes irrelevant, and the orbiting body moves into deep space, so
> that Hubble expansion must become the dominant part of its motion. My
> question is, How do we model such an orbit and where may I better study
> it?

How about studying the Schwarzschild-de Sitter solution, which is an
exact vacuum solution where the expansion is due to a cosmological
constant?

I'm afraid I personally know nothing about this solution, but I expect
people have worked it all out in great detail.

[Oh No]

Thus spake Greg Egan <gregegan@netspace.net.au>
>Oh No wrote:
>> As I understand it we can think of the Schwarzschild geometry
>> surrounding an isolated star as a static perturbation within an
>> expanding flat or near flat space-time. I would much appreciate
>> correction if I am wrong.
>>
>> We can model elliptical orbits with a Newtonian approximation at
>> moderate distances from the star, and in this case expansion has no
>> meaning beyond a change of coordinates. That seems fine.
>>
>> But if we model a hyperbolic orbit then at some point the gravity of the
>> star becomes irrelevant, and the orbiting body moves into deep space, so
>> that Hubble expansion must become the dominant part of its motion. My
>> question is, How do we model such an orbit and where may I better study
>> it?
>
>How about studying the Schwarzschild-de Sitter solution, which is an
>exact vacuum solution where the expansion is due to a cosmological
>constant?

Thanks. That does not seem like a bad starting point.
>
>I'm afraid I personally know nothing about this solution, but I expect
>people have worked it all out in great detail.
>
Yes, a quick glance at recent literature shows it has attracted
attention since the advent of the concordance cosmology, though I will
need to start with papers written about 1918 I think.


Regards

--
Charles Francis
substitute charles for NotI to email