- #1
- 1,931
- 1
Hi,
to pick up on marcus' https://www.physicsforums.com/showpost.php?p=2015482&postcount=67" in the sticky thread:
Recently, I tried to see where this analogy could bring us if evaluated properly, and I was surprised to find that it is actually far better than what marcus pessimistically states here. Here's a short sketch:
If we constrain motion to the surface of the balloon but not otherwise - i.e. if the surface is slippery and does not somehow drag along the particles on it - we get quite a good model of the universe. And that's not a coincidence.
We consider radial motion only (along the surface, but otherwise straight from point to point), because curvature does not really fit in. All derivatives are taken in cosmological time, the time an observer "at rest" with the surface would measure.
We then get from conservation of angular momentum [tex]R\, p = const.[/tex], R being the radius of the Balloon and p being the (transversal) momentum. This yields immediately [tex]E = const./R[/tex] for photons, aka cosmological redshift.
For massive bodies, admitting forces, we have [tex]\dot L = M = R\, F[/tex] or, explicitly, [tex]d/dt (R \gamma v)=R\, F/m[/tex]. Applying the chain rule and sorting out, this gives:
[tex]R\,\ddot \varphi = R^2 \dot R \dot \varphi^3 - 2 \dot R \dot \varphi +F/(\gamma^3 m)[/tex]
which is exactly the relativistic equation of motion in an expanding FRW spacetime - without the usual lethal dosis of Christoffel symbols and the like. Clearly a reason to employ this model to educate aspiring cosmologists without the proper background in GR.
Why does it work? The balloon shares a crucial symmetry with FRW spacetimes: What is invariance under rotation here, is invariance under the operation [tex]\chi \rightarrow \chi + d\chi[/tex] there. Without talking about spacelike Killing vectors, that is where this ominous http://arxiv.org/abs/0808.1552" stems from. If we concentrate on radial motion only, even the metric is the same. We get a glimpse of how geometry affects motion in cosmology.
Where are the drawbacks?
1. The balloon has a definite curvature, independent of [tex]\ddot R[/tex], which does not generally fit the FRW value.
2. The balloon extremely emphasizes cosmological (homogeneous) coordinates. An analysis in flat "private space" gets extremely tedious and looks quite unnatural. But IMHO that is what one must do to get a handle on cosmology.
Any comments? Maybe this work has been done before in 19.., I would appreciate any links.
to pick up on marcus' https://www.physicsforums.com/showpost.php?p=2015482&postcount=67" in the sticky thread:
marcus said:
Recently, I tried to see where this analogy could bring us if evaluated properly, and I was surprised to find that it is actually far better than what marcus pessimistically states here. Here's a short sketch:
If we constrain motion to the surface of the balloon but not otherwise - i.e. if the surface is slippery and does not somehow drag along the particles on it - we get quite a good model of the universe. And that's not a coincidence.
We consider radial motion only (along the surface, but otherwise straight from point to point), because curvature does not really fit in. All derivatives are taken in cosmological time, the time an observer "at rest" with the surface would measure.
We then get from conservation of angular momentum [tex]R\, p = const.[/tex], R being the radius of the Balloon and p being the (transversal) momentum. This yields immediately [tex]E = const./R[/tex] for photons, aka cosmological redshift.
For massive bodies, admitting forces, we have [tex]\dot L = M = R\, F[/tex] or, explicitly, [tex]d/dt (R \gamma v)=R\, F/m[/tex]. Applying the chain rule and sorting out, this gives:
[tex]R\,\ddot \varphi = R^2 \dot R \dot \varphi^3 - 2 \dot R \dot \varphi +F/(\gamma^3 m)[/tex]
which is exactly the relativistic equation of motion in an expanding FRW spacetime - without the usual lethal dosis of Christoffel symbols and the like. Clearly a reason to employ this model to educate aspiring cosmologists without the proper background in GR.
Why does it work? The balloon shares a crucial symmetry with FRW spacetimes: What is invariance under rotation here, is invariance under the operation [tex]\chi \rightarrow \chi + d\chi[/tex] there. Without talking about spacelike Killing vectors, that is where this ominous http://arxiv.org/abs/0808.1552" stems from. If we concentrate on radial motion only, even the metric is the same. We get a glimpse of how geometry affects motion in cosmology.
Where are the drawbacks?
1. The balloon has a definite curvature, independent of [tex]\ddot R[/tex], which does not generally fit the FRW value.
2. The balloon extremely emphasizes cosmological (homogeneous) coordinates. An analysis in flat "private space" gets extremely tedious and looks quite unnatural. But IMHO that is what one must do to get a handle on cosmology.
Any comments? Maybe this work has been done before in 19.., I would appreciate any links.
Last edited by a moderator: