Equivalence of Universe Accelerating Expansion?

[Mathchallenged]

TL;DR Summary

Does the accelerating expansion of the universe with dark energy behave the same way if it was inside a large spherical shell with a very large mass with gravitational attraction but no dark energy?

Without suggesting that there actually is a massive spherical shell enclosing the universe, I am trying to explore whether the observed expansion of the universe would be the same if there was a spherical shell with a diameter greater than the universe, with some arbitrary mass producing a gravitational force acting on the enclosed universe. I might have expressed this idea badly. A little help would be appreciated.

 

[PeroK]

No. The equations of motion are entirely different.

The expansion is uniformly linear with distance. Motion under gravitational attraction is not.

 

[Mathchallenged]

Please point me at a tutorial comparing the two sets of equations. I want to understand how they are different. Thanks.

 

[Orodruin]

The solution inside the spherical shell is essentially a spherical cut out of Minkowski space.

 

[PeroK]

Please point me at a tutorial comparing the two sets of equations. I want to understand how they are different. Thanks.

You may or may not find something comparing the two. The accerating universe expansion is modeled by the Friedmann equation, with the appropriate vacuum/dark energy density.

You should be able to find lots on that.

Gravitational attraction, on the other hand, is an inverse square law. Which is quite different.

 

[Ibix]

The solution inside the spherical shell is essentially a spherical cut out of Minkowski space.

...but filled with a uniform expanding fluid, if we want to model the universe. I think that would be a bit different, and wouldn't it also depend on how the sphere is behaving? Is it also expanding so the mass inside is constant, or what?

@Mathchallenged - The mainstream model of the universe is called the Friedmann-Lemaitre-Robertson-Walker spacetime (although many sources don't include Lemaitre) and its evolution is governed by the Friedmann equations. Chapter 8 of Sean Carroll's GR notes covers it (free to download), and I think the wiki article isn't too bad.

 

[PeterDonis]

spherical shell with a diameter greater than the universe, with some arbitrary mass producing a gravitational force acting on the enclosed universe.

A spherical shell of matter produces zero gravitational force on anything inside the shell. So no, it is not at all the same as the effect of dark energy.

 

[PeterDonis]

The solution inside the spherical shell is essentially a spherical cut out of Minkowski space.

If there's no matter inside the shell. But even if there is matter inside the shell, so the solution inside is not Minkowski, the shell itself has zero effect on the matter inside the shell, because of the shell theorem. That is the key fact that rules out the kind of model described in the OP.

 

[Orodruin]

If there's no matter inside the shell. But even if there is matter inside the shell, so the solution inside is not Minkowski, the shell itself has zero effect on the matter inside the shell, because of the shell theorem. That is the key fact that rules out the kind of model described in the OP.

Sure, I was just focusing on the part of replacing dark energy with a large sphere.

 

[kimbyd]

Summary:: Does the accelerating expansion of the universe with dark energy behave the same way if it was inside a large spherical shell with a very large mass with gravitational attraction but no dark energy?

Without suggesting that there actually is a massive spherical shell enclosing the universe, I am trying to explore whether the observed expansion of the universe would be the same if there was a spherical shell with a diameter greater than the universe, with some arbitrary mass producing a gravitational force acting on the enclosed universe. I might have expressed this idea badly. A little help would be appreciated.

The idea that the accelerated expansion of the universe can potentially be explained by us being near the center of a large underdense region, with more dense universe beyond that, is a serious idea that has been considered by cosmologists. It was very hotly and actively debated around something like 10-20 years ago, with lots of back-and-forth.

I believe the resolution is that the "void models" of expansion here just don't fit with the data. This paper is an example:
https://arxiv.org/abs/1007.3725

So, yes, it might have been possible for the accelerated expansion to have been due to us just being very near the center of an underdense region (a "void", hence the name). But the data doesn't appear to fit.