Changing curvature as a universe evolves

[Orodruin]

A homogeneous and isotropic universe is spherically-symmetric, at every location. We know from that symmetry and Gauss's Law that whatever sphere of the Universe we're looking at at any given time, the behavior of the system outside has no impact on the dynamics inside the sphere. From this we get a result which states how the density of that sphere will change over time. Since we could have used any origin and any size sphere to get this result, it applies to the entire universe.

This common belief is wrong as the requirements for only the inside the sphere to govern the dynamics are not satisfied. The requirement is that the boundary conditions at infinity do not spoil the symmetry - which is true when you consider a potential that tends to zero at infinity. However, the Poisson equation with a constant density term is incompatible with any solution where the symmetry is maintained. This implies external fields - givenby the boundary conditions - also exist inside the sphere. Compare with adding an external electric field to the field of a soherical charge distribution. The external field will certainly affect the dynamics of a test charge inside the sphere.

I can only point to the lectures that are available online from Leonard Susskind. In those Cosmology lectures, the Friedmann equations were derived entirely using Newtonian physics. GR was only used at the very end and just to show that the same equations appear. I think @kimbyd has made this point.



The ability to derive the Friedmann equations from Newtonian physics may be a little surprising but it's not unlike the derivation of an object that was called a "dark star" and has many of the same properties for the object we now describe as a Black hole using GR.

Those are usually heuristic arguments. Regardless, it is not the same Friedmann equation as it is based on Newtonian physics. There is no a priori reason to expect the exact same result in a FLRW universe.

 

[kimbyd]

To Kimbyd,

Thanks again and I meant what I said. There's nothing wrong with what you've written. I'm just a stubborn student trying to understand why things have to be a certain way.Why? This is an assertion not a proof.

This is very true. The full derivation is very complicated, and I haven't done it for a number of years. But basically when you try to describe a homogeneous and isotropic universe, you perform an integral at some point which results in a constant of integration. That constant turns out to be the spatial curvature.

If you tried to make it not constant, you wouldn't have a valid solution to Einstein's equations any longer.

 

[kimbyd]

This common belief is wrong as the requirements for only the inside the sphere to govern the dynamics are not satisfied. The requirement is that the boundary conditions at infinity do not spoil the symmetry - which is true when you consider a potential that tends to zero at infinity. However, the Poisson equation with a constant density term is incompatible with any solution where the symmetry is maintained. This implies external fields - givenby the boundary conditions - also exist inside the sphere. Compare with adding an external electric field to the field of a soherical charge distribution. The external field will certainly affect the dynamics of a test charge inside the sphere.

I seem to remember this. My take-away is generally just that the application of Gauss's Law might not have worked in this situation, due to the diverging gravitational potential at infinity. But it does: Newtonian gravity with this construction produces the exact same result as General Relativity. Thus the effects at infinity didn't spoil the solution when using a spherical construction.

My guess is that it's probably down to the fact that the spherical construction respects the homogeneity and isotropy conditions.

 

[PeterDonis]

the application of Gauss's Law might not have worked in this situation, due to the diverging gravitational potential at infinity. But it does

Doesn't the Gauss's Law integral for this case diverge?

 

[Orodruin]

I seem to remember this. My take-away is generally just that the application of Gauss's Law might not have worked in this situation, due to the diverging gravitational potential at infinity. But it does: Newtonian gravity with this construction produces the exact same result as General Relativity. Thus the effects at infinity didn't spoil the solution when using a spherical construction.

My guess is that it's probably down to the fact that the spherical construction respects the homogeneity and isotropy conditions.

The condition broken is homogeneity. By construction the spherical construction is isotropic. It is also not Gauss law that breaks down but the spherical shell theorem. That it by chance reproduces the correct result is not necessarily a sign of the derivation showing the same.

It may be possible to derive the same dynamic due to tidal forces regardless of the boundary condition at infinity (I have not had the time to look into this). This seems plausible to me as the volume change due to tidal forces should be proportional to a second derivative of the potential and the only such derivative is the Laplace operator. However, this is still a heuristic argument for obtaining the result, it does not show that the same equation is obtained in GR, and the solution is not necessarily isotropic and homogeneous (even if the density is).

 

[Will Learn]

Concerning post #28 in general.
I seem to have caused a problem or offence and I can only apologise for that. Perhaps I could explain the motivation for my actions so that you might see that no harm was intended and how easily a new foum member can get things wrong.

I was looking for a place to discuss some Physics and a Google search came up with Physics Forum. I skim read some of the terms and background for this website and found statements like this -

Our mission is to provide a place for people (whether students, professional scientists, or others interested in science) to learn and discuss science as it is currently generally understood and practiced by the professional scientific community

So I started to use the site. When this thread was created it was my intention to discuss a topic and learn as I went and I tried to make that clear in the OP and early posts. I thought a forum like this one might work as something comparable to a "study group" or just a coffee shop inside a place of learning, where one person can discuss both their ideas and their difficulties in understanding with others. That is what I have tried to do.

I have listened to all of the posts from others, engaged with every person that took the time to discuss anything and openly admitted where they were right and I had been wrong. See post #6 and #7 as a short example of an exchange between @Orodruin and myself. If I hadn't done this it would have been wrong and that's not the sort of person you would want in a study group or any discussion. However, you wouldn't want a person in a discussion or study group if they don't pull their weight either. This seemed to be especially true in a forum where you are the original poster. For example, in post #5 I got a message from a moderator that seemed to be saying something along those lines.

Just saying you can't see any reason why not is not suffcient; you would actually need to find a solution that has the property.

So I increased the amount of writing in my own comments and replies and set about demonstrating why a solution should exist. In later posts, I continued writing more and learned how LaTeX works in this forum. In all likelihood I went too far, my posts were too long and may have seemed like some direct challenge backed up with half-correct references and nonsense mathematics. There were then comments from others along those lines. There is a phrase we use locally, "you're damned if you do and you're damned if you don't". Making a discussion without any references or avoiding Mathematics wouldn't have been good either. The situation is understood, I hope, when you see that I was just trying to discuss things as if in a study group.

I have made a similar mistake while trying to keep a forum thread on topic and updated. I thought it was generally good to stay on the main topic and drifting off topic is often considered as "hijacking". In my limited experience of forums I've also noticed that very old and very long threads with many replies are almost impossible to join in any constructive way. It is not possible to read and appreciate all the replies that have gone before and to see what the current state of the discussion has become. It is useful if someone, often the OP, can occassionally update the thread somehow. This is the sort of thing I was trying to do in post #23. I seem to have got this completely wrong and attracted comments along those lines.

Perhaps that was the purpose of your OP in this thread, but it's not what the purpose of this thread has evolved into.

It is a narrow line to walk, learning to use a forum and new members will get it wrong. My sincerest apologies.

None of us get paid for this

I did not know but suspected that may be the case. I have always been and remain grateful for the time spent. You have never been under an obligation to reply. One of the ways in which this thread has failed is that it has failed to attract attention from people in a similar position to myself and instead consumed time from you. I am very sorry, it is not working as I hoped and I will cease. I cannot offer to pay you for your time since that seems to conflict with the terms and conditions of use, however there are many banners indicating ways in which I can support PF as a whole and I will be looking into those.

I can only repeat the message from post #23:

I'm making good progress answering my own question by looking at ..(stuff)... I've bored everyone else enough already and probably won't write anymore about that. I'm very grateful for all the time, attention and replies I have received on Physics Forums.

 

[PeterDonis]

I seem to have caused a problem or offence

You haven't caused any offense.

you wouldn't want a person in a discussion or study group if they don't pull their weight

It's not at all a matter of "pulling your weight". Obviously we don't expect you to have the same understanding of the topic under discussion as members who have studied the topic for years. If you feel like you need to somehow display that level of knowledge in order to continue posting, that is not at all the case. In fact it's the opposite: as a newcomer to a topic, we would rather you did not try to guess or hypothesize or theorize about it at all. That's why PF has rules about personal theories and personal speculations being off limits.