A cosmological expansion problem too difficult for me

[Buzz Bloom]

I began thinking about the problem I describe below from trying to understand the discussions in another thread.

https://www.physicsforums.com/threads/concordance-cosmology-paper.909603​

This problem is about the expansion behavior of a “simple” universe model that might demonstrate a distinction between an expansion based on the average mass density, and an expansion based on a distribution of mass which is mostly vacuum.

I start with a finite vacuum universe with five point masses equidistant form each other (a 4-simplex), and each having a fixed position with respect to a co-moving coordinate system. This universe being finite, it is closed with a positive curvature ~1/R(t) at places far from a point mass. The approximate volume of the universe is then

V(t) ~= 2 π² R(t)³.​

If each point mass as mass M, then the density

ρ(t) ~= 5M/V(t).​

(The reason this is not exact is that the point masses distort the space and therefore the total volume.

If this mass density were instead uniformly distributed, then the Friedmann Equation

see https://en.wikipedia.org/wiki/Friedmann_equations​

could be used to calculate the Hubble “constant” value H(t), and then numerically R(t).

The problem is to calculate an estimate for the difference between the uniform density Friedmann solution, and a solution which is based on the five point masses.

 

[Chalnoth]

Probably a better way to tackle this problem would be to imagine an infinite, uniform grid of point masses as compared to the uniform fluid.

 

[Buzz Bloom]

Probably a better way to tackle this problem would be to imagine an infinite, uniform grid of point masses as compared to the uniform fluid.

Hi Chalnoth:
Both approaches seem way over my head, so I have no intuition about which would be easier for an adequately prepared investigator.

Can you help me understand why the GM equations would be easier to work with for the infinite grid rather than five point masses?

Regards,
Buzz

 

[PeterDonis]

a solution which is based on the five point masses.

You're assuming there even is such a solution. What if there isn't?

You are also assuming that such a solution (if it exists) has similar global properties to a standard FRW solution, just with a few tweaks because the matter distribution is not continuous. But a solution with five point masses already breaks the two key assumptions that are used in standard FRW cosmology: homogeneity and isotropy. So I would be very wary of making any assumptions at all about its properties.

 

[Buzz Bloom]

You're assuming there even is such a solution. What if there isn't?

Hi Peter:
You are right. That is my assumption, but since I am full of ignorance I have no basis for believing there would no be a solution.

Can you suggest a reason why the GR equations would not be able to be reduced to an equation, perhaps very complicated, relating dR/dt and t, perhaps including higher order differentials? Is there any reason to believe that whatever expansion (or contraction) there might be, that the five point masses will remain equal distant from each other? That is the spatial configuration always remains unchanged? That seems intuitively (for whatever value that might have) to be a reasonable assumption given the symmetry of the configuration.

Regards,
Buzz

 

[PeterDonis]

I have no basis for believing there would no be a solution.

That's backwards. You have to actually try to find a solution, not just assume there is one unless you can think of a reason why not.

If finding a solution is beyond you, that is an indication that you need to spend more time understanding how the EFE is solved.

However, if you want a basis for doubt, here it is: there is no known exact solution to the EFE even for the case of two point masses, let alone five. Such cases are solved numerically, with a computer.

 

[Buzz Bloom]

However, if you want a basis for doubt, here it is: there is no known exact solution to the EFE even for the case of two point masses, let alone five. Such cases are solved numerically, with a computer.

Hi Peter:
I apologize for my confusion regarding terminology.

When I used the term "solution" I intended that to include numerical solutions, rather than it be limited to closed form solutions. What I do not have the skills to do is to derive from GR a differential equation that express the relationship between (1) the curvature at a point half way between two point masses as a function of time, and (2) the initial conditions consisting of the hypothetical universe I described with 5 symmetrically placed point masses of sufficient mass to cause closure.universe. If I had the DE, I think I could calculate a numerical solution.

Regards,
Buzz

 

[PeterDonis]

When I used the term "solution" I intended that to include numerical solutions, rather than it be limited to closed form solutions.

And yet you wrote a bunch of closed-form equations in your OP. Which you now appear to be admitting you just conjured out of thin air. See below.

What I do not have the skills to do is to derive from GR a differential equation that express the relationship between (1) the curvature at a point half way between two point masses as a function of time, and (2) the initial conditions consisting of the hypothetical universe I described with 5 symmetrically placed point masses of sufficient mass to cause closure.universe.

In other words, you wrote down a bunch of equations in your OP which you do not have the skills to derive. And you labeled this thread "A", for advanced, i.e., graduate level knowledge of the subject matter. Which would include the knowledge of what the Einstein Field Equation is and how to derive solutions for it, whether closed form or numerical. I'm sorry, but we can't have a productive discussion on that basis.

This thread is closed.