Mathamatical Description of the Cosmological Principle

[Haths]

I think this belongs here, while not relativistic, is is about objects moving relative to one another.

Summery:
This stems from the description of 'Dark Energy' in a mathamatical context to show that you can place a 'center of the universe' at any point in space, and the accelaration vector between two bodies is inderpendant of the position vectors to this 'arbitary center of the universe'.

Body:
The presumption is that there exsits a 'center of the universe', and that from it we can draw a diagram


A
...\
...\___________________
...B......C

Where A is our 'center of the universe', B is one body, C is the other body. There is a radial distances r_B and r_C respectfully, and the magnitude of radial accelaration from this 'center of the universe' is proportional to these distances.

From B, an oberserver notes that C is accelerating away from him at some accelaration a'_C and reversely, from C an observer notes that B is accelerating away from him at some accelaration a'_B

How do I show that no matter where I place A, the magnitude of accelaration is not dependent on the 'center of the universe'?

I thought you could describe a'_C and a'_B as;

a'_C = r_CK*cos( 1/2 n ) - r_BK
a'_B = r_BK*cos( 1/2 n ) - r_CK

Where K is the constant of proportionality and n is the angle between the bodies. Using the geometry between them. I guess that the cosine formula is now somehow incorperated into this;

a²=b²+c²-2bc*CosA

Because there are simmilarities in both equations, but I'm not sure where to go from here in the proof...

Haths

 

[Mentz114]

You can model the principles of isotropy and homogeneity by setting up a line element that defines the distance measure in the cosmos by all observers. By ansatz we say that every observer will be able to construct a spherically symmetric coordinate system around themselves. Spherically symmetric implies no preferred direction (isotropy) and all observers can do this ( homogeneity).

The line element is one of the Robertson-Walker-Lemaitre varieties and the simplest form looks like this

$$ds^2 = c^2dt^2-a(t)^2d\sigma^2$$

see Wiki on 'Robertson-Walker metric'.